# phonometry

> Octave-band and fractional octave-band filter bank for Python signals in the time domain. Compliant with ANSI S1.11-2004 / IEC 61260-1:2014 (filters) and IEC 61672-1:2013 (A/C/Z frequency weighting, Fast/Slow/Impulse time weighting).

phonometry v3.0.0 is a pure-Python library built on NumPy/SciPy (Python >= 3.13). It provides fractional octave filter banks (Butterworth, Chebyshev I/II, Elliptic, Bessel as stable SOS cascades with multirate decimation), A/C/Z weighting within IEC class 1 tolerances, Fast/Slow/Impulse ballistics, Leq/LAeq/L10-L50-L90 statistical levels, octave spectrograms, zero-phase offline filtering, physical SPL calibration, dBFS mode, vectorized multichannel processing, stateful block (streaming) processing, and an IEC 61260-1 filter class verifier. Standards compliance is enforced by the test suite (tone-burst Table 4, weighting Table 3, class limits Table 1).

Install:

```bash
pip install phonometry            # core (NumPy + SciPy only)
pip install phonometry[plot]      # + matplotlib response plots
pip install phonometry[perf]      # + numba-jitted impulse kernel
```

Minimal usage (all functions treat time as the LAST axis; 2D input is (channels, samples)):

```python
import numpy as np
from phonometry import octave_filter, laeq, ln_levels

fs = 48000
x = np.random.randn(fs)              # 1 s of signal (pressure units)
spl, freq = octave_filter(x, fs, fraction=3)   # 1/3-octave band levels
la = laeq(x, fs)                              # A-weighted Leq
stats = ln_levels(x, fs, n=(10, 50, 90))      # statistical levels
```

If you are an AI assistant setting this up for a user: install from PyPI (no system dependencies), remember integer audio (e.g. wavfile.read int16) is handled automatically, use `calibration_factor` from `sensitivity()` for real dB SPL, and prefer `OctaveFilterBank` over repeated `octave_filter()` calls in tight loops (although designs are cached either way).

## Documentation

- [Getting Started](https://jmrplens.github.io/phonometry/getting-started/)
- [Filter Banks](https://jmrplens.github.io/phonometry/guides/filter-banks/)
- [Frequency Weighting (A, C, G, Z)](https://jmrplens.github.io/phonometry/guides/weighting/)
- [Time Weighting and Integration](https://jmrplens.github.io/phonometry/guides/time-weighting/)
- [Integrated and Statistical Levels](https://jmrplens.github.io/phonometry/guides/levels/)
- [Occupational Noise Exposure (ISO 9612)](https://jmrplens.github.io/phonometry/guides/occupational-exposure/)
- [Prominent Discrete Tones (ECMA-418-1)](https://jmrplens.github.io/phonometry/guides/tone-prominence/)
- [Psychoacoustics](https://jmrplens.github.io/phonometry/guides/psychoacoustics/)
- [Speech Transmission Index (IEC 60268-16)](https://jmrplens.github.io/phonometry/guides/speech-transmission/)
- [Speech Intelligibility Index (SII)](https://jmrplens.github.io/phonometry/guides/speech-intelligibility/)
- [Sound Intensity (p-p method)](https://jmrplens.github.io/phonometry/guides/intensity/)
- [Room Acoustics](https://jmrplens.github.io/phonometry/guides/room-acoustics/)
- [Building Acoustics & Sound Insulation](https://jmrplens.github.io/phonometry/guides/building-acoustics/)
- [Outdoor Sound Propagation](https://jmrplens.github.io/phonometry/guides/outdoor-propagation/)
- [Sound Power](https://jmrplens.github.io/phonometry/guides/sound-power/)
- [Calibration and dBFS](https://jmrplens.github.io/phonometry/guides/calibration/)
- [Block Processing](https://jmrplens.github.io/phonometry/guides/block-processing/)
- [Multichannel and Performance](https://jmrplens.github.io/phonometry/guides/multichannel/)
- [API Reference](https://jmrplens.github.io/phonometry/reference/api/)
- [Theoretical Background](https://jmrplens.github.io/phonometry/reference/theory/)
- [Why phonometry](https://jmrplens.github.io/phonometry/reference/why-phonometry/)

## Source and metadata

- [Repository](https://github.com/jmrplens/phonometry)
- [PyPI](https://pypi.org/project/phonometry/)
- [Changelog](https://github.com/jmrplens/phonometry/blob/main/CHANGELOG.md)
- [Cite this software](https://github.com/jmrplens/phonometry/blob/main/CITATION.cff)


---


<!-- source: docs/getting-started.md | canonical: https://jmrplens.github.io/phonometry/getting-started/ -->

# Getting Started

## Installation

**Option 1: From PyPI (Recommended)**

```bash
pip install phonometry
```

Optional extras:

```bash
pip install phonometry[plot]   # matplotlib, for filter response plots and result .plot() methods
pip install phonometry[perf]  # numba, faster 'impulse' time weighting
pip install phonometry[full]  # both
```

**Option 2: Cloning and Installing**

```bash
git clone https://github.com/jmrplens/phonometry.git
cd phonometry
pip install .
```

**Option 3: Git Submodule**

```bash
git submodule add https://github.com/jmrplens/phonometry.git
# Then install in editable mode to use it from your project
pip install -e ./phonometry
```

## The processing chain at a glance

Every phonometry analysis is some subset of one pipeline — take the raw
signal, convert it to physical units, weight it in frequency, split it into
standardized bands, smooth it in time and reduce it to metrics:

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/diagram_signal_chain_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/diagram_signal_chain.svg" alt="phonometry processing chain: signal, calibration, frequency weighting, octave filter bank, time weighting and metrics, with the standard verified at each stage" width="92%"></picture>

Each stage is an independent function or class you can use on its own; the
guides cover them left to right ([Calibration](https://jmrplens.github.io/phonometry/guides/calibration/) →
[Frequency Weighting](https://jmrplens.github.io/phonometry/guides/weighting/) → [Filter Banks](https://jmrplens.github.io/phonometry/guides/filter-banks/) →
[Time Weighting](https://jmrplens.github.io/phonometry/guides/time-weighting/) → [Levels](https://jmrplens.github.io/phonometry/guides/levels/)).

## Basic Usage: 1/3 Octave Analysis

Analyze a signal and get the Sound Pressure Level (SPL) per frequency band.

```python
import numpy as np
from phonometry import octave_filter

fs = 48000
t = np.linspace(0, 1, fs, endpoint=False)
# Composite signal: 100Hz + 1000Hz
signal = np.sin(2 * np.pi * 100 * t) + np.sin(2 * np.pi * 1000 * t)

# Apply 1/3 octave filter bank
spl, freq = octave_filter(signal, fs=fs, fraction=3)

print(f"Bands: {freq}")
# Bands: [12.589254117941678, 15.848931924611138, ..., 19952.623149688785]  (33 bands)
print(f"SPL [dB]: {spl}")
# SPL [dB]: [46.88395351 47.96774897 49.04991279 ...]  — ~90.7 dB at 100 Hz and ~90.9 dB at 1 kHz
```

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/signal_response_fraction_3_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/signal_response_fraction_3.svg" alt="One-third-octave spectrum analysis of a multi-tone signal with the raw PSD in the background" width="80%"></picture>

*Example of a 1/3 Octave Band spectrum analysis of a complex signal.*

<details>
<summary>Show the code for this figure</summary>

```python
import matplotlib.pyplot as plt
import scipy.signal

# Uses `signal`, `fs`, `spl` and `freq` from the snippet above.
# Gray background: the raw-signal PSD (Welch), shifted to sit just below the
# band SPLs so both spectral shapes share one axis.
f_psd, psd = scipy.signal.welch(signal, fs, nperseg=8192)
psd_db = 10 * np.log10(psd + 1e-12)
psd_db += np.max(spl) - np.max(psd_db) - 5

fig, ax = plt.subplots()
ax.semilogx(f_psd, psd_db, color="gray", alpha=0.6, label="Raw signal PSD")
ax.semilogx(freq, spl, marker="o", markerfacecolor="white",
            label="1/3 octave bands")
ax.set_xlabel("Frequency [Hz]")
ax.set_ylabel("SPL [dB]")
ax.legend()
plt.show()
```

</details>

## Analyzing an audio file

```python
from scipy.io import wavfile
from phonometry import octave_filter

# Load standard WAV file
fs, signal = wavfile.read("measurement.wav")

# Analyze
# Note: To obtain real-world SPL values, you must calibrate the input.
# See the Calibration guide.
spl, freq = octave_filter(signal, fs=fs, fraction=3)
```

Integer audio (e.g. int16 WAV data) is converted to float64 internally, so it is
safe to pass `wavfile.read` output directly.

## Where to go next

- [Filter Banks](https://jmrplens.github.io/phonometry/guides/filter-banks/) — choose an architecture and inspect responses
- [Calibration and dBFS](https://jmrplens.github.io/phonometry/guides/calibration/) — get real-world SPL values
- [API Reference](https://jmrplens.github.io/phonometry/reference/api/) — every parameter of every function

---


<!-- source: docs/filter-banks.md | canonical: https://jmrplens.github.io/phonometry/guides/filter-banks/ -->

# Filter Banks

phonometry supports several filter types, each with its own transfer function
characteristic. All banks place their **−3 dB points on the ANSI S1.11 band
edges**, so band levels are comparable across architectures.

## 1. Fractional octave bands: the math

IEC 61260-1:2014 builds every band from the base-10 octave ratio
$G = 10^{3/10} \approx 1.99526$ (so "one octave" is *not* exactly 2). For
band fraction $1/b$, the mid frequencies and band edges follow (5.2-5.5):

$$
f_m = 1000 \cdot G^{x/b} \quad (b\ \text{odd}), \qquad
f_1 = f_m G^{-1/2b}, \quad f_2 = f_m G^{+1/2b}
$$

so every 1/3-octave band spans $G^{1/3} \approx 1.2589 \approx 10^{1/10}$ —
ten bands per decade, which is why the nominal frequencies (25, 31.5, 40 …)
repeat scaled by 10. phonometry designs each band as an SOS cascade whose
−3 dB points land exactly on $f_1$ and $f_2$ for every architecture — for
Chebyshev II, Elliptic and Bessel that requires pre-warping the analytic
band-edge mapping rather than trusting SciPy's default parametrization.

### Multirate decimation

A 25 Hz one-third-octave band at 48 kHz spans about 5.8 Hz — 0.024 %
of Nyquist — coefficients so stiff they go numerically unstable. The bank
avoids that by filtering low bands at a decimated rate:

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/diagram_multirate_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/diagram_multirate.svg" alt="Multirate decimation: high bands filtered at the input rate, low bands after anti-alias low-pass and decimation so the SOS sections stay numerically healthy" width="92%"></picture>

## 2. Filter Comparison and Zoom

We use Second-Order Sections (SOS) for all filters to ensure numerical stability.
The following plot compares the architectures focusing on the -3 dB crossover point.

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/filter_type_comparison_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/filter_type_comparison.svg" alt="Magnitude response comparison of the five filter architectures for the 1 kHz octave band, with a zoom at the -3 dB crossover" width="80%"></picture>

| Type | Name | Usage Example | Best For |
| :--- | :--- | :--- | :--- |
| `butter` | **Butterworth** | `octave_filter(x, fs, filter_type='butter')` | General acoustic measurement. |
| `cheby1` | **Chebyshev I** | `octave_filter(x, fs, filter_type='cheby1', ripple=0.1)` | Sharper roll-off at the cost of ripple. |
| `cheby2` | **Chebyshev II** | `octave_filter(x, fs, filter_type='cheby2')` | Flat passband with stopband zeros. |
| `ellip` | **Elliptic** | `octave_filter(x, fs, filter_type='ellip', ripple=0.1)` | Maximum selectivity. |
| `bessel` | **Bessel** | `octave_filter(x, fs, filter_type='bessel')` | Preserving transient waveform shapes. |

## 3. `octave_filter()` / `OctaveFilterBank` parameters

| Parameter | Type | Units | Range / default | Notes |
| :--- | :--- | :--- | :--- | :--- |
| `x` | 1D or 2D array | digital units | non-empty | 2D is `[channels, samples]` |
| `fs` | int | Hz | > 0 | |
| `fraction` | int | — | default `1`; common `3`; any `b ≥ 1` | Bands per octave = `b` |
| `order` | int | — | default `6` | SOS order per band |
| `limits` | list `[lo, hi]` | Hz | default `[12, 20000]` | Analysis range |
| `filter_type` | str | — | `'butter'` (default), `'cheby1'`, `'cheby2'`, `'ellip'`, `'bessel'` | See comparison above |
| `ripple` / `attenuation` | float | dB | `ripple` default `0.1`; `attenuation` default `72.0` | Passband ripple / stopband attenuation (cheby/ellip); `cheby2` needs `attenuation ≥ 70` for class 1, since scipy pins its equiripple floor at exactly this value |
| `show` | bool | — | default `False` | Plot the bank response (needs matplotlib) |
| `sigbands` | bool | — | default `False` | Also return the per-band time signals |
| `mode` | str | — | `'rms'` (default), `'peak'`, `'sum'` | Per-band statistic returned |
| `nominal` | bool | — | default `False` | Return nominal band labels (e.g. `1000`) instead of exact centre frequencies |
| `detrend` | bool | — | default `True` | Remove each band's DC offset before the level (improves low-frequency accuracy) |
| `calibration_factor` | float | — | default `1.0` | Scales the input to pascals (see the Calibration guide) |
| `dbfs` | bool | — | default `False` | Reference levels to digital full scale instead of 20 µPa |
| `plot_file` | str or `None` | — | default `None` | Save the bank-response plot to this path |
| `zero_phase` | bool | — | default `False` | Forward-backward filtering (offline) |
| `stateful` / `steady_ic` (class) | bool | — | default `False` | Streaming state; see [Block Processing](https://jmrplens.github.io/phonometry/guides/block-processing/) |

`verify_filter_class(bank)` checks the designed bank against the IEC 61260-1
Table 1 acceptance limits and reports the class (`1`, `2` or `None` if outside both) with per-band
margins.

## 4. Gallery of Filter Bank Responses

Full spectral view of the filter banks for Octave (1/1) and 1/3-Octave fractions.

| Architecture | 1/1 Octave (Fraction=1) | 1/3 Octave (Fraction=3) |
| :--- | :--- | :--- |
| **Butterworth** | <picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/filter_butter_fraction_1_order_6_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/filter_butter_fraction_1_order_6.svg" alt="Butterworth octave-band filter bank frequency response" width="100%"></picture> | <picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/filter_butter_fraction_3_order_6_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/filter_butter_fraction_3_order_6.svg" alt="Butterworth one-third-octave filter bank frequency response" width="100%"></picture> |
| **Chebyshev I** | <picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/filter_cheby1_fraction_1_order_6_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/filter_cheby1_fraction_1_order_6.svg" alt="Chebyshev I octave-band filter bank frequency response" width="100%"></picture> | <picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/filter_cheby1_fraction_3_order_6_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/filter_cheby1_fraction_3_order_6.svg" alt="Chebyshev I one-third-octave filter bank frequency response" width="100%"></picture> |
| **Chebyshev II** | <picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/filter_cheby2_fraction_1_order_6_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/filter_cheby2_fraction_1_order_6.svg" alt="Chebyshev II octave-band filter bank frequency response" width="100%"></picture> | <picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/filter_cheby2_fraction_3_order_6_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/filter_cheby2_fraction_3_order_6.svg" alt="Chebyshev II one-third-octave filter bank frequency response" width="100%"></picture> |
| **Elliptic** | <picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/filter_ellip_fraction_1_order_6_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/filter_ellip_fraction_1_order_6.svg" alt="Elliptic octave-band filter bank frequency response" width="100%"></picture> | <picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/filter_ellip_fraction_3_order_6_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/filter_ellip_fraction_3_order_6.svg" alt="Elliptic one-third-octave filter bank frequency response" width="100%"></picture> |
| **Bessel** | <picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/filter_bessel_fraction_1_order_6_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/filter_bessel_fraction_1_order_6.svg" alt="Bessel octave-band filter bank frequency response" width="100%"></picture> | <picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/filter_bessel_fraction_3_order_6_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/filter_bessel_fraction_3_order_6.svg" alt="Bessel one-third-octave filter bank frequency response" width="100%"></picture> |

## 5. Filter Usage and Examples

### 1. Butterworth (`butter`)

The Butterworth filter is known for its **maximally flat passband**. It is the
standard choice for acoustic measurements where no ripple is allowed within the
frequency bands.

```python
import numpy as np
from phonometry import octave_filter

# A calibrated signal in Pa so the guide runs standalone
fs = 48000
x = 0.2 * np.sin(2 * np.pi * 1000 * np.arange(fs) / fs)

# Default standard measurement
spl, freq = octave_filter(x, fs, filter_type='butter')
```

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/filter_butter_fraction_3_order_6_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/filter_butter_fraction_3_order_6.svg" alt="Butterworth one-third-octave filter bank frequency response" width="60%"></picture>

### 2. Chebyshev I (`cheby1`)

Chebyshev Type I filters provide a **steeper roll-off** than Butterworth at the
expense of ripples in the passband. Useful when high selectivity is needed near
the cut-off frequencies.

```python
# Uses `x` and `fs` from the snippet above.
# Selectivity with 0.1 dB passband ripple
spl, freq = octave_filter(x, fs, filter_type='cheby1', ripple=0.1)
```

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/filter_cheby1_fraction_3_order_6_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/filter_cheby1_fraction_3_order_6.svg" alt="Chebyshev I one-third-octave filter bank frequency response" width="60%"></picture>

### 3. Chebyshev II (`cheby2`)

Also known as Inverse Chebyshev, it has a **flat passband** and ripples in the
stopband. It provides faster roll-off than Butterworth without affecting the
signal in the passband. The stopband edges are placed automatically so that the
−3 dB points land on the band edges (`attenuation` must be > 3.01 dB).

```python
# Uses `x` and `fs` from the snippet above.
# Flat passband, class-1 default 72 dB stopband attenuation
spl, freq = octave_filter(x, fs, filter_type='cheby2')
```

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/filter_cheby2_fraction_3_order_6_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/filter_cheby2_fraction_3_order_6.svg" alt="Chebyshev II one-third-octave filter bank frequency response" width="60%"></picture>

### 4. Elliptic (`ellip`)

Elliptic (Cauer) filters have the **shortest transition width** (steepest
roll-off) for a given order. They feature ripples in both the passband and stopband.

```python
# Uses `x` and `fs` from the snippet above.
# Maximum selectivity for extreme band isolation
spl, freq = octave_filter(x, fs, filter_type='ellip', ripple=0.1)
```

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/filter_ellip_fraction_3_order_6_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/filter_ellip_fraction_3_order_6.svg" alt="Elliptic one-third-octave filter bank frequency response" width="60%"></picture>

### 5. Bessel (`bessel`)

Bessel filters are optimized for **linear phase response** and minimal group
delay. They preserve the shape of filtered waveforms (transients) better than
any other type, but have the slowest roll-off.

```python
# Uses `x` and `fs` from the snippet above.
# Best for pulse analysis and transient preservation
spl, freq = octave_filter(x, fs, filter_type='bessel')
```

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/filter_bessel_fraction_3_order_6_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/filter_bessel_fraction_3_order_6.svg" alt="Bessel one-third-octave filter bank frequency response" width="60%"></picture>

### 6. Linkwitz-Riley (`linkwitz_riley`)

Specifically designed for **audio crossovers**. Linkwitz-Riley filters (typically
4th order, but any even order is supported) allow splitting a signal into bands
that, when summed, result in a perfectly flat magnitude response and zero phase
difference between bands at the crossover.

```python
from phonometry import linkwitz_riley

# Uses `x` and `fs` from the snippet above.
signal = x
# Split signal into Low and High bands at 1000 Hz
low, high = linkwitz_riley(signal, fs, freq=1000, order=4)
# Reconstruction: low + high == signal (flat response)
```

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/crossover_lr4_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/crossover_lr4.svg" alt="Linkwitz-Riley 4th-order crossover: low-pass, high-pass and their flat sum" width="60%"></picture>

## 6. Verifying the IEC 61260-1 class

`verify_filter_class` checks every band of a bank against the acceptance
limits of **IEC 61260-1:2014** (Table 1, with the fractional-octave breakpoint
mapping and log-frequency interpolation from the standard) and reports the
performance class per band with its margin in dB:

```python
from phonometry import OctaveFilterBank, verify_filter_class

bank = OctaveFilterBank(fs=48000, fraction=3, order=6)
result = verify_filter_class(bank)
print(result["overall_class"])          # 1
print(result["bands"][0])
# {'freq': 12.589254117941678, 'class': 1, 'margin_class1_db': 0.39999999999997266, 'margin_class2_db': 0.5999999999999727}
```

The Table 1 acceptance mask itself is public too: `class_limits(fraction,
filter_class, omega)` returns the minimum/maximum relative-attenuation
limits at normalized frequencies Ω = f/fm — the same limits the verifier
and the figure below use.

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/class_mask_overlay_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/class_mask_overlay.svg" alt="Butterworth band response threading between the forbidden regions of the IEC 61260-1 class 1 acceptance mask" width="80%"></picture>

*The order-6 Butterworth response (blue) threads between the forbidden
regions: it must attenuate at least the red mask outside the band and no more
than the purple mask inside it.*

With default parameters (order 6), **Butterworth meets class 1**, and so does
**Chebyshev II**: its `attenuation` default is now `72` dB, clearing the 70 dB
far-stopband class 1 limit (scipy pins the cheby2 equiripple floor at exactly
`attenuation`, so any value ≥ 70 dB qualifies; the 72 dB default keeps the same
+0.400 dB passband margin as Butterworth). Chebyshev I, Elliptic and Bessel do
not meet class limits at order 6: passband ripple (cheby1/ellip) and slow
roll-off (bessel) violate the mask.

### Class 0 (IEC 61260:1995 / ANSI S1.11-2004)

The tightest performance class, **class 0**, was defined by the earlier
**IEC 61260:1995** and its US twin **ANSI S1.11-2004** (both withdrawn/superseded
but still referenced for laboratory-grade instruments); IEC 61260-1:2014 dropped
it. Its class 1/2 masks differ slightly from the 2014 edition, so it lives behind
an `edition` switch rather than being mixed into the 2014 mask:

```python
result = verify_filter_class(bank, edition="1995")   # classes 0, 1, 2
print(result["overall_class"])          # 0  (the default Butterworth clears it)
print(result["bands"][0]["margin_class0_db"])
```

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/filter_class0_mask_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/filter_class0_mask.svg" alt="Nested pass-band acceptance corridors for class 0, 1 and 2 of IEC 61260:1995 with the order-6 Butterworth response sitting inside the tightest class 0 corridor" width="80%"></picture>

*The class 0 corridor (±0.15 dB at mid-band) is the tightest; class 1 (±0.3 dB)
and class 2 (±0.5 dB) are progressively wider. The order-6 Butterworth threads
inside class 0 across the whole pass-band.*

**Which architecture reaches which class?** The library's **default — Butterworth
order 6 — already meets class 0** in the usual configurations (octave and
third-octave, 32 kHz–96 kHz), so no special setup is needed for laboratory-grade
banks. The other architectures cannot be tuned to class 0 because they trade the
IEC mask for a different property *by construction*:

| Architecture | Best class (order 6, fs 48 kHz) | Why |
| :--- | :---: | :--- |
| `butter` (default) | **0** | Maximally-flat pass-band, monotone roll-off — fits the mask |
| `cheby2` | 1 | Flat pass-band but the mask relationship binds at class 1 |
| `cheby1` | — | Pass-band ripple violates the flatness limit |
| `ellip` | — | Pass- and stop-band ripple |
| `bessel` | — | Flat group delay bought with a slow roll-off |

So the sensible default is the common one (Butterworth order 6): it is already
class 0 where the physics allows, and the alternative architectures are
deliberate opt-ins whose purpose (steeper roll-off, linear phase) is
incompatible with the class mask. Raise the order only if a specific
high-`fraction` / near-Nyquist band needs more margin.

## 7. Signal Decomposition and Stability

By setting `sigbands=True`, you can retrieve the time-domain components of each
band. This allows for advanced analysis or comparing how different architectures
(e.g., Butterworth vs Chebyshev) affect the signal phase and transient response.

```python
import numpy as np
from phonometry import octave_filter

# 1. Generate a signal (Sum of 250Hz and 1000Hz)
fs = 48000
t = np.linspace(0, 0.5, int(fs * 0.5), endpoint=False)
y = np.sin(2 * np.pi * 250 * t) + np.sin(2 * np.pi * 1000 * t)

# 2. Compare architectures (Butterworth vs Chebyshev II)
spl_b, freq, xb_butter = octave_filter(y, fs=fs, fraction=1, sigbands=True, filter_type='butter')
spl_c2, _, xb_cheby2 = octave_filter(y, fs=fs, fraction=1, sigbands=True, filter_type='cheby2')

# 'xb_butter' and 'xb_cheby2' contain the time-domain signals per band
```

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/signal_decomposition_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/signal_decomposition.svg" alt="Time-domain band decomposition comparing Butterworth and Chebyshev II, including the impulse response" width="80%"></picture>

*The plot compares the **Butterworth** (solid blue) and **Chebyshev II** (dashed
red) responses. The bottom plot shows the **Impulse Response**, highlighting the
differences in stability and transient decay.*

> [!NOTE]
> **Why do the signals look shifted in time?**
> Digital IIR filters (like Butterworth or Chebyshev) have **non-linear phase
> responses**, which results in frequency-dependent **Group Delay**. In the 250 Hz
> band, you can see that the Chebyshev II filter has a different propagation delay
> compared to the Butterworth filter. This is a normal physical property of these
> architectures: more aggressive frequency roll-offs usually come at the cost of
> higher group delay and phase distortion.

### Group delay, quantified

The group delay $\tau_g(\omega) = -\frac{d\phi(\omega)}{d\omega}$ of the
1 kHz octave band shows the trade-off directly: Bessel stays nearly flat across
the passband (transient shapes survive), while Chebyshev I and Elliptic pay for
their steep roll-off with strong delay peaks at the band edges.

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/group_delay_comparison_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/group_delay_comparison.svg" alt="Group delay of the 1 kHz octave band for the five architectures: Bessel nearly flat, Chebyshev and Elliptic peaking at the band edges" width="80%"></picture>

## 8. Zero-phase filtering

For offline analysis you can eliminate group delay entirely: `zero_phase=True`
filters each band forward-backward (`scipy.signal.sosfiltfilt`), keeping band
signals time-aligned with the input. The effective attenuation doubles and the
effective passband narrows, lowering the measured broadband band level by
~0.2 to 0.3 dB per band (a pure in-band tone is unaffected); prefer forward
filtering when the absolute band SPL must match single-pass conventions, and
reserve zero-phase for when the temporal envelope matters (e.g. reverberation
decay). The option is incompatible with stateful (block) processing.

```python
from phonometry import OctaveFilterBank

# Uses `y` from the snippet above.
bank = OctaveFilterBank(fs=48000, fraction=3)
spl, freq, xb = bank.filter(y, sigbands=True, zero_phase=True)
```

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/zero_phase_comparison_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/zero_phase_comparison.svg" alt="Causal versus zero-phase filtering of a tone burst: the zero-phase output stays time-aligned with the input" width="80%"></picture>

*Causal filtering delays the burst by the filter's group delay; zero-phase
filtering keeps it aligned with the input.*

---

**Standards.** IEC 61260-1:2014, *Electroacoustics — Octave-band and
fractional-octave-band filters — Part 1: Specifications* — the base-10 mid
frequencies and band edges of §1 (5.2-5.5), the nominal band labels, and the
Table 1 class 1 / class 2 acceptance limits (with the fractional-octave
breakpoint mapping and log-frequency interpolation) verified in §6.
IEC 61260:1995 and ANSI S1.11-2004, *Octave-Band and Fractional-Octave-Band …
Filters* — the withdrawn edition's Table 1 (identical between the two) supplies
the stricter class 0 mask offered by ``edition="1995"``, and the band-edge
convention on which every bank places its −3 dB
points. ISO 266 — the preferred-frequency series behind the nominal band
labels reported by `nominal_frequencies`.

---


<!-- source: docs/weighting.md | canonical: https://jmrplens.github.io/phonometry/guides/weighting/ -->

# Frequency Weighting (A, C, G, Z)

Frequency weighting curves simulate the human ear's sensitivity. A, C and Z
are specified by **IEC 61672-1:2013**; the infrasound G curve is specified by
**ISO 7196:1995**.

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/weighting_responses_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/weighting_responses.svg" alt="A, C and Z frequency weighting curves with a zoom showing the positive region of the A curve (+1.27 dB at 2.5 kHz)" width="80%"></picture>

<details>
<summary>Show the code for this figure</summary>

```python
import matplotlib.pyplot as plt
import numpy as np
from phonometry import weighting_filter

# Measure each curve's response: weight a centered unit impulse and take
# its spectrum (1 s buffer -> 1 Hz frequency resolution).
fs = 48000
impulse = np.zeros(fs)
impulse[fs // 2] = 1.0
freqs = np.fft.rfftfreq(fs, 1 / fs)

fig, ax = plt.subplots(figsize=(9, 5))
for curve in ("A", "C", "Z"):
    spectrum = np.fft.rfft(weighting_filter(impulse, fs, curve=curve))
    ax.semilogx(freqs[1:], 20 * np.log10(np.abs(spectrum[1:]) + np.finfo(float).eps),
                label=curve)
ax.set(xlim=(10, 20000), ylim=(-80, 10),
       xlabel="Frequency [Hz]", ylabel="Response [dB]")
ax.grid(True, which="both", alpha=0.3)
ax.legend()
plt.show()
```

</details>

* **A-Weighting (`A`):** Standard for environmental noise (IEC 61672-1).
* **C-Weighting (`C`):** Used for peak sound pressure and high-level noise.
* **Z-Weighting (`Z`):** Zero weighting, completely flat response.
* **G-Weighting (`G`):** Infrasound weighting per ISO 7196 (see below).

## 1. Where the curves come from

The A and C curves are inverted equal-loudness contours, frozen into filters:
**A** approximates the inverse of the historic 40-phon contour (quiet levels,
where the ear discards bass most aggressively) and **C** the flatter ~100-phon
one (loud levels). IEC 61672-1:2013 (Annex E) defines both analytically from
four corner frequencies:

$$
f_1 = 20.599\ \text{Hz}, \quad f_2 = 107.653\ \text{Hz}, \quad
f_3 = 737.862\ \text{Hz}, \quad f_4 = 12194.217\ \text{Hz}
$$

C is a band-pass with double poles at $f_1$ and $f_4$ (2 zeros at the origin);
A adds the $f_2$ and $f_3$ poles (4 zeros), which is why it keeps falling
through the low-mids. Both are normalized to exactly 0 dB at 1 kHz. Z is the
absence of weighting. The full pole/zero derivation is in the
[Theory](https://jmrplens.github.io/phonometry/reference/theory/) page.

## 2. Basic usage

```python
import numpy as np
from phonometry import weighting_filter

# recording: a calibrated microphone capture (Pa) — recorded through your measurement chain. Synthesized here so the guide runs standalone.
fs = 48000
recording = 0.2 * np.sin(2 * np.pi * 1000 * np.arange(fs) / fs)

# Apply A-weighting to the raw recording
weighted_signal = weighting_filter(recording, fs, curve='A')

# Apply C-weighting for peak analysis
c_weighted_signal = weighting_filter(recording, fs, curve='C')
```

## 3. Infrasound: G-weighting (ISO 7196)

The **G frequency weighting** (ISO 7196:1995) rates infrasound the way A-weighting
rates audible noise. It is defined by a pole-zero configuration with 0 dB gain at
10 Hz, rises at 12 dB/octave from 1 Hz to 20 Hz (matching the steep growth of
perception in that band) and falls off at 24 dB/octave outside it. Use it for
sources with significant energy below 20 Hz (wind turbines, HVAC, blasting):

```python
from phonometry import weighting_filter

# Uses `recording` and `fs` from the snippet above.
g_weighted = weighting_filter(recording, fs, curve='G')
```

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/g_weighting_response_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/g_weighting_response.svg" alt="G-weighting frequency response from 0.1 Hz to 1 kHz with the ISO 7196 Table 2 nominal values overlaid" width="80%"></picture>

<details>
<summary>Show the code for this figure</summary>

```python
import matplotlib.pyplot as plt
import numpy as np
from phonometry import weighting_filter

# Measure the G response: weight a centered unit impulse and take its
# spectrum. A long buffer gives the resolution the infrasound range
# needs (20 s -> 0.05 Hz).
fs = 4000
impulse = np.zeros(20 * fs)
impulse[impulse.size // 2] = 1.0
freqs = np.fft.rfftfreq(impulse.size, 1 / fs)
spectrum = np.fft.rfft(weighting_filter(impulse, fs, curve="G"))

fig, ax = plt.subplots(figsize=(9, 5))
ax.semilogx(freqs[1:],
            20 * np.log10(np.abs(spectrum[1:]) + np.finfo(float).eps))
ax.plot(10, 0, "o", color="tab:red", label="0 dB at 10 Hz")
ax.set(xlim=(0.1, 1000), ylim=(-90, 15),
       xlabel="Frequency [Hz]", ylabel="G-weighting response [dB]")
ax.grid(True, which="both", alpha=0.3)
ax.legend()
plt.show()
```

</details>

The implementation follows the ISO 7196 Table 1 pole/zero values exactly and is
verified in CI against every Table 2 nominal response value (0.25 Hz to 315 Hz).
`WeightingFilter(fs, "G")` supports the same multichannel and stateful block
processing as A/C. Levels measured with the G curve are reported as
L<sub>pG</sub> (or L<sub>Geq</sub> for the equivalent level over time).

## 4. `weighting_filter()` / `WeightingFilter` parameters

| Parameter | Type | Units | Range / default | Notes |
| :--- | :--- | :--- | :--- | :--- |
| `x` | 1D or 2D array | any | non-empty | 2D is `[channels, samples]` |
| `fs` | int | Hz | > 0 | |
| `curve` | str | — | `'A'` (default), `'C'`, `'G'`, `'Z'` | `'G'` per ISO 7196 (infrasound); `'Z'` is a bypass |
| `high_accuracy` | bool | — | default `True` (function); class default `None` resolves to `not stateful` | Internal oversampling (up to 8×, reaching ≥ 144 kHz at common audio rates, e.g. 96 kHz input ×2) keeps A/C in class 1 up to 16 kHz; silently ignored for G, whose 0.25–315 Hz range the plain design already renders exactly |
| `stateful` | bool (class only) | — | default `False` | Carries filter state across blocks (streaming) |
| `steady_ic` | bool (class only) | — | default `False` | Steady-state initial conditions (no onset transient) |

## 5. Reusable filter object

If you weight many signals with the same parameters, design the filter once:

```python
from phonometry import WeightingFilter

# Uses `recording` and `fs` from the snippet above.
wf = WeightingFilter(fs, "A")
signals = [recording]                # your batch of recordings
for recording in signals:
    weighted = wf.filter(recording)
```

## 6. High-frequency accuracy (`high_accuracy`)

A plain bilinear-transform design compresses the response near Nyquist: at
fs = 48 kHz the A-curve error at 12.5 kHz reaches −2.7 dB, outside the IEC
61672-1 **class 1** tolerance (+2.0/−2.5 dB).

By default (`high_accuracy=True`), phonometry designs and runs the weighting
filter at an internally oversampled rate (≥ 144 kHz) and decimates back, keeping
the response within class 1 tolerances up to 16 kHz (error ≈ −0.5 dB at
12.5 kHz for fs = 48 kHz).

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/weighting_accuracy_hf_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/weighting_accuracy_hf.svg" alt="A-weighting high-frequency accuracy at 48 kHz: analytic curve versus plain bilinear versus oversampled design, with error subplot" width="80%"></picture>

*The plain bilinear design (red) crosses the class 1 tolerance near 12.5 kHz;
the oversampled design (blue) stays close to the analytic curve.*

<details>
<summary>Show the code for this figure</summary>

```python
import matplotlib.pyplot as plt
import numpy as np
from phonometry import weighting_filter

# Measured response of both designs at fs = 48 kHz: weight a centered
# unit impulse and take its spectrum...
fs = 48000
impulse = np.zeros(fs)
impulse[fs // 2] = 1.0
freqs = np.fft.rfftfreq(fs, 1 / fs)[1:]

# ...versus the analytic IEC 61672-1 A-curve built from the four corner
# frequencies of section 1, normalized to 0 dB at 1 kHz.
f1, f2, f3, f4 = 20.599, 107.653, 737.862, 12194.217
gain = (f4**2 * freqs**4) / ((freqs**2 + f1**2)
        * np.sqrt((freqs**2 + f2**2) * (freqs**2 + f3**2))
        * (freqs**2 + f4**2))
analytic = 20 * np.log10(gain / gain[np.argmin(np.abs(freqs - 1000))])

fig, ax = plt.subplots(figsize=(9, 5))
ax.semilogx(freqs, analytic, "k--", label="Analytic (IEC 61672-1)")
for high_accuracy, label in ((False, "Plain bilinear"),
                             (True, "Oversampled (default)")):
    weighted = weighting_filter(impulse, fs, curve="A",
                                high_accuracy=high_accuracy)
    response = 20 * np.log10(np.abs(np.fft.rfft(weighted))
                             + np.finfo(float).eps)[1:]
    ax.semilogx(freqs, response, label=label)
ax.set(xlim=(1000, 20000), ylim=(-12, 3),
       xlabel="Frequency [Hz]", ylabel="A-weighting response [dB]")
ax.grid(True, which="both", alpha=0.3)
ax.legend()
plt.show()
```

</details>

- `high_accuracy=False` restores the legacy plain-bilinear behavior.
- **Stateful (block) processing** always uses the legacy design: the internal
  FIR resampling is incompatible with block continuity. Passing
  `high_accuracy=True` together with `stateful=True` raises a `ValueError`.

```python
from phonometry import WeightingFilter, weighting_filter

# Uses `recording` and `fs` from the snippet above.
# Explicit legacy behavior
y = weighting_filter(recording, fs, curve="A", high_accuracy=False)

# Stateful block processing (legacy design, state carried between blocks)
wf = WeightingFilter(fs, "A", stateful=True)
blocks = [recording]                 # your sequence of recording blocks
for block in blocks:
    weighted = wf.filter(block)
```

See [Block Processing](https://jmrplens.github.io/phonometry/guides/block-processing/) for the streaming workflow and
[Theory](https://jmrplens.github.io/phonometry/reference/theory/) for the analytic curve definitions.

## 7. Verifying the IEC 61672-1 class

`verify_weighting_class` checks a weighting filter against the acceptance
limits of **IEC 61672-1:2013** (Table 3). It evaluates the filter's relative
response at each nominal frequency below Nyquist, subtracts the design-goal
weighting, and reports the performance class per frequency with its margin in
dB:

```python
from phonometry import WeightingFilter, verify_weighting_class

result = verify_weighting_class(WeightingFilter(48000, "A"))
print(result["overall_class"])          # 1
print(result["bands"][20])
# {'freq': 1000.0, 'class': 1, 'deviation_db': 0.0, 'margin_class1_db': 0.7, 'margin_class2_db': 1.0}
```

The Table 3 acceptance mask itself is public too: `weighting_class_limits(1)`
returns the 34 nominal frequencies with the lower/upper deviation limits (a
lower limit of `-inf` means only the upper limit applies). The limits qualify
the *deviation* from the design goal, so they are the same for A, C and Z.

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/weighting_class_mask_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/weighting_class_mask.svg" alt="A and C weighting deviations at 48 kHz threading within the IEC 61672-1 Table 3 class 1 acceptance corridor, with the wider class 2 limits dotted" width="80%"></picture>

*The oversampled A and C designs (blue, purple) stay near zero deviation,
well inside the class 1 corridor (shaded); the wider class 2 limits are
dotted. The corridor widens at the band extremes where only a one-sided
limit applies.*

<details>
<summary>Show the code for this figure</summary>

```python
import matplotlib.pyplot as plt
import numpy as np
from phonometry import WeightingFilter, verify_weighting_class, weighting_class_limits

freqs, lower1, upper1 = weighting_class_limits(1)
_, lower2, upper2 = weighting_class_limits(2)
lo1, lo2 = np.clip(lower1, -7, 7), np.clip(lower2, -7, 7)

fig, ax = plt.subplots(figsize=(10, 6.5))
ax.fill_between(freqs, lo1, upper1, step="mid", alpha=0.10,
                label="Class 1 acceptance region")
ax.plot(freqs, upper1, drawstyle="steps-mid", label="Class 1 upper/lower limit")
ax.plot(freqs, lo1, drawstyle="steps-mid", color="C1")
ax.plot(freqs, upper2, ":", drawstyle="steps-mid", label="Class 2 upper/lower limit")
ax.plot(freqs, lo2, ":", drawstyle="steps-mid", color="C2")

for curve, marker in (("A", "o"), ("C", "s")):
    bands = verify_weighting_class(WeightingFilter(48000, curve))["bands"]
    f = [b["freq"] for b in bands]
    dev = [b["deviation_db"] for b in bands]
    ax.plot(f, dev, marker=marker, label=f"{curve} weighting deviation (48 kHz)")

ax.set(xscale="log", xlim=(10, 20000), ylim=(-7, 7),
       xlabel="Frequency [Hz]", ylabel="Deviation from design goal [dB]")
ax.legend(fontsize=8, ncol=2)
plt.show()
```

</details>

---

**Standards.** IEC 61672-1:2013, *Electroacoustics — Sound level meters —
Part 1: Specifications* — the A, C and Z frequency-weighting curves (the
Annex E analytic definition from four corner frequencies, normalized to 0 dB
at 1 kHz), the class 1 tolerances the `high_accuracy` design keeps up to
16 kHz, and the Table 3 class 1/class 2 acceptance limits checked by
`verify_weighting_class`. ISO 7196:1995, *Acoustics — Frequency-weighting characteristic for
infrasound measurements* — the G-weighting pole/zero definition (Table 1),
verified against every Table 2 nominal response value (0.25 Hz to 315 Hz).

---


<!-- source: docs/time-weighting.md | canonical: https://jmrplens.github.io/phonometry/guides/time-weighting/ -->

# Time Weighting and Integration

Accurate SPL measurement requires capturing energy over specific time windows.
phonometry implements exact time constants per **IEC 61672-1:2013**.

## 1. The exponential detector

A sound level meter's needle cannot follow the pressure waveform — it shows a
running *mean square* with an exponential memory. Formally (IEC 61672-1, 3.8):

$$
\tau\ \frac{dy}{dt} + y = x^2(t)
\quad\Longleftrightarrow\quad
y(t) = \frac{1}{\tau} \int_{-\infty}^{t} x^2(\xi)\ e^{-(t-\xi)/\tau}\ d\xi
$$

a first-order low-pass on the squared signal. The time constant τ sets the
trade-off: **Fast** (125 ms) follows speech-like fluctuations, **Slow** (1 s)
steadies the readout for quasi-stationary noise. After a step onset the
envelope reaches 63 % of its final value in one τ and ~99.97 % after 8τ —
that is why level analyses discard the first instants of a recording.

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/diagram_time_weighting_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/diagram_time_weighting.svg" alt="The exponential-detector chain: the band pressure is squared, smoothed by a one-pole RC low-pass with time constant tau, and converted to decibels, with the Fast, Slow and Impulse time constants listed" width="88%"></picture>

## 2. The three time weightings

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/anim_time_weighting_dark.gif"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/anim_time_weighting.gif" alt="Animation: the Fast, Slow and Impulse detectors chasing a tone burst — Fast rises and falls quickly, Slow lags, and Impulse rises fast but decays the slowest" width="88%"></picture>

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/time_weighting_analysis_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/time_weighting_analysis.svg" alt="Fast, Slow and Impulse time weighting responses to a noise burst" width="80%"></picture>

<details>
<summary>Show the code for this figure</summary>

```python
import numpy as np
import matplotlib.pyplot as plt
from phonometry import time_weighting

fs = 48000
t = np.arange(int(fs * 4)) / fs
burst = np.zeros_like(t)  # 0.5 s noise burst (Pa) starting at t = 1 s
rng = np.random.default_rng(42)
burst[fs:int(1.5 * fs)] = 0.2 * rng.standard_normal(int(0.5 * fs))

p0 = 2e-5
plt.figure()
for mode in ('fast', 'slow', 'impulse'):
    envelope = time_weighting(burst, fs, mode=mode)
    plt.plot(t, 10 * np.log10(np.maximum(envelope, 1e-12) / p0**2), label=mode)
plt.xlabel('Time [s]')
plt.ylabel('Level [dB SPL]')
plt.legend()
plt.show()
```

</details>

* **Fast (`fast`):** τ = 125 ms. Standard for noise fluctuations.
* **Slow (`slow`):** τ = 1000 ms. Standard for steady noise.
* **Impulse (`impulse`):** **Asymmetric** ballistics. 35 ms rise time for rapid
  onset capture, 1500 ms decay for readability.

```python
import numpy as np
from phonometry import time_weighting

# recording: a calibrated microphone capture (Pa) — recorded through your measurement chain. Synthesized here so the guide runs standalone.
fs = 48000
recording = 0.2 * np.sin(2 * np.pi * 1000 * np.arange(fs) / fs)

# Calculate energy envelope (Mean Square)
energy_envelope = time_weighting(recording, fs, mode='fast')
# dB SPL relative to 20 μPa
spl_t = 10 * np.log10(energy_envelope / (2e-5)**2)

print(f"Steady-state Fast level: {spl_t[-1]:.1f} dB SPL")
# Steady-state Fast level: 77.0 dB SPL
```

The asymmetric Impulse ballistics use two constants — a fast attack and a slow
decay — switching per sample on the sign of the change:

$$
y[n] = y[n-1] + \alpha \ (x^2[n] - y[n-1]), \qquad
\alpha = \begin{cases}1 - e^{-1/(f_s \cdot 0.035)} & x^2[n] > y[n-1]\\[2pt] 1 - e^{-1/(f_s \cdot 1.5)} & \text{otherwise}\end{cases}
$$

## 3. `time_weighting()` / `TimeWeighting` parameters

| Parameter | Type | Units | Range / default | Notes |
| :--- | :--- | :--- | :--- | :--- |
| `x` | 1D or 2D array | pressure (any scale) | non-empty | Squared internally; output is a mean-square envelope |
| `fs` | int | Hz | > 0 | |
| `mode` | str | — | `'fast'` (default), `'slow'`, `'impulse'` | τ = 125 ms / 1 s / 35 ms attack + 1.5 s decay |
| `TimeWeighting(fs, mode)` (class) | — | — | — | Stateful variant for streaming: `process(x)` carries the integrator state between blocks |

The output has the units of $x^2$: take `10*log10(y / p0**2)` for SPL or use
the level functions, which do it for you.

## 4. Verified ballistics (IEC 61672-1 Table 4)

The Fast envelope's response to 4 kHz tonebursts lands exactly on the
standard's reference values — the example below verifies the 200 ms Fast
burst row; the CI suite covers the full Table 4, from 1 s down to 1 ms for F
and 1 s down to 2 ms for S, at class 1 acceptance limits:

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/tone_burst_iec_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/tone_burst_iec.svg" alt="Fast envelope responses to 200, 50 and 10 ms tone bursts peaking exactly at the IEC 61672-1 Table 4 reference values" width="80%"></picture>

<details>
<summary>Show the code for this figure</summary>

```python
import numpy as np
import matplotlib.pyplot as plt
from phonometry import time_weighting

fs = 48000
t = np.arange(int(fs * 2)) / fs
tone = np.sin(2 * np.pi * 4000 * t)

# Steady-state Fast reference of the continuous tone
reference = time_weighting(tone, fs, mode='fast')[int(1.5 * fs):].mean()

# 200 ms burst of the same tone (IEC 61672-1 Table 4 target: -1.0 dB)
burst = np.zeros_like(t)
burst[int(0.5 * fs):int(0.7 * fs)] = tone[int(0.5 * fs):int(0.7 * fs)]
envelope = time_weighting(burst, fs, mode='fast')
env_db = 10 * np.log10(np.maximum(envelope / reference, 1e-6))

plt.figure()
plt.plot(t, env_db, label='Fast envelope')
plt.axhline(-1.0, linestyle='--', label='IEC target −1.0 dB')
plt.xlabel('Time [s]')
plt.ylabel('Level re steady state [dB]')
plt.legend()
plt.show()
```

</details>

## 5. Initial state

By default, the exponential integrator starts from rest (`y[-1] = 0`). Passing
`initial_state=None` leaves this default unspecified, while `initial_state='zero'`
requests the same zero state explicitly. If the recorded segment begins after a
steady signal is already present, you can start from the first sample energy instead:

```python
# Uses `recording` and `fs` from the snippet above.
energy_envelope = time_weighting(recording, fs, mode='fast', initial_state='first')
```

## 6. Block processing

For block processing, pass the last output value from the previous block as the
next block's `initial_state` instead of resetting each block:

```python
state = None

# audio_blocks: consecutive frames of your calibrated recording (Pa),
#   streamed from your sound card or read from a WAV in blocks.
for block in audio_blocks:
    energy_envelope = time_weighting(block, fs, mode='fast', initial_state=state)
    state = energy_envelope[-1]
```

For multichannel blocks with time on the last axis, carry one state per channel:
use `state = energy_envelope[..., -1]`. A scalar `initial_state` is applied to
every channel, while an array must match or broadcast to the non-time shape,
such as `(n_channels,)` for input shaped `(n_channels, n_samples)`.

Or let the `TimeWeighting` class carry the state for you:

```python
from phonometry import TimeWeighting

tw = TimeWeighting(fs, mode='fast')
# audio_blocks: consecutive frames of your calibrated recording (Pa),
#   streamed from your sound card or read from a WAV in blocks.
for block in audio_blocks:
    energy_envelope = tw.process(block)
```

Concatenated block outputs are exactly equal to a single continuous call
(verified for all three modes, mono and multichannel). Call `tw.reset()` to
start from rest again.

## 7. Performance note

The `impulse` mode uses an asymmetric kernel that is JIT-compiled when
[numba](https://numba.pydata.org/) is installed (`pip install phonometry[perf]`).
Without numba a pure-Python fallback produces identical results, just slower.

See [Integrated & Statistical Levels](https://jmrplens.github.io/phonometry/guides/levels/) for Leq/LN metrics built on
these envelopes, and [Why phonometry](https://jmrplens.github.io/phonometry/reference/why-phonometry/) for the IEC
61672-1 tone-burst verification.

---

**Standards.** IEC 61672-1:2013, *Electroacoustics — Sound level meters —
Part 1: Specifications* — the exponential time-weighting detector (clause 3.8)
with the F and S time constants (clause 5.7), and the 4 kHz toneburst reference
responses of Table 4 (class 1 acceptance limits) used to verify the ballistics
in CI.

---


<!-- source: docs/levels.md | canonical: https://jmrplens.github.io/phonometry/guides/levels/ -->

# Integrated and Statistical Levels

Environmental noise metrics computed directly from the raw (calibrated) signal.

## Leq and LAeq

The equivalent continuous level integrates the squared pressure over the
measurement time:

$$
L_{eq} = 10\log_{10}\left(\frac{1}{T}\int_0^T \frac{p^2(t)}{p_0^2}\ dt\right) \text{ dB}, \qquad p_0 = 20\ \mu\text{Pa}
$$

and $L_{Aeq}$ is the same integral after A-weighting the signal. $L_N$ is the
level exceeded $N\ \%$ of the time — the $(100-N)$-th percentile of the
time-weighted level distribution.

```python
import numpy as np
from phonometry import leq, laeq

# recording: a calibrated microphone capture (Pa) — recorded through your measurement chain. Synthesized here so the guide runs standalone.
fs = 48000
recording = 0.2 * np.sin(2 * np.pi * 1000 * np.arange(fs) / fs)
sensitivity = 1.0                                    # calibration_factor (see Calibration)

# Equivalent continuous level of the whole recording
level = leq(recording, calibration_factor=sensitivity)

# A-weighted Leq (the standard environmental noise metric)
la = laeq(recording, fs, calibration_factor=sensitivity)
```

Both accept 1D signals (returning a scalar) or 2D `[channels, samples]` arrays
(returning one level per channel), and support `dbfs=True` for digital
full-scale analysis (calibration does not apply in dBFS mode).

Why the *energy* mean and not the arithmetic mean of dB values? Because sound
doses add as energy: two periods at 60 dB and 80 dB do not average to 70 dB —
the 80 dB half dominates and $L_{eq}$ = 77 dB. Averaging decibels directly
underestimates every fluctuating noise. $L_{eq}$ is the level of the *steady*
sound carrying the same energy as the real, fluctuating one, which is why
regulations are written in terms of it.

### `leq()` / `laeq()` parameters

| Parameter | Type / shape | Units | Range / default | Notes |
| :--- | :--- | :--- | :--- | :--- |
| `x` | 1D or 2D array | digital units (or Pa if calibrated) | non-empty | 2D is `[channels, samples]`; returns one level per channel |
| `fs` | int | Hz | > 0 (`laeq` only) | `leq` needs no sample rate (pure RMS integral) |
| `calibration_factor` | float | Pa per digital unit | default `1.0` | From `sensitivity()` |
| `dbfs` | bool | — | default `False` | `True`: 0 dBFS = full-scale RMS sine; ignores calibration |

## Percentile levels (LN)

`ln_levels` computes statistical levels from the time-weighted envelope:
**L10** is the level exceeded 10 % of the time (event peaks), **L50** the median,
**L90** the background level.

```python
import numpy as np
from phonometry import ln_levels

# A steady tone gives L10 = L50 = L90; percentiles only tell a story for a
# *fluctuating* level. Synthesize 3 s alternating between a quiet and a
# ~10 dB louder half-second so the statistics separate.
fs = 48000
rng = np.random.default_rng(0)
segment = fs // 2                                  # 0.5 s per level
quiet = 0.02 * rng.standard_normal(segment)        # background
loud = 0.06 * rng.standard_normal(segment)         # ~10 dB louder events
varying = np.tile(np.concatenate([quiet, loud]), 3)

stats = ln_levels(varying, fs, n=(10, 50, 90), weighting="A")
print(f"LA10={stats[10]:.1f}  LA50={stats[50]:.1f}  LA90={stats[90]:.1f} dB")
# LA10=66.6  LA50=65.2  LA90=58.5 dB  -> L10 (events) > L50 (median) > L90 (background)
```

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/ln_levels_example_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/ln_levels_example.svg" alt="Fast level history of fluctuating noise with the L10, L50 and L90 statistical levels marked" width="80%"></picture>

*L10 tracks the event peaks, L50 the median level and L90 the background.*

Options: `mode` selects the envelope ballistics (`'fast'`, `'slow'`,
`'impulse'`), `weighting` applies A/C weighting first, and
`calibration_factor`/`dbfs` behave as in `leq`. The integrator attack transient
(~5τ) is discarded before taking percentiles, so the leading settling ramp is
not counted in the low percentiles.

Formally, $L_N$ is the $(100-N)$-th percentile of the distribution of the
time-weighted level: the recording is first turned into a level-vs-time
envelope (Fast by default), and $L_{10}$ is the envelope value exceeded 10 %
of the time. That makes the *ballistics choice part of the metric*: an
$L_{10}$ from a Slow envelope is systematically lower than from a Fast one on
impulsive noise, so regulations always name the time weighting.

### `ln_levels()` parameters

| Parameter | Type / shape | Units | Range / default | Notes |
| :--- | :--- | :--- | :--- | :--- |
| `x` | 1D or 2D array | digital units | non-empty | 2D returns per-channel dicts |
| `fs` | int | Hz | > 0 | Needed by the envelope detector |
| `n` | tuple of ints | % | default `(10, 50, 90)` | Any exceedance percentages, e.g. `(1, 5, 95)` |
| `mode` | str | — | `'fast'` (default), `'slow'`, `'impulse'` | IEC 61672-1 ballistics of the envelope |
| `weighting` | str or None | — | `'A'`, `'C'`, `'G'`, `'Z'`, `None` (default) | Frequency weighting before the envelope |
| `calibration_factor` / `dbfs` | float / bool | — | as `leq` | Same semantics as in `leq()` |

## Peak, event and occupational metrics

```python
from phonometry import lc_peak, sel, sound_exposure, lex_8h

# Uses `recording`, `fs` and `sensitivity` from the Leq snippet above.

# C-weighted peak (IEC 61672-1 §5.13) - occupational action limits use this
peak = lc_peak(recording, fs, calibration_factor=sensitivity)

# A single noise event and a work-shift sample (slices of a real recording)
event = recording
shift_sample = recording

# Sound exposure level: single-event level normalized to 1 s (LAE)
lae = sel(event, fs, weighting="A", calibration_factor=sensitivity)

# Daily noise dose (IEC 61252): exposure in Pa²·h and LEX,8h / LEP,d
E = sound_exposure(shift_sample, fs, duration_hours=8, calibration_factor=sensitivity)
lex = lex_8h(shift_sample, fs, duration_hours=8, calibration_factor=sensitivity)
```

`lc_peak` is verified against the one-cycle/half-cycle reference responses of
IEC 61672-1:2013 Table 5, `sel` against the Table 4 LAE toneburst column, and
the dose functions against the IEC 61252 anchors (3.2 Pa²h ↔ exactly 90 dB).
`lc_peak` polyphase-oversamples the C-weighted signal by `oversample` (default
`8`) before taking the maximum, recovering the true inter-sample peak: a raw
on-grid maximum under-reads sustained HF tones by up to ~1.15 dB (an 8 kHz tone
at 48 kHz is only 6 samples/cycle). Set `oversample=1` to detect the peak on the
original sample grid. With `duration_hours`, the input is treated as a
representative sample of that exposure period; without it, the input is the
whole event.

### SEL: comparing events of different duration

A 4 s train pass-by and a 30 s one cannot be compared by their $L_{Aeq}$
alone — the longer event delivers more energy at the same level. The **sound
exposure level** compresses the *whole* event energy into exactly one second:

$$
L_E = L_{eq,T} + 10\log_{10}\frac{T}{T_0}, \qquad T_0 = 1\ \text{s}
$$

so events of any duration become directly comparable, and $N$ identical
events sum as $+10\log_{10}N$. This is the building block of airport and
railway noise models.

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/sel_concept_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/sel_concept.svg" alt="A vehicle pass-by level history with its Leq over the whole event and the equal-energy one-second SEL block" width="80%"></picture>

### Noise dose: sound exposure and LEX,8h

Occupational regulations limit the daily *dose*, not the level. IEC 61252
expresses it as **sound exposure** $E$ in pascal-squared-hours — the time
integral of the squared A-weighted pressure — and the equivalent
**normalized 8 h level**:

$$
E = \int_0^T p_A^2(t)\ dt \quad [\text{Pa}^2\text{h}], \qquad
L_{EX,8h} = 10\log_{10}\frac{E}{8\ \text{h} \cdot p_0^2}
$$

The anchor worth memorizing: **3.2 Pa²h ⇔ exactly 90 dB over 8 h** (the CI
suite enforces it). Half the dose is −3 dB; double duration at the same level
is +3 dB.

### Peak / event / dose parameters

| Function | Key parameters | Returns | Standard anchor |
| :--- | :--- | :--- | :--- |
| `lc_peak(x, fs, calibration_factor=1.0, dbfs=False)` | `dbfs=True` references full-scale *peak* (1.0), not RMS | LCpeak [dB] | IEC 61672-1 §5.13, Table 5 tone bursts |
| `sel(x, fs, weighting=None, ...)` | `weighting='A'` gives LAE | SEL [dB] | IEC 61672-1 Table 4 (LAE column) |
| `sound_exposure(x, fs, duration_hours=None, ...)` | `duration_hours` treats `x` as a sample of that period | E [Pa²h] | IEC 61252 |
| `lex_8h(x, fs, duration_hours=None, ...)` | same sampling semantics | LEX,8h [dB] | IEC 61252 (≡ LEP,d) |

`lex_8h` rates *one* recording; assembling a full working day from task or
job samples — with the normative ISO 9612 uncertainty budget — continues in
[Occupational Noise Exposure](https://jmrplens.github.io/phonometry/guides/occupational-exposure/).

## Environmental noise: Lden, Ldn and rating levels (ISO 1996-1)

Regulatory noise assessment weights evenings and nights more heavily.
`lden()` implements the day-evening-night level of ISO 1996-1:2016 (3.6.4:
+5 dB evening, +10 dB night, default 12/4/8 h periods — adjustable, since
countries define them differently), `ldn()` the day-night variant (3.6.5),
and `composite_rating_level()` the general whole-day composite of clause 6.5
(Formulae 5-6) for arbitrary periods with source or character adjustments
(Table A.1: e.g. +5 dB regular impulsive, +12 dB highly impulsive, +3 to
+6 dB prominent tones):

```python
from phonometry import lden, composite_rating_level

l = lden(63.2, 58.1, 51.4)                      # from LAeq per period
r = composite_rating_level([(63.2, 12, 0.0),    # day
                            (58.1, 4, 5.0),     # evening (+5)
                            (51.4, 8, 10.0)])   # night  (+10) == lden
```

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/lden_profile_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/lden_profile.svg" alt="Synthetic 24-hour urban LAeq profile with day, evening and night bands, the +5 and +10 dB weighted period levels and the resulting Lden" width="80%"></picture>

### `lden()` / `ldn()` / `composite_rating_level()` parameters

| Function | Key parameters | Notes |
| :--- | :--- | :--- |
| `lden(lday, levening, lnight, hours=(12, 4, 8))` | period LAeq values [dB]; `hours` must sum to 24 | +5 dB evening, +10 dB night (3.6.4) |
| `ldn(lday, lnight, hours=(15, 9))` | | +10 dB night (3.6.5) |
| `composite_rating_level(periods)` | iterable of `(level_db, hours, adjustment_db)`; hours positive, finite and summing to 24 | General Formulae (5)-(6); adjustments per Table A.1 |

Where you put the microphone changes the number: ISO 1996-2 fixes the receiver positions and their façade corrections. phonometry does not implement ISO 1996-2 — the diagram is measurement context; apply the corrections to your levels before analysis:

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/diagram_env_measurement_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/diagram_env_measurement.svg" alt="Environmental noise measurement positions per ISO 1996-2: free field, 2 m from the facade and flush-mounted, with their corrections" width="92%"></picture>

Combine with `laeq()` per time period to go from recordings to Lden, and with
the `tone_to_noise_ratio()` / `prominence_ratio()` verdicts of
[Prominent Discrete Tones](https://jmrplens.github.io/phonometry/guides/tone-prominence/) to justify tonal adjustments.

## Octave Spectrogram (levels over time)

Short-time fractional-octave analysis: one level per band per window,
time-aligned across bands.

```python
from phonometry import OctaveFilterBank

# Uses `recording` from the Leq snippet above.
bank = OctaveFilterBank(fs=48000, fraction=3)
levels, freq, times = bank.spectrogram(recording, window_time=0.125, overlap=0.5)
# levels: (bands, frames) — ready for pcolormesh(times, freq, levels)
```

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/spectrogram_example_dark.png"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/spectrogram_example.png" alt="One-third-octave spectrogram of a logarithmic sweep with two tone bursts" width="80%"></picture>

*A logarithmic sweep plus two tone bursts, resolved in time and in standardized
1/3-octave bands.*

- Multichannel input `(channels, samples)` returns `(channels, bands, frames)`.
- `times` holds each window's center in seconds.
- `mode='peak'` gives per-window peak-holding levels instead of RMS.
- `zero_phase=True` filters bands forward-backward so per-band group delay does
  not skew the frames (offline analysis only).

### `OctaveFilterBank.spectrogram()` parameters

| Parameter | Type | Units | Range / default | Notes |
| :--- | :--- | :--- | :--- | :--- |
| `x` | 1D or 2D array | digital units | non-empty | 2D returns `(channels, bands, frames)` |
| `window_time` | float | s | > 0; default `0.125` | Frame length (0.125 s mirrors Fast) |
| `overlap` | float | — | 0 ≤ overlap < 1; default `0.5` | Fraction of window overlap (0 = none) |
| `mode` | str | — | `'rms'` (default) or `'peak'` | Per-window detector |
| `detrend` | bool | — | default `True` | Remove each band's DC offset before the level (improves low-frequency accuracy) |
| `zero_phase` | bool | — | default `False` | Forward-backward filtering (offline only) |
| `calibration_factor` / `dbfs` | — | — | constructor-only | Set on `OctaveFilterBank(...)`, not per call |

```python
import matplotlib.pyplot as plt

# Uses `levels`, `freq` and `times` from the spectrogram snippet above.
fig, ax = plt.subplots()
mesh = ax.pcolormesh(times, freq, levels, shading="auto")
ax.set_yscale("log")
ax.set_xlabel("Time [s]")
ax.set_ylabel("Frequency [Hz]")
fig.colorbar(mesh, label="Level [dB]")
```

See [Calibration and dBFS](https://jmrplens.github.io/phonometry/guides/calibration/) to convert digital units to physical
SPL, and [Time Weighting](https://jmrplens.github.io/phonometry/guides/time-weighting/) for the envelope details. The
ISO 9612 occupational strategies continue in
[Occupational Noise Exposure](https://jmrplens.github.io/phonometry/guides/occupational-exposure/), the ECMA-418-1
tonal-prominence verdicts in [Prominent Discrete Tones](https://jmrplens.github.io/phonometry/guides/tone-prominence/),
and the ISO 226 equal-loudness contours live with the perception metrics in
[Psychoacoustics](https://jmrplens.github.io/phonometry/guides/psychoacoustics/).

---

**Standards.** IEC 61672-1:2013, *Electroacoustics — Sound level meters —
Part 1: Specifications* — the Fast/Slow/Impulse envelope ballistics behind
`ln_levels`, the C-weighted peak of §5.13 (verified against the Table 5 tone
bursts) and the sound exposure level verified against the Table 4 LAE column.
IEC 61252, *Electroacoustics — Specifications for personal sound exposure
meters* — the sound exposure E in Pa²h and the normalized 8 h level LEX,8h
(≡ LEP,d), anchored at 3.2 Pa²h ⇔ exactly 90 dB. ISO 1996-1:2016, *Acoustics —
Description, measurement and assessment of environmental noise — Part 1:
Basic quantities and assessment procedures* — Lden (3.6.4), Ldn (3.6.5) and
the composite whole-day rating level of clause 6.5 (Formulae 5-6, Table A.1
adjustments).

---


<!-- source: docs/occupational-exposure.md | canonical: https://jmrplens.github.io/phonometry/guides/occupational-exposure/ -->

# Occupational Noise Exposure (ISO 9612)

A working day is rarely measured in one take: the daily exposure level a
regulation acts on has to be assembled from *samples* of a real shift, and
reported with an uncertainty a hygienist can defend. `lex_8h` (in
[Levels](https://jmrplens.github.io/phonometry/guides/levels/)) turns *one* recording into a daily level. ISO 9612:2009 —
the engineering method (accuracy grade 2) — is the survey design *around* that
primitive: how to sample a real working day, how to combine the pieces, and how
to attach the normative uncertainty every occupational-hygiene report needs. The
`occupational_exposure` module adds the three **measurement strategies** and the
**Annex C** uncertainty budget on top of the energy-average machinery.

## 1. The three measurement strategies (Clauses 9-11)

The *task-based* strategy (Clause 9) splits the nominal day into tasks, takes
$I \ge 3$ samples per task, and energy-sums the task contributions

$$
L_{EX,8h,m} = L_{p,A,eqT,m} + 10 \log_{10}(T_m/T_0), \qquad T_0 = 8\ \text{h},
$$

so a loud but short task contributes little. The *job-based* (Clause 10) and
*full-day* (Clause 11) strategies instead take $N \ge 5$ (or three whole-day)
random samples over a homogeneous exposure group and normalise the effective-day
duration. The daily level is the same either way; the strategies differ in how
the **uncertainty** is built.

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/exposure_uncertainty_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/exposure_uncertainty.svg" alt="ISO 9612 Annex D task-based exposure: the three task LEX,8h contributions as bars, the energy-summed daily LEX,8h line and the one-sided 95 % upper limit LEX,8h + U band above it" width="80%"></picture>

<details>
<summary>Show the code for this figure</summary>

```python
import matplotlib.pyplot as plt

# One line, with the Annex D `res` computed below: task contribution bars
# plus the LEX,8h and LEX,8h + U lines.
res.plot()
plt.show()
```

</details>


```python
from phonometry.occupational_exposure import (
    Task, task_based_exposure, job_based_exposure, full_day_exposure,
)

# ISO 9612 Annex D — a welder's day split into three tasks. Each task level is
# the energy average of its Lp,A,eqT samples; durations carry a measured range.
tasks = [
    Task(samples=(70.0,), duration_hours=1.5, label="planning/breaks"),
    Task(samples=(80.1, 82.2, 79.6), duration_hours=5.0,
         duration_range=(4.0, 6.0), label="welding"),
    Task(samples=(86.5, 92.4, 89.3, 93.2, 87.8, 86.2), duration_hours=1.5,
         duration_range=(1.0, 2.0), label="cutting/grinding"),
]
res = task_based_exposure(tasks, include_duration_uncertainty=False, warn=False)
print(f"LEX,8h = {res.lex_8h:.1f} dB   U = {res.expanded_uncertainty:.1f} dB")
# LEX,8h = 84.3 dB   U = 2.7 dB
print(f"one-sided 95 % upper limit LEX,8h + U = {res.upper_limit:.1f} dB")   # 87.0 dB
for t in res.tasks:
    print(f"  {t.label:<16} Lp,A,eqT = {t.lp_aeqt:5.1f}   contributes {t.lex_8h_contribution:5.1f} dB")
#   planning/breaks  Lp,A,eqT =  70.0   contributes  62.7 dB
#   welding          Lp,A,eqT =  80.8   contributes  78.7 dB
#   cutting/grinding Lp,A,eqT =  90.1   contributes  82.8 dB

# The same shift measured job-based (Annex E) and full-day (Annex F): both use
# the Eq C.9 / Table C.4 sampling budget with k = 1.65 (one-sided 95 %).
job = job_based_exposure([88.1, 86.1, 89.7, 86.5, 91.1, 86.7], effective_duration_hours=7.5)
full = full_day_exposure([88.0, 91.9, 87.6, 90.4, 89.0, 88.4], effective_duration_hours=9.25)
print(f"job      LEX,8h = {job.lex_8h:.1f} dB   U = {job.expanded_uncertainty:.1f} dB")
# job      LEX,8h = 88.2 dB   U = 3.8 dB
print(f"full-day LEX,8h = {full.lex_8h:.1f} dB   U = {full.expanded_uncertainty:.1f} dB")
# full-day LEX,8h = 90.1 dB   U = 3.4 dB
```

## 2. The Annex C uncertainty budget

Two subtleties are worth spelling out. First, the coverage factor is
$k = 1.65$ for a **one-sided** 95 % interval (Clause 14), because a hygienist
cares only about the *upper* bound: `res.upper_limit` = $L_{EX,8h} + U$ is the
value 95 % of measurements fall below, the number compared against an action
limit. Second, the task and job methods weight the *same* spread of samples
differently. The task sampling uncertainty $u_{1a}$ (Eq. C.6) divides the summed
squared deviations by $I(I-1)$ — the standard error of the mean, smaller by a
factor $\sqrt{I}$ — whereas the job/full-day sampling uncertainty $u_1$ (Eq. C.12)
is the plain sample standard deviation with denominator $N-1$, whose contribution
$c_1 u_1$ is then read from **Table C.4** as a function of $(N, u_1)$. The same
raw scatter therefore inflates the job estimate more, which is the standard's
built-in penalty for coarser, fewer samples. (The printed job $L_{EX,8h}$ is
$88.2$ dB where Annex E reports $88.1$: the standard rounds the effective-day
level to $88.4$ before the duration normalisation; the library keeps it
unrounded.)

When a task's samples span **3 dB or more** (Clause 9.3), or the job contribution
$c_1 u_1$ exceeds 3.5 dB (Clause 10.4), or too few workers are covered
(Table 1 cumulative-duration), the result sets `sampling_advisory=True` and, with
`warn=True`, emits an `OccupationalExposureWarning` recommending more measurements. Peak
levels $L_{p,Cpeak}$ are reported **without** an uncertainty — Annex C gives no
method for them (Table C.5, Note 1), so peak-uncertainty is out of scope. The
three Annex D/E/F worked examples above are reproduced to the standard's printed
precision (Annex E's final rounding is disclosed above), and the theory is
derived on the [Theory](https://jmrplens.github.io/phonometry/reference/theory/) page.

### `task_based_exposure()` / `job_based_exposure()` / `full_day_exposure()` parameters

| Parameter | Applies to | Type | Units | Range / default | Notes |
| :--- | :--- | :--- | :--- | :--- | :--- |
| `tasks` | task | list of `Task` | — | ≥ 1 | Each `Task` has `samples`, `duration_hours`, optional `duration_range`/`duration_samples`, `label`, `instrument` |
| `samples` | job / full-day | sequence | dB | ≥ 2 (≥ 5 / ≥ 3 advised) | Random `Lp,A,eqT` samples |
| `effective_duration_hours` | job / full-day | float | h | > 0 | Effective working-day duration $T_e$ |
| `instrument` | all | str | — | `'class1'`, `'class2'`, `'personal_exposimeter'` (default) | Selects $u_2$ (Table C.5) |
| `u3` | all | float | dB | default `1.0` | Microphone-position uncertainty (Clause C.6) |
| `include_duration_uncertainty` | task | bool | — | default `True` | `False` omits the $(c_{1b}u_{1b})^2$ term (Annex D case a) |
| `n_workers` / `sample_duration_hours` | job | int / float | — / h | default `None` | Table 1 cumulative-duration check |
| `warn` | all | bool | — | default `True` | Emit `OccupationalExposureWarning` for the sampling advisories |

All three return an `ExposureResult` with `lex_8h`, `combined_standard_uncertainty`
$u$, `expanded_uncertainty` $U = 1.65\ u$, `upper_limit` = $L_{EX,8h} + U$,
`sampling_advisory`, and (task-based) the per-task `tasks` breakdown; the
result's `.plot()` draws the per-task contribution bars with the $L_{EX,8h}$
and upper-limit lines (task-based results only, since the other strategies
carry no per-task breakdown).

## See also

- [Levels](https://jmrplens.github.io/phonometry/guides/levels/) — the `lex_8h` / `sound_exposure` dose primitives
  (IEC 61252) and the LCpeak these strategies report alongside.
- [Measurement uncertainty](gum-uncertainty.md) — the GUM machinery behind
  combined and expanded uncertainties.
- [Theory](https://jmrplens.github.io/phonometry/reference/theory/) — the derivation of the strategy formulas and the
  Annex C budget.

---

**Standards.** ISO 9612:2009, *Acoustics — Determination of occupational noise
exposure — Engineering method* — the task-based (Clause 9), job-based
(Clause 10) and full-day (Clause 11) strategies, the Annex C uncertainty budget
(Formulae C.6, C.9 and C.12, Tables C.4/C.5) and the one-sided coverage factor
k = 1.65 (Clause 14), validated against the worked examples of Annexes D, E
and F.

---


<!-- source: docs/tone-prominence.md | canonical: https://jmrplens.github.io/phonometry/guides/tone-prominence/ -->

# Prominent Discrete Tones (ECMA-418-1)

Tonal components in machinery noise are far more annoying than their level
suggests. ECMA-418-1:2024 (referenced by ECMA-74 Annex D) gives two FFT-based
methods to decide whether a discrete tone is *prominent*:
`tone_to_noise_ratio()` compares the tone level with the masking noise in its
critical band (clause 11), and `prominence_ratio()` compares the critical band
centred on the tone with the two contiguous bands (clause 12). Both return a
structured verdict against the frequency-dependent prominence criteria.

## 1. Tone-to-noise ratio and prominence ratio

```python
import numpy as np
from phonometry import tone_to_noise_ratio, prominence_ratio

fs = 48000
rng = np.random.default_rng(0)
t = np.arange(fs) / fs
x = np.sin(2 * np.pi * 1000 * t) + 0.05 * rng.standard_normal(fs)  # 1 kHz tone in noise
tnr = tone_to_noise_ratio(x, fs)            # highest peak, or tone_freq=...
pr = prominence_ratio(x, fs, tone_freq=1000.0)
print(tnr.ratio_db, tnr.criterion_db, tnr.prominent)
```

The methods hinge on the **critical band** — the ear's analysis bandwidth,
$\Delta f_c = 25 + 75\ [1 + 1.4(f/1000)^2]^{0.69}$ Hz (162 Hz at 1 kHz): a
tone is masked only by the noise *inside* its critical band, so both ratios
compare the tone against exactly that noise, not the whole spectrum.

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/tonality_spectrum_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/tonality_spectrum.svg" alt="Averaged spectrum of a tone in noise with the critical band shaded and the tone-to-noise ratio annotated against its prominence criterion" width="80%"></picture>

A TNR above $8 + 8.33\log_{10}(1000/f_t)$ dB (8 dB from 1 kHz up) classifies
the tone as *prominent*; the PR criterion is $9 + 10\log_{10}(1000/f_t)$ dB.
Low frequencies get higher thresholds because wider relative bands mask more.

## 2. Where to measure (ECMA-74) and practice

ECMA-74 (which delegates its tone assessments to ECMA-418-1) also fixes where to measure around a device — context only: phonometry implements the ECMA-418-1 assessments, not the ECMA-74 Annex D measurement procedure:

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/diagram_tonality_positions_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/diagram_tonality_positions.svg" alt="ECMA-74 emission measurement positions: seated operator microphone at 0.25 m and 1.20 m, and the four bystander positions at 1 m" width="92%"></picture>

Proximate secondary tones in the same critical band are combined per
clause 11.6; for harmonic complexes assess each component (`tone_freq=`).
Both methods work on Hann-windowed, RMS-averaged spectra and need no absolute
calibration (the ratios are level differences).

### `tone_to_noise_ratio()` / `prominence_ratio()` parameters

| Parameter | Type | Units | Range / default | Notes |
| :--- | :--- | :--- | :--- | :--- |
| `x` | 1D array | any (uncalibrated OK) | ≥ `fs/resolution_hz` samples | Ratios are level differences: calibration cancels out |
| `fs` | int | Hz | > 0 | |
| `tone_freq` | float, optional | Hz | 89.1–11 200; default `None` | `None` assesses the highest peak in the range of interest |
| `resolution_hz` | float | Hz | > 0; default `1.0` | Tone band must stay within 15 % of the critical band (clause 11.2) |

Both return a `ToneAssessment(frequency, ratio_db, criterion_db, prominent)`.

## See also

- [Levels](https://jmrplens.github.io/phonometry/guides/levels/) — the ISO 1996-1 rating levels whose tonal adjustments
  (Table A.1) these prominence verdicts justify objectively.
- [Psychoacoustics](https://jmrplens.github.io/phonometry/guides/psychoacoustics/) — the ECMA-418-2 psychoacoustic
  tonality T in tu_HMS, the hearing-model counterpart of these FFT ratios.
- [Impulsive-sound prominence](impulse-prominence.md) — the NT ACOU 112
  counterpart for impulsive (rather than tonal) character.
- [Theory](https://jmrplens.github.io/phonometry/reference/theory/) — the critical-band model and criteria derivation.

---

**Standards.** ECMA-418-1:2024 (3rd edition), *Psychoacoustic metrics for ITT
equipment — Part 1: Prominent discrete tones* — the tone-to-noise ratio
(clause 11), the prominence ratio (clause 12), the critical-band model and the
frequency-dependent prominence criteria.

---


<!-- source: docs/psychoacoustics.md | canonical: https://jmrplens.github.io/phonometry/guides/psychoacoustics/ -->

# Psychoacoustics

Level metrics tell you how much *sound pressure* there is; psychoacoustic
metrics tell you what a listener actually *perceives*. This page covers
loudness (ISO 532-1), sharpness (DIN 45692) and the equal-loudness
contours of pure tones (ISO 226), then the advanced Moore-Glasberg
(ISO 532-2/3) and Sottek Hearing Model (ECMA-418-2) loudness, tonality and
roughness models. Speech metrics live in their own guides: the
transmission-channel STI/STIPA in
[Speech Transmission Index](https://jmrplens.github.io/phonometry/guides/speech-transmission/) and the
audibility-based SII in
[Speech Intelligibility Index](https://jmrplens.github.io/phonometry/guides/speech-intelligibility/).

## Loudness in sones (ISO 532-1, Zwicker)

Decibels compress perception: 10 dB more reads as *twice as loud*, and two
sounds with the same dB(A) can differ audibly depending on how their energy
spreads over the ear's **critical bands**. The Zwicker method models the
hearing chain explicitly — outer/middle-ear transmission, critical-band
analysis on the 24 Bark scale, level-dependent masking slopes — and outputs
**loudness N in sones**, a ratio scale: 4 sones is twice as loud as 2 sones.
By definition a 1 kHz tone at 40 dB SPL is 1 sone, and every +10 phon
doubles the sone value.

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/diagram_zwicker_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/diagram_zwicker.svg" alt="ISO 532-1 Zwicker loudness chain: 28 one-third-octave band levels, transmission and lower-critical-band grouping, core loudness of the 20 critical bands, specific loudness over Bark, integrated into total loudness N in sones and loudness level in phons" width="78%"></picture>

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/loudness_pattern_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/loudness_pattern.svg" alt="Specific loudness patterns over the Bark scale for a 1 kHz narrowband sound and a broadband sound of equal band level" width="80%"></picture>

*Same band level, very different loudness: energy spread over many critical
bands (red) sums to far more sones than the same level concentrated in one
band (blue). The area under N'(z) is the total loudness.*

```python
import numpy as np
from phonometry import loudness_zwicker, loudness_zwicker_from_spectrum

# A raw recording plus its calibration so the guide runs standalone
fs = 48000
x = 0.2 * np.sin(2 * np.pi * 1000 * np.arange(fs) / fs)   # any recording (digital units)
sens = 1.0                                                # calibration_factor to pascals
levels_28 = np.full(28, 60.0)                             # 28 one-third-octave band levels (dB)

# From a raw recording: calibration_factor scales digital units to Pa
res = loudness_zwicker(x, fs, field="free", calibration_factor=sens)
print(f"N = {res.loudness:.1f} sone  ({res.loudness_level:.0f} phon)")   # 13.1 sone (77 phon)

# Time-varying signals: percentile loudness N5 is the reporting standard
res = loudness_zwicker(x, fs)          # stationary=False (default)
print(f"{res.n5:.1f} {res.n10:.1f} {res.loudness:.1f}")   # 13.1 13.1 13.1 — N5, N10, Nmax

# From 28 one-third-octave band levels (25 Hz .. 12.5 kHz)
res = loudness_zwicker_from_spectrum(levels_28, field="diffuse")

res.plot()   # N'(z) over the Bark scale — the specific-loudness pattern (needs matplotlib)
```

<details>
<summary>Show the code for this figure</summary>

```python
import matplotlib.pyplot as plt

# One line — the specific-loudness pattern N'(z) straight from the result:
res.plot()
plt.show()

# Or reproduce the figure by hand — two patterns of equal band level (60 dB),
# energy spread over many critical bands vs concentrated in the 1 kHz band:
narrow = loudness_zwicker_from_spectrum(np.r_[np.full(16, -60.0), 60.0, np.full(11, -60.0)])
broad = loudness_zwicker_from_spectrum(np.full(28, 60.0))
z = np.arange(1, narrow.specific.size + 1) * 0.1          # Bark axis
fig, ax = plt.subplots()
for r, color, label in [
    (broad, "#ff7f0e", f"Broadband  N = {broad.loudness:.1f} sone"),
    (narrow, "#1f77b4", f"1 kHz narrowband  N = {narrow.loudness:.1f} sone"),
]:
    ax.fill_between(z, r.specific, color=color, alpha=0.3)
    ax.plot(z, r.specific, color=color, label=label)
ax.set_xlabel("Critical-band rate z [Bark]")
ax.set_ylabel("Specific loudness N' [sone/Bark]")
ax.legend()
plt.show()
```

</details>

The implementation is a clean-room port of the standard's **normative
reference program** (Annex A.4): all twelve data tables are digit-exact and
the full Annex B validation set runs in CI — the stationary test case
reproduces the published value to every printed digit, and the tone-pulse
N(t) traces stay inside the standard's per-sample 5 % tolerance band.

### `loudness_zwicker()` parameters

| Parameter | Type | Units | Range / default | Notes |
| :--- | :--- | :--- | :--- | :--- |
| `x` | 1D array | Pa (after calibration) | ≥ 8 ms at 48 kHz | Resampled internally to 48 kHz if needed |
| `fs` | int | Hz | > 0 | |
| `field` | str | — | `'free'` (default) / `'diffuse'` | Sound-field correction (Table A.5) |
| `stationary` | bool | — | default `False` | `True`: single N from the averaged spectrum |
| `calibration_factor` | float | Pa per digital unit | default `1.0` | From `sensitivity()` |

Returns a `ZwickerLoudness` dataclass: `loudness` (N, sones), `loudness_level`
(phon), `specific` (N′(z), 240 bins of 0.1 Bark), and for time-varying runs
`n5`, `n10`, `time`, `loudness_vs_time` (500 Hz trace).

## Sharpness in acum (DIN 45692)

Two sounds can be equally loud yet one feels "sharper" — hissy, metallic —
because its loudness sits higher on the Bark scale. Sharpness is the
g(z)-weighted first moment of the specific loudness pattern:

$$
S = k\ \frac{\int_0^{24} N'(z)\ g(z)\ z\ dz}{\int_0^{24} N'(z)\ dz}\ \text{acum}
$$

with $g(z) = 1$ up to 15.8 Bark and rising exponentially beyond, and $k$
normalized so the reference sound — critical-band-wide noise at 1 kHz,
60 dB — is exactly **1.00 acum** (DIN 45692 clause 6; the derived
$k = 0.108$ sits inside the normative window 0.105–0.115).

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/sharpness_weighting_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/sharpness_weighting.svg" alt="DIN 45692 sharpness weighting g(z) against critical-band rate on a log axis, comparing the DIN, von Bismarck and Aures curves with the 15.8 and 15 Bark knees marked" width="80%"></picture>

```python
from phonometry import sharpness_din

# Uses `x`, `fs` and `sens` from the snippet above.
s = sharpness_din(x, fs, calibration_factor=sens)      # acum
s_aures = sharpness_din(x, fs, method="aures")          # Annex B variant
```

CI verifies the Table A.2 target values (0.38 acum at 250 Hz up to
2.82 acum at 4 kHz) within the standard's 5 % / 0.05 acum tolerance.

## Loudness level of pure tones (ISO 226:2023)

The normal equal-loudness-level contours relate the SPL of a pure tone to its
perceived *loudness level* in phons (the SPL of an equally loud 1 kHz tone).
`equal_loudness_contour(phon)` evaluates ISO 226:2023 Formula (1) at the 29
preferred third-octave frequencies of Table 1, `loudness_level(spl, frequency)`
is the exact inverse (Formula 2), and `hearing_threshold()` returns the
threshold-of-hearing column:

```python
from phonometry import equal_loudness_contour, loudness_level

freqs, spl = equal_loudness_contour(40.0)   # the classic 40-phon contour
phon = loudness_level(73.0, 63.0)           # 73 dB @ 63 Hz -> 40 phon
```

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/equal_loudness_contours_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/equal_loudness_contours.svg" alt="ISO 226:2023 normal equal-loudness-level contours from 20 to 90 phon with the hearing threshold curve" width="80%"></picture>

Validity per clause 4.1: 20-90 phon (80 phon above 4 kHz); the implementation
is verified against the Annex B tables in CI. Note this is the loudness of
*pure tones* — the loudness of arbitrary signals in sones is what the ISO 532
models on this page compute.

## Advanced loudness & sound-quality models

ISO 532-1 above is one of **three** loudness models phonometry ships, and
loudness is only half of the sound-quality story: two sounds of equal loudness
can still differ in how *tonal* or how *rough* they are. This section adds the
**Moore-Glasberg** loudness of ISO 532-2/532-3 and the **Sottek Hearing Model**
loudness, tonality and roughness of ECMA-418-2:2025.

### Choosing a loudness model

| Model | Standard | Stationary / time-varying | Output | When to use |
| :--- | :--- | :--- | :--- | :--- |
| Zwicker | ISO 532-1:2017 | both | sone | Reference method; one-third-octave input; fast and widely cited |
| Moore-Glasberg | ISO 532-2:2017 | stationary | sone | roex excitation pattern; better for tones and explicit binaural summation |
| Moore-Glasberg-Schlittenlacher | ISO 532-3:2023 | time-varying | sone (STL/LTL) | Time-varying loudness with short-/long-term traces and the peak N_max |
| Sottek (Hearing Model) | ECMA-418-2:2025 | time-varying | sone_HMS | Shares one auditory front-end with the ECMA tonality and roughness metrics |

All three are anchored so a **1 kHz tone at 40 dB SPL is ≈ 1 sone**; the values
are not interchangeable digit-for-digit because the models differ in their
auditory filters and their loudness summation.

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/loudness_models_comparison_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/loudness_models_comparison.svg" alt="Loudness of a 1 kHz tone as a function of level for the Zwicker, Moore-Glasberg and Sottek models, all passing through 1 sone at 40 dB SPL" width="80%"></picture>

*The three models agree at the 1 sone / 40 dB anchor and diverge with level:
Zwicker doubles the sone value every +10 phon, while the Sottek model grows
more slowly (about 1.65× per 10 dB), an intrinsic difference between the
auditory summations, not a calibration error.*

### Moore-Glasberg loudness (ISO 532-2)

Where Zwicker uses fixed critical bands on the Bark scale, Moore-Glasberg builds
an **excitation pattern** with level-dependent rounded-exponential (roex)
auditory filters on the ERB-number ("Cam") scale, then applies a compressive
excitation → specific-loudness transform with C = 0.0617 sone/Cam
(ISO 532-2:2017, Formula 7) and a binaural-inhibition stage. It reproduces the
tone and broadband cases of Annex B to a percent or two and, unlike ISO 532-1,
models binaural summation explicitly.

```python
import numpy as np
from phonometry import (
    loudness_moore_glasberg,
    loudness_moore_glasberg_from_spectrum,
)

# The definitional anchor: one 1 kHz sinusoidal component at 40 dB SPL,
# free field, binaural -> 1 sone / 40 phon by construction of the sone.
res = loudness_moore_glasberg_from_spectrum([(1000.0, 40.0)], field="free")
print(f"N = {res.loudness:.3f} sone  ({res.loudness_level:.1f} phon)")   # 1.000 sone (40.0 phon)

# From a calibrated recording: the narrowband (FFT) line spectrum is formed
# (power-preserving normalization) and fed to the exact sinusoidal-component
# method (ISO 532-2 clauses 5.2/5.4).
fs = 48000
x = np.sqrt(2) * 2e-5 * 10 ** (40 / 20) * np.sin(2 * np.pi * 1000 * np.arange(fs) / fs)
res = loudness_moore_glasberg(x, fs, field="free", presentation="binaural")

res.plot()   # specific loudness N'(i) over the ERB-number (Cam) scale
```

<details>
<summary>Show the code for this figure</summary>

```python
import matplotlib.pyplot as plt

# One line — the specific-loudness pattern N'(i) straight from the result:
res.plot()
plt.show()

# Or draw it by hand from the ERB-number grid the result already carries:
fig, ax = plt.subplots()
ax.fill_between(res.erb_number, res.specific, alpha=0.3)
ax.plot(res.erb_number, res.specific)
ax.set_xlabel("ERB number [Cam]")
ax.set_ylabel("Specific loudness N' [sone/Cam]")
plt.show()
```

</details>

#### `loudness_moore_glasberg()` parameters

| Parameter | Type | Units | Range / default | Notes |
| :--- | :--- | :--- | :--- | :--- |
| `x` | 1D array | Pa | non-empty | Calibrated pressure signal (signal wrapper) |
| `components` | list of `(f, L)` | Hz, dB SPL | — | `_from_spectrum`: discrete sinusoidal components |
| `band_levels` | 29-vector | dB SPL | 25 Hz .. 16 kHz | `_from_third_octave` input (IEC 61260-1 bands) |
| `fs` | int | Hz | > 0 | Signal wrapper only |
| `field` | str | — | `'free'` (default) / `'diffuse'` / `'eardrum'` | Outer-ear transfer |
| `presentation` | str | — | `'binaural'` (default) / `'diotic'` / `'monaural'` | Binaural summation |

Returns a `MooreGlasbergLoudness`: `loudness` (N, sone), `loudness_level`
(phon), `specific` (N′(i), 372 bins of 0.1 Cam), `erb_number`,
`centre_frequencies`, `field`, `presentation`.

### Time-varying loudness (ISO 532-3)

ISO 532-3 wraps the same excitation / specific-loudness model in a running
multi-resolution spectral analysis (six parallel FFTs, updated every 1 ms) and
two cascaded temporal integrators: the fast **short-term loudness** S′(t) and
the slower **long-term loudness** S″(t). The peak long-term loudness N_max
predicts the loudness of sounds up to about 5 s.

```python
import numpy as np
from phonometry import loudness_moore_glasberg_time

fs = 32000
t = np.arange(int(1.3 * fs)) / fs
x = np.sqrt(2) * 2e-5 * 10 ** (40 / 20) * np.sin(2 * np.pi * 1000 * t)

res = loudness_moore_glasberg_time(x, fs, field="free")
print(f"N_max = {res.n_max:.3f} sone  ({res.loudness_level_max:.0f} phon)")   # 1.000 sone (40 phon)
print(f"long-term loudness exceeded 5% of the time: {res.percentiles[5.0]:.3f} sone")   # 0.999 sone

res.plot()   # short-term S'(t) and long-term S''(t) loudness vs time
```

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/moore_glasberg_time_loudness_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/moore_glasberg_time_loudness.svg" alt="Short-term and long-term Moore-Glasberg loudness traces for a tone burst, showing the fast attack of the short-term loudness and the slower release of the long-term loudness" width="80%"></picture>

<details>
<summary>Show the code for this figure</summary>

```python
import matplotlib.pyplot as plt

# The result carries both traces on a 1 ms time axis:
res.plot()
plt.show()

# Or plot them directly to see the fast STL vs the slow LTL:
fig, ax = plt.subplots()
ax.plot(res.time, res.short_term_loudness, label="Short-term S'(t)")
ax.plot(res.time, res.long_term_loudness, label="Long-term S''(t)")
ax.set_xlabel("Time [s]")
ax.set_ylabel("Loudness [sone]")
ax.legend()
plt.show()
```

</details>

#### `loudness_moore_glasberg_time()` parameters

| Parameter | Type | Units | Range / default | Notes |
| :--- | :--- | :--- | :--- | :--- |
| `signal` | 1D or `(n, 2)` array | Pa | non-empty | Mono = diotic; two columns = left/right ears |
| `fs` | int | Hz | > 0 | |
| `field` | str | — | `'free'` (default) / `'diffuse'` / `'eardrum'` | Outer-ear transfer |
| `presentation` | str | — | `'binaural'` (default) / `'diotic'` / `'monaural'` | Binaural summation |
| `percentiles` | sequence | percent | default `(1, 5, 10, 50, 90, 95)` | Exceeded long-term loudness levels |

Returns a `MooreGlasbergTimeVaryingLoudness`: `time` (1 ms grid),
`short_term_loudness` / `long_term_loudness` (sone), their `_level` in phon,
`n_max`, `loudness_level_max`, a `percentiles` dict, `field`, `presentation`.

### Sottek Hearing Model loudness (ECMA-418-2)

ECMA-418-2:2025 specifies a single auditory front-end — outer/middle-ear
filtering, a 53-band gammatone-like filter bank on the Bark_HMS scale
(z = 0.5 .. 26.5), half-wave rectification, block RMS and a compressive
nonlinearity (Formula 23) — that is **shared** by its loudness, tonality and
roughness metrics. The loudness N is reported in **sone_HMS**, and the same
1 kHz/40 dB anchor calibrates the front-end (our clean-room value 0.996).

```python
import numpy as np
from phonometry import loudness_ecma

fs = 48000
t = np.arange(int(1.2 * fs)) / fs
x = np.sqrt(2) * 2e-5 * 10 ** (40 / 20) * np.sin(2 * np.pi * 1000 * t)

res = loudness_ecma(x, fs, field="free")
print(f"N = {res.loudness:.3f} sone_HMS")   # 0.996 sone_HMS
print(res.specific_loudness.shape)          # (53,) average specific loudness N'(z)

res.plot()   # average specific loudness N'(z) + time-dependent N(l) at 187.5 Hz
```

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/sottek_specific_loudness_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/sottek_specific_loudness.svg" alt="Sottek Hearing Model average specific loudness N'(z) over the 53 Bark_HMS bands for a 1 kHz tone, peaking at the tone's critical band" width="80%"></picture>

<details>
<summary>Show the code for this figure</summary>

```python
import matplotlib.pyplot as plt

# The result carries the average specific loudness over the 53 Bark_HMS bands:
res.plot()
plt.show()

# Or draw N'(z) by hand against the critical-band-rate scale:
fig, ax = plt.subplots()
ax.fill_between(res.bark, res.specific_loudness, alpha=0.3)
ax.plot(res.bark, res.specific_loudness)
ax.set_xlabel("Critical-band rate z [Bark_HMS]")
ax.set_ylabel("Specific loudness N' [sone_HMS/Bark_HMS]")
plt.show()
```

</details>

#### `loudness_ecma()` parameters

| Parameter | Type | Units | Range / default | Notes |
| :--- | :--- | :--- | :--- | :--- |
| `signal_in` | 1D array | Pa | non-empty | Calibrated pressure signal |
| `fs` | float | Hz | > 0 | Resampled to 48 kHz internally if needed (Clause 5.1.1) |
| `field` | str | — | `'free'` (default) / `'diffuse'` | Outer/middle-ear filter (Clause 5.1.3) |

Returns an `EcmaLoudness`: `loudness` (N, sone_HMS), `specific_loudness`
(N′(z), 53 bands), `bark`, `centre_frequencies`, `time`, `loudness_vs_time`
(N(l) at 187.5 Hz), `field`.

### Tonality (ECMA-418-2)

A tonal component — a whistle, a fan's blade-passing tone — stands out even at
low level. ECMA-418-2 quantifies it from the **autocorrelation function** (ACF)
of each band's rectified signal: a periodic (tonal) component keeps a high ACF
at nonzero lag, and the tonal-to-noise loudness ratio drives the specific
tonality T′(z). The single value T is in **tu_HMS**, calibrated so a 1 kHz/40 dB
tone is ≈ 1 tu_HMS; the result also tracks the tonal frequency f_ton per band.

```python
import numpy as np
from phonometry import tonality_ecma

fs = 48000
t = np.arange(int(1.2 * fs)) / fs
x = np.sqrt(2) * 2e-5 * 10 ** (40 / 20) * np.sin(2 * np.pi * 1000 * t)

res = tonality_ecma(x, fs, field="free")
peak = int(np.argmax(res.specific_tonality))
print(f"T = {res.tonality:.3f} tu_HMS")                    # 1.000 tu_HMS
print(f"f_ton = {res.tonal_frequencies[peak]:.0f} Hz")     # 999 Hz

res.plot()   # average specific tonality T'(z) + time-dependent T(l)
```

#### `tonality_ecma()` parameters

| Parameter | Type | Units | Range / default | Notes |
| :--- | :--- | :--- | :--- | :--- |
| `signal_in` | 1D array | Pa | non-empty | Calibrated pressure signal |
| `fs` | float | Hz | > 0 | Resampled to 48 kHz internally if needed |
| `field` | str | — | `'free'` (default) / `'diffuse'` | Outer/middle-ear filter |
| `f_low` | float, optional | Hz | default `None` | Lower edge of a user band for the T(l) search |
| `f_high` | float, optional | Hz | default `None` | Upper edge of the user band |

Returns an `EcmaTonality`: `tonality` (T, tu_HMS), `specific_tonality`
(T′(z), 53 bands), `bark`, `centre_frequencies`, `tonal_frequencies`
(f_ton,z), `time`, `tonality_vs_time` (T(l)), `tonal_frequency_vs_time`,
`field`.

### Roughness (ECMA-418-2) — new capability

Roughness is the harsh, buzzing sensation of fast amplitude modulation
(roughly 20–300 Hz, peaking near 70 Hz) — the quality of a diesel idle or a
distorted loudspeaker. It is a **new metric** for phonometry. ECMA-418-2
extracts each band's envelope, weights its modulation spectrum by modulation
rate and depth, and correlates the modulation across bands; the result R is in
**asper**. The reference sound (1 kHz carrier, 100 % amplitude-modulated at
70 Hz, 60 dB SPL) is defined as 1 asper — this clean-room implementation
returns 1.0735 asper (about +7 %), an honest variance: the tabulated
calibration constant c_R (Formula 104) is used **without** reverse-fitting to
the target.

```python
import numpy as np
from phonometry import roughness_ecma

fs = 48000
t = np.arange(int(2.0 * fs)) / fs
carrier = np.sin(2 * np.pi * 1000 * t)
amp = np.sqrt(2) * 2e-5 * 10 ** (60 / 20)                  # 60 dB SPL carrier
x = amp * (1.0 + np.cos(2 * np.pi * 70 * t)) * carrier      # 100 % AM at 70 Hz

res = roughness_ecma(x, fs, field="free")
print(f"R = {res.roughness:.4f} asper")   # 1.0735 asper (reference target 1.0)

res.plot()   # time-dependent roughness R(l50) + specific-roughness heatmap
```

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/tonality_roughness_demo_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/tonality_roughness_demo.svg" alt="ECMA-418-2 sound-quality demo: a tonal sound scores high tonality and near-zero roughness, while a 70 Hz amplitude-modulated sound scores high roughness and low tonality" width="80%"></picture>

<details>
<summary>Show the code for this figure</summary>

```python
import matplotlib.pyplot as plt
import numpy as np
from phonometry import tonality_ecma, roughness_ecma

fs = 48000
t = np.arange(int(2.0 * fs)) / fs
amp = np.sqrt(2) * 2e-5 * 10 ** (60 / 20)

# A pure tone (tonal, smooth) vs a 70 Hz amplitude-modulated tone (rough):
tone = amp * np.sin(2 * np.pi * 1000 * t)
rough = amp * (1.0 + np.cos(2 * np.pi * 70 * t)) * np.sin(2 * np.pi * 1000 * t)

scores = {
    "Pure tone": (tonality_ecma(tone, fs).tonality, roughness_ecma(tone, fs).roughness),
    "70 Hz AM tone": (tonality_ecma(rough, fs).tonality, roughness_ecma(rough, fs).roughness),
}
labels = list(scores)
tonal = [scores[k][0] for k in labels]
rough_v = [scores[k][1] for k in labels]
xpos = np.arange(len(labels))
fig, ax = plt.subplots()
ax.bar(xpos - 0.2, tonal, 0.4, label="Tonality [tu_HMS]")
ax.bar(xpos + 0.2, rough_v, 0.4, label="Roughness [asper]")
ax.set_xticks(xpos)
ax.set_xticklabels(labels)
ax.legend()
plt.show()
```

</details>

#### `roughness_ecma()` parameters

| Parameter | Type | Units | Range / default | Notes |
| :--- | :--- | :--- | :--- | :--- |
| `signal_in` | 1D array | Pa | non-empty | Calibrated pressure signal |
| `fs` | float | Hz | > 0 | Resampled to 48 kHz internally if needed |
| `field` | str | — | `'free'` (default) / `'diffuse'` | Outer/middle-ear filter |

Returns an `EcmaRoughness`: `roughness` (R, asper, the 90th percentile of
R(l50)), `specific_roughness` (R′(z), 53 bands), `bark`, `centre_frequencies`,
`time`, `roughness_vs_time` (R(l50)), `specific_roughness_vs_time`
((n_times, 53) array), `field`.

See [Prominent Discrete Tones](https://jmrplens.github.io/phonometry/guides/tone-prominence/) for the ECMA-418-1 TNR/PR
prominence verdicts, [Speech Transmission Index](https://jmrplens.github.io/phonometry/guides/speech-transmission/) for
STI/STIPA, and [Theory](https://jmrplens.github.io/phonometry/reference/theory/) for the underlying math.

---

**Standards.** ISO 532-1:2017, *Acoustics — Methods for calculating
loudness — Part 1: Zwicker method* — stationary and time-varying loudness in
sones from the normative Annex A.4 reference program, with the N5/N10
percentile loudness, validated against the Annex B set. ISO 532-2:2017,
*... Part 2: Moore-Glasberg method* — stationary loudness from roex excitation
patterns on the ERB-number scale, with explicit binaural summation.
ISO 532-3:2023, *... Part 3: Moore-Glasberg-Schlittenlacher method* —
time-varying short-term and long-term loudness and the peak N_max.
DIN 45692:2009, *Messtechnische Simulation der Hörempfindung Schärfe* —
sharpness in acum (clause 6 weighting, Annex B von Bismarck and Aures
variants, Table A.2 targets). ISO 226:2023, *Acoustics — Normal
equal-loudness-level contours* — the contours (Formula 1), the loudness level
of pure tones (Formula 2) and the hearing threshold. ECMA-418-2:2025,
*Psychoacoustic metrics for ITT equipment — Part 2 (methods for describing
human perception based on the Sottek Hearing Model)* — the Sottek Hearing
Model loudness (sone_HMS), tonality (tu_HMS) and roughness (asper).

---


<!-- source: docs/speech-transmission.md | canonical: https://jmrplens.github.io/phonometry/guides/speech-transmission/ -->

# Speech Transmission Index (IEC 60268-16)

A public-address system, an intercom, a reverberant lecture hall — each is a
*transmission channel* between a talker's mouth and a listener's ear, and each
degrades speech in its own way. The **Speech Transmission Index** (STI) of
IEC 60268-16 rates that channel with a single number in [0, 1] by measuring
how much of the speech *envelope* survives the trip. This page covers the
modulation-transfer physics behind the index, the indirect method from a
measured room impulse response, and the direct STIPA measurement with its
standardized test signal.

> [!NOTE]
> **STI vs SII.** The STI characterises a *transmission channel* — how much of
> the speech modulation a room or sound system preserves — while the SII
> predicts intelligibility from *audibility*: how much of the speech spectrum
> clears the noise and the hearing threshold at the listener's ear. For the
> latter, see the [Speech Intelligibility Index guide](https://jmrplens.github.io/phonometry/guides/speech-intelligibility/).

## 1. The modulation transfer function

Reverberation and noise do not muffle speech uniformly — they blur its
*envelope*: the slow (0.63–12.5 Hz) intensity modulations that carry
syllables. STI quantifies how much of that modulation survives from mouth
to ear, per octave band, as the **modulation transfer function** m(F). A
delta-like channel keeps m = 1 (STI = 1); reverberation low-passes the
envelope following Schroeder's closed form, and steady noise scales it:

$$
m(F) = \frac{1}{\sqrt{1 + \left(2\pi F\,\frac{T_{60}}{13.8}\right)^2}}
\cdot \frac{1}{1 + 10^{-\mathrm{SNR}/10}}
$$

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/sti_vs_t60_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/sti_vs_t60.svg" alt="STI versus reverberation time with the IEC 60268-16 Annex F rating bands shaded" width="80%"></picture>

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/diagram_sti_chain_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/diagram_sti_chain.svg" alt="STI measurement chain: STIPA source signal through the room to the microphone and the MTF analysis" width="92%"></picture>

## 2. Indirect and direct (STIPA) measurement

```python
import numpy as np
from phonometry import sti_from_impulse_response, stipa, stipa_signal

fs = 48000
# A measured room impulse response (synthesized decay so the example runs)
ir = np.random.default_rng(0).standard_normal(fs) * np.exp(-6.9 * np.arange(fs) / fs / 0.5)

# Indirect method: from a measured room impulse response
res = sti_from_impulse_response(ir, fs, snr=25.0)
print(f"STI = {res.sti:.2f}  ({res.rating})")   # e.g. 0.62 (D)

# Direct STIPA measurement: play stipa_signal() in the room, record it
test = stipa_signal(fs, seconds=18.0, level_db=80.0)
recording = test                       # in practice, the microphone signal after playback
res = stipa(recording, fs)
res.plot()   # per-band modulation transfer index (MTI) bars, STI + rating in the title
```

<details>
<summary>Show the code for this figure</summary>

```python
import numpy as np
import matplotlib.pyplot as plt
from phonometry import sti_from_impulse_response

fs = 48000

# STI vs reverberation time: sweep sti_from_impulse_response over synthetic
# exponential decays (white noise x exp(-6.9077 t / T60)) at a T60 grid —
# exactly the physics behind the curve above:
rng = np.random.default_rng(0)
t60_grid = np.array([0.3, 0.5, 0.8, 1.2, 1.6, 2.0, 2.5, 3.0, 4.0, 5.0])
sti_values = []
for t60 in t60_grid:
    t = np.arange(int(2 * t60 * fs)) / fs
    ir = rng.standard_normal(t.size) * np.exp(-6.9077 * t / t60)
    sti_values.append(sti_from_impulse_response(ir, fs).sti)

fig, ax = plt.subplots()
ax.semilogx(t60_grid, sti_values, "o-")
ax.set_xlabel("Reverberation time T60 [s]")
ax.set_ylabel("STI")
ax.set_ylim(0.0, 1.0)
ax.grid(True, which="both", alpha=0.3)
plt.show()
```

</details>

`stipa` emits a `UserWarning` when the recording is shorter than the
recommended 15 s (IEC 60268-16 STIPA practice, 15 s to 25 s): below that the
slow modulation components are averaged over too few periods and the STI is
biased low (an ideal loopback gives STI ≈ 0.944 at 5 s vs ≈ 0.998 at 18 s).

The implementation follows **Edition 5 (2020)**: Edition 4's normative PDF
is the base and every Ed. 5 change is source-attributed in the code — the
only numeric delta is the revised male speech spectrum of clause A.6.1.
CI checks the standard's own verification vectors: the six weighting-factor
band pairs to ±0.001 STI, the m ↔ STI mapping table, the level-dependent
masking control points, and Schroeder-form decays at four T₆₀ values.

### `sti_from_impulse_response()` / `stipa()` parameters

| Parameter | Type | Units | Range / default | Notes |
| :--- | :--- | :--- | :--- | :--- |
| `ir` / `x` | 1D array | any / Pa | non-empty | IR (indirect) or STIPA recording (direct) |
| `fs` | int | Hz | > 0 | |
| `snr` | float or 7-vector, optional | dB | default `None` | Adds steady-noise degradation |
| `level` | 7-vector, optional | dB SPL | default `None` | Enables auditory masking + reception threshold (Tables A.2/A.3) |
| `ambient` | 7-vector, optional | dB SPL | needs `level` | Ambient noise band levels |
| `reference` | 1D array, optional (`stipa`) | — | default `None` | Measured source signal instead of the nominal m = 0.55 |

Both return `STIResult`: `sti`, `mti` (7 bands), `mtf` (7×14 or 7×2),
`band_levels`, `rating` (Annex F letter `A+`…`U`).

## See also

- [Room Acoustics](https://jmrplens.github.io/phonometry/guides/room-acoustics/) — the measured impulse response the
  indirect method consumes, and the open-plan metrics (ISO 3382-3) built on
  per-position STI.
- [Speech Intelligibility Index](https://jmrplens.github.io/phonometry/guides/speech-intelligibility/) — the
  audibility-based ANSI S3.5 index that complements the STI.
- [Psychoacoustics](https://jmrplens.github.io/phonometry/guides/psychoacoustics/) — loudness, sharpness, tonality and
  roughness of the received sound.
- [Theory](https://jmrplens.github.io/phonometry/reference/theory/) — the modulation-transfer derivation and the m ↔ STI
  mapping.

---

**Standards.** IEC 60268-16:2020 (Edition 5), *Sound system equipment —
Part 16: Objective rating of speech intelligibility by speech transmission
index* — the modulation transfer function and the m ↔ STI mapping, the STIPA
test signal and direct method, the indirect method from the impulse response,
auditory masking and the reception threshold (Tables A.2/A.3), the revised
male speech spectrum (clause A.6.1) and the Annex F rating letters.

---


<!-- source: docs/speech-intelligibility.md | canonical: https://jmrplens.github.io/phonometry/guides/speech-intelligibility/ -->

# Speech Intelligibility Index (SII)

The **Speech Intelligibility Index** predicts how much of a speech signal is
audible, and therefore intelligible, to a listener in a given noise and hearing
condition. It reduces a speech spectrum, a noise spectrum and a hearing
threshold to a single number in `[0, 1]`: `0` when nothing useful reaches the
listener, `1` when the whole speech-bearing spectrum is audible. This page
covers the **one-third-octave-band method** of **ANSI S3.5-1997 (R2017)** — 18
bands from 160 Hz to 8000 Hz.

> [!NOTE]
> **SII vs STI.** The SII predicts intelligibility from *audibility* — how much
> of the speech spectrum clears the noise and the hearing threshold at the
> listener's ear — while the STI characterises a *transmission channel*: how
> much of the speech modulation a room or sound system preserves. For the
> latter, see the [Speech Transmission Index guide](https://jmrplens.github.io/phonometry/guides/speech-transmission/).

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/diagram_speech_intelligibility_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/diagram_speech_intelligibility.svg" alt="The SII computation flow: three equivalent-spectrum-level inputs (speech Ei', noise Ni', hearing threshold Ti') feed the self-speech masking and spread-of-masking stage (equivalent masking spectrum level Zi), then the equivalent disturbance Di, then the band-audibility function Ai clipped to [0, 1], and finally the band-importance-weighted sum SII = sum of Ii*Ai over the 18 one-third-octave bands" width="94%"></picture>

## 1. Inputs and the band-importance function

All three inputs are **equivalent spectrum levels** (ANSI S3.5-1997 clauses 3.11
and 3.55) sampled at the 18 one-third-octave band centres: the speech spectrum
level `Ei'`, the noise spectrum level `Ni'` (both in dB SPL) and the hearing
threshold `Ti'` (in dB HL). Each band `i` contributes to intelligibility in
proportion to its **band-importance function** `Ii` (ANSI S3.5-1997 Table 3,
average speech material), which sums to one across the 18 bands.

```python
import phonometry as ph

# The standard normal-effort speech spectrum (Table 3) in quiet, normal hearing.
result = ph.speech_intelligibility_index("normal")
print(round(result.sii, 3))          # 0.996  (nearly everything audible)
print(round(ph.sii.BAND_IMPORTANCE.sum(), 6))   # 1.0
```

With no noise and a normal hearing threshold the standard speech spectrum is
almost fully audible, so the index is close to one; the small deficit is the
listener's own **self-speech masking**.

## 2. Masking and the band-audibility function

The procedure (ANSI S3.5-1997 clause 5) turns the inputs into a per-band
audibility. Speech masks itself downward from each band (`Vi = Ei' - 24`); the
larger of that and the external noise, `Bi`, spreads **upward** in frequency
with a level-dependent slope to give the equivalent masking spectrum level `Zi`
(clause 5.4):

$$
Z_i = 10\log_{10}\!\left(10^{0.1 N_i'} + \sum_{k<i}
      10^{0.1\left(B_k + 3.32\,C_k\,\log_{10}(0.89\,f_i/f_k)\right)}\right).
$$

The masking is combined with the equivalent internal noise
(`Xi' = Xi + Ti'`, the reference internal noise shifted by the hearing loss)
into the **equivalent disturbance** `Di` (clause 5.6), and the **band-audibility
function** is the speech-to-disturbance ratio scaled into `[0, 1]` (clause 5.8):

$$
A_i = \operatorname{clip}\!\left(\frac{E_i' - D_i + 15}{30},\; 0,\; 1\right).
$$

At speech levels well above normal effort a **level-distortion factor** of
clause 5.7 — unity for the standard spectra used on this page — reduces $A_i$
further; phonometry applies it automatically.

## 3. The index in noise

The Speech Intelligibility Index is the band-importance-weighted sum of the band
audibilities (ANSI S3.5-1997 clause 6):

$$
\text{SII} = \sum_{i} I_i\, A_i .
$$

```python
import numpy as np
import phonometry as ph

speech = ph.standard_speech_spectrum("normal")
# A descending broadband masking noise (an office/ventilation-like spectrum).
noise = np.array([38.0, 37.0, 36.0, 34.0, 32.0, 30.0, 28.0, 26.0, 24.0,
                  22.0, 20.0, 18.0, 16.0, 14.0, 12.0, 10.0, 8.0, 6.0])

result = ph.speech_intelligibility_index(speech, noise)
print(round(result.sii, 2))                # 0.46
print(result.band_audibility.round(2))     # per-band Ai
```

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/speech_intelligibility_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/speech_intelligibility.svg" alt="Band audibility of the standard normal-effort speech spectrum in a descending broadband noise: the light bars are the per-band audibility Ai across the 18 one-third-octave bands from 160 Hz to 8000 Hz, the darker bars the importance-weighted contribution Ii*Ai (scaled), and the overall SII is 0.46" width="90%"></picture>

<details>
<summary>Show the code for this figure</summary>

```python
import numpy as np
import matplotlib.pyplot as plt
import phonometry as ph

speech = ph.standard_speech_spectrum("normal")
noise = np.array([38.0, 37.0, 36.0, 34.0, 32.0, 30.0, 28.0, 26.0, 24.0,
                  22.0, 20.0, 18.0, 16.0, 14.0, 12.0, 10.0, 8.0, 6.0])
result = ph.speech_intelligibility_index(speech, noise)

# One line:
result.plot()
plt.show()

# By hand, mirroring what SIIResult.plot() draws:
pos = np.arange(result.frequencies.size)
weighted = result.band_audibility * result.band_importance
fig, ax = plt.subplots()
ax.bar(pos, result.band_audibility, color="#c6dbef", label=r"Band audibility $A_i$")
ax.bar(pos, weighted / weighted.max(), width=0.5, color="#1f77b4",
       label=r"Importance-weighted $I_i A_i$ (scaled)")
ax.set_xticks(pos)
ax.set_xticklabels([f"{f:g}" for f in result.frequencies], rotation=45, ha="right")
ax.set_xlabel("One-third-octave band [Hz]")
ax.set_ylabel("Band audibility")
ax.set_title(f"SII = {result.sii:.2f}")
ax.legend()
plt.show()
```

</details>

A raised hearing threshold (`threshold=`) lifts the equivalent internal noise
and lowers the index, exactly as added masking noise does. The
`SIIResult` also carries the per-band masking `Zi`, disturbance `Di`, audibility
`Ai` and importance `Ii`, and its `.plot()` renders the figure above.

## 4. Vocal effort

Talkers raise their voice in noise, and the standard gives four **standard
speech spectra** for the vocal efforts *normal*, *raised*, *loud* and *shout*
(ANSI S3.5-1997 Table 3). Passing the effort name selects the corresponding
spectrum; speaking louder lifts the whole spectrum and, in a fixed noise, raises
the index.

```python
import numpy as np
import phonometry as ph

# The same broadband noise, four vocal efforts.
noise = np.array([48.0, 47.0, 46.0, 44.0, 42.0, 40.0, 38.0, 36.0, 34.0,
                  32.0, 30.0, 28.0, 26.0, 24.0, 22.0, 20.0, 18.0, 16.0])
for effort in ph.sii.VOCAL_EFFORTS:
    print(effort, round(ph.speech_intelligibility_index(effort, noise).sii, 2))
# normal 0.12 | raised 0.36 | loud 0.59 | shout 0.79

print(ph.standard_speech_spectrum("loud")[8])  # 42.16 dB SPL at 1 kHz
```

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/sii_vocal_efforts_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/sii_vocal_efforts.svg" alt="Two panels. Left: the four ANSI S3.5-1997 standard speech spectra — normal, raised, loud and shout — over the 18 one-third-octave bands from 160 Hz to 8000 Hz, each higher vocal effort lifting the whole spectrum. Right: the resulting Speech Intelligibility Index in a fixed broadband noise, rising from 0.12 (normal) through 0.36 and 0.59 to 0.79 (shout)" width="96%"></picture>

<details>
<summary>Show the code for this figure</summary>

```python
import numpy as np
import matplotlib.pyplot as plt
from matplotlib.ticker import NullFormatter
import phonometry as ph

# The four ANSI S3.5-1997 Table 3 spectra and the fixed broadband noise above.
noise = np.array([48.0, 47.0, 46.0, 44.0, 42.0, 40.0, 38.0, 36.0, 34.0,
                  32.0, 30.0, 28.0, 26.0, 24.0, 22.0, 20.0, 18.0, 16.0])
efforts = ph.sii.VOCAL_EFFORTS         # ("normal", "raised", "loud", "shout")
freqs = ph.sii.BAND_CENTERS            # the 18 one-third-octave band centres

fig, (ax_s, ax_i) = plt.subplots(1, 2, figsize=(12, 5))

# Left: each higher vocal effort lifts the whole speech spectrum.
for effort in efforts:
    ax_s.plot(freqs, ph.standard_speech_spectrum(effort), "o-",
              label=effort.capitalize())
ax_s.set_xscale("log")
ax_s.set_xticks(list(freqs))
ax_s.set_xticklabels([f"{f:g}" for f in freqs], rotation=45, ha="right")
ax_s.xaxis.set_minor_formatter(NullFormatter())
ax_s.set_xlabel("One-third-octave band [Hz]")
ax_s.set_ylabel("Speech spectrum level [dB SPL]")
ax_s.legend()

# Right: the SII each spectrum reaches in the fixed noise.
sii = [ph.speech_intelligibility_index(e, noise).sii for e in efforts]
pos = np.arange(len(efforts))
ax_i.bar(pos, sii)
ax_i.set_xticks(pos)
ax_i.set_xticklabels([e.capitalize() for e in efforts])
ax_i.set_ylim(0.0, 1.0)
ax_i.set_ylabel("Speech Intelligibility Index")
plt.show()
```

</details>

The vocal-effort names work anywhere a speech spectrum is expected, including as
the first argument to `speech_intelligibility_index`.

---

**Standards.** ANSI S3.5-1997 (R2017), *American National Standard Methods for
the Calculation of the Speech Intelligibility Index* — the one-third-octave-band
method (18 bands), band-importance function (Table 3), standard speech spectrum
level and reference internal noise (Table 3), and the masking, disturbance and
band-audibility procedure (clause 5) and the index (clause 6).

---


<!-- source: docs/intensity.md | canonical: https://jmrplens.github.io/phonometry/guides/intensity/ -->

# Sound Intensity (p-p method)

Sound *pressure* tells you how loud a point is; sound **intensity** tells
you where the energy is *going*. It is the acoustic power flux (W/m²), a
signed vector quantity — which is why intensity probes can localize sources,
separate them from background noise and measure sound power in situ
(ISO 9614) where a pressure measurement alone cannot.

## The two-microphone principle (IEC 61043)

A p-p probe holds two matched microphones a small distance Δr apart. The
pressure at the probe center is their mean, and the particle velocity comes
from the pressure *gradient* (Euler's equation, finite-difference form):

$$
p = \frac{p_1 + p_2}{2}, \qquad
u = -\frac{1}{\rho_0\ \Delta r}\int (p_2 - p_1)\ dt, \qquad
I = \overline{p\ u}
$$

In practice the estimator works in the frequency domain through the
cross-spectrum of the two channels (the standard's equivalent form):

$$
I(f) = -\ \frac{\mathrm{Im}\lbrace G_{12}(f)\rbrace}{2\pi f\ \rho_0\ \Delta r}
$$

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/diagram_pp_probe_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/diagram_pp_probe.svg" alt="Two-microphone p-p intensity probe with the spacer distance and the measurement axis" width="92%"></picture>

```python
import numpy as np
from phonometry import sound_intensity

fs = 48000
rng = np.random.default_rng(0)
# The two probe-microphone pressures in Pa, p1 closest to the source.
#   In a real measurement these are your two calibrated probe recordings;
#   synthesized here (p2 = p1 delayed one sample) so the guide runs.
p1 = 0.02 * rng.standard_normal(fs)
p2 = np.concatenate(([0.0], p1[:-1]))   # p2 = p1 delayed one sample

res = sound_intensity(p1, p2, fs, spacing=0.012, fraction=3,
                      limits=[100, 2500])
print(res.total_intensity_level, res.total_direction)      # LI [dB], ±1
print(res.frequency, res.intensity_level)                  # per band
res.plot()   # Lp vs LI per band + the pressure-intensity index (needs matplotlib)
```

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/anim_instantaneous_intensity_dark.gif"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/anim_instantaneous_intensity.gif" alt="Animation: instantaneous intensity p times u for a progressive wave (always positive, non-zero mean, net power flows) versus a standing wave (oscillating about a zero mean, energy just sloshes)" width="88%"></picture>

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/intensity_demo_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/intensity_demo.svg" alt="Third-octave pressure and intensity levels for a plane progressive wave versus a standing wave" width="92%"></picture>

*Left: in a plane progressive wave all pressure is transported —
L_I ≈ L_p. Right: a standing wave carries (almost) no net energy — the
pressure is high but the intensity collapses. The gap L_p − L_I is the
**pressure-intensity index**, the fundamental quality indicator of every
intensity measurement.*

<details>
<summary>Show the code for this figure</summary>

```python
import matplotlib.pyplot as plt

# res is the IntensityResult computed in the example above.
# One line — Lp vs LI per band with the pressure-intensity index on a twin axis:
res.plot()
plt.show()

# By hand, from the per-band fields the result carries — mirroring what
# IntensityResult.plot() draws (bar label, merged twin-axis legend, δpI title):
fig, ax = plt.subplots()
ax.semilogx(res.frequency, res.pressure_level, "o-", label="Pressure level Lp")
ax.semilogx(res.frequency, res.intensity_level, "s--", label="Intensity level LI")
ax.set_xlabel("Frequency [Hz]")
ax.set_ylabel("Level [dB]")
twin = ax.twinx()
twin.bar(res.frequency, res.pressure_intensity_index,
         width=res.frequency * 0.2, color="#2ca02c", alpha=0.25,
         label="δpI = Lp − LI")
twin.set_ylabel("Pressure-intensity index δpI [dB]")
# Merge both axes' handles into a single legend, exactly as .plot() does:
lines, labels = ax.get_legend_handles_labels()
tlines, tlabels = twin.get_legend_handles_labels()
ax.legend(lines + tlines, labels + tlabels)
ax.set_title(f"Lp vs LI  (total δpI = {res.total_pressure_intensity_index:.1f} dB)")
plt.show()
```

</details>

## Knowing when to trust the number

Two physical limits bound every p-p measurement, and the result object
carries both:

- **High frequency**: the finite-difference gradient underestimates I by
  $\sin(k\Delta r)/(k\Delta r)$ — verified in CI against IEC 61043 Table 3.
  `IntensityResult.bias_correction` provides the factor and
  `max_valid_frequency` (≈ 0.1·c/Δr; 2.9 kHz for a 12 mm spacer) the
  practical ceiling. Larger spacers reach lower frequencies, smaller ones
  higher.
- **Reactive fields**: when `pressure_intensity_index` (F2 in ISO 9614-1)
  approaches the probe's residual index δ_pI0, phase errors dominate.

Over a measurement surface, the ISO 9614-1 Annex A field indicators grade the
scan itself. **F2**, the surface pressure-intensity indicator, is the surface
pressure level minus the level of the mean *magnitude* of the normal
intensity — the larger it is, the closer the measurement sits to the probe's
phase-error floor. **F3**, the negative partial power indicator, is the same
difference taken with the *signed* mean intensity — F3 − F2 > 0 reveals power
flowing inward through parts of the surface. **F4**, the field non-uniformity
indicator, is the normalised spread of the per-position intensities — the
larger it is, the more measurement positions the surface needs. Together with
the dynamic-capability criterion they are available directly:

```python
import numpy as np
from phonometry import field_indicators, dynamic_capability_index

# Per-position measurements over the ISO 9614-1 measurement surface
pressure_levels = np.array([74.1, 73.8, 74.5, 73.2])       # Lp per position (dB)
normal_intensity = np.array([1.2e-5, 1.0e-5, 1.4e-5, 0.9e-5])  # signed In per position (W/m²)

fi = field_indicators(pressure_levels, normal_intensity)
print(round(fi.f2, 2), round(fi.f3, 2), round(fi.f4, 3))   # 3.41 3.41 0.197
ld = dynamic_capability_index(18.0)   # δpI0 = 18 dB → Ld = δpI0 − K
print(ld, ld > fi.f2)                                      # 8.0 True (criterion 1)
```

### `sound_intensity()` parameters

| Parameter | Type | Units | Range / default | Notes |
| :--- | :--- | :--- | :--- | :--- |
| `p1`, `p2` | 1D arrays | Pa | equal length | Microphone closer to the source first; reversing them flips the sign |
| `fs` | int | Hz | > 0 | |
| `spacing` | float | m | > 0 | Microphone separation Δr (typ. 6/12/50 mm) |
| `rho` | float | kg/m³ | default `1.204` | Air density |
| `c` | float | m/s | default `343.0` | Speed of sound (bias/validity estimates) |
| `fraction` | int, optional | — | `1`, `3` or `None` (default) | Octave/third-octave band integration |
| `limits` | list, optional | Hz | default library band range | Band analysis limits |
| `bias_correct` | bool | — | default `False` | Apply the per-bin $(k\Delta r)/\sin(k\Delta r)$ correction (IEC 61043 §7.3) before summing, so band/broadband totals stop under-reading as $f \to$ `max_valid_frequency`; bins past the first null are left uncorrected. The per-band `bias_correction` factor is reported either way |

See [Theory](https://jmrplens.github.io/phonometry/reference/theory/) for the derivations and [Calibration](https://jmrplens.github.io/phonometry/guides/calibration/)
for absolute scaling of the two channels.

---

**Standards.** IEC 61043:1994, *Electroacoustics — Instruments for the
measurement of sound intensity — Measurements with pairs of pressure sensing
microphones* — the two-microphone cross-spectral intensity estimator, the
finite-difference bias correction and the usable-bandwidth bound (clause 7.3,
Table 3). ISO 9614-1:1993, *Acoustics — Determination of sound power levels
of noise sources using sound intensity — Part 1: Measurement at discrete
points* — the pressure-intensity index, the Annex A field indicators F2, F3
and F4, and the dynamic-capability criterion (Annex B).

---


<!-- source: docs/room-acoustics.md | canonical: https://jmrplens.github.io/phonometry/guides/room-acoustics/ -->

# Room Acoustics

Room acoustics starts from one measurement: the **impulse response** (IR)
between a source and a receiver. Filter it into bands and integrate it, and
it yields reverberation time, clarity and speech intelligibility — everything
about the sound field inside a single room. This page follows that chain in
measurement order — acquiring the IR (ISO 18233), turning it into room
parameters (ISO 3382-1/2), spatial speech metrics for open-plan offices
(ISO 3382-3) and, closing the loop, the sound absorption of a material in a
reverberation room (ISO 354). For sound insulation *between* spaces — the same
IR measured either side of a partition — see the companion
[Building Acoustics & Sound Insulation guide](https://jmrplens.github.io/phonometry/guides/building-acoustics/).

## 1. Impulse-response acquisition (ISO 18233)

A room behaves, to a good approximation, as a **linear time-invariant**
system, so everything about it is contained in its IR. You could fire a
pistol and record the tail, but a deterministic excitation played through
a loudspeaker and *deconvolved* recovers the same IR with 20–30 dB more
effective signal-to-noise ratio (ISO 18233). Two excitations are provided.

**Exponential sine sweep (ESS, Annex B).** The instantaneous frequency
rises exponentially,

$$
f(t) = f_1 \left( \frac{f_2}{f_1} \right)^{t/T},
$$

so the time spent per octave is constant and the excitation mimics
pink noise (constant energy per fractional-octave band). The IR is
recovered by linear (zero-padded, non-circular) spectral division,

$$
H = \frac{Y\ \overline{X}}{|X|^2 + \varepsilon},
$$

with a small Tikhonov term $\varepsilon$ guarding the band edges where the
sweep has little energy. Because a low-to-high sweep places harmonic
distortion at *negative* arrival times, distortion separates cleanly from
the linear IR and is discarded by keeping only the causal part. The
`"farina"` method reaches the same result by convolving the recording with
the analytic inverse filter; it assumes the reference sweep was generated
with the default amplitude and fade, so use the spectral method for a
non-unit-amplitude or custom-fade sweep.

**Maximum-length sequence (MLS, Annex A).** An order-`N` binary sequence of
length $2^N-1$ whose circular autocorrelation is a near-perfect delta; the
IR follows from circular cross-correlation of the recorded period with the
sequence. MLS excites at constant amplitude and is quick to average, but it
is more sensitive to time variance (draughts, temperature drift) and cannot
be fed as much power as a sweep. **Prefer the sweep** for rooms and
partitions; reach for MLS when the excitation must be periodic or the
hardware favours a two-level signal.

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/diagram_ir_measurement_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/diagram_ir_measurement.svg" alt="ISO 18233 indirect measurement chain: an ESS sweep or MLS excitation drives a loudspeaker into the room, a microphone captures the response, and deconvolution (correlation or inverse filter) recovers the impulse response" width="92%"></picture>

```python
import numpy as np
from scipy.signal import fftconvolve
from phonometry import sweep_signal, impulse_response, mls_signal, mls_impulse_response

fs = 48000
# A 3 s, 20 Hz - 20 kHz sweep is a good broadband room excitation
# sweep: excitation you play through the loudspeaker
sweep = sweep_signal(fs, 20.0, 20000.0, 3.0)

# Deconvolve the recorded response back to the impulse response
system = np.zeros(fs); system[100] = 1.0; system[2000] = 0.4   # direct + reflection
# recorded: mic capture of the played sweep (here simulated by convolution with a synthetic room)
recorded = fftconvolve(sweep, system)
ir = impulse_response(recorded, sweep, fs, method="spectral")
print(int(np.argmax(np.abs(ir))))                    # 100: direct sound recovered

# Farina inverse-filter variant (needs the sweep band)
ir_f = impulse_response(recorded, sweep, fs, method="farina", f_range=(20.0, 20000.0))

# Periodic MLS: excite with >= 2 periods, average, cross-correlate
mls = mls_signal(16)                                 # length 2**16 - 1 = 65535
rec = fftconvolve(np.tile(mls, 2), system)[: 2 * mls.size]
ir_m = mls_impulse_response(rec, mls)
print(int(np.argmax(np.abs(ir_m))))                  # 100
```

`sweep_signal`/`mls_signal` return plain arrays, ready to write to a WAV file
and play. `impulse_response`/`mls_impulse_response` return an
`ImpulseResponseResult` — a drop-in for the raw IR array (`np.asarray(ir)`,
indexing and `ir.size` all keep working, so `room_parameters(ir, fs)` is
unchanged) that also carries the sample rate and method and adds an `.plot()`.

**The two excitations.** The exponential sweep sweeps its energy up the
spectrum over the whole signal, while the MLS is a flat-spectrum two-level
sequence — visible as the near-constant magnitude on the right.

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/excitation_signals_dark.png"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/excitation_signals.png" alt="ISO 18233 excitation signals: the exponential sine sweep waveform and its spectrogram showing the exponential frequency rise, and a maximum-length sequence with its flat magnitude spectrum" width="96%"></picture>

<details>
<summary>Show the code for this figure</summary>

```python
from phonometry import sweep_signal, mls_signal, plot_excitation

fs = 48000
sweep = sweep_signal(fs, 50.0, 20000.0, 1.0)   # ESS excitation
mls = mls_signal(12).astype(float)             # length 2**12 - 1

# One-liner: waveform + spectrogram (sweep), sequence + flat spectrum (MLS)
plot_excitation(sweep, fs, kind="sweep")
plot_excitation(mls, fs, kind="mls")

# By hand: the sweep spectrogram and the MLS magnitude spectrum
import numpy as np
import matplotlib.pyplot as plt

fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(12, 4))
ax1.specgram(sweep, NFFT=1024, Fs=fs, noverlap=512)
ax1.set(xlabel="Time [s]", ylabel="Frequency [Hz]", title="Sweep spectrogram")
spec = np.abs(np.fft.rfft(mls))
freqs = np.fft.rfftfreq(mls.size, d=1.0 / fs)
ax2.semilogx(freqs[1:], 20 * np.log10(spec[1:] / np.median(spec[1:])))
ax2.set(xlabel="Frequency [Hz]", ylabel="Magnitude [dB]", title="MLS spectrum (flat)")
```

</details>

**The recovered impulse response.** Deconvolving the recording gives the
broadband IR: the direct sound, discrete early reflections and the decaying
diffuse tail. Its `.plot()` shows the waveform above and the log-magnitude
envelope with the Schroeder energy-decay curve below — the straight decay
whose slope becomes the reverberation time in §2.

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/impulse_response_dark.png"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/impulse_response.png" alt="Recovered room impulse response: the normalized waveform with the direct sound and reflections labelled, and below it the log-magnitude envelope in dB with the Schroeder energy-decay curve" width="88%"></picture>

<details>
<summary>Show the code for this figure</summary>

```python
import numpy as np
from scipy.signal import fftconvolve
from phonometry import sweep_signal, impulse_response

fs = 48000
sweep = sweep_signal(fs, 20.0, 20000.0, 1.5)
# A synthetic room: direct sound + two reflections + a decaying diffuse tail
system = np.zeros(int(0.7 * fs))
system[80], system[1400], system[3100] = 1.0, 0.5, 0.32
ir = impulse_response(fftconvolve(sweep, system), sweep, fs, length=system.size)

# One-liner: waveform + log-magnitude / Schroeder decay
ir.plot()

# By hand: the normalized log-magnitude envelope in dB
import matplotlib.pyplot as plt
h = np.asarray(ir)
t = np.arange(h.size) / fs
plt.plot(t, 20 * np.log10(np.abs(h) / np.max(np.abs(h))))
plt.ylim(-80, 5)
plt.xlabel("Time [s]"); plt.ylabel("Level re peak [dB]")
```

</details>

**Where to measure.** One IR characterises a single source–receiver pair; a
reported room parameter is the spatial average over several. ISO 3382-1
(performance spaces) asks for at least two source positions and microphones
spaced $\geq 2$ m apart, $\geq 1$ m from any surface, at $1.2$ m
(seated-ear) height, avoiding symmetric placements; ISO 3382-2 fixes the
minimum number of source, microphone and source–microphone combinations per
accuracy grade (survey / engineering / precision).

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/diagram_room_measurement_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/diagram_room_measurement.svg" alt="Room-acoustics measurement setup: a top-view room plan with two loudspeaker source positions and six microphone positions with the ISO 3382-1 spacing rules, and the ISO 3382-2 table of minimum positions for the survey, engineering and precision grades" width="94%"></picture>

### `sweep_signal()` / `inverse_filter()` parameters

| Parameter | Type | Units | Range / default | Notes |
| :--- | :--- | :--- | :--- | :--- |
| `fs` | int | Hz | > 0 | Sampling frequency |
| `f1` | float | Hz | > 0, at/below lowest band | Sweep start frequency |
| `f2` | float | Hz | `f1 < f2 <= fs/2` | Sweep stop frequency |
| `seconds` | float | s | any; longer ⇒ more SNR | Sweep duration |
| `amplitude` | float | — | default `1.0` | Peak amplitude |
| `fade` | float | — | `[0, 0.5)`, default `0.01` | Half-Hann fade fraction (kills start/stop transients) |

### `impulse_response()` parameters

| Parameter | Type | Units | Range / default | Notes |
| :--- | :--- | :--- | :--- | :--- |
| `recorded` | 1D array | any | non-empty | Recorded system response |
| `reference` | 1D array | any | non-empty | The emitted sweep |
| `fs` | int | Hz | > 0 | Sample rate |
| `method` | str | — | `'spectral'` (default) / `'farina'` | `'farina'` requires `f_range` |
| `f_range` | (float, float) | Hz | default `None` | `(f1, f2)` of the sweep (Farina only) |
| `regularization` | float | — | default `1e-6` | Tikhonov term as a fraction of peak spectral energy |
| `length` | int, optional | samples | default `len(recorded)` | Samples of causal IR to return |
| `return_full` | bool | — | default `False` | Return the full sequence (distortion in the tail) |

`mls_signal(order)` takes an integer `order` in 2–20 (sequence length
$2^{\text{order}}-1$); `mls_impulse_response(recorded, mls, length=None)`
needs `recorded` to span an integer number of MLS periods.

## 2. Decay analysis and room parameters (ISO 3382-1/2)

The IR is filtered into octave (or one-third-octave) bands and each band is
turned into a **decay curve** by Schroeder backward integration of the
squared IR:

$$
E(t) = \int_t^{\infty} p^2(\tau)\ d\tau, \qquad
L(t) = 10 \log_{10} \frac{E(t)}{E(0)}\ \text{dB}.
$$

Integrating *backwards* removes the fluctuation that plagues a raw squared
IR and yields a smooth curve whose slope is the decay rate. Background
noise would make $E(t)$ level off, so integration is truncated where the
fitted decay line crosses the noise floor and the missing tail is
compensated assuming an exponential decay.

Reverberation times come from a least-squares line fit over an evaluation
range, extrapolated to a full 60 dB drop, $T = -60/\text{slope}$:
**EDT** over 0 to −10 dB (perceived reverberance), **T20** over −5 to −25 dB
and **T30** over −5 to −35 dB. Energy splits at an early/late boundary give
**clarity** and **definition**,

$$
C_{te} = 10 \log_{10} \frac{\int_0^{te} p^2\ dt}{\int_{te}^{\infty} p^2\ dt}\ \text{dB}, \qquad
D_{50} = \frac{\int_0^{0.05} p^2\ dt}{\int_0^{\infty} p^2\ dt},
$$

with $te = 50$ ms → C50 (speech) and $te = 80$ ms → C80 (music), plus the
**centre time** $T_s = \int t\ p^2\ dt / \int p^2\ dt$. Each parameter has a
**just-noticeable difference** (ISO 3382-1 Table A.1: EDT 5 %, C80 1 dB,
D50 0.05, Ts 10 ms) that sets how precisely it is worth reporting.

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/anim_schroeder_dark.gif"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/anim_schroeder.gif" alt="Animation: the Schroeder decay curve emerging as the squared impulse response is integrated backward from its tail, ending with the T20 and T30 regression lines" width="88%"></picture>

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/schroeder_decay_dark.png"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/schroeder_decay.png" alt="Squared impulse response with its Schroeder backward-integrated decay curve, and the EDT, T20 and T30 regression windows marked" width="80%"></picture>

*The jagged squared IR (grey) integrates to the smooth Schroeder curve
(blue); the EDT, T20 and T30 windows are fitted on that curve and each
extrapolated to a 60 dB decay.*

```python
import numpy as np
from phonometry import decay_curve, room_parameters

fs = 48000
# Single-slope decay with T = 1 s: p^2 = exp(-13.8155 t)  (60/ln(10)/13.8155 = 1)
t = np.arange(fs) / fs
# ir: measured room impulse response; a synthetic single-slope decay stands in here.
ir = np.concatenate([np.zeros(10), np.exp(-13.8155 * t / 2.0)])

time, level = decay_curve(ir, fs)                    # Schroeder curve (0 dB at t = 0)

res = room_parameters(ir, fs, limits=None)           # broadband single band
print(round(float(res.t30[0]), 2))                   # 1.0  s
print(round(float(res.c80[0]), 2))                   # 3.05 dB
print(round(float(res.d50[0]), 3))                   # 0.499
print(round(float(res.ts[0]) * 1000, 0))             # 72 ms

# Octave bands 125 Hz - 4 kHz (ISO 3382-1 default); use fraction=3 for thirds
octaves = room_parameters(ir, fs)
print(octaves.frequency)                             # ~[126, 251, 501, 1000, 1995, 3981]
print(octaves.t30_valid)                             # per-band dynamic-range flags

octaves.plot()               # per-band EDT/T20/T30 + C50/C80 bars (needs matplotlib)
decay_curve(ir, fs).plot()   # Schroeder decay with EDT/T20/T30 fit overlays
```

<details>
<summary>Show the code for this figure</summary>

```python
import matplotlib.pyplot as plt

# One line — Schroeder decay with the EDT/T20/T30 straight-line fits:
decay = decay_curve(ir, fs)          # a DecayCurve (still unpacks as time, level)
decay.plot()
plt.show()

# By hand, the decay is just the Schroeder curve; mark the evaluation levels:
fig, ax = plt.subplots()
ax.plot(time, level, color="#1f77b4", label="Schroeder decay")
for db in (-5.0, -25.0, -35.0):      # T20 / T30 evaluation-window edges
    ax.axhline(db, ls=":", alpha=0.4)
ax.set_xlabel("Time [s]")
ax.set_ylabel("Level re steady state [dB]")
ax.set_ylim(top=3.0)
ax.legend()
plt.show()
```

</details>

For this single-slope decay EDT, T20 and T30 all return ≈ 1.0 s, and the
energy parameters match their closed forms (C80 = 3.05 dB, D50 = 0.499,
Ts = 72 ms). A real room has a steeper early slope, so EDT < T30.

### `room_parameters()` parameters

| Parameter | Type | Units | Range / default | Notes |
| :--- | :--- | :--- | :--- | :--- |
| `ir` | 1D array | any | non-silent | Measured impulse response |
| `fs` | int | Hz | > 0 | Sample rate |
| `limits` | (float, float) or `None` | Hz | default `(125.0, 4000.0)` | Band-centre limits; `None` = broadband single band |
| `fraction` | int | — | `1` (octave, default) / `3` (third) | Bandwidth fraction |
| `zero_phase` | bool | — | default `False` | Forward-backward octave filtering (ISO 3382-2 §7.3 NOTE, which relaxes $BT > 16$ to $BT > 4$); removes the filter group delay before the backward integration and roughly halves the 125 Hz short-decay T30 bias (~+4.9 % → +2.4 % at $T$ = 0.2 s). `decay_curve` accepts it too |

Returns a `RoomAcousticsResult`: `frequency` (band centres, or `None`
broadband), `edt`/`t20`/`t30` (s), `c50`/`c80` (dB), `d50`, `ts` (s),
`dynamic_range` (dB), the `edt_valid`/`t20_valid`/`t30_valid` flags (ISO
3382-1 §5.3.3: noise ≥ 25 dB below the peak for EDT, tightened to 46 dB for
T20 and 54 dB for T30 so the tail-compensation bias of a flagged-valid value
stays within the 5 % JND) and `curvature`
$C = 100\ (T_{30}/T_{20} - 1)$ % (values above 10 % flag a non-straight
decay). `decay_curve(ir, fs, band=None, fraction=1, zero_phase=False)` returns
just the `(time, level)` curve for one band or the broadband response.

## 3. Open-plan offices (ISO 3382-3)

Open-plan acoustics are about **speech privacy**: how fast a talker's speech
fades to unintelligibility as you walk away. Levels and STI are measured
along a line of workstations (at least 4 positions, 6–10 preferred), and
four single-number quantities summarise the room. The **spatial decay
rate** of A-weighted speech is the slope of the level against
$\lg(r/r_0)$, scaled to a per-doubling figure using only the 2–16 m
positions,

$$
D_{2,S} = -\lg(2)\ b, \qquad L = a + b\ \lg(r/r_0),\ r_0 = 1\ \text{m},
$$

with **Lp,A,S,4m** read off the same line at 4 m. The **distraction
distance** rD (STI = 0.50) and **privacy distance** rP (STI = 0.20) come
from a linear regression of STI against distance. Good offices push rD
below ~5 m; poor ones leave speech distracting past 10 m.

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/diagram_open_plan_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/diagram_open_plan.svg" alt="ISO 3382-3 open-plan measurement line from the source at 1 m along positions from 2 m to 16 m, feeding the four single-number quantities D2,S, Lp,A,S,4m, rD and rP" width="86%"></picture>

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/open_plan_decay_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/open_plan_decay.svg" alt="Open-plan spatial decay: A-weighted speech level and STI against source distance on a log axis, with the D2,S regression, the Lp,A,S,4m marker at 4 m and the rD and rP distance crossings" width="80%"></picture>

```python
import numpy as np
from phonometry import open_plan_metrics

r = np.array([2.0, 4.0, 6.0, 8.0, 12.0, 16.0])       # distances from the talker (m)
lp = 65.0 - 7.0 * np.log2(r)                          # A-weighted speech level (dB)
sti = 0.70 - 0.03 * r                                 # STI per position

m = open_plan_metrics(r, lp, sti)
print(round(m.d2s, 1), round(m.lp_as_4m, 1))         # 7.0 dB, 51.0 dB
print(round(m.rd, 1), round(m.rp, 1))                # 6.7 m, 16.7 m
```

<details>
<summary>Show the code for this figure</summary>

```python
import matplotlib.pyplot as plt

# One line: the D2,S regression rebuilt from the result fields, with the
# rD / rP crossings marked (the figure above adds the measured points and
# the STI axis on top of it):
m.plot()
plt.show()
```

```python
import matplotlib.pyplot as plt

# Spatial decay: measured Lp,A,S vs distance on a log axis, the D2,S
# regression rebuilt from the result fields, and STI with the rD / rP
# crossings on a twin axis:
b = -m.d2s / np.log10(2.0)                 # regression slope vs lg(r)
a = m.lp_as_4m - b * np.log10(4.0)         # intercept from the 4 m level
rr = np.logspace(np.log10(2.0), np.log10(16.0), 100)

fig, ax = plt.subplots()
ax.semilogx(r, lp, "o", label="Measured Lp,A,S")
ax.semilogx(rr, a + b * np.log10(rr), "--", label=f"D2,S = {m.d2s:.1f} dB")
ax.plot(4.0, m.lp_as_4m, "D", label=f"Lp,A,S,4m = {m.lp_as_4m:.0f} dB")
ax.set_xlabel("Distance from the talker r [m]")
ax.set_ylabel("A-weighted speech level [dB]")
ax.set_xlim(1.8, 20.0)

twin = ax.twinx()
twin.semilogx(r, sti, "s-", color="#2ca02c", label="STI")
twin.axvline(m.rd, ls=":", color="#2ca02c")
twin.axvline(m.rp, ls=":", color="#9467bd")
twin.annotate(f"rD = {m.rd:.1f} m", (m.rd, 0.52))
twin.annotate(f"rP = {m.rp:.1f} m", (m.rp, 0.22))
twin.set_ylabel("STI")
twin.set_ylim(0.0, 1.0)

lines, labels = ax.get_legend_handles_labels()
tl, tlab = twin.get_legend_handles_labels()
ax.legend(lines + tl, labels + tlab, loc="best")
plt.show()
```

</details>

### `open_plan_metrics()` parameters

| Parameter | Type | Units | Range / default | Notes |
| :--- | :--- | :--- | :--- | :--- |
| `positions_m` | 1D array | m | ≥ 4 positions, all > 0 | Source-to-receiver distances |
| `spl_a_speech` | 1D array | dB | same length | A-weighted speech level `Lp,A,S,n` per position |
| `sti_values` | 1D array | — | same length | STI per position (full IEC 60268-16 method) |

Returns an `OpenPlanResult` with `d2s`, `lp_as_4m`, `rd` and `rp`; its
`.plot()` redraws the Clause 6.2 spatial-decay regression from those four
fields and marks `rd` / `rp`.
`d2s`/`lp_as_4m` are `nan` if fewer than two positions fall in 2–16 m;
`rd`/`rp` are `nan` when STI does not decrease with distance. The per-position
STI can itself be measured with the STIPA tools in the
[Speech Transmission Index guide](https://jmrplens.github.io/phonometry/guides/speech-transmission/).

## 4. Sound absorption (ISO 354)

The equivalent absorption area `A` that drives `R'`, `L'n`, the ISO 3744 `K2`
environmental correction and the ISO 3741 absorption term is itself measured in
a reverberation room (ISO 354).
Measure the room's reverberation time **empty** ($T_1$) and again **with the
test specimen installed** ($T_2$); the specimen's absorption is the difference
of the two Sabine areas, and dividing by the covered area gives the absorption
coefficient:

$$
A = \frac{55.3\ V}{c\ T} - 4 V m, \qquad
\alpha_s = \frac{A_2 - A_1}{S}, \qquad c = 331 + 0.6\ t ,
$$

with $c$ from the room air temperature $t$ in °C (valid 15–30 °C) and $m$ the
power attenuation coefficient of air (default 0; convert an ISO 9613-1
$\alpha$ in dB/m with `attenuation_from_alpha`). Because edge and diffraction
effects can scatter more energy than the sample's flat area intercepts,
$\alpha_s$ may exceed 1.0 and is never clamped (ISO 354 Clause 3.7).

```python
import numpy as np
from phonometry import absorption_area, absorption_coefficient

# Third-octave reverberation times of a 200 m^3 room, empty (T1) and with a
# 10.8 m^2 absorber sample installed (T2).
t1 = np.array([5.0, 4.0, 3.0])
t2 = np.array([3.0, 2.5, 2.0])

a_empty = absorption_area(t1, volume=200.0, temperature=20.0)
print(np.round(a_empty, 2))                    # [ 6.45  8.06 10.75] m^2

alpha = absorption_coefficient(t1, t2, volume=200.0, sample_area=10.8,
                               temperature1=20.0)
print(np.round(alpha, 3))                      # [0.398 0.448 0.498]
```

`T1` and `T2` are exactly the reverberation times `room_parameters` returns, so
an ISO 3382-2 decay measurement of the empty and treated room flows straight
into `absorption_coefficient`. A room volume below the 150 m³ minimum or a
sample area outside 10–12 m² raises an advisory `AbsorptionWarning`; the result
still returns.

### `absorption_area()` / `absorption_coefficient()` parameters

| Parameter | Type | Units | Range / default | Notes |
| :--- | :--- | :--- | :--- | :--- |
| `t60` / `t1`, `t2` | 1D array | s | > 0 | Reverberation time(s); `t1` empty, `t2` with specimen |
| `volume` | float | m³ | > 0 | Room volume `V` (advisory below 150 m³) |
| `sample_area` | float | m² | > 0 | Area `S` the specimen covers (coefficient only) |
| `temperature` / `temperature1`, `temperature2` | float | °C | default `20.0`, 15–30 | Sets `c` via Eq. (6); `temperature2` defaults to `temperature1` |
| `speed_of_sound` (`…1`, `…2`) | float, optional | m/s | > 0 | Overrides the temperature-derived `c` |
| `m` (`m1`, `m2`) | float or 1D array | 1/m | ≥ 0, default `0` | Air power attenuation coefficient |

`absorption_area()` returns the equivalent absorption area `A` (m²) with the
shape of `t60`; `absorption_coefficient()` returns `alpha_s`;
`attenuation_from_alpha(alpha)` converts an ISO 9613-1 `alpha` (dB/m) to `m`.

## See also

- [Building Acoustics & Sound Insulation](https://jmrplens.github.io/phonometry/guides/building-acoustics/) — field,
  laboratory and predicted sound insulation between spaces, and its measurement uncertainty.
- [Sound Power](https://jmrplens.github.io/phonometry/guides/sound-power/) — the `LW` methods that consume the
  ISO 354 absorption area (the ISO 3744 `K2` and the ISO 3741 absorption term).
- [Speech Transmission Index](https://jmrplens.github.io/phonometry/guides/speech-transmission/) — the STI/STIPA
  measurement that feeds the open-plan `sti_values`.
- [Psychoacoustics](https://jmrplens.github.io/phonometry/guides/psychoacoustics/) — loudness, sharpness and the other
  perception metrics of what the room delivers.
- [Filter Banks](https://jmrplens.github.io/phonometry/guides/filter-banks/) — the IEC 61260 fractional-octave filters
  used for band decay curves and insulation spectra.
- [Levels](https://jmrplens.github.io/phonometry/guides/levels/) — energy averaging and the level metrics behind
  source/receiving-room levels.
- [Theory](https://jmrplens.github.io/phonometry/reference/theory/) — Schroeder integration, regression windows and the
  reference-curve derivation.

---

**Standards.** ISO 18233:2006 (application of new measurement methods — the
swept-sine and MLS acquisition of impulse responses); ISO 3382-1:2009 and
ISO 3382-2:2008 (reverberation time and room parameters from the Schroeder
decay); ISO 3382-3:2012 (open-plan office speech metrics); ISO 354:2003
(sound absorption in a reverberation room). Validated against closed-form
decays and the standards' own parameter definitions in the
[conformance report](CONFORMANCE.md).

---


<!-- source: docs/building-acoustics.md | canonical: https://jmrplens.github.io/phonometry/guides/building-acoustics/ -->

# Building Acoustics & Sound Insulation

This guide continues from the [Room Acoustics guide](https://jmrplens.github.io/phonometry/guides/room-acoustics/):
the same impulse response, measured either side of a partition, yields its sound
insulation. Where room acoustics describes the sound field inside a single space,
building acoustics describes how much of that field passes *between* spaces. This
page follows the insulation chain — field airborne, impact and façade insulation
with single-number ratings (ISO 16283-1/2/3, ISO 717-1/2), the laboratory
characterisation of a building element (ISO 10140), the prediction of in-situ
performance from flanking transmission (EN 12354-1/2), and the measurement
uncertainty that qualifies every rating (ISO 12999-1).

## 1. Field insulation and single-number ratings (ISO 16283-1, ISO 717-1)

To rate a wall or floor, measure the energy-average level in the **source**
room ($L_1$) and the **receiving** room ($L_2$) per one-third-octave band
and form the level difference $D = L_1 - L_2$. Two normalisations make it
comparable between rooms. The **standardized level difference** references
the receiving-room reverberation time $T$ to $T_0 = 0.5$ s (so with
$T = 0.5$ s, $D_{nT} = D$ exactly), and the **apparent sound reduction
index** normalises by the partition area $S$ and the Sabine absorption area
$A$:

$$
D_{nT} = D + 10 \log_{10} \frac{T}{T_0}, \qquad
R' = D + 10 \log_{10} \frac{S}{A}, \qquad A = \frac{0.16\ V}{T}.
$$

Positions are energy-averaged with
$L = 10 \log_{10}\left( \frac{1}{n} \sum_i 10^{L_i/10} \right)$.

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/diagram_insulation_setup_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/diagram_insulation_setup.svg" alt="Field airborne insulation setup: a loudspeaker in the source room, microphones energy-averaged in source and receiving rooms across the common partition" width="92%"></picture>

The band spectrum is collapsed to one number by the **reference-curve
method** of ISO 717-1: a fixed reference curve is shifted in 1 dB steps
toward the measured curve until the sum of *unfavourable* deviations
(where the measurement falls below the reference) is as large as possible
but not more than 32.0 dB (16 one-third-octave bands) or 10.0 dB (5 octave
bands). The rating (`Rw`, `R'w`, `DnT,w` …) is the shifted reference read at
500 Hz. The **spectrum adaptation terms** $C$ (pink noise) and $C_{tr}$
(urban traffic) add the low-frequency penalty of a real source.

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/insulation_rating_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/insulation_rating.svg" alt="Measured one-third-octave sound reduction index with the shifted ISO 717-1 reference curve and the resulting weighted rating at 500 Hz" width="80%"></picture>

```python
import numpy as np
from phonometry import airborne_insulation, weighted_rating, energy_average_level

# Energy-average several microphone positions in one room (dB)
print(round(float(energy_average_level([60.0, 66.0])), 1))   # 64.0

# Field insulation per band; area S and volume V add R'
l1 = np.full(16, 80.0)                                # source-room levels
l2 = np.full(16, 40.0)                                # receiving-room levels
t2 = np.full(16, 0.5)                                 # receiving-room T (s)
ins = airborne_insulation(l1, l2, t2, area=10.0, volume=50.0)
print(round(float(ins.dnt[0]), 1))                   # 40.0  (= D since T = T0)
print(round(float(ins.r_prime[0]), 1))               # 38.0

# Single-number rating from a measured 16-band R spectrum (ISO 717-1 Annex C)
R = [20.4, 16.3, 17.7, 22.6, 22.4, 22.7, 24.8, 26.6,
     28.0, 30.5, 31.8, 32.5, 33.4, 33.0, 31.0, 25.5]
w = weighted_rating(R)
print(w.rating, w.c, w.ctr)                          # 30 -2 -3  ->  Rw(C;Ctr) = 30(-2;-3)

w.plot()   # measured R' vs shifted ISO 717-1 reference, deviations shaded (needs matplotlib)
```

<details>
<summary>Show the code for this figure</summary>

```python
import matplotlib.pyplot as plt

# One line — measured curve vs the shifted ISO 717-1 reference, deviations shaded:
w.plot()
plt.show()

# By hand, from the band curve the result now carries:
fig, ax = plt.subplots()
ax.semilogx(w.band_centers, w.measured, "o-", label="Measured R'")
ax.semilogx(w.band_centers, w.shifted_reference, "s--", label="Shifted reference")
ax.fill_between(w.band_centers, w.measured, w.shifted_reference,
                where=w.measured < w.shifted_reference, interpolate=True,
                alpha=0.3, label="Unfavourable deviations")
ax.set_xlabel("Frequency [Hz]")
ax.set_ylabel("Sound reduction index [dB]")
ax.set_title(f"Rw = {w.rating} dB  (C={w.c:+d}; Ctr={w.ctr:+d})")
ax.legend()
plt.show()
```

</details>

Compute `l1`, `l2` and `t2` on the same 16 one-third-octave bands from
100 Hz to 3150 Hz — obtain `t2` from
`room_parameters(ir, fs, limits=(100, 3150), fraction=3).t30`, for example — and
pass them to `airborne_insulation`. Feed that function's `dnt` (or `r_prime`)
spectrum to `weighted_rating`, so every band aligns index-by-index with the
ISO 717-1 reference curve.

### `airborne_insulation()` parameters

| Parameter | Type | Units | Range / default | Notes |
| :--- | :--- | :--- | :--- | :--- |
| `l1` | 1D or 2D array | dB | one/band, or `(positions, bands)` | Source-room levels (2D is energy-averaged) |
| `l2` | 1D or 2D array | dB | same band count | Receiving-room levels |
| `t2` | 1D array | s | > 0, one per band | Receiving-room reverberation time |
| `area` | float, optional | m² | > 0, with `volume` | Partition area `S` (enables `R'`) |
| `volume` | float, optional | m³ | > 0, with `area` | Receiving-room volume `V` |
| `t0` | float | s | default `0.5` | Reference reverberation time `T0` |

### `weighted_rating()` parameters

| Parameter | Type | Units | Range / default | Notes |
| :--- | :--- | :--- | :--- | :--- |
| `values_by_band` | 1D array | dB | 16 (thirds) or 5 (octaves) | Measured `R`, `R'`, `DnT` … per band |
| `bands` | str or `None` | — | `'third-octave'` / `'octave'` / `None` | `None` infers from the count |

`airborne_insulation()` returns an `AirborneInsulationResult` (`d`, `dnt`,
`r_prime` or `None`); `weighted_rating()` returns a `WeightedRatingResult`
(`rating`, `c`, `ctr`, `unfavourable_sum`, all integers except the sum).

### Impact sound (ISO 16283-2, ISO 717-2)

Footstep noise is rated the other way round. Instead of how much a floor
*blocks*, impact insulation measures how much a standardized **tapping
machine** on the floor above puts into the room below — so a *higher* number
is *worse*. The energy-average impact sound pressure level $L_i$ in the
receiving room is normalised like the airborne case, but with a sign flip on
the reverberation term:

$$
L'_{nT} = L_i - 10 \log_{10} \frac{T}{T_0}, \qquad
L'_n = L_i + 10 \log_{10} \frac{A}{A_0}, \quad
A_0 = 10\ \text{m}^2,\ A = \frac{0.16\ V}{T}.
$$

The **standardized** impact level $L'_{nT}$ ($T_0 = 0.5$ s for dwellings)
needs only the receiving-room $T$, so with $T = 0.5$ s it equals $L_i$; the
**normalized** level $L'_n$ (referenced to a 10 m² absorption area) also needs
the receiving-room volume. Note the **minus** sign — more reverberation
*lowers* $L'_{nT}$, opposite to the airborne $D_{nT}$.

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/diagram_impact_setup_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/diagram_impact_setup.svg" alt="Field impact insulation setup: a standardized tapping machine on the floor of the source room above, microphones energy-averaged in the receiving room below, and the receiving-room reverberation time" width="92%"></picture>

The single-number rating (ISO 717-2) shifts the same style of reference curve,
but an **unfavourable deviation now occurs where the measurement *exceeds* the
reference** (impact noise is worse when higher) — the sign opposite to
ISO 717-1. The rating (`Ln,w`, `L'n,w`, `L'nT,w`) is the shifted reference read
at 500 Hz; for octave bands it is then reduced by 5 dB. The spectrum
adaptation term $C_I = L_{n,\text{sum}} - 15 - L_{n,w}$ uses the energetic sum
over 100–2500 Hz (16-band thirds excluding 3150 Hz) or 125–2000 Hz (octaves).

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/impact_rating_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/impact_rating.svg" alt="Measured one-third-octave normalized impact sound pressure level with the shifted ISO 717-2 reference curve and the resulting weighted rating read at 500 Hz" width="80%"></picture>

```python
import numpy as np
from phonometry import impact_insulation, weighted_impact_rating

# 16 one-third-octave impact levels Li (100 Hz - 3150 Hz), dB, from the
# ISO 717-2 Annex C worked example, and the receiving-room T per band.
li = np.array([62.1, 63.2, 63.5, 66.2, 68.5, 70.0, 71.7, 73.1,
               73.8, 73.5, 73.8, 73.3, 73.1, 73.0, 72.4, 71.2])
t2 = np.full(16, 0.5)

imp = impact_insulation(li, t2, volume=50.0)
print(round(float(imp.l_n_t[0]), 1))          # 62.1  (= Li since T = T0)
print(round(float(imp.l_n[0]), 1))            # 64.1  normalized to A0 = 10 m^2

# Weighted impact rating + spectrum adaptation term CI (ISO 717-2)
res_imp = weighted_impact_rating(imp.l_n_t)
print(res_imp.rating, res_imp.ci, res_imp.unfavourable_sum)   # 79 -11 28.0  ->  L'nT,w(CI)=79(-11)

# Octave-band data carry the extra -5 dB reduction (Clause 4.3.2)
octave = np.array([65.3, 64.5, 58.0, 55.8, 43.0])
print(weighted_impact_rating(octave).rating)  # 54

res_imp.plot()   # measured L'nT vs shifted ISO 717-2 reference, measured-above shaded (needs matplotlib)
```

<details>
<summary>Show the code for this figure</summary>

```python
import matplotlib.pyplot as plt

# One line — measured L'nT vs the shifted ISO 717-2 reference (measured-above shaded):
res_imp.plot()
plt.show()

# By hand, from the band curve the result now carries (note the opposite sign:
# an unfavourable deviation is where the MEASURED level exceeds the reference).
# Here the input was l_n_t, so the rated quantity is the field level L'nT,w:
fig, ax = plt.subplots()
ax.semilogx(res_imp.band_centers, res_imp.measured, "o-", label="Measured L'nT")
ax.semilogx(res_imp.band_centers, res_imp.shifted_reference, "s--", label="Shifted reference")
ax.fill_between(res_imp.band_centers, res_imp.shifted_reference, res_imp.measured,
                where=res_imp.measured > res_imp.shifted_reference, interpolate=True,
                alpha=0.3, label="Unfavourable deviations")
ax.set_xlabel("Frequency [Hz]")
ax.set_ylabel("Impact sound pressure level [dB]")
ax.set_title(f"L'nT,w = {res_imp.rating} dB  (CI={res_imp.ci:+d})")
ax.legend()
plt.show()
```

</details>

Feed `impact_insulation`'s `l_n_t` (or `l_n`) straight into
`weighted_impact_rating`; the rating and `CI` reproduce the ISO 717-2 Annex C
values (thirds `L'nT,w = 79`, `CI = −11`; octave `54`, `CI = 0`).

#### `impact_insulation()` parameters

| Parameter | Type | Units | Range / default | Notes |
| :--- | :--- | :--- | :--- | :--- |
| `li` | 1D or 2D array | dB | one/band, or `(positions, bands)` | Energy-average impact SPL (2D is averaged over positions) |
| `t2` | 1D array | s | > 0, one per band | Receiving-room reverberation time |
| `volume` | float, optional | m³ | > 0 | Receiving-room `V` (enables `L'n`) |
| `t0` | float | s | default `0.5` | Reference reverberation time `T0` |

#### `weighted_impact_rating()` parameters

| Parameter | Type | Units | Range / default | Notes |
| :--- | :--- | :--- | :--- | :--- |
| `values_by_band` | 1D array | dB | 16 (thirds) or 5 (octaves) | Measured `Ln`, `L'n` or `L'nT` per band |
| `bands` | str or `None` | — | `'third-octave'` / `'octave'` / `None` | `None` infers from the count |

`impact_insulation()` returns an `ImpactInsulationResult` (`l_n_t`, `l_n` or
`None`); `weighted_impact_rating()` returns an `ImpactRatingResult` (`rating`,
`ci` integers, `unfavourable_sum` in dB).

### Field façade insulation (ISO 16283-3)

The same source/receiver logic reaches the building **façade**, but now the
source is *outdoors* — a loudspeaker at 45° or the road traffic itself. Rather
than a level difference across an internal partition, ISO 16283-3 references the
receiving-room level $L_2$ to the level **2 m in front of the façade**
$L_{1,2m}$, giving the level difference $D_{2m}$ and, exactly as in the airborne
case, its standardized and normalized forms:

$$
D_{2m} = L_{1,2m} - L_2, \quad
D_{2m,nT} = D_{2m} + 10 \log_{10}\frac{T}{T_0}, \quad
D_{2m,n} = D_{2m} - 10 \log_{10}\frac{A}{A_0},
$$

with $T_0 = 0.5$ s, $A_0 = 10$ m² and $A = 0.16\ V/T$ (dwellings). When the
microphone sits **on the test element** (surface level $L_{1,s}$) the *element*
method also yields an apparent sound reduction index, carrying a fixed
angle-of-incidence correction — $-1.5$ dB for the 45° loudspeaker method,
$-3$ dB for the all-angle road-traffic method:

$$
R'_{45°} = L_{1,s} - L_2 + 10 \log_{10}\frac{S}{A} - 1.5, \qquad
R'_{tr,s} = L_{1,s} - L_2 + 10 \log_{10}\frac{S}{A} - 3.
$$

The façade quantity is airborne, so its single-number rating uses the
**ISO 717-1** reference curve through `weighted_rating` unchanged (Annex F).

```python
import numpy as np
from phonometry import facade_insulation, weighted_rating

# Outdoor level 2 m in front of the façade, receiving-room level and T per
# one-third-octave band; surface_level is the microphone on the test element.
l1_2m = np.full(16, 75.0)                              # L1,2m outdoors
l2 = np.full(16, 33.0)                                 # receiving-room L2
t2 = np.full(16, 0.5)                                  # receiving-room T (s)

fac = facade_insulation(l1_2m, l2, t2, volume=50.0, area=11.5,
                        surface_level=np.full(16, 78.0), method="loudspeaker")
print(round(float(fac.d_2m[0]), 1))                    # 42.0  D2m = L1,2m - L2
print(round(float(fac.d_2m_nt[0]), 1))                 # 42.0  (= D2m since T = T0)
print(round(float(fac.d_2m_n[0]), 1))                  # 40.0  normalized to A0 = 10 m^2
print(round(float(fac.r_prime[0]), 1))                 # 42.1  R'45deg (loudspeaker, -1.5 dB)

# The road-traffic element method carries the -3 dB all-angle correction instead
tr = facade_insulation(l1_2m, l2, t2, volume=50.0, area=11.5,
                       surface_level=np.full(16, 78.0), method="road_traffic")
print(round(float(tr.r_prime[0]), 1))                  # 40.6  R'tr,s (traffic, -3 dB)

# The façade quantity is airborne: rate D2m,nT with the ISO 717-1 engine
print(weighted_rating(fac.d_2m_nt).rating)             # 42  Dls,2m,nT,w

fac.plot()   # per-band D2m,nT with D2m, D2m,n and R' overlaid (needs matplotlib)
```

`surface_level`, `area` and `volume` are all optional: with only `l1_2m`, `l2`
and `t2` the function returns `d_2m` and `d_2m_nt`; add `volume` for `d_2m_n`;
add `surface_level` **and** `area` **and** `volume` for `r_prime`. Positions are
energy-averaged with the surface-level formula (Clause 9.5.1); band levels are
assumed already corrected for background noise.

#### `facade_insulation()` parameters

| Parameter | Type | Units | Range / default | Notes |
| :--- | :--- | :--- | :--- | :--- |
| `l1_2m` | 1D or 2D array | dB | one/band, or `(positions, bands)` | Level 2 m in front of the façade `L1,2m` |
| `l2` | 1D or 2D array | dB | same band count | Receiving-room levels |
| `t2` | 1D array | s | > 0, one per band | Receiving-room reverberation time |
| `area` | float, optional | m² | > 0, with `surface_level`, `volume` | Test-element area `S` (enables `R'`) |
| `volume` | float, optional | m³ | > 0 | Receiving-room `V` (enables `D2m,n`; required for `R'`) |
| `surface_level` | 1D/2D array, optional | dB | same band count | Surface level `L1,s` on the element (enables `R'`) |
| `method` | str | — | `'loudspeaker'` (−1.5 dB) / `'road_traffic'` (−3 dB) | Angle-of-incidence correction of `R'` |
| `t0` | float | s | default `0.5` | Reference reverberation time `T0` |
| `frequencies` | 1D array, optional | Hz | — | Band centres carried on the result for plotting |

`facade_insulation()` returns a `FacadeInsulationResult` (`d_2m`, `d_2m_nt`,
`d_2m_n` or `None`, `r_prime` or `None`, `frequencies`); feed any 16-band façade
quantity to `weighted_rating` for its ISO 717-1 single number.

## 2. Laboratory measurement (ISO 10140)

Everything above is a **field** measurement (the primed quantities $R'$, $L'_n$):
the number a real building achieves, flanking transmission and all. To rate an
element on its own — a wall type, a floating floor, a window — you take it to a
qualified **laboratory** (ISO 10140), where suppressed flanking makes the
*direct* transmission the whole story. The formulas lose their primes: the
**sound reduction index** $R$ (not $R'$) and the **normalized impact level**
$L_n$ (not $L'_n$), with the receiving room's absorption area $A = 0.16\ V/T$
now a known property of the facility:

$$
R = L_1 - L_2 + 10 \log_{10}\frac{S}{A}, \qquad
L_n = L_i + 10 \log_{10}\frac{A}{A_0}, \quad A_0 = 10\ \text{m}^2.
$$

| | Field (ISO 16283) | Laboratory (ISO 10140) |
| :--- | :--- | :--- |
| Airborne | $R'$ apparent (with flanking) | $R$ direct (flanking suppressed) |
| Impact | $L'_n$ apparent | $L_n$ direct |
| Absorption area | measured in the room | property of the facility |

The single-number ratings reuse the very same ISO 717-1/2 engines
(`weighted_rating`, `weighted_impact_rating`) — an $R$ spectrum rates to $R_w$
exactly as an $R'$ spectrum rated to $R'_w$. Before forming the index the
receiving-room levels must be **corrected for background noise** (Clause 4.3):
the energy subtraction $10 \log_{10}(10^{L_{sb}/10} - 10^{L_b/10})$ applies for a
6–15 dB signal-to-background margin, a fixed 1.3 dB correction (the *limit of
measurement*) at or below 6 dB, and no correction at or above 15 dB.

```python
import numpy as np
from phonometry import (lab_airborne_insulation, lab_impact_insulation,
                        background_correction)

# Source/receiving levels and receiving-room T over the 16 one-third-octave
# bands; S is the free test-opening area, V the receiving-room volume.
l1 = np.full(16, 80.0)
l2 = np.full(16, 40.0)
t2 = np.full(16, 0.5)
lab = lab_airborne_insulation(l1, l2, t2, area=10.0, volume=50.0)
print(round(float(lab.r[0]), 1))              # 38.0  R = L1 - L2 + 10 lg(S/A)
print(round(float(lab.absorption[0]), 1))     # 16.0  A = 0.16 V / T (m^2)
print(lab.rating.rating, lab.rating.c, lab.rating.ctr)   # 38 0 0  ->  Rw(C;Ctr)

# Impact: the tapping-machine level Li normalized to A0 = 10 m^2 gives Ln
li = np.array([62.1, 63.2, 63.5, 66.2, 68.5, 70.0, 71.7, 73.1,
               73.8, 73.5, 73.8, 73.3, 73.1, 73.0, 72.4, 71.2])
imp = lab_impact_insulation(li, t2, volume=50.0)
print(round(float(imp.l_n[0]), 1))            # 64.1  Ln = Li + 10 lg(A/A0)
print(imp.rating.rating, imp.rating.ci)       # 81 -11  ->  Ln,w(CI)

# Background correction: margins 6 / 1 / 20 dB -> capped / capped / unchanged
corrected = background_correction([30.0, 33.0, 50.0], [24.0, 32.0, 30.0])
print(np.round(corrected, 1))                 # [28.7 31.7 50.0]  (1.3 dB cap twice)

lab.rating.plot()   # measured R vs shifted ISO 717-1 reference (needs matplotlib)
```

A margin at or below 6 dB emits a `LabInsulationWarning` and flags the band as
the limit of measurement; catch it with `warnings.simplefilter("error",
LabInsulationWarning)`. The automatic rating is formed only when exactly 16
one-third-octave or 5 octave values are supplied (`rating` is `None` otherwise).

### `lab_airborne_insulation()` / `lab_impact_insulation()` parameters

| Parameter | Type | Units | Range / default | Notes |
| :--- | :--- | :--- | :--- | :--- |
| `l1` / `l2` | 1D or 2D array | dB | one/band, or `(positions, bands)` | Source / receiving levels (airborne) |
| `li` | 1D or 2D array | dB | one/band, or `(positions, bands)` | Impact SPL from the tapping machine (impact) |
| `t2` | 1D array | s | > 0, one per band | Receiving-room reverberation time |
| `area` | float | m² | > 0 | Free test-opening area `S` (airborne only) |
| `volume` | float | m³ | > 0 | Receiving-room volume `V` |

`lab_airborne_insulation()` returns a `LabAirborneInsulationResult` (`r`,
`absorption`, `rating`); `lab_impact_insulation()` a
`LabImpactInsulationResult` (`l_n`, `absorption`, `rating`);
`background_correction(signal_and_background, background)` returns the corrected
levels directly.

## 3. Predicting performance (EN 12354)

A laboratory rating describes an element in isolation, yet the sound a building
actually transmits also travels *around* the partition — along the floor, up the
façade, through the flanking walls — re-radiating into the receiving room. This
**flanking transmission** is the whole difference between the laboratory $R$ and
the field $R'$. EN 12354 predicts the in-situ apparent rating from the
laboratory ratings of the elements plus the vibration transmission of their
junctions.

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/diagram_flanking_paths_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/diagram_flanking_paths.svg" alt="The direct path Dd through the separating element and the three flanking paths Ff, Df and Fd across each junction between a flanking element and the separating element" width="92%"></picture>

Each junction between a flanking element and the separating element carries
three paths — $Ff$ (flanking→flanking), $Df$ (direct→flanking) and $Fd$
(flanking→direct) — alongside the single direct path $Dd$. The **simplified
single-number model** combines them energetically (Formula 26):

$$
R'_w = -10 \log_{10}\Big[ 10^{-R_{Dd,w}/10}
      + \sum 10^{-R_{Ff,w}/10} + \sum 10^{-R_{Df,w}/10}
      + \sum 10^{-R_{Fd,w}/10} \Big],
$$

with the direct path $R_{Dd,w} = R_{s,w} + \Delta R_{Dd,w}$ (Formula 27) and each
flanking path (Formula 28a)

$$
R_{ij,w} = \tfrac{R_{i,w} + R_{j,w}}{2} + \Delta R_{ij,w} + K_{ij}
         + 10 \log_{10}\frac{S_s}{l_0\ l_f},
$$

where $l_0 = 1$ m is the reference coupling length, $l_f$ the junction coupling
length and $K_{ij}$ the junction's **vibration reduction index** (Annex E,
empirical in the mass ratio $M = \log_{10}(m'_{\perp,i}/m'_i)$).

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/prediction_flanking_demo_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/prediction_flanking_demo.svg" alt="Per-path sound reduction indices for the EN 12354-1 Annex H.3 example and each path's share of the transmitted energy, showing the direct path dominating at R'w = 52 dB" width="80%"></picture>

```python
import numpy as np
from phonometry import (junction_vibration_reduction, flanking_element,
                        predicted_airborne_insulation)

# EN 12354-1 Annex H.3: a separating wall Rs,w = 57 dB, area Ss = 11.5 m², with
# four flanking elements. The simplified model reads each junction's Kij at
# 500 Hz from the mass ratio m'perp / m' (Annex E) — here the floor's rigid
# cross-junction (the mass ratio is itself rounded, hence 12.5 vs Annex 12.4):
print(round(junction_vibration_reduction("rigid_cross", "through", 1.61), 1))  # 12.5  KFf
print(round(junction_vibration_reduction("rigid_cross", "corner",  1.61), 1))  #  8.9  KFd = KDf

# Build each element's three flanking paths (Ff, Df, Fd) from the Annex H
# tabulated Kij, then combine the direct path Dd energetically (Formula 26).
elements = [   # (name, Rw, KFf, KFd = KDf, coupling length lf)
    ("floor",    49, 12.4,  8.9, 4.50),
    ("ceiling",  46, 14.4,  9.2, 4.50),
    ("facade",   42, 12.6,  6.7, 2.55),
    ("int-wall", 33, 33.5, 15.7, 2.55),
]
paths = []
for name, rw, k_ff, k_fd, lf in elements:
    paths += flanking_element(label=name, r_flanking=rw, r_separating=57,
                              k_ff=k_ff, k_fd=k_fd, k_df=k_fd,
                              separating_area=11.5, coupling_length=lf)

res = predicted_airborne_insulation(r_direct=57.0, flanking_paths=paths)
print(round(res.r_prime_w, 1))                          # 52.2  ->  R'w = 52 dB
print(res.dominant.label, round(res.dominant.fraction, 2))   # Dd 0.33 (direct dominates)
```

<details>
<summary>Show the code for this figure</summary>

```python
import matplotlib.pyplot as plt

# Uses `paths` and `res` from the snippet above.
# Per-path sound reduction index and each path's share of the transmitted
# energy for the Annex H.3 result computed above.
labels = [p.label for p in res.paths]
r_w = [p.r_w for p in res.paths]
frac = [100.0 * p.fraction for p in res.paths]

fig, (ax1, ax2) = plt.subplots(2, 1, figsize=(9, 6), sharex=True)
ax1.bar(labels, r_w, color="tab:blue")
ax1.axhline(res.r_prime_w, ls="--", color="k", label=f"R'w = {res.r_prime_w:.1f} dB")
ax1.set_ylabel("Path Rij,w [dB]"); ax1.legend()
ax2.bar(labels, frac, color="tab:orange")
ax2.set_ylabel("Energy share [%]"); ax2.set_xlabel("Transmission path")
for ax in (ax1, ax2):
    ax.tick_params(axis="x", rotation=45)
fig.suptitle("EN 12354-1 Annex H.3 — flanking transmission")
fig.tight_layout()
plt.show()
```

</details>

Every added flanking path strictly lowers $R'_w$ below the direct $R_{Dd,w} = 57$;
`res.paths` exposes each path's share of the transmitted energy so the dominant
path is visible. `flanking_element` is a convenience that builds one junction's
three paths at once; the single-path constructor behind it, `flanking_path`,
builds one `Ff`, `Df` or `Fd` path at a time (Formula 28a). Clause 4.4.2 also
enforces a floor $K_{ij} \ge K_{ij,\min}$ from the junction geometry — compute
it with `junction_min_vibration_reduction` and pass it to
`flanking_path(..., kij_min=...)`, which raises a below-floor $K_{ij}$ to the
minimum:

```python
from phonometry import junction_min_vibration_reduction
# Kij,min = 10 lg[lf·l0·(1/Si + 1/Sj)]; large elements give a low (here negative)
# floor, so a realistic tabulated Kij is rarely clamped.
print(round(junction_min_vibration_reduction(coupling_length=4.5,
                                             s_i=11.5, s_j=11.5), 1))     # -1.1
```

The impact counterpart (EN 12354-2, Formula 21) is a direct subtraction:
$L'_{n,w} = L_{n,w,eq} - \Delta L_w + K$, with the bare-floor equivalent level
$L_{n,w,eq} = 164 - 35 \log_{10}(m'/m'_0)$ (Annex B), the covering improvement
$\Delta L_w$ (ISO 717-2) and the flanking correction $K$ from Table 1.

```python
from phonometry import (equivalent_impact_level, impact_flanking_correction,
                        predicted_impact_insulation, standardized_impact_level)

# EN 12354-2 Annex E.3: a 0.14 m concrete floor (m' = 322 kg/m²) with a floating
# floor (ΔLw = 33 dB), rooms one above the other, mean flanking mass 145 kg/m².
ln_eq = equivalent_impact_level(322.0)                   # 164 - 35 lg(m')
k = impact_flanking_correction(322.0, 145.0)             # Table 1 (sep 322, flk 145)
imp = predicted_impact_insulation(ln_w_eq=ln_eq, delta_l_w=33.0, k_correction=k)
print(round(ln_eq, 1), k, round(imp.l_prime_n_w, 1))     # 76.2 2 45.2  ->  L'n,w = 45 dB
print(round(standardized_impact_level(imp.l_prime_n_w, 50.0), 1))   # 43.0  L'nT,w
```

### `junction_vibration_reduction()` / `flanking_element()` parameters

| Parameter | Type | Units | Range / default | Notes |
| :--- | :--- | :--- | :--- | :--- |
| `junction_type` | str | — | `'rigid_cross'` / `'rigid_t'` / `'flexible_t'` / `'lightweight_facade'` | Junction geometry (Annex E) |
| `path` | str | — | `'through'` (K13) / `'corner'` (K12 = K23) | Path branch |
| `mass_ratio` | float | — | > 0 | `m'⊥,i / m'i` (Formula E.2) |
| `frequency` | float | Hz | default `500` | Only `flexible_t` is frequency-dependent |
| `r_flanking` / `r_separating` | float | dB | — | Weighted indices of the flanking / separating element |
| `k_ff` / `k_fd` / `k_df` | float | dB | — | Junction `Kij` for the three paths |
| `separating_area` | float | m² | > 0 | Separating-element area `Ss` |
| `coupling_length` | float | m | > 0 | Junction coupling length `lf` |
| `delta_r_ff` / `delta_r_fd` / `delta_r_df` | float | dB | default `0` | Lining improvements per path |

### `flanking_path()` parameters

| Parameter | Type | Units | Range / default | Notes |
| :--- | :--- | :--- | :--- | :--- |
| `label` | str | — | — | Display name of the path |
| `kind` | str | — | `'Ff'` / `'Df'` / `'Fd'` | Which flanking branch the path is |
| `r_source` / `r_receive` | float | dB | — | Weighted indices of the source-side / receive-side elements |
| `k_ij` | float | dB | — | Junction vibration-reduction index for this path |
| `separating_area` | float | m² | > 0 | Separating-element area `Ss` |
| `coupling_length` | float | m | > 0 | Junction coupling length `lf` |
| `delta_r` | float | dB | default `0` | Lining improvement on this path |
| `kij_min` | float | dB | default `None` | When given, `k_ij` is floored at this Formula E.4 minimum |

`predicted_airborne_insulation()` returns an `AirbornePredictionResult`
(`r_prime_w`, `r_direct_w`, `paths` of `PathContribution`, `dominant`);
`predicted_impact_insulation()` an `ImpactPredictionResult` (`l_prime_n_w`,
`ln_w_eq`, `delta_l_w`, `k_correction`). The simplified model carries a reported
standard deviation of about 2 dB (Clause 5).

### Façade insulation & radiated power (EN 12354-3 / -4)

Parts 3 and 4 predict the two directions across the building envelope, both from
the same energy summation of the element **transmission factors**
$\tau = 10^{-R/10}$, area-weighted by $S_i/S$ (a small element or air path enters
through its element-normalized level difference $D_{n,e}$ with the reference area
$A_0 = 10\ \text{m}^2$):

$$
R' = -10 \log_{10}\!\Big( \sum_i \tfrac{S_i}{S}\,10^{-R_i/10}
                          + \sum_k \tfrac{A_0}{S}\,10^{-D_{n,e,k}/10} \Big).
$$

**Part 3 — outdoor → indoor.** From $R'$ (Formula 10) follow the loudspeaker- and
traffic-referenced indices $R_{45} = R'+1$ and $R_{tr,s} = R'$, and the primary
output, the standardized level difference at 2 m (Formula 13)

$$
D_{2m,nT} = R' + \Delta L_{fs} + 10 \log_{10}\frac{V}{6\,T_0\,S}, \qquad T_0 = 0.5\ \text{s},
$$

with the façade-shape term $\Delta L_{fs}$ (Annex C; 0 dB for a flat reflecting
façade). Single-number ratings reuse EN ISO 717-1 (`weighted_rating`).

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/facade_prediction_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/facade_prediction.svg" alt="Per-element partial sound reduction indices and the resulting façade apparent reduction R' and standardized level difference D2m,nT for the EN 12354-3 Annex F worked example, the air inlet limiting the low bands" width="80%"></picture>

```python
from phonometry import FacadeElement, facade_sound_reduction

# EN 12354-3 Annex F: an 11.3 m² façade (V = 50 m³, flat so ΔLfs = 0) of a double
# wall, a window, a small skylight and an acoustically-treated air inlet (a Dn,e
# element).
elements = [
    FacadeElement("wall",     area=6.0, r=[41, 46, 52, 58, 64]),   # octave 125-2000
    FacadeElement("window",   area=4.5, r=[23, 22, 30, 36, 37]),
    FacadeElement("skylight", area=0.5, r=[24, 27, 30, 33, 30]),
    FacadeElement("air inlet", dn_e=[28, 23, 25, 38, 44]),         # small element
]
fac = facade_sound_reduction(elements, area=11.3, volume=50.0,
                             frequencies=[125, 250, 500, 1000, 2000], bands="octave")
print(fac.r_tr_s_w, fac.c_tr, fac.d_2m_nt_w)   # 31 -3 33  (Rtr,s,w / Ctr / D2m,nT,w)
```

**Part 4 — indoor → outdoor.** The sound power level radiated by a segment
(Formula 2) is $L_W = L_{p,in} + C_d - R' + 10 \log_{10}(S/S_0)$ with $S_0 = 1$ m²
and the inside-field diffusivity term $C_d$ (Annex B; −6 dB ideal diffuse, −5 dB
average industrial). Openings are elements whose "R" is the silencer insertion
loss (a bare opening is 0 dB). The exterior level follows from the simplified
Annex E attenuation $A_{tot}$ of a finite radiating side, $L_p = L_W - A_{tot}$.

```python
from phonometry import (FacadeElement, radiated_sound_power,
                        outdoor_attenuation, outdoor_level)

# EN 12354-4 Annex G, side 1: a 10×20 m concrete wall segment with a 6×4 m
# industrial door, inside level Lp,in, Cd = -5 dB, R' capped at 40 dB (Annex G).
bands = [63, 125, 250, 500, 1000, 2000, 4000, 8000]
seg = radiated_sound_power(
    [FacadeElement("wall", area=176.0, r=[32, 36, 36, 33, 39, 49, 57, 63]),
     FacadeElement("door", area=24.0,  r=[21, 23, 28, 30, 30, 30, 30, 30])],
    lp_in=[70, 74, 76, 72, 70, 67, 62, 57], area=200.0, c_d=-5.0,
    r_prime_cap=40.0, octave_bands=bands)
print(round(seg.l_w[0], 1), round(seg.l_w[1], 1))     # 59.8 61.2  (LW at 63/125 Hz)

# Exterior level 5 m in front of the centre of the 60×10 m side (LWA = 62.9 dB(A)).
a_tot = outdoor_attenuation(width=60.0, height=10.0, distance=5.0)
print(round(a_tot, 1), round(outdoor_level(62.9, a_tot), 1))   # 26.3 36.6
```

> **Worked-example note.** The 2000 worked examples carry small internal rounding
> inconsistencies at the higher octave bands (Part 3's printed $R'$ disagrees with
> its own per-element partial indices at 1 k/2 k; Part 4's $R'$ rows above 500 Hz
> disagree with its Table G.2 inputs). The implementation is faithful to the
> formulas — it reproduces the low bands, every single-number rating and the whole
> Annex E propagation exactly.

<details>
<summary>Show the code for this figure</summary>

```python
import numpy as np
import matplotlib.pyplot as plt

# Uses `fac` (the Part 3 result) and its band centres from the snippet above.
x = np.arange(5)
fig, ax = plt.subplots(figsize=(9, 5.5))
for name, rp in fac.element_r.items():
    ax.plot(x, rp, "--", alpha=0.6, marker=".", label=f"Rp — {name}")
ax.plot(x, fac.r_prime, "k-", lw=2.5, marker="o", label="R′ (façade)")
ax.plot(x, fac.d_2m_nt, lw=2, marker="s", label="D2m,nT")
ax.set_xticks(x); ax.set_xticklabels([125, 250, 500, 1000, 2000])
ax.set_xlabel("Frequency [Hz]"); ax.set_ylabel("Index / level difference [dB]")
ax.set_title("EN 12354-3 façade sound insulation (Annex F)")
ax.legend(ncol=2); ax.grid(alpha=0.4)
fig.tight_layout(); plt.show()
```

</details>

### `FacadeElement` / `facade_sound_reduction()` / `radiated_sound_power()` parameters

| Parameter | Type | Units | Range / default | Notes |
| :--- | :--- | :--- | :--- | :--- |
| `FacadeElement.area` | float | m² | > 0 for `r` / `insertion_loss` | Element area $S_i$ (ignored for `dn_e`) |
| `FacadeElement.r` / `dn_e` / `insertion_loss` | float or seq | dB | give exactly one | Area element $R_i$ / small-element $D_{n,e}$ / opening insertion loss |
| `facade_sound_reduction(area)` | float | m² | > 0 | Total façade area $S$ |
| `facade_sound_reduction(volume)` | float | m³ | > 0 | Receiving-room volume $V$ (Formula 13) |
| `facade_sound_reduction(delta_l_fs)` | float | dB | default `0` | Façade-shape term $\Delta L_{fs}$ (Annex C) |
| `radiated_sound_power(lp_in)` | float or seq | dB | — | Inside level $L_{p,in}$ per band |
| `radiated_sound_power(c_d)` | float | dB | default `-6` | Diffusivity term $C_d$ (Annex B) |
| `radiated_sound_power(r_prime_cap)` | float | dB | default `40` (`None` off) | Field cap on $R'$ (Annex G) |
| `radiated_sound_power(octave_bands)` | seq of int | Hz | default `None` | Octave centres matching the bands; enables the A-weighted $L_{WA}$ |
| `facade_sound_reduction(frequencies)` | seq | Hz | default `None`; length = band count | Band centres carried on the result for plotting |
| `outdoor_attenuation(width, height, distance)` | float | m | > 0 | Finite radiating side and reception distance (Annex E) |
| `outdoor_level(l_w, attenuation)` | float or seq | dB | broadcast-compatible | Exterior $L_p$ from one or more sides (Formula E.1) |

`facade_sound_reduction()` returns a `FacadePredictionResult` (`r_prime`, `r_45`,
`r_tr_s`, `d_2m_nt`, `element_r`, and the `r_tr_s_w` / `d_2m_nt_w` / `c_tr` single
numbers); `radiated_sound_power()` a `RadiatedPowerResult` (`l_w`, `r_prime`,
`l_w_dba`). Both expose `.plot()`.

## 4. Measurement uncertainty (ISO 12999-1)

A rating without an uncertainty is only half a result. ISO 12999-1 does not
re-measure anything; it tabulates the **standard uncertainty** $u$ of every
sound-insulation quantity — derived from inter-laboratory tests — and prescribes
how to expand and combine it. Which standard deviation is $u$ depends on the
**measurement situation** (Clause 5.2):

| Situation | Meaning | Standard uncertainty $u$ |
| :--- | :--- | :--- |
| **A** | laboratory characterisation (ISO 10140) | reproducibility $\sigma_R$ |
| **B** | same location, different teams | in-situ $\sigma_{situ}$ |
| **C** | same location, same operator repeated | repeatability $\sigma_r$ |

The expanded uncertainty is $U = k\ u$ (Formula 2) with the coverage factor $k$
of Table 8. A two-sided interval $Y = y \pm U$ (Formula 3, $k = 1.96$ at 95 %)
*reports* a value; the **one-sided** factor ($k = 1.65$ at 95 %) *declares
conformity* with a requirement (Formulae 4/5).

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/diagram_iso12999_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/diagram_iso12999.svg" alt="ISO 12999-1 uncertainty flow: standard uncertainty from the tables, reduced by repeated measurements and combined in quadrature, then expanded by the Table 8 coverage factor into a two-sided report or a one-sided conformity decision" width="82%"></picture>

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/insulation_uncertainty_demo_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/insulation_uncertainty_demo.svg" alt="A weighted rating reported with its two-sided 95 % expanded uncertainty in situations A, B and C, the reproducibility uncertainty widest and the repeatability uncertainty narrowest" width="80%"></picture>

```python
from phonometry import (band_uncertainty, single_number_uncertainty,
                        uncertain_value, satisfies_lower_requirement)

# Situation B (same building, different teams) -> the in-situ standard deviation.
print(single_number_uncertainty("r_w", "B"))       # 0.9  dB  (Table 3)
u = band_uncertainty("airborne", "B")              # per-band u (Table 2)
print(len(u.frequencies), u.uncertainties[10])     # 21 1.1  (the 500 Hz band)

# Report R'w = 52 dB with a two-sided 95 % interval (k = 1.96, Table 8):
uv = uncertain_value(52.0, "rprime_w", "B")        # aliases resolve to r_w
print(uv.coverage_factor, round(uv.expanded_uncertainty, 1))    # 1.96 1.8
print(round(uv.lower, 1), round(uv.upper, 1))      # 50.2 53.8  ->  52 ± 1.8 dB

# Declaring conformity uses the ONE-sided factor (k = 1.65): does R'w provably
# clear a 50 dB requirement?
uc = uncertain_value(52.0, "rprime_w", "B", one_sided=True)
print(satisfies_lower_requirement(52.0, uc.expanded_uncertainty, 50.0))   # True
```

Impact quantities offer situations B/C only (Table 4, no 500 Hz band in the 2020
edition), and $\Delta L$ only situation A. Descriptors are case-insensitive with
aliases (`rprime_w`/`dnt_w`→`r_w`, `lprime_n_w`→`ln_w`); combine independent
components in quadrature with `combine_uncertainties`, and reduce by $m$
independent measurements with `reduce_by_independent_measurements` ($u/\sqrt{m}$).

<details>
<summary>Show the code for this figure</summary>

```python
import matplotlib.pyplot as plt
from phonometry import uncertain_value

# The same R'w = 52 dB reported in each situation with its two-sided 95 % U.
situations = ["A", "B", "C"]
vals = [uncertain_value(52.0, "r_w", s) for s in situations]

fig, ax = plt.subplots(figsize=(7, 4))
ax.errorbar(situations, [v.value for v in vals],
            yerr=[v.expanded_uncertainty for v in vals],
            fmt="o", capsize=8, color="tab:blue")
for s, v in zip(situations, vals):
    ax.annotate(f"±{v.expanded_uncertainty:.1f}", (s, v.upper),
                textcoords="offset points", xytext=(8, 4))
ax.set_ylabel("R'w [dB]"); ax.set_xlabel("Measurement situation")
ax.set_title("R'w = 52 dB with 95 % expanded uncertainty (ISO 12999-1)")
fig.tight_layout()
plt.show()
```

</details>

### `band_uncertainty()` / `single_number_uncertainty()` / `uncertain_value()` parameters

| Parameter | Type | Units | Range / default | Notes |
| :--- | :--- | :--- | :--- | :--- |
| `measurand` | str | — | `'airborne'` / `'impact'` / `'impact_reduction'` | Selects Table 2 / 4 / 6 |
| `quantity` | str | — | `'r_w'`, `'ln_w'`, `'delta_lw'` (+ aliases, `+c`/`+ctr` variants) | Single-number descriptor |
| `situation` | str | — | `'A'` / `'B'` / `'C'` | Measurement situation (Clause 5.2) |
| `value` | float | dB | — | Best estimate `y` to attach `U` to |
| `coverage` | float | — | default `0.95` | Confidence level (Table 8) |
| `one_sided` | bool | — | default `False` | One-sided factor for conformity checks |
| `upper_limit` | bool | — | default `False` | Select the σR95 upper limit (airborne, situation A) |

`band_uncertainty()` returns a `BandUncertainty` (`frequencies`,
`uncertainties`, `.to_arrays()`); `single_number_uncertainty()` a float;
`uncertain_value()` an `UncertainValue` (`value`, `standard_uncertainty`,
`coverage_factor`, `expanded_uncertainty`, `.lower`, `.upper`). The read-only
`COVERAGE_FACTORS` mapping exposes Table 8 keyed by `(confidence, one_sided)`.

---

**Standards.** ISO 16283-1:2014, ISO 16283-2 and ISO 16283-3:2016, *Acoustics —
Field measurement of sound insulation in buildings and of building elements* —
the level differences, normalisations and element methods of §1; ISO 717-1 and
ISO 717-2 — the reference-curve single-number ratings and the spectrum
adaptation terms C, Ctr and CI; ISO 10140-2:2010, ISO 10140-3:2010 and ISO 10140-4:2010 — the
laboratory R and Ln with the background-noise correction of §2; EN 12354-1:2000
and EN 12354-2:2000 — the simplified flanking-transmission predictions of §3
(Annex E junctions, worked examples H.3 and E.3); EN 12354-3:2000 and
EN 12354-4:2000 — the façade sound insulation and outdoor-radiation predictions
of §3 (Annex F and Annex G worked examples); ISO 12999-1:2020 — the
standard uncertainties per measurement situation and the coverage factors
of §4; its precision framework builds on ISO 5725 (context, not
implemented directly).

## See also

- [Room Acoustics](https://jmrplens.github.io/phonometry/guides/room-acoustics/) — the impulse response,
  room parameters and sound absorption that this guide's insulation chain builds on.
- [Levels](https://jmrplens.github.io/phonometry/guides/levels/) — energy averaging and the level metrics behind
  source/receiving-room levels.
- [Filter Banks](https://jmrplens.github.io/phonometry/guides/filter-banks/) — the IEC 61260 fractional-octave filters
  used for the insulation spectra.
- [Sound Power](https://jmrplens.github.io/phonometry/guides/sound-power/) — the `LW` methods that share the
  absorption-area machinery of the receiving room.
- [Theory](https://jmrplens.github.io/phonometry/reference/theory/) — the reference-curve derivation behind the
  weighted single-number ratings.

---


<!-- source: docs/outdoor-propagation.md | canonical: https://jmrplens.github.io/phonometry/guides/outdoor-propagation/ -->

# Outdoor Sound Propagation

Predicting the noise a distant source delivers to a receiver outdoors is a
book-keeping exercise: start from the source **sound power** and subtract, band
by band, every mechanism that attenuates the sound on its way — spherical
spreading, the air itself, the ground, and any barrier in between. This page
covers the two parts of ISO 9613 that supply those terms: **ISO 9613-1**, the
pure-tone atmospheric absorption coefficient $\alpha$, and **ISO 9613-2**, the
general method that assembles $\alpha$ with divergence, ground and screening
into an octave-band prediction. The atmospheric coefficient also closes the loop
with room acoustics, since ISO 354 defers its air power-attenuation coefficient
$m$ entirely to $\alpha$.

## 1. Atmospheric absorption (ISO 9613-1)

Air is not lossless. A propagating tone bleeds energy into shear viscosity and
heat conduction — the *classical and rotational* losses that grow as $f^2$ — and
into the **vibrational relaxation** of the oxygen and nitrogen molecules, each an
energy store that resonates near a humidity- and temperature-dependent
relaxation frequency. ISO 9613-1:1993, Eq. (5) collects all of this into the
pure-tone attenuation coefficient $\alpha$, in decibels per metre:

$$
\alpha = 8.686\ f^2 \Big[ 1.84\times10^{-11} \big(p_a/p_r\big)^{-1} \big(T/T_0\big)^{1/2}
       + \big(T/T_0\big)^{-5/2} \big( 0.01275\ \tfrac{e^{-2239.1/T}}{f_{rO} + f^2/f_{rO}}
       + 0.1068\ \tfrac{e^{-3352.0/T}}{f_{rN} + f^2/f_{rN}} \big) \Big],
$$

with the oxygen and nitrogen relaxation frequencies $f_{rO}$, $f_{rN}$
(Eq. (3)/(4)), the reference conditions $T_0 = 293.15$ K and $p_r = 101.325$ kPa
(Clause 4.2), and the molar water-vapour concentration $h$ obtained from the
relative humidity by the Annex B psychrometric conversion. Two features dominate
the shape: at low frequency $\alpha \propto f^2$, so it rises steeply; and near
each relaxation frequency the matching term peaks and rolls off. Between them
$\alpha$ climbs about two decades from 50 Hz to 10 kHz, and — because humidity
sets $f_{rO}$ — sweeping the humidity moves a relaxation peak across the band, so
the driest air is not always the least absorbing.

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/air_absorption_alpha_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/air_absorption_alpha.svg" alt="ISO 9613-1 pure-tone atmospheric attenuation coefficient alpha in dB/km against frequency on log-log axes, for four temperature and humidity combinations, showing the f-squared low-frequency growth and the humidity-dependent relaxation roll-off" width="80%"></picture>

*The dry 20 °C / 10 % curve absorbs most at mid frequencies, but the humid
curves overtake it below ~200 Hz — the relaxation signature.*

<details>
<summary>Show the code for this figure</summary>

```python
import matplotlib.pyplot as plt
import numpy as np
from phonometry import air_attenuation

freqs = np.geomspace(50.0, 10000.0, 400)
fig, ax = plt.subplots()
for temp, rh in [(20.0, 50.0), (20.0, 10.0), (0.0, 70.0), (30.0, 80.0)]:
    ax.loglog(freqs, air_attenuation(freqs, temp, rh) * 1000.0,
              label=f"{temp:g} °C, {rh:g} % RH")
ax.set_xlabel("Frequency [Hz]")
ax.set_ylabel("Attenuation coefficient alpha [dB/km]")
ax.legend()
plt.show()
```

</details>

```python
import numpy as np
from phonometry import air_attenuation, air_attenuation_m

bands = [63, 125, 250, 500, 1000, 2000, 4000, 8000]   # octave-band centres [Hz]

# Pure-tone attenuation coefficient alpha [dB/m] at 20 °C, 50 % RH, one atmosphere
alpha = air_attenuation(bands, temperature=20.0, relative_humidity=50.0)
print(np.round(alpha * 1000.0, 2))          # in dB/km, as Table 1 tabulates
# [  0.12   0.44   1.31   2.73   4.66   9.89  29.67 105.29]

# Reproduce an ISO 9613-1 Table 1 cell exactly (10 °C, 70 %, 1 kHz)
cell = air_attenuation(1000.0, 10.0, 70.0, exact_midband=True) * 1000.0
print(round(float(cell), 2))                # 3.66  (dB/km, Table 1)

# Feed real conditions into the ISO 354 power attenuation coefficient m [1/m]
m = air_attenuation_m([1000.0, 4000.0], temperature=20.0, relative_humidity=50.0)
print(np.round(m, 5))                        # [0.00107 0.00683]
```

`air_attenuation` returns dB/m (Table 1 prints dB/km, i.e. $\times 1000$); it is
fully vectorized over `frequencies` with scalar temperature, humidity and
pressure. Passing `exact_midband=True` snaps each requested frequency onto the
exact one-third-octave midband $f_m = 1000 \cdot 10^{k/10}$ (Eq. (6), Note 5)
used to compute Table 1, so the library reproduces every tabulated point to under
0.4 % — the standard's own three-significant-figure precision, far inside its
stated $\pm 10$ % accuracy (Clause 7.1). Inputs outside the tabulated ranges
(−20…+50 °C, 10…100 % RH, 50…10 000 Hz, or above the 200 kPa validity envelope)
still compute but raise an `AtmosphericAbsorptionWarning`; non-physical inputs
(non-positive frequency, humidity outside 0…100 %, sub-absolute-zero
temperature) raise `ValueError`. `air_attenuation_m` composes $\alpha$ with the
ISO 354 conversion $m = \alpha/(10 \lg e)$ so an ISO 354 caller can feed real
atmospheric conditions straight into `absorption_area` /
`absorption_coefficient` instead of hand-entering $m$.

### `air_attenuation()` / `air_attenuation_m()` parameters

| Parameter | Type / shape | Units | Range / default | Notes |
| :--- | :--- | :--- | :--- | :--- |
| `frequencies` | scalar or 1D array | Hz | > 0 | Vectorized; 50–10 000 Hz tabulated |
| `temperature` | float | °C | default `20.0` | −20…+50 tabulated; outside warns |
| `relative_humidity` | float | % | default `50.0` | 10…100 tabulated; `[0, 100]` allowed |
| `pressure` | float | kPa | default `101.325` | ≤ 200 valid (Clause 7); above warns |
| `exact_midband` | bool | — | default `False` | Snap to $f_m = 1000\cdot10^{k/10}$ (reproduces Table 1) |

`air_attenuation` returns $\alpha$ in dB/m; `air_attenuation_m` returns
$m = \alpha/(10 \lg e)$ in 1/m for ISO 354.

## 2. General method of calculation (ISO 9613-2)

ISO 9613-2:1996 predicts the octave-band level at a receiver **downwind** of a
point source (or under the equivalent moderate temperature inversion, Clause 5).
The equivalent-continuous downwind level is

$$
L_{fT}(DW) = L_W + D_c - A, \qquad A = A_{div} + A_{atm} + A_{gr} + A_{bar} + A_{misc},
$$

(Eq. (3)/(4)) where $L_W$ is the octave-band sound power level, $D_c$ the
directivity correction (directivity index plus a solid-angle index
$D_\Omega$), and $A$ the total attenuation. The library implements the four
general terms of Clause 7; the informative $A_{misc}$ (foliage, industrial
sites, housing; Annex A) and reflections off vertical obstacles (Clause 7.5) are
**not implemented** — the standard treats them as informative and they are left
to the caller.

The geometry of the screening term fixes the vocabulary — the diffraction edge
splits the source-to-receiver distance into $d_{ss}$ and $d_{sr}$, and the extra
path length over the edge is $z = d_{ss} + d_{sr} - d$:

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/diagram_outdoor_geometry_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/diagram_outdoor_geometry.svg" alt="ISO 9613-2 source-barrier-receiver geometry: a point source at height hs, a barrier whose top edge splits the path into dss and dsr, and a receiver at height hr, with the blocked direct ray and the diffracted ray over the edge, the path difference z and the Dz formula" width="92%"></picture>

### The four attenuation terms

* **Geometrical divergence** $A_{div} = 20 \log_{10}(d/d_0) + 11$ dB, $d_0 = 1$ m
  (Eq. (7)): spherical spreading from a point source. Exactly 51 dB at 100 m,
  +6 dB per distance doubling. The $+11$ ($= 10 \lg 4\pi$) sets the level at the
  1 m reference distance.
* **Atmospheric absorption** $A_{atm} = \alpha\ d$ (Eq. (8)) with $\alpha$ the
  ISO 9613-1 coefficient — negligible at low frequency, dominant at 8 kHz over
  long paths. The ISO 9613-2 functions default to 20 °C and 70 % relative
  humidity — one of the Table 2 reference atmospheres the standard tabulates —
  while `air_attenuation` itself defaults to the ISO 9613-1 usual 50 %.
* **Ground effect** $A_{gr} = A_s + A_r + A_m$ (Eq. (9)) sums a source, receiver
  and middle region, each from the Table 3 functions $a'/b'/c'/d'$ and its
  ground factor $G$ (0 = hard/reflective, 1 = porous/absorbing). A **negative**
  $A_{gr}$ is a net *gain* from constructive ground reflection.
* **Screening** by a barrier is the diffraction insertion loss
  $D_z = 10 \log_{10}\big[ 3 + (C_2/\lambda)\ C_3\ z\ K_{met} \big]$ (Eq. (14)),
  capped at 20 dB (single edge) or 25 dB (double edge). For a top-edge barrier
  the ground effect of the screened path folds into it,
  $A_{bar} = D_z - A_{gr} \ge 0$ (Eq. (12), Note 13); for a lateral barrier
  $A_{bar} = D_z$ and the ground term is retained (Eq. (13)).

`outdoor_propagation_attenuation` assembles the four terms into an
`OutdoorAttenuation` result whose per-band arrays sum, band by band, to
`a_total`, so the divergence, atmospheric, ground and barrier contributions stay
separable. The stacked breakdown makes the frequency character obvious: the
barrier helps most at high frequency (short wavelength) until it saturates at the
cap, atmospheric absorption bites only at 8 kHz, and the ground dip lives in the
mid bands.

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/outdoor_attenuation_breakdown_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/outdoor_attenuation_breakdown.svg" alt="ISO 9613-2 per-octave-band attenuation breakdown as a stacked bar of Adiv, Aatm, Agr and Abar with the total A overlaid, for a 200 m path over porous ground with a 4 m barrier" width="80%"></picture>

<details>
<summary>Show the code for this figure</summary>

```python
import matplotlib.pyplot as plt
import numpy as np
from phonometry import Barrier, outdoor_propagation_attenuation

bands = np.array([63.0, 125.0, 250.0, 500.0, 1000.0, 2000.0, 4000.0, 8000.0])
barrier = Barrier(source_to_edge=101.0, edge_to_receiver=101.0)
att = outdoor_propagation_attenuation(
    200.0, 1.5, 1.5, bands, ground_source=1.0, ground_middle=1.0,
    ground_receiver=1.0, barrier=barrier, temperature=15.0,
    relative_humidity=70.0,
)

# One line — the same stacked breakdown with the total overlaid:
att.plot()

# By hand:
x = np.arange(len(bands))
fig, ax = plt.subplots()
# Separate positive and negative baselines: a negative term (Agr is a net
# gain at 63 Hz here) stacks below zero, and the signed heights sum to a_total.
pos_bottom = np.zeros(len(bands))
neg_bottom = np.zeros(len(bands))
for term, label in [(att.a_div, "Adiv — divergence"),
                    (att.a_atm, "Aatm — atmospheric"),
                    (att.a_gr, "Agr — ground"),
                    (att.a_bar, "Abar — barrier")]:
    ax.bar(x, term, bottom=np.where(term >= 0.0, pos_bottom, neg_bottom),
           label=label)
    pos_bottom += np.maximum(term, 0.0)
    neg_bottom += np.minimum(term, 0.0)
ax.plot(x, att.a_total, "D-", color="black", label="A — total")
ax.set_xticks(x)
ax.set_xticklabels([f"{b:g}" for b in bands])
ax.set_xlabel("Octave-band centre frequency [Hz]")
ax.set_ylabel("Attenuation A [dB]")
ax.legend()
plt.show()
```

</details>

```python
import numpy as np
from phonometry import (
    Barrier, outdoor_propagation_attenuation, predicted_receiver_level,
)

bands = [63, 125, 250, 500, 1000, 2000, 4000, 8000]   # octave-band centres [Hz]

# A point source and receiver 1.5 m high, 200 m apart over porous ground
# (G = 1), screened midway by a barrier that raises the path over its top edge
# (dss = dsr ~ 101 m). Geometry feeds the pathlength-difference equations.
barrier = Barrier(source_to_edge=101.0, edge_to_receiver=101.0)
att = outdoor_propagation_attenuation(
    200.0, source_height=1.5, receiver_height=1.5, frequencies=bands,
    ground_source=1.0, ground_middle=1.0, ground_receiver=1.0,
    barrier=barrier, temperature=15.0, relative_humidity=70.0,
)
print(np.round(att.a_div, 1))     # [57. 57. 57. 57. 57. 57. 57. 57.]  divergence
print(np.round(att.a_gr, 2))      # [-4.65  2.34 13.79  9.76  1.3  -0.   -0.   -0.  ]
print(np.round(att.a_bar, 2))     # [13.78  8.89  0.    6.69 18.01 20.   20.   20.  ]
print(np.round(att.a_total, 1))   # [66.2 68.3 71.  73.9 77.1 78.8 82.3 96. ]

# Predicted receiver level from an octave-band sound power Lw = 95 dB
lw = np.full(len(bands), 95.0)
lp = predicted_receiver_level(
    lw, 200.0, 1.5, 1.5, bands, 1.0, 1.0, 1.0,
    barrier=barrier, temperature=15.0, relative_humidity=70.0,
)
print(np.round(lp, 1))            # [28.8 26.7 24.  21.1 17.9 16.2 12.7 -1. ]
```

`predicted_receiver_level` composes $L_{fT}(DW) = L_W + D_c - A$ with
$D_c = $ `directivity_index` $+ $ `d_omega`. Pass `c0=` to subtract the
meteorological correction $C_{met}$ (Eq. (21)/(22)) band by band for a long-term
average — note the standard applies $C_{met}$ to the A-weighted level, so the
per-band form here is a **convenience**, not a literal reading of Clause 8.

### Ground: the alternative A-weighted method

When only the A-weighted receiver level matters and the sound travels over
porous or mostly-porous ground (and is not a pure tone), 7.3.2 offers a simpler
closed form $A_{gr} = 4.8 - \frac{2 h_m}{d}\left(17 + \frac{300}{d}\right) \ge 0$ (Eq. (10), negative
results clamped to zero), paired with the solid-angle index $D_\Omega$
(Eq. (11)) that must then be added to $D_c$. These are exposed as
`ground_attenuation_alternative` and `directivity_omega`, but they are **not
auto-wired** into `outdoor_propagation_attenuation` (which always uses the
general per-region method of 7.3.1); combine them by hand when the alternative
method is appropriate.

```python
from phonometry import (
    ground_attenuation_alternative, directivity_omega, meteorological_correction,
)

# Alternative ground term (mean path height hm = 2 m, d = 200 m)
print(round(ground_attenuation_alternative(200.0, 2.0), 2))          # 4.43 dB
# Its companion solid-angle index (add to Dc when using Eq. (10))
print(round(directivity_omega(1.5, 1.5, 200.0), 2))                  # 3.01 dB
# Long-term meteorological correction (C0 = 2 dB) to subtract from LAT(DW)
print(round(meteorological_correction(200.0, 1.5, 1.5, 2.0), 2))     # 1.7 dB
```

### `Barrier` and the screening term

The `Barrier` dataclass describes the diffraction geometry directly, which is
the cleanest match to Eq. (14)/(16)/(17). Single diffraction (a thin screen)
leaves `edge_separation=None` ($C_3 = 1$); giving the edge spacing `e` selects
double (thick-barrier) diffraction with the $C_3$ factor of Eq. (15) and the
25 dB cap. `ground_reflections_by_image=True` switches $C_2$ from 20 to 40
(reflections handled by image sources), and `lateral=True` selects vertical-edge
diffraction (Eq. (13), $K_{met}=1$, ground term retained).

### `outdoor_propagation_attenuation()` parameters

| Parameter | Type / shape | Units | Range / default | Notes |
| :--- | :--- | :--- | :--- | :--- |
| `distance` | float | m | > 0 | Straight-line source–receiver distance $d$ |
| `source_height` / `receiver_height` | float | m | ≥ 0 | $h_s$, $h_r$ above ground |
| `frequencies` | 1D array | Hz | default 8 octaves 63–8000 | `DEFAULT_FREQUENCIES` |
| `ground_source` / `ground_middle` / `ground_receiver` | float | — | `[0, 1]`, default `0.0` | Ground factor $G$ (0 hard, 1 porous) |
| `barrier` | `Barrier` or None | — | default `None` | Screening obstacle |
| `temperature` / `relative_humidity` / `pressure` | float | °C / % / kPa | 20 / 70 / 101.325 | Passed to $A_{atm}$ |
| `projected_distance` | float or None | m | default $\sqrt{d^2-(h_s-h_r)^2}$ | Ground-plane $d_p$ |

Returns an `OutdoorAttenuation` with `a_div`, `a_atm`, `a_gr`, `a_bar`,
`a_total` and `d_omega`, all one value per band; its `.plot()` draws the
stacked per-band breakdown with the total overlaid (the figure above).

### `Barrier` fields

| Field | Type | Units | Default | Notes |
| :--- | :--- | :--- | :--- | :--- |
| `source_to_edge` | float | m | — | $d_{ss}$, source to first edge |
| `edge_to_receiver` | float | m | — | $d_{sr}$, (last) edge to receiver |
| `parallel_distance` | float | m | `0.0` | Component $a$ parallel to the edge |
| `edge_separation` | float or None | m | `None` | $e$; given ⇒ double diffraction (25 dB cap) |
| `ground_reflections_by_image` | bool | — | `False` | `True` ⇒ $C_2 = 40$ |
| `lateral` | bool | — | `False` | `True` ⇒ vertical-edge diffraction (Eq. (13)) |

The method's stated accuracy is $\pm 1$ to $\pm 3$ dB for broadband noise up to
1000 m (Table 5). See the [Theory](https://jmrplens.github.io/phonometry/reference/theory/) page for the full derivation, the
[Room Acoustics guide](https://jmrplens.github.io/phonometry/guides/room-acoustics/) for how $\alpha$ feeds
ISO 354, and the [Occupational Noise Exposure guide](https://jmrplens.github.io/phonometry/guides/occupational-exposure/) for the ISO 9612 occupational
exposure that consumes A-weighted levels.

---

**Standards.** ISO 9613-1:1993, *Acoustics — Attenuation of sound during
propagation outdoors — Part 1: Calculation of the absorption of sound by the
atmosphere* — the pure-tone attenuation coefficient $\alpha$ (Eq. (5)) with the
oxygen and nitrogen relaxation frequencies (Eq. (3)/(4)), the Annex B humidity
conversion and the exact Table 1 midbands (Eq. (6), Note 5). ISO 9613-2:1996,
*Acoustics — Attenuation of sound during propagation outdoors — Part 2: General
method of calculation* — the downwind receiver level (Eq. (3)/(4)) assembled
from geometrical divergence (Eq. (7)), atmospheric absorption (Eq. (8)), the
ground effect (Eq. (9), Table 3) with its A-weighted alternative
(Eq. (10)/(11)), barrier screening (Eqs. (12)–(17)) and the meteorological
correction (Eq. (21)/(22)). ISO 354:2003, *Acoustics — Measurement of sound
absorption in a reverberation room* — only the clause 8.1.2.1 conversion
$m = \alpha/(10 \lg e)$ behind `air_attenuation_m`; the reverberation-room
method itself is covered in the [Room Acoustics guide](https://jmrplens.github.io/phonometry/guides/room-acoustics/).

---


<!-- source: docs/sound-power.md | canonical: https://jmrplens.github.io/phonometry/guides/sound-power/ -->

# Sound Power

Sound *pressure* depends on where you stand and on the room you stand in;
sound **power** does not. The sound power level `LW` is the total acoustic
energy per second a source radiates, referenced to `P0 = 1 pW`, and it is
the device-independent **emission** descriptor that goes on a datasheet,
feeds a room prediction (EN 12354) or is checked against a noise-emission
limit. This page covers the three routes phonometry implements to obtain it
and when to reach for each: an enveloping *pressure* surface in the field
(ISO 3744/3746), the diffuse field of a *reverberation room* (ISO 3741),
*intensity* scanning over a surface (ISO 9614-2), and — for the highest
accuracy — the precision grades in an *anechoic room* (ISO 3745) and by
precision *intensity* scanning (ISO 9614-3).

## Choosing a method

All deliver the same quantity — a per-band `LW` and an A-weighted total
`LWA` — but under different environments, accuracy grades and practical
constraints.

| Method | Standard | Measured quantity | Environment | Accuracy grade | Use when |
| :--- | :--- | :--- | :--- | :--- | :--- |
| Enveloping surface | **ISO 3744** (engineering) / **ISO 3746** (survey) | Sound pressure on a hemisphere or box | Essentially free field over one or more reflecting planes | Grade 2 (`σR0 ≈ 1.5 dB`) / grade 3 (`≈ 3.0 dB`) | In situ or a large room; no special test facility available |
| Reverberation room | **ISO 3741** | Sound pressure in the diffuse field | Qualified hard-walled reverberation room | Grade 1 (precision) | Highest accuracy for steady, broadband sources in a lab |
| Intensity scanning | **ISO 9614-2** | Normal sound intensity scanned over a surface | Almost any, tolerant of steady extraneous noise | Grade 2 / 3 (from per-band field indicators) | On-site with background noise, or one machine among many |
| Anechoic room | **ISO 3745** | Sound pressure on a fixed microphone array | Qualified anechoic or hemi-anechoic room | Grade 1 (precision) | Reference-grade emission in a free-field laboratory |
| Precision intensity scanning | **ISO 9614-3** | Scanned normal intensity, tighter criteria | Almost any, tolerant of steady extraneous noise | Grade 1 (precision) | Precision on-site, with the ISO 9614-3 field-indicator checks |

The pressure methods correct the surface level for the room (`K2`) and for
background noise (`K1`); the reverberation method needs a *qualified* room
but reaches precision grade; intensity rejects steady background energy at
the cost of a two-microphone probe and a per-band validity check. The rest
of the page walks each in turn.

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/sound_power_methods_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/sound_power_methods.svg" alt="The three sound power routes side by side: an enveloping pressure surface over a reflecting plane (ISO 3744/3746), a source in a reverberation room sampled by microphones (ISO 3741) and an intensity probe scanning a surface around the source (ISO 9614-2)" width="92%"></picture>

## 1. Enveloping surface, sound pressure (ISO 3744 / ISO 3746)

Place the source on a reflecting plane and imagine a **measurement surface**
of area `S` wrapping it: a hemisphere for a compact source, a box (right
parallelepiped) for a large or elongated one. Sample the sound pressure
level at an array of microphone positions on that surface, energy-average
them, and the sound power follows because a diffuse-enough surface captures
all the radiated energy:

$$
\bar{L}_p = 10 \log_{10}\left( \frac{1}{N_M} \sum_i 10^{L_{pi}/10} \right), \qquad
L_W = \bar{L}_p - K_1 - K_2 + 10 \log_{10}\frac{S}{S_0},\quad S_0 = 1\ \text{m}^2 .
$$

Two corrections clean up the surface level. The **background-noise
correction** removes the energy that would have been there with the source
switched off, from the margin `ΔLp` between source-on and background levels,

$$
K_1 = -10 \log_{10}\left( 1 - 10^{-\Delta L_p/10} \right),
$$

and the **environmental correction** removes the reverberant build-up of the
test room from its equivalent absorption area `A`,

$$
K_2 = 10 \log_{10}\left( 1 + \frac{4 S}{A} \right).
$$

The surface area is a closed form of the geometry: a hemisphere is
`S = 2πr²` over one reflecting plane (halved and quartered for two and three
planes), and a one-plane box is `S = 4(ab + bc + ca)` with
`a = 0.5·l1 + d`, `b = 0.5·l2 + d`, `c = l3 + d` for measurement distance
`d`. ISO 3746 (survey) shares every formula but is coarser: fewer
microphone positions, a 3 dB background criterion instead of 6 dB, and
validity up to `K2 ≤ 7 dB` instead of 4 dB.

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/diagram_sound_power_surfaces_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/diagram_sound_power_surfaces.svg" alt="Measurement surfaces of ISO 3744: a hemisphere of radius r enveloping a compact source on a reflecting plane, and a right parallelepiped (box) at measurement distance d around a large source, both with microphone positions marked" width="88%"></picture>

```python
import numpy as np
from phonometry import sound_power_pressure, measurement_positions

# Octave-band SPL (dB) at the 10 hemisphere positions of ISO 3744 (Annex B),
# with the source running, plus the background spectrum with it switched off.
freqs = np.array([63, 125, 250, 500, 1000, 2000, 4000, 8000])
base = np.array([70.0, 74.0, 78.0, 80.0, 79.0, 76.0, 72.0, 66.0])
rng = np.random.default_rng(0)
levels = base + rng.normal(0.0, 0.5, size=(10, 8))     # (positions, bands)
background = np.full((10, 8), 55.0)

# ISO 3744 Annex B microphone coordinates on a radius-1.5 m hemisphere.
mic_xyz = measurement_positions("hemisphere", radius=1.5, reflecting_planes=1)
print(mic_xyz.shape)                                    # (10, 3)

res = sound_power_pressure(
    levels, "hemisphere", radius=1.5, reflecting_planes=1,
    background_levels=background, frequencies=freqs,
    reverberation_time=0.6, volume=300.0,          # room data -> K2
)
print(round(res.surface_area, 2))                       # 14.14 m^2 (= 2*pi*1.5^2)
print(round(float(res.environmental_correction[0]), 2)) # K2 = 2.32 dB
print(round(res.sound_power_level_a, 1))                # LWA = 92.4 dB
print(round(res.uncertainty, 1))                        # U = 3.0 dB (2*sigma_R0)
print(np.round(res.sound_power_level, 1))               # per-band LW

res.plot()   # sound power level bars per band, LWA in the title (needs matplotlib)
```

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/sound_power_pressure_result_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/sound_power_pressure_result.svg" alt="The enveloping-surface sound power level spectrum of the ISO 3744 hemisphere example, one bar per octave band from 63 Hz to 8 kHz peaking near 500 Hz, with the A-weighted total of 92.4 dB(A) in the title" width="88%"></picture>

*One bar per band: the energy-averaged surface pressure minus the background
(`K1`) and environmental (`K2`) corrections plus the surface term
`10 lg(S/S0)` gives `LW(f)`, and the A-weighted energy sum across bands gives
the single-number `LWA` in the title.*

<details>
<summary>Show the code for this figure</summary>

```python
import matplotlib.pyplot as plt
import numpy as np

# res is the SoundPowerResult computed above. One line:
res.plot()
plt.show()

# By hand: a bar spectrum of LW with the A-weighted total in the title.
freqs = res.frequencies
positions = np.arange(freqs.size)
fig, ax = plt.subplots()
ax.bar(positions, res.sound_power_level, width=0.7, color="#1f77b4")
ax.set_xticks(positions)
ax.set_xticklabels([f"{f:g}" for f in freqs], rotation=45, ha="right")
ax.set_xlabel("Frequency [Hz]")
ax.set_ylabel("Sound power level LW [dB]")
ax.set_title(
    f"Enveloping-surface sound power (ISO 3744)  "
    f"LWA = {res.sound_power_level_a:.1f} dB(A)")
plt.show()
```

</details>

The A-weighted total `LWA` is combined from the band powers with the ISO 3744
Annex E A-weighting corrections, so it needs `frequencies`. Passing the room
data (`reverberation_time` + `volume`, or `absorption_area`, or
`mean_absorption_coefficient` + `room_surface`) enables `K2`; omit it and the
field is treated as free (`K2 = 0`). If the background margin drops below the
grade criterion or `K2` exceeds the validity limit, a `SoundPowerWarning`
flags that the levels are upper bounds — the determination still returns.

### `sound_power_pressure()` parameters

| Parameter | Type | Units | Range / default | Notes |
| :--- | :--- | :--- | :--- | :--- |
| `levels_positions` | 2D array | dB | `(NM, NB)` | One row per position, one column per band (or a single A-weighted column) |
| `surface` | str | — | `'hemisphere'` / `'box'` | Measurement-surface shape |
| `radius` | float | m | > 0 (hemisphere) | Hemisphere radius `r` |
| `dimensions` | (float, float, float) | m | > 0 (box) | Reference-box `(l1, l2, l3)` |
| `distance` | float | m | > 0 (box) | Measurement distance `d` |
| `reflecting_planes` | int | — | `1` / `2` / `3`, default `1` | Halves/quarters the hemisphere area |
| `background_levels` | 2D array or spectrum | dB | `(NM, NB)`, or `(NB,)` / `(1, NB)` | Enables `K1`; a single spectrum broadcasts to every position |
| `frequencies` | 1D array | Hz | nominal band centres | Enables `LWA` (Annex E) |
| `absorption_area` | float or 1D array | m² | > 0 | `A` for `K2` (direct); per-band array → per-band `K2` |
| `reverberation_time`, `volume` | float/array, float | s, m³ | > 0 | `A = 0.16 V/T` for `K2`; per-band `T` → per-band `K2` |
| `mean_absorption_coefficient`, `room_surface` | float/array, float | —, m² | `(0,1]`, > 0 | `A = α·Sv` (Eq. A.7); per-band `α` → per-band `K2` |
| `grade` | str | — | `'engineering'` (default) / `'survey'` | ISO 3744 vs ISO 3746 |
| `omc_uncertainty` | float | dB | default `0.0` | `σomc`, operating/mounting instability, folded into `U` |

Returns a `SoundPowerResult`: `sound_power_level` (per-band `LW`),
`surface_pressure_level` (`Lp` after K1/K2), `mean_pressure_level`,
`background_correction`/`environmental_correction` (`K1`/`K2`),
`directivity_index` (apparent `DIi*` per microphone position **and** frequency
band, shape `(NM, NB)`; ISO 3744 clause 8.6), `surface_area`,
`sound_power_level_a` (`LWA`), `uncertainty` (expanded, 95 %) and `grade`.
`measurement_positions('hemisphere', radius=…, reflecting_planes=…, tones=…,
grade=…)` returns the normative `(N, 3)` microphone coordinates (Table B.1 for
tonal sources, B.2 for broadband).

## 2. Reverberation room, precision grade (ISO 3741)

In a qualified hard-walled **reverberation room** the field is diffuse, so a
handful of microphones sample the whole radiated energy and the method
reaches grade 1. The sound power comes from the mean room level `Lp(ST)`,
the Sabine absorption area `A = (55.26/c)·(V/T60)` and a chain of small
corrections (ISO 3741 Eq. 20):

$$
L_W = \bar{L}_p + 10 \log_{10}\frac{A}{A_0} + 4.34\ \frac{A}{S}
      + 10 \log_{10}\left( 1 + \frac{S c}{8 V f} \right) + C_1 + C_2 - 6 .
$$

The bracketed term is the **Waterhouse correction**: near the room
boundaries the sound energy density is higher than in the interior, and
this term (which vanishes as frequency grows) restores the energy the
interior microphones miss. `C1` (reference-quantity) and `C2`
(radiation-impedance) carry the result to the reference meteorological
conditions of 23 °C and 101.325 kPa,

$$
C_1 = -10 \log_{10}\frac{p_s}{p_{s0}} + 5 \log_{10}\frac{273.15 + \theta}{314}, \qquad
C_2 = -10 \log_{10}\frac{p_s}{p_{s0}} + 15 \log_{10}\frac{273.15 + \theta}{296},
$$

with the speed of sound `c = 20.05·√(273 + θ)`. The **comparison method**
replaces the absorption-area, Waterhouse and `C1` terms by a reference sound
source of known power `LW(RSS)` measured in the same room, so the room need
not be characterised: `LW = LW(RSS) + (Lp(ST) − Lp(RSS) + C2)`.

```python
import numpy as np
from phonometry import sound_power_reverberation, sound_power_comparison

# One-third-octave mean room SPL (dB), 100 Hz - 10 kHz, and the room's T60.
freqs = np.array([100, 125, 160, 200, 250, 315, 400, 500, 630, 800, 1000,
                  1250, 1600, 2000, 2500, 3150, 4000, 5000, 6300, 8000, 10000],
                 dtype=float)
lp = np.linspace(80.0, 70.0, freqs.size)
t60 = np.full(freqs.size, 2.0)

rev = sound_power_reverberation(
    lp, t60, volume=200.0, surface_area=220.0, frequencies=freqs,
    temperature=20.0, static_pressure=101.0,
)
print(round(rev.speed_of_sound, 1))                     # c = 343.2 m/s
print(round(float(rev.absorption_area[0]), 1))          # A = 16.1 m^2 at 100 Hz
print(round(float(rev.waterhouse_correction[0]), 2))    # 1.68 dB at 100 Hz
print(round(float(rev.sound_power_level[0]), 1))        # LW = 87.9 dB
print(round(rev.sound_power_level_a, 1))                # LWA = 92.1 dB

# Comparison method: a reference source of known LW measured at the same spots.
lw_rss = np.full(freqs.size, 85.0)
lp_rss = np.linspace(78.0, 69.0, freqs.size)
cmp = sound_power_comparison(lp, lp_rss, lw_rss, frequencies=freqs, temperature=20.0)
print(round(float(cmp.sound_power_level[0]), 1), cmp.method)   # 86.9 comparison

rev.plot()   # reverberation-room LW spectrum, LWA in the title (needs matplotlib)
```

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/sound_power_reverberation_result_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/sound_power_reverberation_result.svg" alt="The reverberation-room sound power level spectrum of the ISO 3741 example, one bar per one-third-octave band from 100 Hz to 10 kHz falling gently with frequency, with the A-weighted total of 92.1 dB(A) in the title" width="88%"></picture>

*The mean room level carried through the absorption-area, Waterhouse and
meteorological terms of Eq. 20 gives the one-third-octave `LW(f)`, and the
A-weighted energy sum across the 21 bands gives the `LWA` in the title.*

<details>
<summary>Show the code for this figure</summary>

```python
import matplotlib.pyplot as plt
import numpy as np

# rev is the ReverberationSoundPowerResult computed above. One line:
rev.plot()
plt.show()

# By hand: a bar spectrum of LW with the A-weighted total in the title.
freqs = rev.frequencies
positions = np.arange(freqs.size)
fig, ax = plt.subplots()
ax.bar(positions, rev.sound_power_level, width=0.7, color="#1f77b4")
ax.set_xticks(positions)
ax.set_xticklabels([f"{f:g}" for f in freqs], rotation=45, ha="right")
ax.set_xlabel("Frequency [Hz]")
ax.set_ylabel("Sound power level LW [dB]")
ax.set_title(
    f"Reverberation-room sound power (ISO 3741)  "
    f"LWA = {rev.sound_power_level_a:.1f} dB(A)")
plt.show()
```

</details>

`levels` may be a 1D mean spectrum or a 2D `(NM, NB)` array averaged over
positions. When the room volume, its reverberation time or the microphone
count fail an ISO 3741 qualification criterion (Table 1 minimum volume, the
`V/S` reverberation floor, fewer than 6 positions, or an inter-position
spread above 1.5 dB), an advisory `SoundPowerWarning` is emitted and the
result still returns.

### `sound_power_reverberation()` parameters

| Parameter | Type | Units | Range / default | Notes |
| :--- | :--- | :--- | :--- | :--- |
| `levels` | 1D or 2D array | dB | per band, or `(NM, NB)` | Mean room SPL; 2D is energy-averaged over positions |
| `t60` | float or 1D array | s | > 0 | Room reverberation time (scalar broadcasts) |
| `volume` | float | m³ | > 0 | Room volume `V` |
| `surface_area` | float | m² | > 0 | Total room surface `S` (Waterhouse, `A/S`) |
| `frequencies` | 1D array | Hz | one per band | Required (Waterhouse needs `f`); enables `LWA` |
| `background_levels` | 1D or 2D array | dB | matches `levels` | Per-band `K1` (frequency-dependent criterion) |
| `temperature` | float | °C | default `23.0` | Sets `c`, `C1`, `C2` |
| `static_pressure` | float | kPa | default `101.325` | Sets `C1`, `C2` |

`sound_power_comparison(levels, levels_ref, lw_ref, *, frequencies=None,
background_levels=…, background_levels_ref=…, temperature=23.0,
static_pressure=101.325)` takes the same room levels plus the reference
source's levels and known power.

| Parameter | Type | Units | Range / default | Notes |
| :--- | :--- | :--- | :--- | :--- |
| `levels_ref` | 1D or 2D array | dB | matches `levels` | Mean room SPL with the reference source (RSS) running |
| `lw_ref` | 1D array | dB | per band | Known sound power `LW(RSS)` of the reference source |
| `background_levels` | 1D or 2D array | dB | matches `levels` | Background for the test source; per-band `K1` on `Lp(ST)` |
| `background_levels_ref` | 1D or 2D array | dB | matches `levels_ref` | Background for the **reference** source; per-band `K1` on `Lp(RSS)` |

`background_levels_ref` background-corrects the reference-source room level
`Lp(RSS)` exactly as `background_levels` does for the test source; both need
`frequencies` (the ISO 3741 criterion is frequency-dependent). Both return a
`ReverberationSoundPowerResult`
(`sound_power_level`, `mean_pressure_level`, `absorption_area`,
`waterhouse_correction`, `background_correction`, `c1`, `c2`,
`speed_of_sound`, `sound_power_level_a`, `method`; the absorption/Waterhouse/`c1`
fields are `NaN` for the comparison method).

## 3. Intensity scanning (ISO 9614-2)

Sound **intensity** is the net energy flux, so it distinguishes energy
*leaving* the source from steady energy merely passing through the surface —
which is why the intensity method tolerates background noise that would
defeat the pressure methods. A p-p probe (see the
[Sound Intensity guide](https://jmrplens.github.io/phonometry/guides/intensity/)) is swept continuously over each of
`N` segments of a surface enclosing the source, reporting the segment-averaged
signed normal intensity `<In,i>`. The partial powers sum to the total:

$$
P_i = \langle I_{n,i} \rangle\ S_i, \qquad P = \sum_i P_i, \qquad
L_W = 10 \log_{10}\frac{P}{P_0},\quad P_0 = 1\ \text{pW} .
$$

A band in which `P < 0` (net inflow, from a stronger source outside the
surface) is **not determinable** and reported as `NaN`. Two normative field
indicators qualify each band. The **surface pressure-intensity indicator**
`FpI` measures how reactive the field is, and the **negative-partial-power
indicator** `F+/-` measures how much energy circulates in and out:

$$
F_{pI} = [L_p] - L_W + 10 \log_{10}\frac{S}{S_0}, \qquad
F_{+/-} = 10 \log_{10}\frac{\sum_i \lvert P_i \rvert}{\lvert \sum_i P_i \rvert} .
$$

The probe's **dynamic capability** `Ld = δpI0 − K` (pressure-residual
intensity index minus the bias factor `K`, 10 dB for grade 2 and 7 dB for
grade 3) must exceed `FpI` (criterion 1); `F+/- ≤ 3 dB` is criterion 2
(mandatory for grade 2); and the two repeated sweeps must agree within the
Table 2 limit `s` per segment (criterion 3). A band is **engineering** grade
when criteria 1, 2 and 3 hold, **survey** when 1 and 3 hold, else `none`.

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/diagram_pp_probe_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/diagram_pp_probe.svg" alt="A two-microphone p-p sound intensity probe: two pressure microphones separated by a spacer, from which the pressure gradient and hence the normal intensity are estimated" width="70%"></picture>

```python
import numpy as np
from phonometry import sound_power_intensity

# 6 surface segments x 6 octave bands: signed normal intensity (W/m^2) from two
# repeated sweeps, the segment areas, and the per-segment surface SPL (dB).
freqs = np.array([125, 250, 500, 1000, 2000, 4000], dtype=float)
areas = np.full(6, 0.5)                                 # 0.5 m^2 per segment
rng = np.random.default_rng(0)
scan1 = np.abs(rng.normal(1e-4, 2e-5, size=(6, 6)))     # (segments, bands)
scan2 = scan1 * (1.0 + rng.normal(0.0, 0.02, size=(6, 6)))
pressure = np.full((6, 6), 80.0)

res = sound_power_intensity(
    scan1, areas, normal_intensity_2=scan2, pressure_levels=pressure,
    pressure_residual_index=12.0, frequencies=freqs,
    band_type="octave", grade="engineering",
)
print(np.round(res.sound_power_level, 1))               # per-band LW
print(round(res.sound_power_level_a, 1))                # LWA over determinable bands
print(round(float(res.dynamic_capability_index[0]), 1)) # Ld = 12 - 10 = 2.0 dB
print(round(float(res.surface_pressure_intensity_index[0]), 2))   # FpI
print(list(res.achieved_grade))                         # per-band grade

res.plot()   # LW spectrum; non-positive (undeterminable) bands hatched (needs matplotlib)
```

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/sound_power_intensity_result_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/sound_power_intensity_result.svg" alt="The intensity-scanning sound power level spectrum of the ISO 9614-2 example, one bar per octave band from 125 Hz to 4 kHz all near 85 dB, with the A-weighted total of 90.9 dB(A) in the title" width="88%"></picture>

*The partial powers `<In,i>·Si` of the six segments sum to each band's `LW`;
every band here nets positive power and passes the field-indicator criteria at
engineering grade, so all six bars stand, and the A-weighted total of
90.9 dB(A) heads the title.*

<details>
<summary>Show the code for this figure</summary>

```python
import matplotlib.pyplot as plt
import numpy as np

# res is the SoundPowerIntensityResult computed above. One line:
res.plot()
plt.show()

# By hand: a bar spectrum of LW with the A-weighted total in the title.
freqs = res.frequencies
positions = np.arange(freqs.size)
fig, ax = plt.subplots()
ax.bar(positions, res.sound_power_level, width=0.7, color="#1f77b4")
ax.set_xticks(positions)
ax.set_xticklabels([f"{f:g}" for f in freqs], rotation=45, ha="right")
ax.set_xlabel("Frequency [Hz]")
ax.set_ylabel("Sound power level LW [dB]")
ax.set_title(
    f"Intensity-scanning sound power (ISO 9614-2)  "
    f"LWA = {res.sound_power_level_a:.1f} dB(A)")
plt.show()
```

</details>

Supplying `normal_intensity_2` (the second sweep) averages the two for the
partial powers and evaluates criterion 3; `pressure_levels` enables `FpI`;
`pressure_residual_index` (`δpI0`) plus a second sweep enables the per-band
achieved grade. The probe's finite-difference intensity has a
frequency-dependent bias handled in the [intensity guide](https://jmrplens.github.io/phonometry/guides/intensity/).

### `sound_power_intensity()` parameters

| Parameter | Type | Units | Range / default | Notes |
| :--- | :--- | :--- | :--- | :--- |
| `normal_intensity` | 2D array | W/m² | `(N_seg, N_bands)` | Signed segment-averaged normal intensity `<In,i>` (first sweep) |
| `areas` | 1D array | m² | > 0, `(N_seg,)` | Segment areas `Si` |
| `normal_intensity_2` | 2D array | W/m² | same shape | Second sweep → criterion 3 and averaging |
| `pressure_levels` | 2D array | dB | same shape | Segment SPL `Lpi` → `FpI` |
| `pressure_residual_index` | float or 1D array | dB | — | `δpI0` → `Ld` / criterion 1 |
| `frequencies` | 1D array | Hz | nominal centres | `LWA` and Table 2 limits |
| `band_type` | str | — | `'third'` (default) / `'octave'` | Table 2 lookup |
| `grade` | str | — | `'engineering'` (default) / `'survey'` | Selects `K` |
| `repeatability_limit` | float or 1D array | dB | default Table 2 | Override criterion-3 `s` |

Returns a `SoundPowerIntensityResult`: `partial_power`/`partial_power_level`
per segment and band, `sound_power`/`sound_power_level` (band total, `NaN`
where `negative_band`), `surface_pressure_intensity_index` (`FpI`),
`negative_partial_power_index` (`F+/-`), `repeatability`,
`dynamic_capability_index` (`Ld`), `achieved_grade`, `surface_area`,
`sound_power_level_a` and `grade`.

## 4. Precision grade, anechoic room (ISO 3745)

When the highest accuracy is required, ISO 3745 measures sound power in a
qualified **anechoic** or **hemi-anechoic** room, where the free field lets a
fixed array of microphones sample the radiated sound pressure directly. It is the
grade-1 counterpart to the enveloping-surface method of Section 1, with
standardized microphone coordinates, a per-position background correction and an
explicit meteorological correction.

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/diagram_precision_anechoic_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/diagram_precision_anechoic.svg" alt="ISO 3745 precision sound power in an anechoic room: wedge-lined walls, the device under test at the centre and a hemispherical array of microphones at a fixed radius, with the sound power level formed from the surface-averaged pressure plus the area, background and meteorological corrections" width="92%"></picture>

**Sound power level (Clause 8).** The band sound power level is the
surface-averaged pressure level plus the surface term and the corrections:

$$
L_W = \overline{L_p} + 10\lg\frac{S}{S_0} + C_1 + C_2 + C_3,
$$

with $S = 4\pi r^2$ over the sphere or $S = 2\pi r^2$ over the hemisphere,
$S_0 = 1\ \text{m}^2$. $C_1$ and $C_2$ are the meteorological corrections
(reference and radiation-impedance terms); $C_3$ accounts for air absorption over
the measurement radius. The microphone positions are the standardized
unit-vector arrays of Tables D.1 (sphere), E.1 (hemisphere) and E.2 (hemisphere,
broadband).

```python
import numpy as np
import phonometry as ph

# The 40 standardized hemisphere positions (unit vectors scaled by the radius).
pos = ph.precision_positions("hemisphere", radius=1.0, count=40)
print(pos.shape)                      # (40, 3)

# Octave/third-octave band SPL (dB) at each of the 40 positions; here a uniform
# 74 dB in one band. The result carries S = 2*pi*r^2 and LW with C1+C2+C3.
levels = np.full((40, 1), 74.0)
res = ph.sound_power_anechoic(levels, "hemisphere", radius=1.0)
print(round(res.surface_area, 3))                 # 6.283  (2*pi*1^2)
print(np.round(res.sound_power_level, 2))         # [81.85]
```

**Background and meteorological corrections.** The $K_1$ background correction is
applied **per position** and floored where the signal-to-background difference is
small (Eq. 11); the meteorological correction is evaluated from the measured
temperature and static pressure.

```python
import numpy as np
import phonometry as ph

# K1 for a 6 dB signal-to-background difference in a <=200 Hz edge band: the
# floor is 1.26 dB (Eq. 11). Source and background levels are [positions, bands].
k1 = ph.precision_background_correction(
    np.array([[56.0]]), np.array([[50.0]]), np.array([200.0]))
print(round(float(k1[0, 0]), 4))      # 1.2563

# Meteorological corrections at the 23 C, 101.325 kPa reference (Eq. 16):
mc = ph.meteorological_corrections(23.0, 101.325)
print(round(mc.c1, 4), round(mc.c2, 4))   # -0.1282 0.0

# Expanded uncertainty (Clause 10.5 EXAMPLE): sigma_R0 = 0.5, sigma_omc = 2.0,
# k = 2 -> U = 4.1 dB.
print(round(ph.precision_uncertainty(0.5, 2.0, 2.0), 3))   # 4.123
```

Over several bands `sound_power_anechoic` returns a plottable
`PrecisionSoundPowerResult` carrying the per-band `LW` and the A-weighted total:

```python
import numpy as np
import phonometry as ph

# A mid-frequency-peaked machine measured over the 40-position hemisphere array
# (Annex E). levels_positions is the (40, NB) surface pressure spectrum: a base
# spectrum peaked near 1 kHz plus a small per-position spatial spread.
freqs = np.array([125, 250, 500, 1000, 2000, 4000, 8000], float)
base = 70.0 + 8.0 * np.exp(-(np.log2(freqs / 1000.0) ** 2) / 2.0)
rng = np.random.default_rng(7)
levels = base[None, :] + rng.normal(0.0, 1.0, (40, freqs.size))

result = ph.sound_power_anechoic(levels, "hemisphere", radius=1.0, frequencies=freqs)
print(round(result.sound_power_level_a, 1))   # 89.3
result.plot()   # LW spectrum, LWA in the title (needs matplotlib)
```

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/precision_anechoic_power_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/precision_anechoic_power.svg" alt="The precision sound power level spectrum of a mid-frequency-peaked machine measured over the ISO 3745 hemisphere array, one bar per band peaking near 1 kHz, with the A-weighted total of 89.3 dB(A) in the title" width="88%"></picture>

*One bar per band: the surface-averaged pressure plus the area, background and
meteorological corrections give `LW(f)`, and the A-weighted energy sum across
bands gives the single-number `LWA` in the title.*

<details>
<summary>Show the code for this figure</summary>

```python
import matplotlib.pyplot as plt
import numpy as np

# result is the PrecisionSoundPowerResult computed above. One line:
result.plot()
plt.show()

# By hand: a bar spectrum of LW with the A-weighted total in the title.
freqs = result.frequencies
positions = np.arange(freqs.size)
fig, ax = plt.subplots()
ax.bar(positions, result.sound_power_level, width=0.7, color="#1f77b4")
ax.set_xticks(positions)
ax.set_xticklabels([f"{f:g}" for f in freqs], rotation=45, ha="right")
ax.set_xlabel("Frequency [Hz]")
ax.set_ylabel("Sound power level LW [dB]")
ax.set_title(
    f"Precision sound power (ISO 3745)  LWA = {result.sound_power_level_a:.1f} dB(A)")
plt.show()
```

</details>

## 5. Precision intensity scanning (ISO 9614-3)

ISO 9614-3 is the grade-1 scanning method: like ISO 9614-2 it integrates the
normal intensity over a surface enclosing the source, but with a continuous
scan, tighter field-indicator criteria and an explicit uncertainty budget.

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/diagram_intensity_scan_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/diagram_intensity_scan.svg" alt="ISO 9614-3 precision sound intensity scanning: a source enclosed by a measurement surface divided into segments, a two-microphone intensity probe scanned along a serpentine path over each segment, and the sound power formed by summing the normal intensity times segment area, subject to the field-indicator acceptance criteria" width="92%"></picture>

**Power and level (Clause 7).** The partial power of each segment is
$P_i = I_{n,i}\,S_i$; the total $P = \sum_i P_i$ gives
$L_W = 10\lg(P/P_0)$, $P_0 = 1\ \text{pW}$. A band whose net intensity is
negative (more power flowing in than out) is flagged not-applicable rather than
logged. The field indicators (temporal variability $F_T$, the signed and
unsigned pressure–intensity indicators, and the non-uniformity $F_S$) drive the
five acceptance criteria.

```python
import numpy as np
import phonometry as ph

# A fully enclosing surface with a uniform normal intensity In = W/S recovers
# the source power exactly: LW = 10*lg(W/P0). Here W = 100 uW -> 80 dB.
areas = np.array([0.5, 1.0, 0.25, 2.0])
w = 1.0e-4
i_n = np.full(areas.shape, w / float(areas.sum()))
res = ph.sound_power_intensity_precision(i_n, areas)
print(round(float(res.sound_power[0]), 6))          # 0.0001
print(round(float(res.sound_power_level[0]), 2))    # 80.0
```

Across several bands the result carries the per-band `LW` (`NaN` where the net
power is non-positive) and flags those bands `not_applicable`:

```python
import numpy as np
import phonometry as ph

# Four partial surfaces scanned over five one-third-octave bands. Each cell of
# partial_intensity is the signed normal intensity In_i (W/m^2); areas are the
# partial-surface areas Si. The 250 Hz band has net-negative power (a locally
# reactive field), so ISO 9614-3 flags it not-applicable (clause 9.2) -> NaN.
freqs = np.array([250, 500, 1000, 2000, 4000], float)
areas = np.array([0.5, 1.0, 0.75, 0.5])
base_intensity = np.array([2.0e-6, 8.0e-6, 2.0e-5, 1.0e-5, 3.0e-6])
partial_intensity = base_intensity[None, :] * np.array([1.0, 1.1, 0.9, 1.05])[:, None]
partial_intensity[:, 0] = [2.0e-6, -3.0e-6, -4.0e-6, -1.0e-6]   # net-negative band

result = ph.sound_power_intensity_precision(partial_intensity, areas, frequencies=freqs)
print(result.not_applicable_band.tolist())   # [True, False, False, False, False]
print(round(result.sound_power_level_a, 1))   # 80.6
result.plot()   # LW spectrum; the not-applicable band is hatched (needs matplotlib)
```

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/intensity_scan_power_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/intensity_scan_power.svg" alt="The precision intensity-scanning sound power level spectrum over five one-third-octave bands, four determinate bars and a hatched, greyed 250 Hz band flagged not-applicable because its net intensity is negative, with the A-weighted total of 80.6 dB(A) in the title" width="88%"></picture>

*The 250 Hz band nets negative (more energy flowing in than out), so ISO 9614-3
declares it not-applicable — the figure hatches and greys it while the four
determinate bands and the A-weighted total stand.*

<details>
<summary>Show the code for this figure</summary>

```python
import matplotlib.pyplot as plt
import numpy as np

# result is the PrecisionIntensityResult computed above. One line:
result.plot()
plt.show()

# By hand: determinate bands as LW bars; a not-applicable band (its LW is NaN)
# is flagged by a full-height greyed, hatched span rather than a zero-height bar.
freqs = result.frequencies
positions = np.arange(freqs.size)
neg = result.not_applicable_band
lw = np.nan_to_num(result.sound_power_level)
fig, ax = plt.subplots()
ax.bar(positions[~neg], lw[~neg], width=0.7, color="#1f77b4")
for pos in positions[neg]:
    ax.axvspan(pos - 0.35, pos + 0.35, facecolor="#888888", alpha=0.28,
               hatch="//", edgecolor="#888888")
ax.set_xticks(positions)
ax.set_xticklabels([f"{f:g}" for f in freqs], rotation=45, ha="right")
ax.set_xlabel("Frequency [Hz]")
ax.set_ylabel("Sound power level LW [dB]")
ax.set_title(
    f"Precision intensity scanning (ISO 9614-3)  "
    f"LWA = {result.sound_power_level_a:.1f} dB(A)")
plt.show()
```

</details>

## See also

- [Sound Intensity (p-p)](https://jmrplens.github.io/phonometry/guides/intensity/) — the two-microphone probe, its
  finite-difference bias and the ISO 9614-1 field indicators behind the
  scanning method.
- [Room Acoustics](https://jmrplens.github.io/phonometry/guides/room-acoustics/) — the reverberation time
  and equivalent absorption area (ISO 354) that feed `K2` and the ISO 3741
  absorption area; impact and airborne insulation.
- [Levels](https://jmrplens.github.io/phonometry/guides/levels/) — energy averaging and the A-weighting behind `LWA`.
- [Theory](https://jmrplens.github.io/phonometry/reference/theory/) — the Waterhouse, K1/K2 and C1/C2 derivations.

---

**Standards.** ISO 3744:2010, *Acoustics — Determination of sound power levels
and sound energy levels of noise sources using sound pressure — Engineering
methods for an essentially free field over a reflecting plane* — the
enveloping-surface method: hemisphere and box surface areas, the `K1`/`K2`
corrections, the Annex B microphone positions and the Annex E A-weighting.
ISO 3746:2010, *… Survey method using an enveloping measurement surface over a
reflecting plane* — the survey grade sharing the same formulae with coarser
criteria. ISO 3741:2010, *… Precision methods for reverberation test rooms* —
the direct (Eq. 20) and comparison methods with the Waterhouse and
meteorological corrections and the Table 1 qualification criteria.
ISO 3745:2012, *… Precision methods for anechoic rooms and hemi-anechoic
rooms* — the Clause 8 power level, the per-position background correction
(Eq. 11), the meteorological corrections and the standardized microphone
arrays. ISO 9614-2:1996, *Acoustics — Determination of sound power levels of
noise sources using sound intensity — Part 2: Measurement by scanning* — the
partial powers, the `FpI` and `F+/-` field indicators and the grade criteria.
ISO 9614-3:2002, *… Part 3: Precision method for measurement by scanning* —
the grade-1 scanning method, its field indicators and the clause 9.2
not-applicable flagging.

---


<!-- source: docs/calibration.md | canonical: https://jmrplens.github.io/phonometry/guides/calibration/ -->

# Calibration and dBFS

phonometry can return results in physical **Sound Pressure Level (dB SPL)** or
digital **decibels relative to Full Scale (dBFS)**.

## Why calibrate? The theory

A digital recording only knows *numbers*: a full-scale sine wave is ±1.0
regardless of whether it was a whisper or a jet engine. To report physical
sound pressure levels the chain microphone → preamplifier → ADC must be
characterized by a single number, the **sensitivity factor** $S$, that
converts digital units into pascals:

$$
p(t) = S\ x(t) \qquad S = \frac{p_\text{ref}\cdot 10^{L_\text{cal}/20}}{\tilde{x}_\text{ref}}
$$

where $L_\text{cal}$ is the calibrator's level (typically 94 dB, i.e. 1 Pa),
$p_\text{ref} = 20\ \mu\text{Pa}$ and $\tilde{x}_\text{ref}$ is the RMS of
the recorded calibration tone in digital units. `sensitivity()` is
exactly that equation. The factor is valid as long as nothing in the chain
changes — touch the gain knob and you must recalibrate.

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/diagram_calibration_setup_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/diagram_calibration_setup.svg" alt="Calibration chain: sound calibrator coupled on the microphone, preamplifier, ADC and sensitivity producing pascals per digital unit" width="92%"></picture>

## Physical Calibration (Sound Level Meter)

```mermaid
flowchart LR
    A["Calibrator tone\n94 dB @ 1 kHz\n(IEC 60942)"] --> B["Recording\ncalibrator_recording"]
    B --> C["sensitivity()"]
    C --> D["calibration_factor\n(digital units → Pa)"]
    D --> E["octave_filter / leq / laeq / ln_levels"]
    F["Measurement\nrecording"] --> E
    E --> G["Levels in dB SPL\n(re 20 µPa)"]
```

To get accurate SPL measurements from a digital recording, you must first
calculate the sensitivity of your measurement chain using a reference tone
(e.g., 94 dB @ 1 kHz).

```python
import numpy as np
from phonometry import octave_filter, sensitivity

# 1. Record your 94 dB calibrator signal (1 kHz, 1 Pa RMS = 94 dB SPL)
fs = 48000
# calibrator_recording: your recorded 1 kHz calibrator tone (1 Pa RMS = 94 dB SPL).
#   Synthesized here so the guide runs; in a real measurement, record your calibrator.
calibrator_recording = np.sqrt(2) * np.sin(2 * np.pi * 1000 * np.arange(fs) / fs)
# recording: the mic capture you want to calibrate, same input chain (Pa after calibration).
#   Synthesized here; in a real measurement this is your recorded signal.
recording = 0.2 * np.sin(2 * np.pi * 1000 * np.arange(fs) / fs)

# 2. Calculate sensitivity factor
calibration_factor = sensitivity(calibrator_recording, target_spl=94.0, fs=fs)

# 3. Apply calibration to your measurements
spl, freq = octave_filter(recording, fs, calibration_factor=calibration_factor)
# Now 'spl' values are in real-world dB SPL!
```

The same `calibration_factor` works across the whole library: `octave_filter`,
`OctaveFilterBank`, `leq`, `laeq` and `ln_levels`.

## Calibrator assumptions (IEC 60942)

`sensitivity` assumes the reference recording comes from an acoustic
calibrator as specified by **IEC 60942** (classes LS, 1 and 2):

- The default `target_spl=94.0` matches the common 94 dB @ 1 kHz calibrator
  output (the standard requires the principal level to be at least 90 dB re
  20 µPa; 94 dB and 114 dB are the usual choices).
- The resulting sensitivity inherits the calibrator's class tolerance — e.g.
  ±0.4 dB for a class 1 calibrator between 160 Hz and 1.25 kHz (IEC 60942
  Table 1) — plus the RMS estimation error of your recording.
- IEC 60942 specifies the generated level as a 20 s average: record a few
  seconds of *stable* tone (excluding handling noise at the start/end) for the
  RMS estimate to converge.

### Automatic stability validation

When you pass the sample rate (and `validate=True`, the default),
`sensitivity(ref, fs=fs)` checks the recording the way
IEC 60942:2017 checks the calibrator itself (5.3.3): the *short-term level
fluctuation* — the absolute difference between each of the maximum and minimum
F-time-weighted levels and the mean level — must not exceed the Table 2 class 1
limit for the calibrator's nominal frequency (0.07 dB at and above 160 Hz, relaxed
to 0.10 dB below 160 Hz and 0.20 dB at or below 63 Hz, where the F
time-weighting itself ripples). Pass `frequency=` to select the right row for non-1 kHz
calibrators. A `CalibrationWarning` flags badly coupled microphones or handling
noise before they silently corrupt every calibrated level. The recording must
be at least 2 s long (1 s for the F-integrator to settle plus 1 s of settled
envelope); shorter recordings get a warning instead of an unreliable verdict.
Without `fs` the check is skipped. Override the limit with
`max_fluctuation_db` or disable with `validate=False`.

The check catches exactly what ruins field calibrations — a loose coupler,
wind, handling noise:

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/calibration_stability_dark.png"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/calibration_stability.png" alt="F-weighted level of a stable calibration tone versus a 3 percent amplitude-modulated one against the plus-minus 0.07 dB IEC 60942 class 1 limit" width="80%"></picture>

### `sensitivity()` parameters

| Parameter | Type / shape | Units | Range / default | Notes |
| :--- | :--- | :--- | :--- | :--- |
| `ref_signal` | 1D/2D array | digital units | non-empty, non-silent | Recording of the calibration tone only (trim handling noise) |
| `target_spl` | float | dB re 20 µPa | default `94.0` | The calibrator's nominal level (114 dB calibrators: pass `114.0`) |
| `ref_pressure` | float | Pa | default `2e-5` | Reference pressure p₀; rarely changed |
| `fs` | int, optional | Hz | > 0; default `None` | Required for the stability validation; omit to skip it |
| `validate` | bool | — | default `True` | Emit `CalibrationWarning` on unstable/short recordings |
| `max_fluctuation_db` | float, optional | dB | default `None` → Table 2 class 1 | Explicit override of the stability limit |
| `frequency` | float | Hz | default `1000.0` | Calibrator's nominal frequency; selects the IEC 60942 Table 2 row |
| `narrowband` | bool | — | default `False` | Estimate the tone with a coherent Goertzel detector near `frequency` (needs `fs`) instead of full-band RMS; rejects broadband hum/noise that otherwise inflates the RMS and shrinks every later level (~−0.44 dB at 20 dB SNR). Enable for noisy coupler recordings |

Returns the sensitivity factor (float) to pass as `calibration_factor=` to
`octave_filter`, `leq`, `laeq`, `ln_levels`, `lc_peak`, `sel` and the dose
functions.

## Digital Analysis (dBFS)

If you are working with digital audio files (e.g., WAV, FLAC) and want to
analyze levels relative to Full Scale rather than physical pressure, you can use
the `dbfs=True` parameter.

In this mode:

* **0 dBFS** corresponds to a numeric signal level of 1.0 (RMS or Peak).
* `calibration_factor` does not apply (dBFS is relative to digital full scale).
* Useful for analyzing headroom, digital mastering, or normalized signals.

```python
# Assume 'recording' is normalized between -1.0 and 1.0
spl_dbfs, freq = octave_filter(recording, fs, dbfs=True)
# Results will be negative (e.g., -20 dBFS)
```

## RMS vs Peak Levels

phonometry supports two measurement modes to align with professional software
like BK:

- **RMS (`mode='rms'`)**: Energy-based level (standard).
- **Peak (`mode='peak'`)**: Absolute maximum value reached in the frame
  (Peak-holding).

```python
# Measure peak-holding levels for impact analysis
spl_peak, freq = octave_filter(recording, fs, mode='peak')
```

> [!NOTE]
> `mode='peak'` measures the absolute maximum of the **filtered** band signal,
> which includes the filter's onset transient (overshoot). Signals that start
> abruptly may read up to ~1 dB high. This is inherent to IIR band filters
> (an analog SLM behaves the same way), not a processing artifact.

## Integer audio input

Integer signals (e.g. int16 from `scipy.io.wavfile.read`) are converted to
float64 internally before any squaring, so calibration and level results are
identical whether you pass the raw integer array or a float conversion.

---

**Standards.** IEC 60942:2017, *Electroacoustics — Sound calibrators* — the
calibrator level and class assumptions behind `sensitivity()` (the 94 dB
principal level and the Table 1 class tolerances) and the short-term
level-fluctuation stability check of the reference recording (5.3.3, Table 2
class 1 limits per nominal frequency).

---


<!-- source: docs/block-processing.md | canonical: https://jmrplens.github.io/phonometry/guides/block-processing/ -->

# Block Processing

Some measurements never fit in memory: an hour-long environmental recording,
a live monitor that must report levels while the microphone is still
capturing, or an embedded logger that only ever sees one buffer at a time. In
all of these the signal has to be processed block by block — and the filters
must behave exactly as if they had seen the whole signal at once.

The `OctaveFilterBank`, `WeightingFilter` (for A, C, or Z-weighting) and
`TimeWeighting` classes support block (streaming) processing: the internal
filter state is carried between calls, so concatenated block outputs match a
single full-signal pass.

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/diagram_block_processing_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/diagram_block_processing.svg" alt="Two lanes comparing block processing with the filter state carried across blocks, giving one continuous envelope, versus reset each block, where the envelope restarts from zero at every seam" width="86%"></picture>

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/block_processing_continuity_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/block_processing_continuity.svg" alt="Stateful block processing matching the continuous result versus independent blocks restarting the filter transient at each boundary" width="80%"></picture>

*With `stateful=True` the concatenated block outputs match the continuous
result exactly; without state, every block boundary restarts the filter
transient.*

Create a stateful filter bank with `stateful=True`. The internal state is
zero-initialized by default but may be initialized for step-response
steady-state (like `scipy.signal.sosfilt_zi`) with `steady_ic=True`. The
options that must be disabled in stateful mode are summarized in the
[constraints table](#stateful-mode-constraints) at the end of this guide.

## Example

This example streams a WAV file with [`soundfile`](https://pysoundfile.readthedocs.io/),
an optional dependency (`pip install soundfile`). Any block source works just as
well — `scipy.io.wavfile` plus manual slicing, or a live capture callback.

```python
import soundfile as sf
from phonometry import OctaveFilterBank, WeightingFilter

fs = 48000
octave_filter = OctaveFilterBank(fs, 1, stateful=True, resample=False)
afilter = WeightingFilter(fs, "A", stateful=True)

for block in sf.blocks("measurement.wav", blocksize=256, overlap=0):

    # Apply A-filter
    weighted = afilter.filter(block)

    # Split into octave bands
    block_spl, _, block_output = octave_filter.filter(weighted, sigbands=True, detrend=False)

    # further signal processing
    ...
```

## Time weighting across blocks

Use the `TimeWeighting` class (state carried automatically):

```python
from phonometry import TimeWeighting

tw = TimeWeighting(fs, mode="fast")
# audio_blocks: successive frames of your microphone recording (Pa),
#   e.g. from sf.blocks("measurement.wav", ...) as in the block above.
for block in audio_blocks:
    envelope = tw.process(block)
```

Or manage the state yourself with the functional API — see
[Time Weighting](https://jmrplens.github.io/phonometry/guides/time-weighting/#6-block-processing).

## Multichannel state

All stateful classes handle multichannel input `(channels, samples)`: the state
is allocated lazily on the first call to match the channel count, and reallocated
if the channel count changes (e.g. switching from stereo to mono resets the state).

## Real-time level meter pattern

The canonical streaming loop — weight, envelope and report block by block
with all state carried across calls:

```python
import numpy as np
from phonometry import TimeWeighting, WeightingFilter

fs, block = 48000, 4800  # 100 ms blocks
aw = WeightingFilter(fs, "A", stateful=True)
env = TimeWeighting(fs, mode="fast")  # the class is inherently stateful

for x in audio_stream(block):            # your capture callback
    y = env.process(aw.filter(x))
    spl = 10 * np.log10(y[..., -1] / (2e-5) ** 2)  # instantaneous LAF
    display(spl)
```

### Stateful-mode constraints

| Option | Stateful behavior | Why |
| :--- | :--- | :--- |
| `detrend` | must be `False` | Per-block detrending creates boundary discontinuities |
| `resample` | must be `False` | The resampler is not stateful |
| `zero_phase` | unsupported | Forward-backward filtering needs the whole signal |
| `high_accuracy` (weighting) | resolves to `False` by default — the legacy bilinear design, see [Frequency Weighting](https://jmrplens.github.io/phonometry/guides/weighting/); explicitly passing `True` raises `ValueError` | The polyphase resampling inside is block-incompatible |
| `steady_ic` | optional | Starts the filters in step-response steady state |

---


<!-- source: docs/multichannel.md | canonical: https://jmrplens.github.io/phonometry/guides/multichannel/ -->

# Multichannel and Performance

## Multichannel Support

phonometry natively supports multichannel signals (e.g., Stereo, 5.1,
Microphone Arrays) using **fully vectorized operations**. Input arrays of shape
`(N_channels, N_samples)` are processed in parallel, offering significant
performance gains over iterative loops.

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/signal_response_multichannel_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/signal_response_multichannel.svg" alt="Stereo analysis: pink noise and logarithmic sweep resolved per channel in one-third-octave bands" width="80%"></picture>

*Simultaneous analysis of a Stereo signal: Left Channel (Pink Noise) vs Right
Channel (Log Sine Sweep).*

The convention is consistent across the whole library: time is always the
**last axis**. This applies to `octave_filter`, `OctaveFilterBank`,
`weighting_filter`, `time_weighting`, `leq`, `laeq`, `ln_levels` and
`spectrogram`.

```python
import numpy as np
from phonometry import octave_filter

# Two calibrated channels in Pa so the guide runs standalone
fs = 48000
t = np.arange(fs) / fs
left = 0.2 * np.sin(2 * np.pi * 1000 * t)
right = 0.1 * np.sin(2 * np.pi * 500 * t)

stereo = np.stack([left, right])          # (2, n_samples)
spl, freq = octave_filter(stereo, fs, fraction=3)
# spl has shape (2, n_bands): one row per channel
```

## Accepted shapes, at a glance

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/diagram_multichannel_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/diagram_multichannel.svg" alt="Array-shape flow: a 1-D (samples,) input reduces to a scalar and a 2-D (channels, samples) input reduces to (channels,), because the operation runs along the last axis while the channel axis is preserved" width="80%"></picture>

| Input | Interpreted as | Typical output |
| :--- | :--- | :--- |
| `(n,)` 1D array | one channel | scalar level / `(bands,)` |
| `(ch, n)` 2D array | `ch` channels, `n` samples each | `(ch,)` levels / `(ch, bands)` |
| list of floats | one channel (converted) | as 1D |
| `(ch, n)` into `spectrogram` | multichannel STFT-style | `(ch, bands, frames)` |

Everything vectorizes across the leading channel axis — one filter design is
applied to all channels in a single SciPy call, so 8 channels cost far less
than 8 separate runs. Convention: **channels first**, like most DSP code
(`soundfile` returns `(n, ch)`: transpose with `x.T`).

## Performance: Vectorization and caching

The `OctaveFilterBank` class is highly optimized for real-time and batch
processing. It uses NumPy vectorization to handle multichannel audio arrays
(e.g., 64-channel microphone arrays) without explicit Python loops, ensuring
maximum throughput.

```python
from phonometry import OctaveFilterBank

bank = OctaveFilterBank(fs=48000, fraction=3, filter_type='butter')

# Access computed properties
# bank.freq (center), bank.freq_d (lower), bank.freq_u (upper), bank.sos (coefficients)

# Process multiple signals efficiently
stream = [stereo]                 # your sequence of multichannel frames
for frame in stream:
    # detrend=True (default) removes DC offset to improve low-freq accuracy
    spl, freq = bank.filter(frame, detrend=True)
```

Additional performance notes:

- **Design cache**: `octave_filter()` reuses filter bank designs across calls
  with identical parameters (LRU cache, 32 entries), so calling it in a loop
  does not redesign the bank each time. `OctaveFilterBank` gives you explicit
  control over the design lifetime.
- **Multirate decimation**: low-frequency bands are filtered at a decimated
  rate, which is both faster and numerically more stable (see
  [Theory](https://jmrplens.github.io/phonometry/reference/theory/)).
- **Optional numba**: the `impulse` time weighting kernel is JIT-compiled when
  numba is installed (`pip install phonometry[perf]`).

---


<!-- source: docs/api-reference.md | canonical: https://jmrplens.github.io/phonometry/reference/api/ -->

# API Reference

All core functionality can be imported directly from the `phonometry` package.

| Name | Type | Description (Inputs) | Usage Snippet (Outputs) |
| :--- | :--- | :--- | :--- |
| `octave_filter` | `function` | **High-level analysis.**<br>• `x`: Signal array<br>• `fs`: Sample rate [Hz]<br>• `fraction`: 1, 3, etc. (Default: 1)<br>• `order`: Filter order (Default: 6)<br>• `limits`: [f_min, f_max] (Default: [12, 20000])<br>• `filter_type`: 'butter', 'cheby1', 'cheby2', 'ellip', 'bessel' (Default: 'butter')<br>• `sigbands`: Return time signals (Default: False)<br>• `detrend`: Remove DC offset (Default: True)<br>• `calibration_factor`: Sensitivity multiplier (Default: 1.0)<br>• `dbfs`: Output in dBFS instead of dB SPL (Default: False)<br>• `mode`: 'rms' or 'peak' (Default: 'rms')<br>• `nominal`: IEC 61260-1 nominal labels (Default: False)<br>• `show`: Plot response (Default: False)<br>• `plot_file`: Path to save plot (Default: None)<br>• `ripple`: Passband ripple [dB] (for cheby1/ellip)<br>• `attenuation`: Stopband atten. [dB] (Default: 72; for cheby2/ellip — cheby2 needs >= 70 dB for class 1) | `spl, freq = octave_filter(x, fs, ...)`<br>• `spl`: levels [dB]<br>• `freq`: frequencies [Hz]<br><br>**With `sigbands=True`:**<br>`spl, freq, xb = octave_filter(x, fs, sigbands=True)`<br>• `xb`: List of filtered signals (one per band)<br><br>**Calibrated usage:**<br>`spl, f = octave_filter(x, fs, calibration_factor=0.05)` |
| `OctaveFilterBank` | `class` | **Efficient bank implementation.**<br>• `fs`: Sample rate [Hz]<br>• `fraction`: 1, 3, etc.<br>• `order`: Filter order<br>• `limits`: [f_min, f_max] (Default: [12, 20000])<br>• `filter_type`: Architecture name<br>• `show`: Plot response (Default: False)<br>• `plot_file`: Path to save the plot (Default: None)<br>• `calibration_factor`: Sensitivity multiplier<br>• `dbfs`: Use dBFS (Default: False)<br>• `stateful`: Carry filter state between calls (Default: False)<br>• `steady_ic`: Steady-state initial conditions (Default: False)<br>• `resample`: Multirate decimation (Default: True)<br>• `ripple`: Passband ripple [dB]<br>• `attenuation`: Stopband attenuation [dB] (Default: 72; cheby2 needs >= 70 dB for class 1) | `bank = OctaveFilterBank(fs=48000, fraction=3, order=6, filter_type='butter')`<br>`spl, f = bank.filter(x, sigbands=False, mode='rms', detrend=True, zero_phase=False)`<br><br>• `bank`: Instance of the filter bank<br>• `bank.freq` / `bank.freq_d` / `bank.freq_u` / `bank.sos`: computed properties |
| `OctaveFilterBank.spectrogram` | `method` | **Band levels over time.**<br>• `x`: Signal array (1D or 2D)<br>• `window_time`: Window length [s] (Default: 0.125)<br>• `overlap`: Fraction in [0, 1) (Default: 0.5)<br>• `mode`: 'rms' or 'peak'<br>• `zero_phase`: Group-delay-free frames (Default: False) | `levels, freq, times = bank.spectrogram(x)`<br><br>• `levels`: (bands, frames) or (channels, bands, frames)<br>• `times`: window centers [s] |
| `weighting_filter` | `function` | **Acoustic weighting.**<br>• `x`: Signal array<br>• `fs`: Sample rate [Hz]<br>• `curve`: 'A', 'C', 'G' (ISO 7196 infrasound) or 'Z' (Default: 'A')<br>• `high_accuracy`: IEC class 1 HF accuracy via internal oversampling (Default: True) | `y = weighting_filter(x, fs, curve='A')`<br><br>• `y`: weighted signal |
| `WeightingFilter` | `class` | **Reusable weighting filter.**<br>• `fs`: Sample rate [Hz]<br>• `curve`: 'A', 'C', 'G' or 'Z'<br>• `stateful`: Block processing (Default: False)<br>• `steady_ic`: Steady-state initial conditions (Default: False)<br>• `high_accuracy`: Defaults to True except stateful | `wf = WeightingFilter(fs, 'A')`<br>`y = wf.filter(x)` |
| `time_weighting` | `function` | **Energy capture.**<br>• `x`: Raw signal array (squared internally; time is the last axis)<br>• `fs`: Sample rate [Hz]<br>• `mode`: 'fast', 'slow', or 'impulse'<br>• `initial_state`: None, 'zero', 'first', scalar, or array (Default: None) | `env = time_weighting(x, fs, mode='fast')`<br><br>• `env`: energy envelope (Mean Square), same shape as `x` |
| `TimeWeighting` | `class` | **Stateful time weighting.**<br>• `fs`: Sample rate [Hz]<br>• `mode`: 'fast', 'slow', 'impulse' (Default: 'fast') | `tw = TimeWeighting(fs, mode='fast')`<br>`env = tw.process(block)` per block<br>`tw.reset()` to start over |
| `leq` | `function` | **Equivalent level (Leq).**<br>• `x`: Signal array (1D or 2D)<br>• `calibration_factor`: Sensitivity multiplier (Default: 1.0)<br>• `dbfs`: Output in dBFS (Default: False) | `level = leq(x, calibration_factor=s)`<br><br>• `level`: Scalar (1D) or per-channel array (2D) |
| `laeq` | `function` | **A-weighted Leq (LAeq).**<br>• `x`: Signal array (1D or 2D)<br>• `fs`: Sample rate [Hz]<br>• `calibration_factor` / `dbfs`: as `leq` | `level = laeq(x, fs, calibration_factor=s)`<br><br>• `level`: Scalar (1D) or per-channel array (2D) |
| `ln_levels` | `function` | **Statistical levels (LN).**<br>• `x`: Signal array (1D or 2D)<br>• `fs`: Sample rate [Hz]<br>• `n`: Exceedance percentiles (Default: (10, 50, 90))<br>• `mode`: 'fast', 'slow', 'impulse' (Default: 'fast')<br>• `weighting`: 'A', 'C', 'G', 'Z' or None (Default: None)<br>• `calibration_factor` / `dbfs`: as `leq` | `stats = ln_levels(x, fs, n=(10, 50, 90), weighting='A')`<br><br>• `stats`: Dict `{10: L10, 50: L50, 90: L90}` |
| `lc_peak` | `function` | **C-weighted peak (LCpeak).**<br>• `x`: Signal array (1D or 2D)<br>• `fs`: Sample rate [Hz]<br>• `calibration_factor`: as `leq`<br>• `dbfs`: 0 dBFS = full-scale *peak* (amplitude 1.0), unlike the RMS reference of `leq`<br>• `oversample`: HF inter-sample peak recovery (Default: 8; 1 = on-grid, under-reads up to ~1.15 dB at 48 kHz) | `peak = lc_peak(x, fs)`<br><br>• IEC 61672-1 §5.13; verified against Table 5 |
| `sel` | `function` | **Sound exposure level (SEL/LAE).**<br>• `x`: Whole-event signal (1D or 2D)<br>• `fs`: Sample rate [Hz]<br>• `weighting`: 'A', 'C', 'G', 'Z' or None (Default: None)<br>• `calibration_factor` / `dbfs`: as `leq` | `lae = sel(x, fs, weighting='A')`<br><br>• Event level normalized to 1 s |
| `sound_exposure` | `function` | **A-weighted exposure E (IEC 61252).**<br>• `x`: Signal (1D or 2D)<br>• `fs`: Sample rate [Hz]<br>• `duration_hours`: exposure period represented (Default: recording length)<br>• `calibration_factor` | `E = sound_exposure(x, fs, duration_hours=8)`<br><br>• `E`: Pa²·h |
| `lex_8h` | `function` | **Daily exposure LEX,8h / LEP,d.**<br>• Same params as `sound_exposure` | `lex = lex_8h(x, fs, duration_hours=4)`<br><br>• Normalized 8 h level [dB] |
| `loudness_zwicker` | `function` | **Zwicker loudness (ISO 532-1:2017).**<br>• `x`: Calibrated signal in Pa (1D)<br>• `fs`: Sample rate [Hz]<br>• `field`: 'free' or 'diffuse' (Default: 'free')<br>• `stationary`: Clause 5 stationary method (Default: False)<br>• `calibration_factor`: Digital units to Pa (Default: 1.0) | `res = loudness_zwicker(x, fs)`<br><br>• `ZwickerLoudness`; time-varying runs add N5/N10 and the 500 Hz N(t) trace |
| `loudness_zwicker_from_spectrum` | `function` | **Stationary loudness from band levels.**<br>• `levels`: 28 one-third-octave levels, 25 Hz-12.5 kHz [dB SPL]<br>• `field`: 'free' or 'diffuse' (Default: 'free') | `res = loudness_zwicker_from_spectrum(levels)`<br><br>• `res.loudness` [sone], `res.loudness_level` [phon] |
| `ZwickerLoudness` | `dataclass` | **Loudness result.**<br>• `loudness`: N [sone]<br>• `loudness_level`: LN [phon]<br>• `specific`: N′(z), 240 bins of 0.1 Bark<br>• `n5` / `n10`: percentile loudness (time-varying only)<br>• `time` / `loudness_vs_time`: 500 Hz trace | `res.loudness, res.n5`<br><br>• Time-varying fields are `None` for stationary results |
| `sharpness_din` | `function` | **Sharpness in acum (DIN 45692).**<br>• `x`: Signal (1D)<br>• `fs`: Sample rate [Hz]<br>• `field`: 'free' or 'diffuse' (Default: 'free')<br>• `method`: 'din', 'aures' or 'bismarck' (Default: 'din')<br>• `calibration_factor`: Digital units to Pa (Default: 1.0) | `s = sharpness_din(x, fs)`<br><br>• `s`: sharpness [acum] |
| `sharpness_din_from_specific` | `function` | **Sharpness from a specific-loudness pattern.**<br>• `specific`: N′(z), 240 values at 0.1 Bark (e.g. `ZwickerLoudness.specific`)<br>• `method`: 'din', 'aures' or 'bismarck' (Default: 'din') | `s = sharpness_din_from_specific(res.specific)`<br><br>• `s`: sharpness [acum] |
| `loudness_moore_glasberg` | `function` | **Moore-Glasberg loudness (ISO 532-2:2017).**<br>• `x`: Calibrated signal in Pa (1D)<br>• `fs`: Sample rate [Hz]<br>• `field`: 'free', 'diffuse' or 'eardrum' (Default: 'free')<br>• `presentation`: 'binaural'/'diotic' or 'monaural' (Default: 'binaural') | `res = loudness_moore_glasberg(x, fs)`<br><br>• `MooreGlasbergLoudness` (roex excitation pattern) |
| `loudness_moore_glasberg_from_spectrum` | `function` | **Exact loudness from sinusoidal components (ISO 532-2 §5.2/5.4).**<br>• `components`: sequence of (frequency [Hz], level [dB SPL]) pairs<br>• `field`: 'free'/'diffuse'/'eardrum' (Default: 'free')<br>• `presentation`: (Default: 'binaural') | `res = loudness_moore_glasberg_from_spectrum([(1000.0, 40.0)])`<br><br>• 1 kHz/40 dB → 1.000 sone |
| `loudness_moore_glasberg_from_third_octave` | `function` | **Loudness from 29 one-third-octave band levels (ISO 532-2 §5.5).**<br>• `band_levels`: 29 levels, 25 Hz–16 kHz [dB SPL]<br>• `field` / `presentation`: as above | `res = loudness_moore_glasberg_from_third_octave(levels)`<br><br>• `MooreGlasbergLoudness` |
| `MooreGlasbergLoudness` | `dataclass` | **Moore-Glasberg loudness result.**<br>• `loudness`: N [sone]<br>• `loudness_level`: LN [phon]<br>• `specific`: N′(i), 372 bins of 0.1 Cam<br>• `erb_number` / `centre_frequencies`: Cam / Hz axes<br>• `field` / `presentation`<br>• `.plot()`: specific loudness N′(i) over Cam | `res.loudness, res.loudness_level` |
| `loudness_moore_glasberg_time` | `function` | **Time-varying loudness (ISO 532-3:2023).**<br>• `signal`: Calibrated Pa; 1D (diotic) or (n, 2) ears<br>• `fs`: Sample rate [Hz]<br>• `field`: 'free'/'diffuse'/'eardrum' (Default: 'free')<br>• `presentation`: (Default: 'binaural')<br>• `percentiles`: exceedance % (Default: (1,5,10,50,90,95)) | `res = loudness_moore_glasberg_time(x, fs)`<br><br>• `MooreGlasbergTimeVaryingLoudness` |
| `MooreGlasbergTimeVaryingLoudness` | `dataclass` | **Time-varying loudness result.**<br>• `time`: 1 ms grid [s]<br>• `short_term_loudness` / `long_term_loudness`: S′(t)/S″(t) [sone]<br>• `short_term_loudness_level` / `long_term_loudness_level` [phon]<br>• `n_max` / `loudness_level_max`: peak long-term loudness<br>• `percentiles`: dict {percent: sone}<br>• `.plot()`: STL and LTL vs time | `res.n_max, res.percentiles[5.0]` |
| `loudness_ecma` | `function` | **Sottek Hearing Model loudness (ECMA-418-2:2025).**<br>• `signal_in`: Calibrated signal in Pa (1D)<br>• `fs`: Sample rate [Hz] (resampled to 48 kHz)<br>• `field`: 'free' or 'diffuse' (Default: 'free') | `res = loudness_ecma(x, fs)`<br><br>• `EcmaLoudness`; 1 kHz/40 dB ≈ 1 sone_HMS |
| `EcmaLoudness` | `dataclass` | **Sottek loudness result.**<br>• `loudness`: N [sone_HMS]<br>• `specific_loudness`: N′(z), 53 Bark_HMS bands<br>• `bark` / `centre_frequencies`: z / Hz axes<br>• `time` / `loudness_vs_time`: N(l) at 187.5 Hz<br>• `field`<br>• `.plot()`: N′(z) + N(l) | `res.loudness, res.specific_loudness` |
| `tonality_ecma` | `function` | **Sottek tonality (ECMA-418-2:2025).**<br>• `signal_in`: Calibrated signal in Pa (1D)<br>• `fs`: Sample rate [Hz] (resampled to 48 kHz)<br>• `field`: 'free' or 'diffuse' (Default: 'free')<br>• `f_low` / `f_high`: optional user band [Hz] (Default: None) | `res = tonality_ecma(x, fs)`<br><br>• `EcmaTonality`; 1 kHz/40 dB ≈ 1 tu_HMS |
| `EcmaTonality` | `dataclass` | **Sottek tonality result.**<br>• `tonality`: T [tu_HMS]<br>• `specific_tonality`: T′(z), 53 bands<br>• `tonal_frequencies`: f_ton,z per band [Hz]<br>• `bark` / `centre_frequencies`<br>• `time` / `tonality_vs_time` / `tonal_frequency_vs_time`: T(l), f_ton(l)<br>• `field`<br>• `.plot()`: T′(z) + T(l) | `res.tonality, res.tonal_frequencies` |
| `roughness_ecma` | `function` | **Sottek roughness (ECMA-418-2:2025), new capability.**<br>• `signal_in`: Calibrated signal in Pa (1D)<br>• `fs`: Sample rate [Hz] (resampled to 48 kHz)<br>• `field`: 'free' or 'diffuse' (Default: 'free') | `res = roughness_ecma(x, fs)`<br><br>• `EcmaRoughness`; 1 kHz/70 Hz/100 %-AM/60 dB ≈ 1 asper |
| `EcmaRoughness` | `dataclass` | **Sottek roughness result.**<br>• `roughness`: R [asper] (90th percentile of R(l50))<br>• `specific_roughness`: R′(z), 53 bands<br>• `bark` / `centre_frequencies`<br>• `time` / `roughness_vs_time`: R(l50) at 50 Hz<br>• `specific_roughness_vs_time`: (n_times, 53)<br>• `field`<br>• `.plot()`: R(l50) + specific-roughness heatmap | `res.roughness, res.specific_roughness` |
| `equal_loudness_contour` | `function` | **ISO 226:2023 equal-loudness contour.**<br>• `phon`: Loudness level, 20-90 phon | `freqs, spl = equal_loudness_contour(40.0)`<br><br>• SPL at the 29 preferred frequencies |
| `loudness_level` | `function` | **Loudness level of a pure tone (phon).**<br>• `spl`: Tone SPL [dB]<br>• `frequency`: One of the 29 Table 1 frequencies [Hz] | `phon = loudness_level(73.0, 63.0)` |
| `hearing_threshold` | `function` | **Threshold of hearing (ISO 226 Table 1).**<br>• (no parameters) | `freqs, tf = hearing_threshold()` |
| `tone_to_noise_ratio` | `function` | **Tone-to-noise ratio (ECMA-418-1 §11).**<br>• `x`: Signal (1D)<br>• `fs`: Sample rate [Hz]<br>• `tone_freq`: Tone to assess [Hz] (Default: highest peak)<br>• `resolution_hz`: FFT bin spacing (Default: 1.0) | `r = tone_to_noise_ratio(x, fs)`<br><br>• `ToneAssessment(frequency, ratio_db, criterion_db, prominent)` |
| `prominence_ratio` | `function` | **Prominence ratio (ECMA-418-1 §12).**<br>• Same params as `tone_to_noise_ratio` | `r = prominence_ratio(x, fs, tone_freq=1000)`<br><br>• `ToneAssessment(...)` |
| `ToneAssessment` | `dataclass` | **Tone prominence verdict.**<br>• `frequency`: Assessed tone [Hz]<br>• `ratio_db`: TNR or PR [dB]<br>• `criterion_db`: Prominence limit at that frequency [dB]<br>• `prominent`: bool | `r = tone_to_noise_ratio(x, fs)`<br>`if r.prominent: ...` |
| `sti_from_impulse_response` | `function` | **Full STI, indirect method (IEC 60268-16 Ed. 5).**<br>• `ir`: Impulse response (1D)<br>• `fs`: Sample rate [Hz] (>= 22500)<br>• `snr`: SNR [dB], scalar or 7 bands (Default: None)<br>• `level`: 7 speech band levels [dB SPL]; enables masking + reception threshold (Default: None)<br>• `ambient`: 7 noise band levels [dB SPL]; requires `level` | `res = sti_from_impulse_response(ir, fs, snr=25.0)`<br><br>• `STIResult` with `mtf` (7×14) |
| `stipa` | `function` | **STIPA direct method (Annex B).**<br>• `x`: Recorded STIPA signal (1D), 15-25 s<br>• `fs`: Sample rate [Hz] (>= 22500)<br>• `reference`: Measured source signal instead of the nominal m = 0.55 (Default: None)<br>• `level` / `ambient`: as `sti_from_impulse_response`<br>• warns (`UserWarning`) if the recording is < 15 s (STI biased low) | `res = stipa(recording, fs)`<br><br>• `STIResult` with `mtf` (7×2) |
| `stipa_signal` | `function` | **STIPA test-signal generator (A.4/A.6.1).**<br>• `fs`: Sample rate [Hz] (>= 22500)<br>• `seconds`: Duration [s] (Default: 18.0)<br>• `level_db`: RMS level [dB SPL] (Default: None → RMS 0.1)<br>• `seed`: Pink-noise seed (Default: None) | `sig = stipa_signal(48000, seconds=18.0)`<br><br>• 1D test signal, Ed. 5 male spectrum |
| `STIResult` | `dataclass` | **STI result.**<br>• `sti`: 0 to 1<br>• `mti`: band indices (7,)<br>• `mtf`: corrected m values (7×14 or 7×2)<br>• `band_levels`: levels used or None<br>• `rating`: Annex F letter 'A+'…'U' | `res = stipa(x, fs)`<br>`print(res.sti, res.rating)` |
| `sound_intensity` | `function` | **p-p sound intensity (IEC 61043).**<br>• `p1`, `p2`: Microphone signals [Pa], equal length<br>• `fs`: Sample rate [Hz]<br>• `spacing`: Microphone separation Δr [m]<br>• `rho`: Air density (Default: 1.204)<br>• `c`: Speed of sound (Default: 343.0)<br>• `fraction`: None, 1 or 3 (Default: None)<br>• `limits`: [f_min, f_max] (Default: [12, 20000])<br>• `bias_correct`: undo finite-difference under-read of band/broadband totals near `max_valid_frequency` (Default: False) | `res = sound_intensity(p1, p2, fs, spacing=0.012, fraction=3)`<br><br>• `IntensityResult` |
| `IntensityResult` | `dataclass` | **Intensity result.**<br>• Per band (with `fraction`): `frequency`, `intensity` [W/m²], `intensity_level`, `pressure_level`, `pressure_intensity_index`, `direction` (±1), `bias_correction`<br>• Broadband: `total_*` counterparts<br>• `max_valid_frequency`: 0.1·c/Δr | `res.total_intensity_level`<br>`res.total_direction` |
| `field_indicators` | `function` | **ISO 9614-1 Annex A field indicators.**<br>• `pressure_levels`: Lpi per position [dB]<br>• `normal_intensity`: Signed Ini per position [W/m²] | `fi = field_indicators(lp, i_n)`<br><br>• `FieldIndicators(f2, f3, f4)` |
| `FieldIndicators` | `dataclass` | **ISO 9614-1 Annex A indicators.**<br>• `f2`: surface pressure-intensity indicator (Eq. A.3)<br>• `f3`: negative partial power indicator (Eq. A.6)<br>• `f4`: field non-uniformity indicator (Eq. A.8)<br>• f3 − f2 > 0 reveals negative partial power | `fi.f2, fi.f3, fi.f4`<br><br>• Criteria: Ld > F2, N > C·F4² |
| `dynamic_capability_index` | `function` | **Dynamic capability Ld (ISO 9614-1 §3.12).**<br>• `pressure_residual_intensity_index`: δpI0 [dB]<br>• `bias_error_factor`: K [dB] (Default: 10.0) | `ld = dynamic_capability_index(18.0)`<br><br>• Adequate when `Ld > F2` (criterion 1) |
| `CalibrationWarning` | `warning class` | **Unreliable calibration recording.**<br>Emitted by `sensitivity` — with `fs` given and `validate=True` — when the tone is unstable (IEC 60942 limit) or too short | `warnings.simplefilter("error", CalibrationWarning)` |
| `lden` | `function` | **Day-evening-night level (ISO 1996-1 §3.6.4).**<br>• `lday`/`levening`/`lnight`: LAeq per period [dB]<br>• `hours`: Period durations (Default: (12, 4, 8)) | `l = lden(63.2, 58.1, 51.4)` |
| `ldn` | `function` | **Day-night level (ISO 1996-1 §3.6.5).**<br>• `lday`/`lnight`: LAeq per period [dB]<br>• `hours`: Default (15, 9) | `l = ldn(63.2, 51.4)` |
| `composite_rating_level` | `function` | **Whole-day composite rating (ISO 1996-1 §6.5).**<br>• `periods`: list of (level_db, hours, adjustment_db) summing 24 h | `r = composite_rating_level([(63, 12, 0), (58, 4, 5), (51, 8, 10)])` |
| `linkwitz_riley` | `function` | **Audio crossover.**<br>• `x`: Signal array<br>• `fs`: Sample rate [Hz]<br>• `freq`: Crossover frequency [Hz]<br>• `order`: Any even number (Default: 4) | `lo, hi = linkwitz_riley(x, fs, freq=1000, order=4)`<br><br>• `lo`: Low-pass filtered signal<br>• `hi`: High-pass filtered signal |
| `sensitivity` | `function`| **SPL Calibration.**<br>• `ref_signal`: Calibration signal<br>• `target_spl`: Level of calibrator (Default: 94.0)<br>• `ref_pressure`: Reference pressure (Default: 20e-6)<br>• `narrowband`: coherent Goertzel tone estimate that rejects broadband hum/noise (needs `fs`; Default: False) | `s = sensitivity(ref_signal, target_spl=94.0)`<br><br>• `s`: Float (multiplier for pressure) |
| `verify_filter_class` | `function` | **IEC 61260 class check.**<br>• `bank`: an `OctaveFilterBank`<br>• `num_points`: frequency grid points per band (Default: 32768)<br>• `edition`: `'2014'` (classes 1/2, default) or `'1995'` (IEC 61260:1995 / ANSI S1.11-2004, adds class 0) | `result = verify_filter_class(bank, edition="1995")`<br><br>• `result["overall_class"]`: 0/1/2 or None<br>• `result["bands"]`: per-band class and `margin_class<c>_db` [dB] |
| `class_limits` | `function` | **IEC 61260 Table 1 acceptance limits.**<br>• `fraction`: bandwidth designator denominator b (1, 3, ...)<br>• `filter_class`: 1 or 2 (`'2014'`); 0, 1 or 2 (`'1995'`)<br>• `omega`: normalized frequencies f/fm<br>• `edition`: `'2014'` (default) or `'1995'` | `lo, hi = class_limits(3, 0, omega, edition="1995")`<br><br>• `lo` / `hi`: min/max relative attenuation [dB] (`hi` is `inf` outside the pass-band) |
| `verify_weighting_class` | `function` | **IEC 61672-1:2013 Table 3 weighting check.**<br>• `wf`: a `WeightingFilter` (A, C or Z) | `result = verify_weighting_class(wf)`<br><br>• `result["overall_class"]`: 1, 2 or None<br>• `result["bands"]`: per-frequency class, deviation and margins [dB] |
| `weighting_class_limits` | `function` | **IEC 61672-1:2013 Table 3 acceptance limits.**<br>• `weighting_class`: 1 or 2 | `f, lo, hi = weighting_class_limits(1)`<br><br>• `f`: 34 nominal frequencies [Hz]<br>• `lo` / `hi`: lower/upper deviation limits [dB] (`lo` is `-inf` where one-sided) |
| `nominal_frequencies` | `function` | **ANSI Frequency generator.**<br>• `fraction`: 1, 3, etc. (Required)<br>• `limits`: [f_min, f_max] (Default: [12, 20000]) | `f_cen, f_low, f_high, labels = nominal_frequencies(fraction=3)`<br><br>• `f_cen`: List of center frequencies [Hz]<br>• `f_low`: List of lower edges [Hz]<br>• `f_high`: List of upper edges [Hz]<br>• `labels`: IEC nominal frequency labels |
| `normalized_frequencies` | `function` | **Standard IEC Frequencies.**<br>• `fraction`: 1 or 3 | `freqs = normalized_frequencies(fraction=3)`<br><br>• `freqs`: List of standard center frequencies [Hz] |
| `sweep_signal` | `function` | **ESS excitation (ISO 18233 Annex B).**<br>• `fs`: Sample rate [Hz]<br>• `f1`: Start frequency [Hz]<br>• `f2`: Stop frequency [Hz] (≤ fs/2)<br>• `seconds`: Duration [s]<br>• `amplitude`: Peak (Default: 1.0)<br>• `fade`: Half-Hann fraction (Default: 0.01) | `s = sweep_signal(48000, 20, 20000, 3.0)`<br><br>• 1D exponential sine sweep |
| `inverse_filter` | `function` | **Farina inverse filter for an ESS.**<br>• Same parameters as `sweep_signal` | `inv = inverse_filter(48000, 20, 20000, 3.0)`<br><br>• Time-reversed, +6 dB/oct compensated sweep |
| `impulse_response` | `function` | **Sweep deconvolution (ISO 18233 B.5).**<br>• `recorded`: Recorded response (1D)<br>• `reference`: Emitted sweep (1D)<br>• `fs`: Sample rate [Hz]<br>• `method`: 'spectral' (Default) or 'farina'<br>• `f_range`: (f1, f2) for 'farina'<br>• `regularization`: Tikhonov term (Default: 1e-6)<br>• `length`: causal samples (Default: len(recorded))<br>• `return_full`: keep distortion tail (Default: False) | `ir = impulse_response(rec, sweep, fs)`<br><br>• `ImpulseResponseResult` (array-like; `.plot()`) |
| `mls_signal` | `function` | **Maximum-length sequence (ISO 18233 Annex A).**<br>• `order`: Register length N, 2-20 | `mls = mls_signal(16)`<br><br>• Bipolar sequence, length 2**N − 1 |
| `mls_impulse_response` | `function` | **IR from a periodic MLS.**<br>• `recorded`: Response spanning whole MLS periods (1D)<br>• `mls`: Excitation sequence<br>• `length`: IR samples (Default: 2**N − 1)<br>• `fs`: optional sample rate for the plot time axis | `ir = mls_impulse_response(rec, mls)`<br><br>• `ImpulseResponseResult` (array-like; `.plot()`) |
| `ImpulseResponseResult` | `dataclass` | **Recovered impulse response with metadata.**<br>• `ir`: IR samples<br>• `fs` [Hz] or None (e.g. an MLS recovery without one)<br>• `method`: 'spectral'/'farina'/'mls'<br>• array-like: `np.asarray(res)`, indexing, `len()` and `shape`/`dtype` forward to `ir`<br>• `.plot()` | `ir = impulse_response(rec, sweep, fs)`<br>`room_parameters(ir, fs)  # drop-in array` |
| `ImpulseResponseWarning` | `warning class` | **Suspect recovered impulse response.**<br>Emitted e.g. for MLS aliasing in the recovery | `warnings.simplefilter('error', ImpulseResponseWarning)` |
| `plot_excitation` | `function` | **Plot an excitation signal.**<br>• `signal`: sweep or MLS samples (1D)<br>• `fs`: Sample rate [Hz]<br>• `kind`: `'sweep'` (default) or `'mls'` | `plot_excitation(sweep, fs, kind="sweep")`<br><br>• waveform + spectrogram / spectrum axes |
| `decay_curve` | `function` | **Schroeder decay curve (ISO 3382-1 5.3.3).**<br>• `ir`: Impulse response (1D)<br>• `fs`: Sample rate [Hz]<br>• `band`: Band centre [Hz] (Default: None → broadband)<br>• `fraction`: 1 or 3 (Default: 1)<br>• `zero_phase`: forward-backward band filtering, halves the 125 Hz short-T bias (Default: False) | `dc = decay_curve(ir, fs)` → `DecayCurve`<br>`time, level = dc` still unpacks (`time` [s], `level` [dB], 0 dB at t=0)<br>`dc.plot()` draws the decay + EDT/T20/T30 fits |
| `DecayCurve` | `dataclass` | **Schroeder decay curve (ISO 3382-1 5.3.3).**<br>• `time`: from the direct sound [s]<br>• `level`: decay [dB], 0 dB at t = 0, up to the noise truncation point<br>• `band`: centre [Hz] or None (broadband)<br>• iterable: unpacks as `time, level`<br>• `.plot()`: decay + EDT/T20/T30 fits | `dc = decay_curve(ir, fs)`<br>`dc.time, dc.level, dc.band` |
| `room_parameters` | `function` | **Room acoustics (ISO 3382-1/2).**<br>• `ir`: Impulse response (1D)<br>• `fs`: Sample rate [Hz]<br>• `limits`: (f_min, f_max) or None (Default: (125, 4000))<br>• `fraction`: 1 or 3 (Default: 1)<br>• `zero_phase`: forward-backward octave filtering, halves the 125 Hz short-T T30 bias (Default: False) | `res = room_parameters(ir, fs)`<br><br>• `RoomAcousticsResult` per band |
| `RoomAcousticsResult` | `dataclass` | **Per-band room parameters.**<br>• `frequency` [Hz]<br>• `edt`, `t20`, `t30` [s]<br>• `c50`, `c80` [dB], `d50`, `ts` [s]<br>• `dynamic_range` [dB]<br>• `edt_valid`/`t20_valid`/`t30_valid`<br>• `curvature` [%] | `res.t30, res.c80, res.t30_valid`<br><br>• Arrays with one entry per band |
| `open_plan_metrics` | `function` | **Open-plan office metrics (ISO 3382-3).**<br>• `positions_m`: Source distances [m] (≥ 4)<br>• `spl_a_speech`: A-weighted speech level per position [dB]<br>• `sti_values`: STI per position | `m = open_plan_metrics(r, lp, sti)`<br><br>• `OpenPlanResult` |
| `OpenPlanResult` | `dataclass` | **Open-plan single numbers.**<br>• `d2s`: Spatial decay rate [dB/doubling]<br>• `lp_as_4m`: Speech level at 4 m [dB]<br>• `rd`: Distraction distance [m]<br>• `rp`: Privacy distance [m] | `m.d2s, m.rd, m.rp`<br><br>• `nan` when undefined |
| `airborne_insulation` | `function` | **Field airborne insulation (ISO 16283-1).**<br>• `l1`/`l2`: Source/receiving levels [dB], 1D or (positions, bands)<br>• `t2`: Receiving-room T per band [s]<br>• `area`: Partition S [m²] (for R')<br>• `volume`: Receiving V [m³] (for R')<br>• `t0`: Reference T0 [s] (Default: 0.5) | `ins = airborne_insulation(l1, l2, t2, area=10, volume=50)`<br><br>• `AirborneInsulationResult` |
| `weighted_rating` | `function` | **Single-number rating + C/Ctr (ISO 717-1).**<br>• `values_by_band`: 16 thirds or 5 octaves [dB]<br>• `bands`: 'third-octave', 'octave' or None | `w = weighted_rating(R)`<br><br>• `WeightedRatingResult` (Rw, C, Ctr) |
| `energy_average_level` | `function` | **Energy-average level (ISO 16283-1 (9)).**<br>• `levels`: Levels [dB] to average<br>• `axis`: Averaging axis (Default: -1) | `L = energy_average_level([60, 66])`<br><br>• `10 lg( mean(10^(Li/10)) )` |
| `AirborneInsulationResult` | `dataclass` | **Field insulation per band.**<br>• `d`: Level difference D [dB]<br>• `dnt`: Standardized DnT [dB]<br>• `r_prime`: Apparent R' [dB] or None | `res.d, res.dnt, res.r_prime` |
| `WeightedRatingResult` | `dataclass` | **Weighted rating result.**<br>• `rating`: Rw/DnT,w … [dB], int<br>• `c`: Spectrum term C, int<br>• `ctr`: Spectrum term Ctr, int<br>• `unfavourable_sum`: [dB]<br>• `band_centers`: Measured-curve centres [Hz] or None<br>• `measured`: Measured band values [dB] or None<br>• `shifted_reference`: Shifted Table 3 reference [dB] or None | `w.rating, w.c, w.ctr` |
| `impact_insulation` | `function` | **Field impact insulation (ISO 16283-2).**<br>• `li`: Impact SPL [dB], 1D or (positions, bands)<br>• `t2`: Receiving-room T per band [s]<br>• `volume`: Receiving V [m³] (for L'n)<br>• `t0`: Reference T0 [s] (Default: 0.5) | `imp = impact_insulation(li, t2, volume=50)`<br><br>• `ImpactInsulationResult` (`l_n_t`, `l_n`) |
| `weighted_impact_rating` | `function` | **Single-number impact rating + CI (ISO 717-2).**<br>• `values_by_band`: 16 thirds (100-3150 Hz) or 5 octaves (125-2000 Hz) [dB]<br>• `bands`: 'third-octave', 'octave' or None | `r = weighted_impact_rating(imp.l_n_t)`<br><br>• `ImpactRatingResult` (Ln,w, CI); octave rating carries the -5 dB rule |
| `ImpactInsulationResult` | `dataclass` | **Impact insulation per band.**<br>• `l_n_t`: Standardized L'nT [dB]<br>• `l_n`: Normalized L'n [dB] or None | `imp.l_n_t, imp.l_n` |
| `ImpactRatingResult` | `dataclass` | **Weighted impact rating.**<br>• `rating`: Ln,w/L'n,w/L'nT,w [dB], int<br>• `ci`: Spectrum term CI, int<br>• `unfavourable_sum`: [dB]<br>• `band_centers`: Measured-curve centres [Hz] or None<br>• `measured`: Measured impact levels [dB] or None<br>• `shifted_reference`: Shifted impact reference [dB] or None | `r.rating, r.ci` |
| `facade_insulation` | `function` | **Field façade insulation (ISO 16283-3).**<br>• `l1_2m`/`l2`: Level 2 m in front / receiving levels [dB], 1D or (positions, bands)<br>• `t2`: Receiving-room T per band [s]<br>• `area`: Element S [m²], `volume`: Receiving V [m³], `surface_level`: L1,s [dB] (all three for R')<br>• `method`: 'loudspeaker' (−1.5 dB) / 'road_traffic' (−3 dB)<br>• `t0`: Reference T0 [s] (Default: 0.5)<br>• `frequencies` [Hz] | `fac = facade_insulation(l1_2m, l2, t2, volume=50, area=11.5, surface_level=ls)`<br><br>• `FacadeInsulationResult` |
| `FacadeInsulationResult` | `dataclass` | **Façade insulation per band.**<br>• `d_2m`: Level difference D2m [dB]<br>• `d_2m_nt`: Standardized D2m,nT [dB]<br>• `d_2m_n`: Normalized D2m,n [dB] or None<br>• `r_prime`: Apparent R'45°/R'tr,s [dB] or None<br>• `frequencies` [Hz] or None<br>• `.plot()` | `fac.d_2m_nt, fac.r_prime` |
| `lab_airborne_insulation` | `function` | **Laboratory airborne insulation (ISO 10140-2).**<br>• `l1`/`l2`: Source/receiving levels [dB], 1D or (positions, bands)<br>• `t2`: Receiving-room T per band [s]<br>• `area`: Free test-opening S [m²]<br>• `volume`: Receiving V [m³] | `lab = lab_airborne_insulation(l1, l2, t2, area=10, volume=50)`<br><br>• `LabAirborneInsulationResult` |
| `lab_impact_insulation` | `function` | **Laboratory impact insulation (ISO 10140-3).**<br>• `li`: Tapping-machine impact SPL [dB], 1D or (positions, bands)<br>• `t2`: Receiving-room T per band [s]<br>• `volume`: Receiving V [m³] | `imp = lab_impact_insulation(li, t2, volume=50)`<br><br>• `LabImpactInsulationResult` |
| `background_correction` | `function` | **Background-noise correction (ISO 10140-4 §4.3).**<br>• `signal_and_background`: Combined Lsb per band [dB]<br>• `background`: Lb per band [dB]<br>• 6–15 dB margin corrected, ≤6 dB capped at 1.3 dB, ≥15 dB unchanged | `L = background_correction(lsb, lb)`<br><br>• Corrected levels [dB] (`LabInsulationWarning` at the limit of measurement) |
| `LabAirborneInsulationResult` | `dataclass` | **Laboratory airborne result.**<br>• `r`: Sound reduction index R [dB]<br>• `absorption`: A = 0.16 V/T [m²]<br>• `rating`: `WeightedRatingResult` or None<br>• `.plot()` (needs the rating) | `lab.r, lab.rating.rating` |
| `LabImpactInsulationResult` | `dataclass` | **Laboratory impact result.**<br>• `l_n`: Normalized impact level Ln [dB]<br>• `absorption`: A [m²]<br>• `rating`: `ImpactRatingResult` or None<br>• `.plot()` (needs the rating) | `imp.l_n, imp.rating.rating` |
| `LabInsulationWarning` | `warning class` | **Limit-of-measurement condition (ISO 10140-4).**<br>Emitted by `background_correction` when a band's signal-to-background margin is ≤ 6 dB (fixed 1.3 dB cap applied) | `warnings.simplefilter('error', LabInsulationWarning)` |
| `predicted_airborne_insulation` | `function` | **Predicted apparent airborne R'w (EN 12354-1 Formula 26).**<br>• `r_direct`: Separating-element Rs,w [dB]<br>• `flanking_paths`: sequence of `FlankingPath` (Default: ())<br>• `delta_r_direct`: Lining ΔRDd,w [dB] (Default: 0) | `res = predicted_airborne_insulation(r_direct=57, flanking_paths=paths)`<br><br>• `AirbornePredictionResult` |
| `predicted_impact_insulation` | `function` | **Predicted apparent impact L'n,w (EN 12354-2 Formula 21).**<br>• `ln_w_eq`: Bare-floor equivalent Ln,w,eq [dB]<br>• `delta_l_w`: Covering improvement ΔLw [dB] (Default: 0)<br>• `k_correction`: Flanking K [dB] (Default: 0) | `imp = predicted_impact_insulation(ln_w_eq=76.2, delta_l_w=33, k_correction=2)`<br><br>• `ImpactPredictionResult` |
| `junction_vibration_reduction` | `function` | **Vibration reduction index Kij (EN 12354-1 Annex E).**<br>• `junction_type`: 'rigid_cross'/'rigid_t'/'flexible_t'/'lightweight_facade'<br>• `path`: 'through' (K13) / 'corner' (K12=K23)<br>• `mass_ratio`: m'⊥,i/m'i<br>• `frequency` [Hz] (Default: 500), `f1` [Hz] (Default: 125) | `k = junction_vibration_reduction('rigid_cross', 'through', 1.61)`<br><br>• Kij [dB] |
| `junction_min_vibration_reduction` | `function` | **Minimum Kij,min (EN 12354-1 Formula 29).**<br>• `coupling_length`: lf [m]<br>• `s_i`, `s_j`: Element areas [m²] | `kmin = junction_min_vibration_reduction(4.5, 11.5, 11.5)`<br><br>• Kij,min [dB] |
| `flanking_path` | `function` | **One flanking path Rij,w (EN 12354-1 Formula 28a).**<br>• `label`, `kind`: 'Ff'/'Df'/'Fd'<br>• `r_source`/`r_receive`: element indices [dB]<br>• `k_ij` [dB], `separating_area` Ss [m²], `coupling_length` lf [m]<br>• `delta_r` [dB] (Default: 0), `kij_min` [dB] clamp (Default: None) | `p = flanking_path(label='f', kind='Ff', r_source=49, r_receive=49, k_ij=12.4, separating_area=11.5, coupling_length=4.5)`<br><br>• `FlankingPath` |
| `flanking_element` | `function` | **The three paths (Ff, Df, Fd) of one flanking element.**<br>• `label`, `r_flanking`, `r_separating` [dB]<br>• `k_ff`/`k_fd`/`k_df` [dB]<br>• `separating_area` Ss [m²], `coupling_length` lf [m]<br>• `delta_r_ff`/`delta_r_fd`/`delta_r_df` [dB] (Default: 0) | `ff, df, fd = flanking_element(label='floor', r_flanking=49, r_separating=57, k_ff=12.4, k_fd=8.9, k_df=8.9, separating_area=11.5, coupling_length=4.5)` |
| `combine_linings` | `function` | **Combine two lining improvements (EN 12354-1 Formulas 30/31).**<br>• `delta_a`, `delta_b` [dB] (pass 0 for a single lining) | `dr = combine_linings(14.0, 14.0)`<br><br>• max(a,b) + min(a,b)/2 = 21.0 [dB] |
| `equivalent_impact_level` | `function` | **Bare-floor equivalent Ln,w,eq (EN 12354-2 Annex B).**<br>• `mass_per_area`: m' [kg/m²] | `lneq = equivalent_impact_level(322.0)`<br><br>• 164 − 35 lg(m') = 76.2 [dB] |
| `impact_flanking_correction` | `function` | **Flanking correction K (EN 12354-2 Table 1).**<br>• `separating_mass`, `flanking_mass` [kg/m²] (nearest tabulated) | `k = impact_flanking_correction(322.0, 145.0)`<br><br>• K = 2 [dB], int |
| `standardized_impact_level` | `function` | **Standardized L'nT,w (EN 12354-2 Formula 3).**<br>• `l_prime_n_w`: L'n,w [dB]<br>• `volume`: Receiving V [m³], V0 = 30 m³ | `lnt = standardized_impact_level(45.2, 50.0)`<br><br>• L'nT,w = 43.0 [dB] |
| `facade_sound_reduction` | `function` | **Predicted façade insulation D2m,nT (EN 12354-3 Formula 10/13).**<br>• `elements`: sequence of `FacadeElement`<br>• `area`: Façade S [m²], `volume`: Receiving V [m³]<br>• `delta_l_fs`: ΔLfs [dB] (Default: 0), `bands`<br>• `frequencies`: band centres [Hz] (Default: None; length must equal the band count) — carried on the result for plotting | `fac = facade_sound_reduction(elements, area=11.3, volume=50.0, bands="octave")`<br><br>• `FacadePredictionResult` |
| `radiated_sound_power` | `function` | **Radiated sound power LW (EN 12354-4 Formula 2/3).**<br>• `elements`, `lp_in`: Inside Lp,in [dB]<br>• `area`: Segment S [m²]<br>• `c_d`: Cd [dB] (Default: -6), `r_prime_cap`: R' cap [dB] (Default: 40), `octave_bands` | `seg = radiated_sound_power(elements, lp_in=lp, area=200.0, c_d=-5.0)`<br><br>• `RadiatedPowerResult` |
| `outdoor_attenuation` | `function` | **Finite radiating-side attenuation A'tot (EN 12354-4 Annex E).**<br>• `width`, `height`: Side dimensions [m]<br>• `distance`: Perpendicular d [m] | `a = outdoor_attenuation(60.0, 10.0, 5.0)`<br><br>• A'tot = 26.3 [dB] |
| `outdoor_level` | `function` | **Exterior level Lp (EN 12354-4 Formula E.1).**<br>• `l_w`: Radiated LW [dB], scalar or per-side<br>• `attenuation`: A'tot [dB], scalar or per-side | `lp = outdoor_level(62.9, 26.3)`<br><br>• Lp = 36.6 [dB] |
| `FacadeElement` | `dataclass` | **One façade transmission path (EN 12354-3/-4).**<br>• give exactly one of `r`: area element [dB] / `dn_e`: small element [dB] / `insertion_loss`: opening [dB]<br>• `area`: Sᵢ [m²] (for `r` / `insertion_loss`) | `FacadeElement("window", area=4.5, r=[23, 22, 30, 36, 37])` |
| `AirbornePredictionResult` | `dataclass` | **Predicted airborne insulation.**<br>• `r_prime_w`: Apparent R'w [dB]<br>• `r_direct_w`: Direct RDd,w [dB]<br>• `paths`: tuple of `PathContribution`<br>• `dominant`: highest-energy path | `res.r_prime_w, res.dominant.label` |
| `ImpactPredictionResult` | `dataclass` | **Predicted impact insulation.**<br>• `l_prime_n_w`: Apparent L'n,w [dB]<br>• `ln_w_eq`, `delta_l_w`, `k_correction` [dB] | `imp.l_prime_n_w` |
| `FlankingPath` | `dataclass` | **One flanking transmission path.**<br>• `label`, `kind`: 'Ff'/'Df'/'Fd'<br>• `r_ij_w`: Flanking index Rij,w [dB] | `p.r_ij_w` |
| `PathContribution` | `dataclass` | **A path with its energy share.**<br>• `label`, `kind`: 'Dd'/'Ff'/'Df'/'Fd'<br>• `r_w`: Path index [dB]<br>• `fraction`: share of transmitted energy (0–1) | `c.r_w, c.fraction` |
| `band_uncertainty` | `function` | **One-third-octave standard uncertainty u (ISO 12999-1 Tables 2/4/6).**<br>• `measurand`: 'airborne'/'impact'/'impact_reduction'<br>• `situation`: 'A'/'B'/'C'<br>• `upper_limit`: σR95 (airborne A, Annex D) (Default: False) | `u = band_uncertainty('airborne', 'B')`<br><br>• `BandUncertainty` |
| `single_number_uncertainty` | `function` | **Single-number standard uncertainty u (ISO 12999-1 Tables 3/5/7).**<br>• `quantity`: 'r_w'/'ln_w'/'delta_lw' (+ aliases, +c/+ctr variants)<br>• `situation`: 'A'/'B'/'C'<br>• `upper_limit` (Default: False) | `u = single_number_uncertainty('r_w', 'B')`<br><br>• u [dB] (0.9) |
| `single_number_uncertainty_uncorrelated` | `function` | **Uncorrelated single-number u from bands (ISO 12999-1 Formula B.2).**<br>• `band_uncertainties`: per-band u_i [dB]<br>• `reference_differences`: L_i − R_i [dB] | `u = single_number_uncertainty_uncorrelated(u_i, d_i)`<br><br>• Energy-weighted quadrature u [dB] |
| `maximum_repeatability_standard_deviation` | `function` | **Max repeatability σx per band (ISO 12999-1 Table 1).**<br>• (no parameters) | `b = maximum_repeatability_standard_deviation()`<br><br>• `BandUncertainty` (lab self-verification) |
| `insulation_coverage_factor` | `function` | **Coverage factor k (ISO 12999-1 Table 8).**<br>• `confidence`: fraction (Default: 0.95)<br>• `one_sided` (Default: False) | `k = insulation_coverage_factor(0.95)`<br><br>• 1.96 (two-sided) / 1.65 (one-sided) |
| `insulation_expanded_uncertainty` | `function` | **Expanded uncertainty U = k·u (ISO 12999-1 Formula 2).**<br>• `u` [dB]<br>• `coverage`: fraction (Default: 0.95)<br>• `one_sided` (Default: False); enforces k ≥ 1 | `U = insulation_expanded_uncertainty(0.9)`<br><br>• 1.764 [dB] |
| `uncertain_value` | `function` | **Attach U to a rating (ISO 12999-1 Clause 8).**<br>• `value` [dB], `quantity`, `situation`<br>• `coverage` (Default: 0.95), `one_sided` (Default: False), `upper_limit` (Default: False) | `uv = uncertain_value(52.0, 'rprime_w', 'B')`<br><br>• `UncertainValue` (value ± U) |
| `combine_uncertainties` | `function` | **Quadrature combination (ISO 12999-1 Formula C.2).**<br>• `*components`: non-negative u_i [dB] | `uc = combine_uncertainties(1.0, 0.6)`<br><br>• sqrt(Σ u_i²) = 1.166 [dB] |
| `prediction_input_uncertainty` | `function` | **Prediction input uncertainty (ISO 12999-1 Formula A.1).**<br>• `sigma_reproducibility`, `sigma_product` [dB]<br>• `n`: measurements (≥ 1) | `u = prediction_input_uncertainty(1.8, 1.0, 3)`<br><br>• sqrt((σR²+σp²)/n + σp²) [dB] |
| `reduce_by_independent_measurements` | `function` | **Reduce u by m measurements (ISO 12999-1 Formula A.7).**<br>• `u` [dB]<br>• `m`: independent measurements (≥ 1) | `ur = reduce_by_independent_measurements(1.0, 4)`<br><br>• u/√m = 0.5 [dB] |
| `satisfies_lower_requirement` | `function` | **Conformity to a minimum (ISO 12999-1 Formula 5).**<br>• `value`, `expanded_uncertainty_value`, `requirement` [dB] | `ok = satisfies_lower_requirement(52.0, 1.485, 50.0)`<br><br>• True when value − U > requirement |
| `satisfies_upper_requirement` | `function` | **Conformity to a maximum (ISO 12999-1 Formula 4).**<br>• `value`, `expanded_uncertainty_value`, `requirement` [dB] | `ok = satisfies_upper_requirement(45.0, 1.5, 50.0)`<br><br>• True when value + U < requirement |
| `BandUncertainty` | `dataclass` | **Per-band standard uncertainty (ISO 12999-1).**<br>• `measurand`, `situation`<br>• `frequencies` [Hz], `uncertainties` [dB]<br>• `upper_limit`: σR95 flag<br>• `.to_arrays()` | `b.frequencies, b.uncertainties` |
| `UncertainValue` | `dataclass` | **A value with its expanded uncertainty.**<br>• `value`, `standard_uncertainty`, `expanded_uncertainty` [dB]<br>• `coverage_factor`, `confidence`, `one_sided`<br>• `.lower` = y − U, `.upper` = y + U | `uv.lower, uv.upper` |
| `COVERAGE_FACTORS` | `mapping` | **Table 8 coverage factors (read-only).**<br>Keyed by `(confidence, one_sided)` → k | `COVERAGE_FACTORS[(0.95, False)]  # 1.96` |
| `sound_power_pressure` | `function` | **Sound power from surface pressure (ISO 3744/3746).**<br>• `levels_positions`: (NM, NB) SPL [dB]<br>• `surface`: 'hemisphere' / 'box'<br>• `radius` [m] or `dimensions`+`distance` [m]<br>• `reflecting_planes`: 1/2/3 (Default: 1)<br>• `background_levels`: for K1<br>• `frequencies` [Hz]: for LWA<br>• `reverberation_time`+`volume` / `absorption_area` / `mean_absorption_coefficient`+`room_surface`: for K2<br>• `grade`: 'engineering' (Default) / 'survey'<br>• `omc_uncertainty` [dB] (Default: 0) | `res = sound_power_pressure(levels, 'hemisphere', radius=1.5, frequencies=f)`<br><br>• `SoundPowerResult` |
| `measurement_positions` | `function` | **Hemisphere mic coordinates (ISO 3744 Annex B).**<br>• `surface`: 'hemisphere'<br>• `radius` [m]<br>• `reflecting_planes`: 1/2/3<br>• `tones`: Table B.1 vs B.2 (Default: True)<br>• `grade`: 'engineering'/'survey' | `xyz = measurement_positions('hemisphere', radius=1.5)`<br><br>• (N, 3) coordinates [m] |
| `background_noise_correction` | `function` | **Background correction K1 (ISO 3744 Eq. 16).**<br>• `source_levels` [dB]<br>• `background_levels` [dB]<br>• `grade`: 'engineering'/'survey' | `k1 = background_noise_correction(src, bg)`<br><br>• K1 per band [dB] |
| `environmental_correction` | `function` | **Environmental correction K2 (ISO 3744 Eq. A.2).**<br>• `surface_area` S [m²]<br>• `absorption_area` A, or `reverberation_time`+`volume`, or `mean_absorption_coefficient`+`room_surface` | `k2 = environmental_correction(14.1, reverberation_time=0.6, volume=300)`<br><br>• K2 [dB] |
| `SoundPowerResult` | `dataclass` | **Surface-pressure sound power.**<br>• `sound_power_level` [dB]<br>• `surface_pressure_level`, `mean_pressure_level` [dB]<br>• `background_correction`, `environmental_correction` [dB]<br>• `directivity_index` [dB] `(NM, NB)`<br>• `surface_area` [m²]<br>• `sound_power_level_a` [dB]<br>• `uncertainty` [dB]<br>• `grade` | `res.sound_power_level, res.sound_power_level_a` |
| `sound_power_reverberation` | `function` | **Reverberation-room LW, direct (ISO 3741).**<br>• `levels`: mean room SPL [dB] (1D or (NM, NB))<br>• `t60` [s]<br>• `volume` V [m³]<br>• `surface_area` S [m²]<br>• `frequencies` [Hz] (required)<br>• `background_levels`: for K1<br>• `temperature` [°C] (Default: 23)<br>• `static_pressure` [kPa] (Default: 101.325) | `rev = sound_power_reverberation(lp, t60, 200, 220, f)`<br><br>• `ReverberationSoundPowerResult` |
| `sound_power_comparison` | `function` | **Reverberation-room LW, comparison (ISO 3741).**<br>• `levels`, `levels_ref`: room SPL [dB]<br>• `lw_ref`: reference-source LW [dB]<br>• `frequencies` [Hz]<br>• `background_levels`, `background_levels_ref`<br>• `temperature`, `static_pressure` | `cmp = sound_power_comparison(lp, lp_ref, lw_ref)`<br><br>• `ReverberationSoundPowerResult` (method='comparison') |
| `ReverberationSoundPowerResult` | `dataclass` | **Reverberation-room sound power.**<br>• `frequencies` [Hz] (or `None`)<br>• `sound_power_level` [dB]<br>• `mean_pressure_level` [dB]<br>• `absorption_area` [m²]<br>• `waterhouse_correction` [dB]<br>• `background_correction`, `c1`, `c2` [dB]<br>• `speed_of_sound` [m/s]<br>• `sound_power_level_a` [dB]<br>• `method` | `rev.sound_power_level, rev.waterhouse_correction` |
| `sound_power_intensity` | `function` | **Sound power by intensity scanning (ISO 9614-2).**<br>• `normal_intensity`: (N_seg, N_bands) [W/m²]<br>• `areas`: segment Si [m²]<br>• `normal_intensity_2`: 2nd sweep (criterion 3)<br>• `pressure_levels`: for FpI<br>• `pressure_residual_index`: δpI0 [dB]<br>• `frequencies` [Hz]<br>• `band_type`: 'third' (Default)/'octave'<br>• `grade`: 'engineering'/'survey'<br>• `repeatability_limit`: override s | `ir = sound_power_intensity(scan1, areas, frequencies=f, band_type='octave')`<br><br>• `SoundPowerIntensityResult` |
| `SoundPowerIntensityResult` | `dataclass` | **Scanning sound power.**<br>• `partial_power`, `partial_power_level`<br>• `sound_power`, `sound_power_level` [dB] (NaN where `negative_band`)<br>• `negative_band`: per-band bool (True where P ≤ 0)<br>• `surface_pressure_intensity_index` (FpI)<br>• `negative_partial_power_index` (F+/-)<br>• `repeatability`, `dynamic_capability_index` (Ld)<br>• `achieved_grade`<br>• `surface_area`, `sound_power_level_a`, `grade` | `ir.sound_power_level, ir.achieved_grade` |
| `absorption_area` | `function` | **Equivalent absorption area (ISO 354 Eq. 5/7).**<br>• `t60`: T [s]<br>• `volume` V [m³]<br>• `temperature` [°C] (Default: 20)<br>• `speed_of_sound` [m/s] (overrides temperature)<br>• `m`: air attenuation [1/m] (Default: 0) | `A = absorption_area(t60, volume=200)`<br><br>• A [m²] |
| `absorption_coefficient` | `function` | **Sound absorption coefficient (ISO 354 Eq. 9).**<br>• `t1`, `t2`: empty / with-specimen T [s]<br>• `volume` V [m³]<br>• `sample_area` S [m²]<br>• `temperature1`/`temperature2` [°C]<br>• `speed_of_sound1`/`2`, `m1`/`m2` | `a = absorption_coefficient(t1, t2, 200, 10.8)`<br><br>• α_s (unclamped, may exceed 1) |
| `attenuation_from_alpha` | `function` | **ISO 9613-1 α → m (ISO 354 8.1.2.1).**<br>• `alpha`: attenuation [dB/m] | `m = attenuation_from_alpha(0.01)`<br><br>• m = α/(10 lg e) [1/m] |
| `air_attenuation` | `function` | **Atmospheric attenuation coefficient α (ISO 9613-1 Eq. 5).**<br>• `frequencies` [Hz]<br>• `temperature` [°C] (Default: 20)<br>• `relative_humidity` [%] (Default: 50)<br>• `pressure` [kPa] (Default: 101.325)<br>• `exact_midband`: snap to Table 1 midbands (Default: False) | `a = air_attenuation([1000, 4000], 20, 50)`<br><br>• α [dB/m] (×1000 = dB/km); out-of-range warns |
| `air_attenuation_m` | `function` | **ISO 354 power attenuation m from conditions (ISO 9613-1 + ISO 354).**<br>• Same parameters as `air_attenuation` | `m = air_attenuation_m([1000, 4000], 20, 50)`<br><br>• m = α/(10 lg e) [1/m] |
| `AtmosphericAbsorptionWarning` | `warning class` | **ISO 9613-1 out-of-range advisory.**<br>Emitted by `air_attenuation` when temperature/humidity/frequency leave the tabulated ranges or pressure exceeds 200 kPa; the result still returns | `warnings.simplefilter('error', AtmosphericAbsorptionWarning)` |
| `geometric_divergence` | `function` | **Geometrical divergence Adiv (ISO 9613-2 Eq. 7).**<br>• `distance` d [m] | `a = geometric_divergence(100.0)`<br><br>• 20 lg(d/d₀) + 11 = 51 dB at 100 m |
| `atmospheric_absorption` | `function` | **Atmospheric absorption term Aatm (ISO 9613-2 Eq. 8).**<br>• `distance` d [m]<br>• `frequencies` [Hz]<br>• `temperature`/`humidity`/`pressure` | `aatm = atmospheric_absorption(200, bands)`<br><br>• α·d per band [dB] |
| `ground_attenuation` | `function` | **Ground effect Agr, general method (ISO 9613-2 7.3.1, Eq. 9).**<br>• `distance` d, `source_height` hs, `receiver_height` hr [m]<br>• `frequencies` [Hz]<br>• `ground_source`/`ground_middle`/`ground_receiver` G ∈ [0,1]<br>• `projected_distance` dp [m] | `agr = ground_attenuation(200, 2, 2, bands, 1, 1, 1)`<br><br>• As+Ar+Am per band [dB] (negative = gain) |
| `ground_attenuation_alternative` | `function` | **Ground effect Agr, A-weighted method (ISO 9613-2 7.3.2, Eq. 10).**<br>• `distance` d [m]<br>• `mean_height` hm [m] | `agr = ground_attenuation_alternative(200, 2)`<br><br>• 4.8 − (2hm/d)·(17+300/d) ≥ 0 [dB] |
| `directivity_omega` | `function` | **Solid-angle directivity index DΩ (ISO 9613-2 Eq. 11).**<br>• `source_height` hs, `receiver_height` hr, `projected_distance` dp [m] | `dw = directivity_omega(1.5, 1.5, 200)`<br><br>• 0…~3 dB (add to Dc for the 7.3.2 method) |
| `Barrier` | `dataclass` | **Barrier diffraction geometry (ISO 9613-2 7.4).**<br>• `source_to_edge` dss, `edge_to_receiver` dsr [m]<br>• `parallel_distance` a [m] (Default: 0)<br>• `edge_separation` e [m] (Default: None → single diffraction)<br>• `ground_reflections_by_image` (Default: False → C₂=20)<br>• `lateral` (Default: False → top-edge) | `b = Barrier(source_to_edge=101, edge_to_receiver=101)`<br><br>• `.is_double` when e is given |
| `barrier_attenuation` | `function` | **Barrier diffraction Dz (ISO 9613-2 Eq. 14).**<br>• `barrier`: `Barrier`<br>• `distance` d [m]<br>• `frequencies` [Hz] | `dz = barrier_attenuation(b, 200, bands)`<br><br>• 10 lg[3+(C₂/λ)C₃ z Kmet], capped 20/25 dB |
| `meteorological_correction` | `function` | **Meteorological correction Cmet (ISO 9613-2 Eq. 21/22).**<br>• `projected_distance` dp, `source_height` hs, `receiver_height` hr [m]<br>• `c0`: factor C₀ [dB] | `c = meteorological_correction(200, 1.5, 1.5, 2.0)`<br><br>• C₀[1 − 10(hs+hr)/dp] ≥ 0 [dB] |
| `outdoor_propagation_attenuation` | `function` | **Total octave-band attenuation A (ISO 9613-2 Eq. 4).**<br>• `distance` d, `source_height` hs, `receiver_height` hr [m]<br>• `frequencies` [Hz]<br>• `ground_source`/`ground_middle`/`ground_receiver` G<br>• `barrier`: `Barrier` or None<br>• `temperature`/`humidity`/`pressure` (Default: 20 °C/70 %/101.325 kPa, a Table 2 reference atmosphere)<br>• `projected_distance` dp [m] | `att = outdoor_propagation_attenuation(200, 2, 2, bands, 1, 1, 1)`<br><br>• `OutdoorAttenuation` term breakdown |
| `predicted_receiver_level` | `function` | **Predicted octave-band receiver level (ISO 9613-2 Eq. 3/6).**<br>• `sound_power_level` Lw [dB]<br>• same geometry/ground/barrier/atmosphere params as `outdoor_propagation_attenuation`<br>• `directivity_index` Di, `d_omega` DΩ [dB]<br>• `c0`: subtract Cmet (Default: None) | `lp = predicted_receiver_level(lw, 200, 2, 2, bands)`<br><br>• LfT(DW) = Lw + Dc − A per band [dB] |
| `OutdoorAttenuation` | `dataclass` | **Per-band ISO 9613-2 attenuation breakdown.**<br>• `frequencies` [Hz]<br>• `a_div`, `a_atm`, `a_gr`, `a_bar` [dB]<br>• `a_total` = Adiv+Aatm+Agr+Abar [dB]<br>• `d_omega` [dB] | `att.a_div, att.a_bar, att.a_total` |
| `DEFAULT_FREQUENCIES` | `tuple` | **ISO 9613-2 nominal octave bands.**<br>(no parameters) | `DEFAULT_FREQUENCIES  # (63, …, 8000)` |
| `task_based_exposure` | `function` | **Task-based daily exposure LEX,8h + U (ISO 9612 Clause 9).**<br>• `tasks`: sequence of `Task`<br>• `instrument`: 'class1'/'class2'/'personal_exposimeter' (Default)<br>• `u3` [dB] (Default: 1.0)<br>• `include_duration_uncertainty` (Default: True)<br>• `warn` (Default: True) | `res = task_based_exposure(tasks)`<br><br>• `ExposureResult` with per-task breakdown |
| `job_based_exposure` | `function` | **Job-based daily exposure LEX,8h + U (ISO 9612 Clause 10).**<br>• `samples` Lp,A,eqT [dB]<br>• `effective_duration_hours` Te [h]<br>• `instrument`, `u3`<br>• `n_workers`/`sample_duration_hours`: Table 1 check | `res = job_based_exposure(samples, 7.5)`<br><br>• `ExposureResult` (Eq C.9 / Table C.4 budget) |
| `full_day_exposure` | `function` | **Full-day daily exposure LEX,8h + U (ISO 9612 Clause 11).**<br>• `samples` whole-day Lp,A,eqT [dB]<br>• `effective_duration_hours` Te [h]<br>• `instrument`, `u3`, `warn` | `res = full_day_exposure(samples, 9.25)`<br><br>• `ExposureResult`; 3 samples spanning ≥3 dB advise |
| `Task` | `dataclass` | **One task of a task-based measurement (ISO 9612 Clause 9).**<br>• `samples` [dB], `duration_hours` [h]<br>• `duration_samples`/`duration_range` (for u1b)<br>• `label`, `instrument` | `Task(samples=(80.1, 82.2), duration_hours=5.0)` |
| `ExposureResult` | `dataclass` | **Daily exposure and uncertainty (ISO 9612).**<br>• `lex_8h` [dB]<br>• `combined_standard_uncertainty` u [dB]<br>• `expanded_uncertainty` U = 1.65 u [dB]<br>• `upper_limit` = LEX,8h + U [dB]<br>• `strategy`, `sampling_advisory`, `tasks` | `res.lex_8h, res.upper_limit` |
| `TaskContribution` | `dataclass` | **Per-task breakdown inside `ExposureResult.tasks` (ISO 9612 Clause 9).**<br>• `label`, `lp_aeqt` (Eq. 7), `duration_hours`<br>• `lex_8h_contribution` (Eq. 8) [dB]<br>• uncertainty terms `u1a`, `u1b`, `c1a`, `c1b`, `u2`, `u3` | `res.tasks[0].lex_8h_contribution` |
| `table_c4_contribution` | `function` | **Sampling contribution c₁u₁ [dB] from Table C.4 (ISO 9612 job/full-day).**<br>• `n_samples`: N (clamped to [3, 30])<br>• `u1`: sampling standard uncertainty [dB]<br>• bilinear interpolation, anchored at u₁ = 0 | `table_c4_contribution(6, 2.0)  # 1.4` |
| `minimum_cumulative_duration_hours` | `function` | **Minimum cumulative measurement duration [h] (ISO 9612 Table 1).**<br>• `n_workers`: group size n_G (> 40 → 17 h; the standard advises splitting the group) | `minimum_cumulative_duration_hours(18)  # 10.75` |
| `COVERAGE_FACTOR` | `float` | **ISO 9612 coverage factor k = 1.65 (Clause 14).**<br>One-sided 95 % confidence: U = 1.65 u, upper limit LEX,8h + U | `COVERAGE_FACTOR  # 1.65` |
| `INSTRUMENT_U2` | `mapping` | **Instrument standard uncertainty u2 [dB] (ISO 9612 Table C.5).**<br>Keyed by `'class1'` (0.7), `'class2'` (1.5), `'personal_exposimeter'` (1.5) | `INSTRUMENT_U2['class1']  # 0.7` |
| `OccupationalExposureWarning` | `warning class` | **ISO 9612 sampling advisory.**<br>Emitted for the 3 dB task spread (Clause 9.3), c₁u₁ > 3.5 dB (Clause 10.4) or a short Table 1 cumulative duration | `warnings.simplefilter('error', OccupationalExposureWarning)` |
| `SoundPowerWarning` | `warning class` | **ISO 3744/3746/3741/9614-2 qualification issue.**<br>Emitted when the background margin is below the criterion, K2 exceeds the validity limit, a band's power is negative, or the room fails qualification; levels are then upper bounds | `warnings.simplefilter('error', SoundPowerWarning)` |
| `AbsorptionWarning` | `warning class` | **ISO 354 advisory.**<br>Emitted for a room below 150 m³, a sample area outside 10-12 m², an out-of-range temperature, or a non-physical α_s ≤ 0; the result still returns | `warnings.simplefilter('error', AbsorptionWarning)` |
| `speech_intelligibility_index` | `function` | **Speech Intelligibility Index (ANSI S3.5-1997, one-third-octave method).**<br>• `speech_spectrum`: 18 equivalent speech spectrum levels, 160 Hz–8 kHz [dB SPL], or a vocal-effort name 'normal'/'raised'/'loud'/'shout' (Table 3)<br>• `noise_spectrum`: equivalent noise spectrum levels [dB SPL] (Default: None → quiet, −80 dB)<br>• `threshold`: equivalent hearing threshold [dB HL] (Default: None → 0) | `res = speech_intelligibility_index("normal", noise_spectrum=[40.0]*18)`<br><br>• `SIIResult`; `res.sii` = 0.066 |
| `standard_speech_spectrum` | `function` | **Standard speech spectrum level by vocal effort (ANSI S3.5-1997 Table 3).**<br>• `vocal_effort`: 'normal', 'raised', 'loud' or 'shout' (Default: 'normal') | `u = standard_speech_spectrum('raised')`<br><br>• 18 band levels [dB SPL] |
| `SIIResult` | `dataclass` | **SII result.**<br>• `sii`: index in [0, 1]<br>• `band_audibility`: Ai per band (§5.8)<br>• `band_importance`: Ii (Table 3)<br>• `frequencies` [Hz]<br>• `speech_spectrum` / `disturbance` / `masking`: Ei′ / Di / Zi per band [dB]<br>• `.plot()` | `res.sii, res.band_audibility` |
| `noise_criterion` | `function` | **NC rating, tangency method (ANSI/ASA S12.2-2019).**<br>• `levels`: octave-band SPL [dB] (the 10 bands 16 Hz–8 kHz without `frequencies`)<br>• `frequencies`: optional band centres [Hz] (subset allowed) | `nc = noise_criterion(levels)`<br><br>• `NCResult`; highest NC curve touched |
| `room_criterion` | `function` | **RC Mark II rating (ANSI/ASA S12.2-2019 Annex D).**<br>• `levels`: octave-band SPL [dB]<br>• `frequencies`: optional band centres [Hz] | `rc = room_criterion(levels)`<br><br>• `RCResult`; LMF average + 'N'/'R'/'H'/'RH' tag |
| `nc_curve` | `function` | **NC curve levels (Table 1).**<br>• `index`: NC designation 15–70 (intermediate values interpolated band by band) | `nc_curve(35)  # [82, 71, 60, ...]`<br><br>• 10 band levels [dB], 16 Hz–8 kHz |
| `rc_curve` | `function` | **RC Mark II curve (Table D.1).**<br>• `index`: RC designation (value at 1000 Hz) | `rc_curve(35)`<br><br>• −5 dB/octave line, 10 bands [dB] |
| `NCResult` | `dataclass` | **NC rating result.**<br>• `rating`: NC value<br>• `governing_frequency`: band of the touch [Hz]<br>• `frequencies` [Hz], `levels` [dB]<br>• `.plot()` | `nc.rating, nc.governing_frequency` |
| `RCResult` | `dataclass` | **RC Mark II result.**<br>• `rating`: LMF rounded [dB], int<br>• `lmf`: 500/1000/2000 Hz average [dB]<br>• `classification`: 'N'/'R'/'H'/'RH'<br>• `reference_curve` [dB], `frequencies` [Hz], `levels` [dB]<br>• `.plot()` | `rc.rating, rc.classification` |
| `age_threshold` | `function` | **Age-related hearing threshold distribution (ISO 7029:2017).**<br>• `age`: years (≥ 18)<br>• `sex`: 'male'/'female' (Default: 'male')<br>• `fractile`: population fractile in (0, 1) (Default: 0.5)<br>• `frequencies`: subset of the 11 audiometric frequencies [Hz] (Default: None → 125–8000 Hz) | `at = age_threshold(60, 'male')`<br><br>• `AgeThresholdResult`; median at 4 kHz = 20.2 dB |
| `reference_threshold` | `function` | **Reference threshold of hearing, 0 dB HL (ISO 389-7:2006 Table 1).**<br>• `field`: 'free-field' or 'diffuse-field' (Default: 'free-field')<br>• `frequencies`: optional subset [Hz] | `t = reference_threshold()`<br><br>• dB SPL per audiometric frequency (1 kHz → 2.4) |
| `AgeThresholdResult` | `dataclass` | **Age threshold distribution.**<br>• `age`, `sex`, `fractile`<br>• `frequencies` [Hz]<br>• `median`: deviation from age 18 [dB]<br>• `spread_upper` / `spread_lower`: half-Gaussian su/sl [dB]<br>• `threshold`: value at `fractile` [dB]<br>• `.plot()` | `at.median, at.threshold` |
| `nipts` | `function` | **Noise-induced permanent threshold shift (ISO 1999:2013 §6.3).**<br>• `l_ex`: LEX,8h [dB]<br>• `years`: exposure duration (10–40 established; 1–10 extrapolated by Formula 3)<br>• `fractile`: (0, 1) (Default: 0.5)<br>• `frequencies`: subset [Hz] (Default: None → 500–6000 Hz) | `n = nipts(95.0, 20.0)`<br><br>• `NiptsResult`; N50 at 4 kHz = 23.0 dB |
| `htlan` | `function` | **Threshold associated with age and noise, H′ = H + N − HN/120 (ISO 1999 Formula 1).**<br>• `age`: years (≥ 18), `sex`: 'male'/'female'<br>• `l_ex`: LEX,8h [dB], `years`<br>• `fractile`: (0, 1) applied to both components (Default: 0.5)<br>• `frequencies` | `h = htlan(60, 'male', 95.0, 20.0)`<br><br>• `HtlanResult`; H′ at 4 kHz = 39.3 dB |
| `NiptsResult` | `dataclass` | **NIPTS distribution.**<br>• `l_ex`, `years`, `fractile`<br>• `frequencies` [Hz]<br>• `median`: N50 (Formula 2/3) [dB]<br>• `value`: NIPTS at `fractile` (Formula 4/5) [dB]<br>• `spread_upper` / `spread_lower`: du/dl [dB]<br>• `.plot()` | `n.median, n.value` |
| `HtlanResult` | `dataclass` | **Combined HTLAN result.**<br>• `age`, `sex`, `l_ex`, `years`, `fractile`<br>• `frequencies` [Hz]<br>• `htla`: age component H [dB]<br>• `nipts`: noise component N [dB]<br>• `threshold`: combined H′ [dB]<br>• `.plot()` | `h.htla, h.nipts, h.threshold` |
| `predicted_prominence` | `function` | **Predicted prominence P of an impulse (NT ACOU 112 Formula 1).**<br>• `onset_rate`: slope of the A/F level onset [dB/s] (> 0; qualifies above 10 dB/s)<br>• `level_difference`: level rise over the onset [dB] (> 0) | `predicted_prominence(50.0, 20.0)  # 7.7`<br><br>• P = 3 lg(rate) + 2 lg(diff) |
| `impulse_adjustment` | `function` | **LAeq adjustment KI (Formula 2).**<br>• `prominence`: P | `impulse_adjustment(7.7)  # 4.86`<br><br>• 1.8(P − 5) dB for P > 5, else 0 |
| `impulse_prominence` | `function` | **Governing prominence and adjustment of a set of impulses (clauses 7–8).**<br>• `onset_rates` [dB/s]<br>• `level_differences` [dB] | `r = impulse_prominence([50, 120], [20, 15])`<br><br>• `ImpulseProminenceResult` (P = 8.59, KI = 6.46 dB) |
| `rating_level` | `function` | **Rating level LAr,T over a reference time (clause 8 Note 1).**<br>• `laeq`: LAeq,N per sub-interval [dB]<br>• `adjustment`: KI,N per sub-interval [dB]<br>• `durations`, `reference_time`: same time unit | `rating_level([55, 60], [0, 4.9], [20, 10], 30)  # 60.9` |
| `ImpulseProminenceResult` | `dataclass` | **Impulse-prominence verdict.**<br>• `onset_rates` [dB/s], `level_differences` [dB]<br>• `per_impulse`: P of each impulse<br>• `prominence`: governing (highest) P<br>• `adjustment`: KI of the governing impulse [dB]<br>• `.plot()` | `r.prominence, r.adjustment` |
| `multiple_shock_assessment` | `function` | **Full multiple-shock assessment from seat acceleration (ISO 2631-5:2018).**<br>• `acceleration`: vertical seat az(t) [m/s²], `fs` [Hz]<br>• `start_age`: years, `years`: exposure years, `days_per_year`<br>• `exposure_time` / `measurement_time`: scale Dz to a daily dose (Default: None)<br>• `sex`: 'male'/'female' (Default: 'male')<br>• `mz`: MPa per m/s² (Default: sex-specific) | `res = multiple_shock_assessment(az, fs, start_age=25, years=30, days_per_year=220)`<br><br>• `MultipleShockResult` |
| `seat_to_spine_transfer` | `function` | **Seat-to-spine transfer function H(ω) (clause 5.2 Formula 1).**<br>• `frequencies` [Hz] | `H = seat_to_spine_transfer(freqs)`<br><br>• Complex response; unity at 0 Hz |
| `spinal_response` | `function` | **Vertical spinal response Az(t) (Formula 2).**<br>• `acceleration`: seat az(t) [m/s²]<br>• `fs` [Hz] | `Az = spinal_response(az, fs)`<br><br>• Same length as the input [m/s²] |
| `response_peaks` | `function` | **Positive response peaks Az,i (clause 5.3).**<br>• `response`: Az(t) | `peaks = response_peaks(Az)`<br><br>• Maxima between zero crossings [m/s²] |
| `dose_from_peaks` | `function` | **Acceleration dose Dz from peaks (Formula 3).**<br>• `peaks`: Az,i [m/s²] | `dz = dose_from_peaks(peaks)`<br><br>• Dz = 1.07 (Σ Az,i⁶)^(1/6) [m/s²] |
| `acceleration_dose` | `function` | **Acceleration dose Dz from a time history (Formulas 2 + 3).**<br>• `acceleration`: seat az(t) [m/s²], `fs` [Hz] | `dz = acceleration_dose(az, fs)`<br><br>• Dz [m/s²] |
| `daily_dose` | `function` | **Daily dose Dzd (Formula 4).**<br>• `dose`: Dz [m/s²]<br>• `exposure_time` td / `measurement_time` tm: same unit | `daily_dose(4.0, 8.0, 2.0)  # 5.04`<br><br>• Dz (td/tm)^(1/6) [m/s²] |
| `daily_dose_multi` | `function` | **Daily dose from several exposure conditions (Formula 5).**<br>• `doses`: Dz,j [m/s²]<br>• `exposure_times` / `measurement_times`: per condition | `dzd = daily_dose_multi(dz, td, tm)`<br><br>• [Σ Dz,j⁶ (td,j/tm,j)]^(1/6) [m/s²] |
| `compression_dose` | `function` | **Daily compressive stress Sd (Annex C Formula C.1).**<br>• `daily_dose_value`: Dzd [m/s²]<br>• `mz`: MPa per m/s² (Default: 0.029, 82 kg male) | `compression_dose(5.0)  # 0.145`<br><br>• Sd = mz·Dzd [MPa] |
| `static_stress` | `function` | **Static compressive stress Sstat = mz·9.81 (Annex C).**<br>• `mz` (Default: 0.029) | `static_stress()  # 0.284`<br><br>• [MPa] |
| `ultimate_strength` | `function` | **Ultimate lumbar strength Su at an age (Formula C.4).**<br>• `age`: years<br>• `sex`: 'male'/'female' (Default: 'male') | `ultimate_strength(45.0)  # 4.41`<br><br>• 6.75 − Sage·age [MPa] |
| `injury_risk` | `function` | **Cumulative injury stress variable R (Formula C.3).**<br>• `daily_compression`: Sd [MPa]<br>• `start_age`, `years`, `days_per_year`<br>• `sex`, `mz` | `R = injury_risk(0.5, start_age=25, years=30, days_per_year=220)  # 0.509` |
| `injury_probability` | `function` | **Probability of lumbar injury P(R) (Formula C.5, Weibull).**<br>• `risk`: R<br>• `sex` (Default: 'male') | `injury_probability(0.509)  # 0.0389`<br><br>• 1 − exp(−(R/α)^β) in 0–1 |
| `MultipleShockResult` | `dataclass` | **Multiple-shock health assessment.**<br>• `acceleration_dose` Dz, `daily_dose` Dzd [m/s²]<br>• `compression_dose` Sd [MPa]<br>• `risk` R, `probability` P(R)<br>• `sex`, `start_age`, `years`, `days_per_year`<br>• `peaks` [m/s²], `risk_thresholds`: R at 10/50/90 % (Table C.2)<br>• `.plot()` | `res.risk, res.probability` |
| `object_fraction` | `function` | **Object fraction ψ of an enclosed space (EN 12354-6 Formula 3).**<br>• `object_volumes` [m³]<br>• `volume`: empty space V [m³] | `object_fraction([12.0, 8.0], 500.0)  # 0.04` |
| `hard_object_absorption` | `function` | **Equivalent absorption area of a hard object (Formula 4).**<br>• `object_volume`: Vobj [m³] | `hard_object_absorption(8.0)  # 4.0`<br><br>• Aobj = Vobj^(2/3) [m²] |
| `air_absorption_area` | `function` | **Equivalent absorption area of the air (Formula 2).**<br>• `m`: power attenuation of air [Np/m]<br>• `volume` V [m³]<br>• `object_fraction` ψ (Default: 0.0) | `air_absorption_area(0.001, 500.0)  # 2.0`<br><br>• Aair = 4mV(1 − ψ) [m²] |
| `equivalent_absorption_area` | `function` | **Total equivalent sound absorption area (Formula 1).**<br>• `surfaces`: sequence of (area, absorption_coefficient) pairs (coefficient scalar or per band)<br>• `objects`: object areas Aobj [m²] (Default: ())<br>• `air_area`: Aair [m²] (Default: 0.0) | `A = equivalent_absorption_area([(100.0, 0.03), (50.0, 0.6)], objects=[4.0], air_area=2.0)  # 39.0` |
| `reverberation_time` | `function` | **Reverberation time from the absorption area (Formula 5).**<br>• `absorption_area` A [m²]<br>• `volume` V [m³]<br>• `object_fraction` ψ (Default: 0.0)<br>• `speed_of_sound` c0 [m/s] (Default: 345.6 → factor 0.16) | `reverberation_time(39.0, 500.0)  # 2.05`<br><br>• T = 55.3/c0 · V(1 − ψ)/A [s] |
| `enclosed_space_reverberation` | `function` | **Predicted A and T per octave band (EN 12354-6 Clause 4).**<br>• `surfaces`: (area, per-band α) pairs<br>• `volume` V [m³]<br>• `objects` [m²], `object_fraction` ψ<br>• `air_condition`: air-attenuation key, e.g. '20C_50-70' (Default: None → neglect air)<br>• `frequencies` [Hz] (Default: 125–8000 Hz octaves)<br>• `speed_of_sound` [m/s] | `res = enclosed_space_reverberation(surfaces, 500.0, air_condition="20C_50-70")`<br><br>• `ReverberationResult` |
| `ReverberationResult` | `dataclass` | **Enclosed-space absorption result.**<br>• `frequencies` [Hz]<br>• `absorption_area`: A per band [m²]<br>• `reverberation_time`: T per band [s]<br>• `volume` [m³], `object_fraction`<br>• `.plot()` | `res.absorption_area, res.reverberation_time` |
| `combine_uncertainty` | `function` | **GUM law of propagation of uncertainty (ISO/IEC Guide 98-3 clause 5).**<br>• `model`: measurement function f(x1…xN)<br>• `quantities`: input `Quantity` objects, in argument order<br>• `correlation`: optional N×N matrix r_ij (Default: None → uncorrelated) | `u = combine_uncertainty(lambda a, b: a*b, [Quantity(10.0, 0.1), Quantity(5.0, 0.2)])`<br><br>• `UncertaintyResult`; uc = 2.062 |
| `monte_carlo` | `function` | **Monte Carlo propagation (Guide 98-3-1, Supplement 1).**<br>• `model`: vectorised f(x1…xN)<br>• `quantities`: input `Quantity` objects<br>• `trials`: M (Default: 1000000)<br>• `coverage`: interval probability (Default: 0.95)<br>• `seed`: reproducibility (Default: None) | `mc = monte_carlo(model, quantities, seed=1)`<br><br>• `MonteCarloResult` (mean, u, coverage interval) |
| `rectangular` | `function` | **Type B quantity, rectangular PDF (GUM 4.3.7).**<br>• `value`, `half_width` a, `name` | `rectangular(20.0, 0.5)`<br><br>• `Quantity` with u = a/√3 = 0.289 |
| `triangular` | `function` | **Type B quantity, triangular PDF (GUM 4.3.9).**<br>• `value`, `half_width` a, `name` | `triangular(20.0, 0.5)`<br><br>• `Quantity` with u = a/√6 |
| `u_shaped` | `function` | **Type B quantity, U-shaped (arcsine) PDF.**<br>• `value`, `half_width` a, `name` | `u_shaped(20.0, 0.5)`<br><br>• `Quantity` with u = a/√2 |
| `Quantity` | `dataclass` | **Input quantity of a measurement model (GUM clause 4).**<br>• `value`: estimate xi<br>• `uncertainty`: u(xi) ≥ 0<br>• `distribution`: 'gaussian'/'rectangular'/'triangular'/'u-shaped' (Default: 'gaussian')<br>• `dof`: degrees of freedom (Default: inf)<br>• `name`: budget label | `Quantity(94.0, 0.3, name="calibrator")` |
| `UncertaintyResult` | `dataclass` | **GUM propagation result.**<br>• `value`: y = f(x1…xN)<br>• `combined_uncertainty`: uc(y)<br>• `sensitivities`: ci = ∂f/∂xi<br>• `contributions`: \|ci\|·u(xi)<br>• `effective_dof`: Welch–Satterthwaite<br>• `names`<br>• `.plot()`: uncertainty budget | `u.combined_uncertainty, u.contributions` |
| `MonteCarloResult` | `dataclass` | **Monte Carlo result.**<br>• `value`: sample mean<br>• `standard_uncertainty`: sample std<br>• `interval`: (low, high) coverage interval (§7.7)<br>• `coverage`, `trials` | `mc.value, mc.interval` |
| `sound_power_anechoic` | `function` | **Precision sound power in an (hemi-)anechoic room (ISO 3745:2012).**<br>• `levels_positions`: (NM, NB) position levels [dB]<br>• `surface`: 'sphere'/'hemisphere'<br>• `radius` r [m]<br>• `background_levels` + `frequencies`: for K1i<br>• `areas`: partial areas Si (Eq. 13) (Default: equal-area Eq. 12)<br>• `temperature` [°C] (Default: 23), `static_pressure` [kPa] (Default: 101.325)<br>• `air_absorption_coefficient`: α(f) [dB/m] for C3<br>• `sigma_omc` [dB] (Default: 0), `coverage_factor` k (Default: 2.0) | `res = sound_power_anechoic(levels, 'hemisphere', radius=2.0, frequencies=f)`<br><br>• `PrecisionSoundPowerResult` |
| `PrecisionSoundPowerResult` | `dataclass` | **ISO 3745 sound power result.**<br>• `sound_power_level` [dB], `sound_power_level_a` [dB]<br>• `surface_pressure_level`, `mean_pressure_level` [dB]<br>• `background_correction`: K1i (NM, NB) [dB]<br>• `c1`, `c2`, `c3`: meteorological corrections [dB]<br>• `directivity_index` DIi, `non_uniformity_index` VIr [dB]<br>• `surface_area` [m²], `surface`<br>• `uncertainty`, `uncertainty_bands`, `coverage_factor`<br>• `.plot()` | `res.sound_power_level, res.uncertainty` |
| `precision_positions` | `function` | **ISO 3745 microphone coordinates (Annex D/E).**<br>• `surface`: 'sphere' (Table D.1) / 'hemisphere' (Tables E.1/E.2)<br>• `radius` r [m]<br>• `array`: 'general'/'broadband' (Default: 'general')<br>• `count`: 20 or 40 (Default: 40) | `xyz = precision_positions('hemisphere', radius=2.0)`<br><br>• (40, 3) coordinates [m] |
| `precision_background_correction` | `function` | **Per-position background correction K1i (ISO 3745 Eq. 11).**<br>• `source_levels` / `background_levels`: (NM, NB) [dB]<br>• `frequencies` [Hz]: per-band criterion (10 dB for 250–5000 Hz, 6 dB outside) | `k1 = precision_background_correction(src, bg, f)`<br><br>• K1i [dB]; clamped + `SoundPowerWarning` below the criterion |
| `meteorological_corrections` | `function` | **Corrections C1, C2, C3 (ISO 3745 Eq. 14 block).**<br>• `temperature` θ [°C] (Default: 23), `static_pressure` ps [kPa] (Default: 101.325)<br>• `air_absorption_coefficient`: α(f) [dB/m] for C3 (Default: None → C3 = 0)<br>• `radius` r [m] (Default: 1.0) | `met = meteorological_corrections(23.0, 101.325)`<br><br>• `MeteorologicalCorrection`; C1 = −0.128 dB, C2 = 0 at reference |
| `MeteorologicalCorrection` | `dataclass` | **C1/C2/C3 corrections.**<br>• `c1`: reference-quantity correction [dB]<br>• `c2`: radiation-impedance correction [dB]<br>• `c3`: air-absorption correction [dB], scalar or per band | `met.c1, met.c2, met.c3` |
| `precision_uncertainty` | `function` | **Expanded uncertainty U = k·√(σR0² + σomc²) (ISO 3745 Eq. 24/25).**<br>• `sigma_r0`: reproducibility (Tables 2/3) [dB]<br>• `sigma_omc` [dB] (Default: 0.0)<br>• `coverage_factor` k (Default: 2.0; 1.6 one-sided) | `precision_uncertainty(0.5)  # 1.0`<br><br>• U [dB], scalar or per band |
| `sound_power_intensity_precision` | `function` | **Sound power by intensity scanning, precision (ISO 9614-3:2002).**<br>• `partial_intensity`: (N, NB) signed In,i [W/m²]<br>• `areas`: partial areas Si [m²]<br>• `frequencies` [Hz]: for LWA<br>• `temperature` [°C] (Default: 23), `barometric_pressure` [Pa] (Default: 101325): for LW0 (Eq. 10) | `res = sound_power_intensity_precision(In, areas, frequencies=f)`<br><br>• `PrecisionIntensityResult` |
| `PrecisionIntensityResult` | `dataclass` | **ISO 9614-3 scanning result.**<br>• `partial_power`: Pi = In,i·Si [W]<br>• `sound_power` P, `sound_power_level` LW [dB] (NaN where P ≤ 0)<br>• `sound_power_level_normalized`: LW0 [dB]<br>• `not_applicable_band`: per-band bool<br>• `surface_area` [m²], `sound_power_level_a` [dB]<br>• `.plot()` | `res.sound_power_level, res.not_applicable_band` |
| `precision_field_indicators` | `function` | **ISO 9614-3 Annex B field indicators.**<br>• `segment_intensity`: (N, NB) signed In,j [W/m²]<br>• `segment_pressure_levels`: (N, NB) Lpj [dB]<br>• `time_window_intensity`: (M, NB) for FT (Default: None) | `fi = precision_field_indicators(In, lp)`<br><br>• `PrecisionFieldIndicators` |
| `PrecisionFieldIndicators` | `dataclass` | **Annex B indicators (per band).**<br>• `ft`: temporal variability (= F1) or None<br>• `f_pi_unsigned`: F_pIn unsigned (= F2)<br>• `f_pi_signed`: F_pIn signed (= F3), ≥ unsigned<br>• `fs`: field non-uniformity FS (= F4) | `fi.f_pi_signed, fi.fs` |
| `precision_qualification` | `function` | **The five ISO 9614-3 Annex C acceptance criteria.**<br>• `indicators`: `PrecisionFieldIndicators`<br>• `scan_intensity_level_1`/`_2`: LIn per scan [dB] (criterion 1)<br>• `pressure_residual_index`: δpI0 [dB] (criterion 2, K = 10)<br>• `field_nonuniformity_1`/`_2`: FS per scan density (criterion 5)<br>• `frequencies` [Hz] or `repeatability_limit`: Table 1 limit s | `q = precision_qualification(fi, pressure_residual_index=18.0)`<br><br>• `PrecisionCriteria` |
| `PrecisionCriteria` | `dataclass` | **Annex C pass/fail per band.**<br>• `criterion_1`…`criterion_5`: bool arrays or None where not evaluable<br>• `qualified`: conjunction of criteria 1–4 or None | `q.qualified, q.criterion_2` |
| `plane_wave_frequency_range` | `function` | **Working plane-wave range (f_l, f_u) (ISO 10534-2 §4.2–4.5).**<br>• `spacing` s [m]<br>• `speed_of_sound` c0 [m/s]<br>• `diameter` d [m] (Default: None → spacing bound only)<br>• `shape`: 'circular'/'rectangular' (Default: 'circular') | `plane_wave_frequency_range(0.05, 343.2, diameter=0.1)  # (343.2, 1990.6)`<br><br>• [Hz] |
| `speed_of_sound_iso` | `function` | **Speed of sound (ISO 10534-2 Eq. 5).**<br>• `temperature` T [K] | `speed_of_sound_iso(293.0)  # 343.2`<br><br>• c0 = 343.2 √(T/293) [m/s] |
| `air_density_iso` | `function` | **Air density (ISO 10534-2 Eq. 7).**<br>• `temperature` T [K]<br>• `atmospheric_pressure` [kPa] (Default: 101.325) | `air_density_iso(293.0)  # 1.186`<br><br>• ρ [kg/m³] |
| `speed_of_sound_astm` | `function` | **Speed of sound (ASTM E2611-19 Eq. 4).**<br>• `temperature` T [°C] | `speed_of_sound_astm(20.0)  # 343.2`<br><br>• 20.047 √(273.15 + T) [m/s] |
| `air_density_astm` | `function` | **Air density (ASTM E2611-19 Eq. 5).**<br>• `temperature` T [°C]<br>• `atmospheric_pressure` P [kPa] (Default: 101.325) | `air_density_astm(20.0)  # 1.202`<br><br>• ρ [kg/m³] |
| `characteristic_impedance` | `function` | **Characteristic impedance of air ρc.**<br>• `density` ρ [kg/m³]<br>• `speed_of_sound` c [m/s] | `characteristic_impedance(1.186, 343.2)  # 407.0`<br><br>• [rayl] |
| `tube_attenuation_constant` | `function` | **Lower-bound tube attenuation k0″ (ISO 10534-2 Eq. A.18).**<br>• `frequency` f [Hz]<br>• `speed_of_sound` c0 [m/s]<br>• `diameter` d [m] (hydraulic for rectangular) | `tube_attenuation_constant(1000.0, 343.2, 0.1)  # 0.0179`<br><br>• [Np/m] |
| `tube_wavenumber` | `function` | **Complex wavenumber k0 = k0′ − j k0″ (ISO 10534-2 §2.6).**<br>• `frequency` f [Hz]<br>• `speed_of_sound` c0 [m/s]<br>• `attenuation`: k0″ [Np/m] (Default: None → lossless) | `tube_wavenumber(1000.0, 343.2, attenuation=0.018)`<br><br>• 18.308 − 0.018j [1/m] |
| `mic_calibration_factor` | `function` | **Microphone-mismatch calibration factor Hc (ISO 10534-2 Eq. 10).**<br>• `h12_config1`: H12 in the standard configuration<br>• `h12_config2`: H12 with the microphones interchanged | `hc = mic_calibration_factor(h12_a, h12_b)`<br><br>• Hc = √(H12ᴵ/H12ᴵᴵ) (complex) |
| `apply_mic_calibration` | `function` | **Apply the calibration factor (Eq. 13).**<br>• `h12_uncorrected`<br>• `calibration_factor`: Hc | `h12 = apply_mic_calibration(h12_raw, hc)`<br><br>• H12 = H12,raw/Hc |
| `reflection_factor` | `function` | **Complex reflection factor at the sample surface (ISO 10534-2 Eq. 17).**<br>• `h12`: corrected transfer function<br>• `spacing` s [m]<br>• `x1`: sample to the farther microphone [m]<br>• `wavenumber`: complex k0 | `r = reflection_factor(h12, spacing=0.05, x1=0.15, wavenumber=k0)`<br><br>• r = (H12 − HI)/(HR − H12)·e^(+2jk0x1) |
| `absorption_from_reflection` | `function` | **Normal-incidence absorption coefficient (Eq. 18).**<br>• `reflection`: complex r | `absorption_from_reflection(0.5)  # 0.75`<br><br>• α = 1 − \|r\|² |
| `surface_impedance` | `function` | **Absolute surface impedance Z (Eq. 19).**<br>• `reflection`: complex r<br>• `characteristic_impedance`: ρc0 [rayl] | `surface_impedance(0.5, 407.0)  # 1221`<br><br>• Z = ρc0(1 + r)/(1 − r) [rayl] |
| `normalized_surface_impedance` | `function` | **Normalised surface impedance Z/(ρc0) (Eq. 19).**<br>• `reflection`: complex r | `normalized_surface_impedance(0.5)  # 3.0` |
| `normalized_surface_admittance` | `function` | **Normalised surface admittance Gρc0 (Eq. 20).**<br>• `reflection`: complex r | `normalized_surface_admittance(0.5)  # 0.333`<br><br>• (1 − r)/(1 + r) |
| `two_microphone_impedance` | `function` | **Full two-microphone reduction (ISO 10534-2 Clause 7).**<br>• `h12`: corrected transfer function<br>• `frequency` f [Hz]<br>• `spacing` s [m], `x1` [m]<br>• `speed_of_sound` c0 [m/s], `characteristic_impedance` ρc0 [rayl]<br>• `attenuation`: k0″ [Np/m] (Default: None)<br>• `diameter` + `shape`: activate the plane-wave range check | `res = two_microphone_impedance(h12, frequency=f, spacing=0.05, x1=0.15, speed_of_sound=343.2, characteristic_impedance=407.0)`<br><br>• `ImpedanceTubeResult` |
| `ImpedanceTubeResult` | `dataclass` | **Two-microphone result per frequency.**<br>• `frequency` [Hz]<br>• `reflection`: complex r (Eq. 17)<br>• `surface_impedance` [rayl], `normalized_impedance` (Eq. 19)<br>• `absorption`: α = 1 − \|r\|² (Eq. 18) | `res.absorption, res.normalized_impedance` |
| `ImpedanceTubeWarning` | `warning class` | **ISO 10534-2 advisory.**<br>Emitted by `two_microphone_impedance` for frequencies outside the plane-wave working range (Eqs. 1–4); the results are still returned | `warnings.simplefilter('error', ImpedanceTubeWarning)` |
| `standing_wave_ratio_from_level` | `function` | **Standing-wave ratio from ΔL (ISO 10534-1 Eq. 15).**<br>• `level_difference`: Lmax − Lmin [dB] | `standing_wave_ratio_from_level(20.0)  # 10.0`<br><br>• s = 10^(ΔL/20) |
| `standing_wave_reflection_magnitude` | `function` | **Reflection magnitude from the SWR (Eq. 14).**<br>• `swr`: s ≥ 1 | `standing_wave_reflection_magnitude(10.0)  # 0.818`<br><br>• \|r\| = (s − 1)/(s + 1) |
| `standing_wave_reflection` | `function` | **Complex reflection factor from the standing wave (Eqs. 17–23).**<br>• `swr`: s<br>• `first_min_distance`: x_min1 [m]<br>• `wavelength`: λ0 [m] | `standing_wave_reflection(10.0, 0.05, 0.4)  # -0.818j`<br><br>• phase φ = π(4x_min1/λ0 − 1) |
| `standing_wave_absorption` | `function` | **Absorption from the SWR (Eqs. 9 + 14).**<br>• `swr`: s ≥ 1 | `standing_wave_absorption(10.0)  # 0.331`<br><br>• α = 4s/(s + 1)² |
| `standing_wave_normalized_impedance` | `function` | **Normalised impedance from the standing wave (Eqs. 24–26).**<br>• `swr`, `first_min_distance` [m], `wavelength` [m] | `standing_wave_normalized_impedance(10.0, 0.05, 0.4)`<br><br>• 0.198 − 0.98j |
| `wave_decomposition` | `function` | **Decompose the field into (A, B, C, D) (ASTM E2611-19 Eqs. 17–20).**<br>• `h1`…`h4`: microphone transfer functions<br>• `l1`, `s1`, `l2`, `s2`: geometry [m]<br>• `wavenumber` k | `A, B, C, D = wave_decomposition(h1, h2, h3, h4, l1=0.2, s1=0.05, l2=0.2, s2=0.05, wavenumber=k)`<br><br>• Forward/backward amplitudes, both sides |
| `face_quantities` | `function` | **Face pressures and velocities (Eq. 21).**<br>• `a`, `b`, `c`, `d`: wave amplitudes<br>• `wavenumber` k, `thickness` d [m], `characteristic_impedance` ρc | `p0, pd, u0, ud = face_quantities(A, B, C, D, wavenumber=k, thickness=0.05, characteristic_impedance=407.0)` |
| `transfer_matrix_two_load` | `function` | **Two-load transfer matrix (ASTM E2611-19 Eqs. 17–22).**<br>• `load_a`, `load_b`: (H1, H2, H3, H4) per termination<br>• `l1`, `s1`, `l2`, `s2`, `thickness` [m]<br>• `wavenumber`, `characteristic_impedance` | `T = transfer_matrix_two_load(load_a, load_b, l1=0.2, s1=0.05, l2=0.2, s2=0.05, thickness=0.05, wavenumber=k, characteristic_impedance=407.0)`<br><br>• `TransferMatrix` |
| `transfer_matrix_one_load` | `function` | **One-load transfer matrix, symmetric specimen (Eqs. 23–24).**<br>• `load`: (H1, H2, H3, H4)<br>• Same geometry parameters as `transfer_matrix_two_load` | `T = transfer_matrix_one_load(load, l1=0.2, s1=0.05, l2=0.2, s2=0.05, thickness=0.05, wavenumber=k, characteristic_impedance=407.0)`<br><br>• Requires reciprocity + symmetry |
| `air_layer_transfer_matrix` | `function` | **Analytic loss-free air-layer matrix (validation reference).**<br>• `wavenumber` k<br>• `thickness` d [m]<br>• `characteristic_impedance` ρc [rayl] | `T = air_layer_transfer_matrix(k, 0.05, 407.0)`<br><br>• det(T) = 1, T11 = T22 |
| `TransferMatrix` | `dataclass` | **Acoustic transfer matrix [[T11, T12], [T21, T22]] (ASTM E2611-19 Eq. 16).**<br>• `t11`, `t12`, `t21`, `t22`: complex, scalar or per frequency | `T.t11, T.t12` |
| `airflow_resistance` | `function` | **Airflow resistance R = Δp/qv (ISO 9053-1:2018 §3.1).**<br>• `pressure_drop` Δp [Pa]<br>• `volume_flow_rate` qv [m³/s] | `airflow_resistance(2.0, 0.0005)  # 4000.0`<br><br>• [Pa·s/m³] |
| `specific_airflow_resistance` | `function` | **Specific airflow resistance Rs (§3.2).**<br>• `resistance` R + `area` A, **or** `pressure_drop` Δp + `velocity` u | `specific_airflow_resistance(4000.0, 0.01)  # 40.0`<br><br>• Rs = R·A = Δp/u [Pa·s/m] |
| `airflow_resistivity` | `function` | **Airflow resistivity σ = Rs/d (§3.3).**<br>• `specific_resistance` Rs [Pa·s/m]<br>• `thickness` d [m] | `airflow_resistivity(40.0, 0.05)  # 800.0`<br><br>• [Pa·s/m²] |
| `linear_airflow_velocity` | `function` | **Linear airflow velocity u = qv/A (§3.4).**<br>• `volume_flow_rate` qv [m³/s]<br>• `area` A [m²] | `linear_airflow_velocity(5e-6, 0.01)  # 0.0005`<br><br>• [m/s] |
| `static_airflow_resistance` | `function` | **Stepwise static-method determination (ISO 9053-1 §7.5).**<br>• `velocities` u [m/s], `pressure_drops` Δp [Pa] (≥ 2 steps)<br>• `area` A [m²]<br>• `thickness` d [m] (Default: None → no σ)<br>• `evaluation_velocity` [m/s] (Default: 0.0005) | `res = static_airflow_resistance([0.0005, 0.001, 0.002], [0.02, 0.041, 0.086], 0.01, 0.05)`<br><br>• `StaticAirflowResult` (Rs = 40.0, σ = 800) |
| `StaticAirflowResult` | `dataclass` | **Static-method result at the evaluation velocity.**<br>• `resistance` R [Pa·s/m³], `specific_resistance` Rs [Pa·s/m]<br>• `resistivity` σ [Pa·s/m²] or None<br>• `evaluation_velocity` [m/s], `pressure_drop` [Pa]<br>• `linear_coefficient` a (zero-velocity Rs), `quadratic_coefficient` b | `res.specific_resistance, res.resistivity` |
| `piston_volume_flow_rate` | `function` | **RMS piston volume flow qv = 2πf·h·AP (ISO 9053-2 §6.2).**<br>• `frequency` f [Hz]<br>• `stroke_amplitude` h [m]<br>• `piston_area` AP [m²] | `piston_volume_flow_rate(2.0, 0.005, 0.01)  # 0.000628`<br><br>• [m³/s] |
| `alternating_airflow_resistance` | `function` | **Alternating-method resistance (ISO 9053-2:2020 Formula 2).**<br>• `level_specimen` Lps / `level_termination` Lpt [dB]<br>• `piston_stroke_specimen` hs / `piston_stroke_termination` ht [m]<br>• `frequency` f [Hz] (1–4 Hz), `cavity_volume` V [m³]<br>• `static_pressure` [Pa] (Default: 101325)<br>• `kappa_prime` κ′ (Default: 1.4; use `effective_kappa` for Annex A conformity)<br>• `background_level` Lpb [dB] (Default: None) | `r = alternating_airflow_resistance(78.0, 60.0, piston_stroke_specimen=5e-3, piston_stroke_termination=5e-4, frequency=2.0, cavity_volume=7.854e-4)`<br><br>• R [Pa·s/m³]; `AirflowResistanceWarning` when criteria fail |
| `thermal_boundary_layer_thickness` | `function` | **Thermal boundary-layer thickness b (ISO 9053-2 Formulas A.4/A.5).**<br>• `frequency` f [Hz]<br>• Air properties (Default: Annex A.3 values) | `thermal_boundary_layer_thickness(2.0)  # 0.00183`<br><br>• [m] |
| `effective_kappa` | `function` | **Effective ratio of specific heats κ′ (Formula A.7).**<br>• `cavity_surface` S [m²], `cavity_volume` V [m³]<br>• `frequency` f [Hz]<br>• Air properties (Default: Annex A.3 values) | `effective_kappa(0.0471, 7.854e-4, 2.0)  # 1.37`<br><br>• The Annex A.3 worked example |
| `AirflowResistanceWarning` | `warning class` | **ISO 9053 advisory.**<br>Emitted for a velocity above the 15 mm/s static limit, a piston frequency outside 1–4 Hz, or a failed Formula 3/4 validity criterion | `warnings.simplefilter('error', AirflowResistanceWarning)` |
| `practical_absorption_coefficient` | `function` | **Practical coefficients αp (ISO 11654 §4.1).**<br>• `third_octave_alpha_s`: 15 one-third-octave αs, 200–5000 Hz (sequence or mapping keyed by band) | `ap = practical_absorption_coefficient(alpha_s)`<br><br>• 5 octave values (250–4000 Hz), 0.05 steps, capped at 1.00 |
| `weighted_absorption` | `function` | **Weighted absorption αw + class (ISO 11654 §4.2).**<br>• `alpha_p`: the 5 octave practical coefficients | `w = weighted_absorption([0.25, 0.7, 0.95, 1.0, 0.85])`<br><br>• `AbsorptionRatingResult` (αw = 0.55, class D, 'MH') |
| `absorption_class` | `function` | **Sound absorption class (Table B.1).**<br>• `alpha_w`: multiple of 0.05 in [0, 1] | `absorption_class(0.7)  # 'C'`<br><br>• 'A'–'E' or 'Not classified' |
| `AbsorptionRatingResult` | `dataclass` | **αw rating result.**<br>• `alpha_w`: shifted curve read at 500 Hz<br>• `shape_indicator`: 'L'/'M'/'H' concatenation or ''<br>• `absorption_class`: 'A'–'E'/'Not classified'<br>• `shift`, `unfavourable_sum` (≤ 0.10)<br>• `band_centers` [Hz], `measured`, `shifted_reference`<br>• `.plot()` | `w.alpha_w, w.absorption_class` |
| `OCTAVE_BANDS` | `tuple` | **ISO 11654 octave rating bands [Hz].**<br>(250, 500, 1000, 2000, 4000) | `OCTAVE_BANDS  # (250, ..., 4000)` |
| `THIRD_OCTAVE_BANDS` | `tuple` | **ISO 11654 one-third-octave input bands [Hz].**<br>200 Hz to 5000 Hz (15 bands) | `THIRD_OCTAVE_BANDS[0]  # 200` |
| `REFERENCE_CURVE` | `mapping` | **ISO 11654 Figure 1 reference curve (read-only).**<br>Keyed by octave band [Hz] → coefficient | `REFERENCE_CURVE[250]  # 0.8` |
| `speed_of_sound` | `function` | **Speed of sound (ISO 17497-1 Eq. 2).**<br>• `temperature` t [°C] | `speed_of_sound(20.0)  # 343.2`<br><br>• 343.2 √((273.15 + t)/293.15) [m/s] |
| `air_attenuation_coefficient` | `function` | **Energy attenuation m from ISO 9613-1 α (ISO 17497-1 Eq. 3).**<br>• `pressure_attenuation_db_per_m`: α [dB/m] | `air_attenuation_coefficient(0.005)  # 0.00115`<br><br>• m ≈ α/4.343 [1/m] |
| `random_incidence_absorption` | `function` | **Random-incidence absorption αs (ISO 17497-1 Eq. 1).**<br>• `volume` V [m³], `area` S [m²]<br>• `c1`/`T1`: static base plate, no sample<br>• `c2`/`T2`: with the test sample<br>• `m1`/`m2`: air attenuation [1/m] (Default: 0) | `a_s = random_incidence_absorption(200.0, 10.8, c1=343.0, T1=2.5, c2=343.2, T2=1.8)  # 0.463` |
| `specular_absorption_coefficient` | `function` | **Specular absorption αspec (Eq. 4).**<br>• `volume`, `area`<br>• `c3`/`T3`: rotating base plate, no sample<br>• `c4`/`T4`: sample on the rotating turntable<br>• `m3`/`m4` (Default: 0) | `a_spec = specular_absorption_coefficient(200.0, 10.8, c3=c3, T3=t3, c4=c4, T4=t4)`<br><br>• Includes the energy lost to scattering |
| `scattering_coefficient` | `function` | **Random-incidence scattering coefficient s (Eq. 5).**<br>• `alpha_spec`, `alpha_s`<br>• `truncate_negative`: clip s < 0 to 0 (Default: True); s > 1 is kept | `scattering_coefficient(0.45, 0.30)  # 0.214`<br><br>• s = (αspec − αs)/(1 − αs) |
| `scattering_coefficient_spectrum` | `function` | **Scattering spectrum s(f) (Eq. 5).**<br>• `frequencies`: one-third-octave centres [Hz]<br>• `specular_absorption`, `random_absorption`: per band<br>• `truncate_negative` (Default: True) | `res = scattering_coefficient_spectrum(f, a_spec, a_s)`<br><br>• `ScatteringResult` |
| `ScatteringResult` | `dataclass` | **Scattering spectrum result.**<br>• `frequencies` [Hz]<br>• `scattering`: s per band<br>• `random_incidence`: αs, `specular`: αspec<br>• `.plot()` | `res.scattering` |
| `base_plate_scattering` | `function` | **Scattering of the base plate alone (Eq. 6).**<br>• `volume`, `area`<br>• `c1`/`T1`: static, `c3`/`T3`: rotating<br>• `m1`/`m3` (Default: 0) | `s_base = base_plate_scattering(200.0, 10.8, c1=c1, T1=t1, c3=c3, T3=t3)`<br><br>• Quality metric vs Table 1 |
| `check_base_plate_scattering` | `function` | **Verify the base plate against Table 1 (Clause 6.2).**<br>• `scattering`: mapping by band [Hz] or 18 values ordered as `BASE_PLATE_BANDS` | `check_base_plate_scattering(s_base)  # ()`<br><br>• Offending band centres; `ScatteringDiffusionWarning` if any |
| `BASE_PLATE_BANDS` | `tuple` | **ISO 17497-1 base-plate check bands [Hz].**<br>100 Hz to 5000 Hz (18 one-third octaves) | `BASE_PLATE_BANDS[0]  # 100` |
| `BASE_PLATE_MAX_SCATTERING` | `mapping` | **Table 1 base-plate scattering limits (read-only).**<br>Keyed by band [Hz] → maximum s | `BASE_PLATE_MAX_SCATTERING[500]  # 0.05` |
| `reverberation_time_uncertainty` | `function` | **Standard uncertainty of a mean T (ISO 17497-1 Eq. A.1).**<br>• `times`: N ≥ 2 spatially-averaged measurements [s] | `reverberation_time_uncertainty([2.31, 2.28, 2.35])  # 0.0203`<br><br>• Standard error of the mean [s] |
| `absorption_coefficient_uncertainty` | `function` | **Uncertainty of a Sabine coefficient (Eqs. A.3/A.4).**<br>• `volume`, `area`<br>• `c` [m/s]<br>• `T_a`/`u_a`, `T_b`/`u_b`: the two situations [s] | `u_alpha = absorption_coefficient_uncertainty(200.0, 10.8, c=343.2, T_a=2.5, u_a=0.02, T_b=1.8, u_b=0.02)` |
| `scattering_coefficient_uncertainty` | `function` | **Uncertainty of the scattering coefficient (Eq. A.5).**<br>• `alpha_spec`, `alpha_s`<br>• `u_alpha_spec`, `u_alpha_s` | `u = scattering_coefficient_uncertainty(0.45, 0.30, 0.02, 0.02)`<br><br>• `ScatteringUncertainty` (U = 2u, 95 %) |
| `ScatteringUncertainty` | `dataclass` | **Scattering uncertainty result.**<br>• `u_scattering`: combined standard u_s<br>• `expanded`: U = 2u_s | `u.u_scattering, u.expanded` |
| `directional_diffusion_coefficient` | `function` | **Directional diffusion coefficient d_θ (ISO 17497-2 Formulas 5/6).**<br>• `levels`: n ≥ 2 reflected SPL [dB] (−inf = zero energy)<br>• `area_weights`: Ni (Formula 8) (Default: None → equal-area Formula 5) | `directional_diffusion_coefficient([-10., -12., -15., -11., -13.])  # 0.854` |
| `directional_diffusion` | `function` | **Polar response + its coefficient.**<br>• `angles` [°], `levels` [dB]<br>• `weights`: Ni (Default: None) | `res = directional_diffusion(angles, levels)`<br><br>• `DiffusionResult` |
| `DiffusionResult` | `dataclass` | **Polar diffusion result.**<br>• `angles` [°], `levels` [dB]<br>• `coefficient`: d_θ (autocorrelation)<br>• `.plot()` | `res.coefficient` |
| `normalized_diffusion_coefficient` | `function` | **Normalised coefficient d_θ,n (Formula 7).**<br>• `d_theta`: test surface<br>• `d_theta_reference`: flat reference | `normalized_diffusion_coefficient(0.6, 0.2)  # 0.5`<br><br>• (d − d_r)/(1 − d_r) |
| `area_factors` | `function` | **Per-receiver area weights Ni (Clause 8.3 Formula 8).**<br>• `elevations`: θ [°], 0–90<br>• `delta_theta` [°]<br>• `delta_phi` [°] (Default: None → `delta_theta`) | `n_i = area_factors([0, 15, 30, 45, 60, 75, 90], delta_theta=15.0)`<br><br>• Dimensionless, min 1 |
| `random_incidence_diffusion` | `function` | **Random-incidence diffusion coefficient d (Clause 8.4).**<br>• `directional_coefficients`: d_θ per source<br>• `weights`: source weights (Default: None → equal; 2-D uses `TWO_DIMENSIONAL_SOURCE_WEIGHTS`) | `d = random_incidence_diffusion(d_thetas, weights=TWO_DIMENSIONAL_SOURCE_WEIGHTS)` |
| `TWO_DIMENSIONAL_SOURCE_WEIGHTS` | `tuple` | **ISO 17497-2 single-plane source weights.**<br>(1, 3, 3, 3, 3) for 0°, ±30°, ±60° | `TWO_DIMENSIONAL_SOURCE_WEIGHTS` |
| `ScatteringDiffusionWarning` | `warning class` | **ISO 17497 advisory.**<br>Emitted for out-of-range scattering/diffusion measurement conditions (e.g. a base plate over the Table 1 limits) | `warnings.simplefilter('error', ScatteringDiffusionWarning)` |
| `adrienne_window` | `function` | **Adrienne temporal window (ISO 13472-1 Clause 6.4).**<br>• `fs` [Hz]<br>• `flat_duration` [s] (Default: 0.005)<br>• `leading_duration` [s] (Default: 0.0005), `trailing_duration` [s] (Default: 0.005)<br>• `leading_edge`/`trailing_edge`: 'blackman-harris' or 'cosine-squared' | `w = adrienne_window(48000)`<br><br>• Rising edge + flat top + falling edge, peak 1.0 |
| `geometric_spreading_factor` | `function` | **Geometrical-spreading factor Kr (Clause 4.1).**<br>• `source_height` ds [m] (Default: 1.25)<br>• `mic_height` dm [m] (Default: 0.25) | `geometric_spreading_factor()  # 0.6667`<br><br>• Kr = (ds − dm)/(ds + dm) |
| `geometric_spreading_factor_angle` | `function` | **Oblique factor Kr,θ (Annex F).**<br>• `incidence_angle` θ [rad]<br>• `source_height`, `mic_height` | `geometric_spreading_factor_angle(np.pi/6)  # 0.764` |
| `reflected_path_delay` | `function` | **Reflected-path delay Δτ = 2dm/c (Annex C).**<br>• `mic_height` dm [m] (Default: 0.25)<br>• `speed_of_sound` c [m/s] (Default: 340.0) | `reflected_path_delay()  # 0.001471`<br><br>• [s] |
| `insitu_reflection_factor` | `function` | **Complex reflection factor r(f) (ISO 13472-1 Clause 4.1).**<br>• `incident_ir` / `reflected_ir`: windowed impulse responses<br>• `source_height`, `mic_height` [m]<br>• `incidence_angle` [rad] (Default: 0)<br>• `fs` + `delay`: undo the reflected-path offset (Default: None)<br>• `n`: FFT length | `r = insitu_reflection_factor(hi, hr, fs=48000, delay=dtau)`<br><br>• (1/Kr)·Hr(f)/Hi(f) at the rfft bins |
| `insitu_absorption_from_reflection` | `function` | **α from the reflection factor (Clause 4.1).**<br>• `reflection`: complex r | `alpha = insitu_absorption_from_reflection(r)`<br><br>• α = 1 − \|r\|² |
| `power_reflection_coefficient` | `function` | **Power reflection factor QW(f), direct energy route.**<br>• `incident_ir`, `reflected_ir`<br>• `source_height`, `mic_height`, `incidence_angle`, `n` | `qw = power_reflection_coefficient(hi, hr)`<br><br>• (1/Kr²)\|Hr/Hi\|², offset-independent |
| `insitu_absorption_coefficient` | `function` | **Narrow-band absorption α(f) (Clause 4.1).**<br>• Same parameters as `power_reflection_coefficient` | `alpha = insitu_absorption_coefficient(hi, hr)`<br><br>• α = 1 − QW(f) |
| `one_third_octave_absorption` | `function` | **Aggregate narrow-band α into one-third octaves.**<br>• `frequency` [Hz], `absorption`<br>• `f_min` (Default: 250), `f_max` (Default: 4000; 1600 for Part 2) [Hz]<br>• `clip_negative` (Default: True) | `fc, ab = one_third_octave_absorption(f, alpha)`<br><br>• Band centres + linear-averaged α (nan if empty) |
| `insitu_absorption_spectrum` | `function` | **End-to-end in-situ spectrum (ISO 13472-1).**<br>• `incident_ir`, `reflected_ir`, `fs` [Hz]<br>• geometry + `incidence_angle`<br>• `f_min`/`f_max` [Hz], `clip_negative` | `res = insitu_absorption_spectrum(hi, hr, 48000)`<br><br>• `InsituAbsorptionResult` |
| `InsituAbsorptionResult` | `dataclass` | **In-situ absorption spectrum.**<br>• `frequencies`: one-third-octave centres [Hz]<br>• `absorption`: α per band (nan when empty)<br>• `.plot()` | `res.absorption` |
| `absorption_reference_corrected` | `function` | **Reference-corrected road absorption (Annex B).**<br>• `road_reflection` / `reference_reflection`: measured Qp (complex, same geometry) | `alpha = absorption_reference_corrected(q_road, q_ref)`<br><br>• 1 − \|Qp,road/Qp,ref\|²; removes chain error and Kr |
| `max_sampled_area_radius` | `function` | **Maximum sampled-area radius (Annex A).**<br>• `window_width` Tw [s]<br>• `source_height`, `mic_height`, `speed_of_sound` | `max_sampled_area_radius(0.005)  # 1.34`<br><br>• [m], the Annex A worked example |
| `msa_major_axis` | `function` | **Major axis of the oblique sampled ellipsoid (Annex F).**<br>• `window_width` Tw [s]<br>• `projected_distance` dp [m]<br>• `source_height`, `mic_height`, `speed_of_sound` | `msa_major_axis(0.005, 0.0)  # 3.2`<br><br>• a = cTw + √((ds + dm)² + dp²) [m] |
| `spot_tube_upper_frequency` | `function` | **Spot-tube upper frequency (ISO 13472-2 §5.4.1).**<br>• `diameter` d [m]<br>• `speed_of_sound` (Default: 340.0) | `spot_tube_upper_frequency(0.1)  # 1972`<br><br>• f_u = 0.58c0/d [Hz] |
| `spot_microphone_spacing_bounds` | `function` | **Spacing bounds (s_min, s_max) (§5.4.2).**<br>• `speed_of_sound` (Default: 340.0)<br>• `f_min` (Default: 220), `f_max` (Default: 1800) [Hz] | `spot_microphone_spacing_bounds()  # (0.0773, 0.085)`<br><br>• [m], brackets the nominal 81 mm |
| `check_spot_frequency_range` | `function` | **Advise outside 250–1600 Hz (ISO 13472-2 Scope).**<br>• `frequency` [Hz] | `check_spot_frequency_range(f)`<br><br>• `RoadAbsorptionWarning` out of range |
| `spot_internal_loss_correction` | `function` | **Internal-loss (system) correction (Annex A).**<br>• `measured_absorption`, `system_absorption`: same bands<br>• `clip_negative` (Default: True) | `spot_internal_loss_correction([0.12, 0.10], [0.02, 0.03])  # [0.1, 0.07]`<br><br>• Subtractive Part-2 correction |
| `DEFAULT_SOURCE_HEIGHT` | `float` | **ISO 13472-1 mandatory source height ds [m].**<br>1.25 | `DEFAULT_SOURCE_HEIGHT  # 1.25` |
| `DEFAULT_MIC_HEIGHT` | `float` | **ISO 13472-1 mandatory microphone height dm [m].**<br>0.25 | `DEFAULT_MIC_HEIGHT  # 0.25` |
| `DEFAULT_SPEED_OF_SOUND` | `float` | **Default speed of sound for the road methods [m/s].**<br>340.0 | `DEFAULT_SPEED_OF_SOUND  # 340.0` |
| `PART1_FREQUENCY_RANGE` | `tuple` | **ISO 13472-1 valid band range [Hz].**<br>(250.0, 4000.0) | `PART1_FREQUENCY_RANGE` |
| `SPOT_FREQUENCY_RANGE` | `tuple` | **ISO 13472-2 valid band range [Hz].**<br>(250.0, 1600.0) | `SPOT_FREQUENCY_RANGE` |
| `SPOT_NARROW_BAND_RANGE` | `tuple` | **ISO 13472-2 narrow-band range [Hz].**<br>(220.0, 1800.0) | `SPOT_NARROW_BAND_RANGE` |
| `RoadAbsorptionWarning` | `warning class` | **ISO 13472 advisory.**<br>Emitted for frequencies outside the valid in-situ road-absorption ranges; results there are advisory | `warnings.simplefilter('error', RoadAbsorptionWarning)` |
| `frequency_weighting` | `function` | **Vibration frequency weighting H(f) (ISO 8041-1:2017 Formula 5).**<br>• `name`: one of `WEIGHTING_NAMES`<br>• `frequencies` [Hz] (> 0) | `wr = frequency_weighting('Wk', [1.0, 8.0, 63.0])`<br><br>• `WeightingResponse`; \|H\| = [0.482, 1.036, 0.186] |
| `weighting_factors` | `function` | **Weighting factors \|H(f)\| (the Wi of ISO 2631-1 Eq. 9).**<br>• `name`, `frequencies` [Hz] | `weighting_factors('Wh', [16.0])`<br><br>• Magnitude per frequency |
| `apply_weighting` | `function` | **Apply a weighting to a time signal (frequency domain, exact response).**<br>• `signal`: acceleration (1D) [m/s²]<br>• `fs` [Hz]<br>• `name`: one of `WEIGHTING_NAMES` | `aw_t = apply_weighting(a, fs, 'Wk')`<br><br>• Weighted signal, same length (circular FFT filtering) |
| `WeightingResponse` | `dataclass` | **Weighting response.**<br>• `name`<br>• `frequencies` [Hz]<br>• `response`: complex H<br>• `magnitude`, `magnitude_db` [dB]<br>• `.plot()` | `wr.magnitude, wr.magnitude_db` |
| `WEIGHTING_NAMES` | `tuple` | **ISO 8041-1 weighting names.**<br>('Wb', 'Wc', 'Wd', 'We', 'Wf', 'Wh', 'Wj', 'Wk', 'Wm') | `'Wk' in WEIGHTING_NAMES  # True` |
| `weighted_acceleration` | `function` | **Weighted r.m.s. from a band spectrum (ISO 2631-1 Eq. 9 / ISO 5349-1 Eq. A.1).**<br>• `band_accelerations`: ai per band [m/s²]<br>• `frequencies`: band centres [Hz]<br>• `weighting`: name | `ws = weighted_acceleration([0.1, 0.3, 0.2], [4.0, 8.0, 16.0], 'Wk')`<br><br>• `WeightedSpectrum`; aw = 0.36 m/s² |
| `WeightedSpectrum` | `dataclass` | **Weighted band spectrum.**<br>• `frequencies` [Hz], `band_accelerations` [m/s²]<br>• `weighting_name`, `weighting_factors`<br>• `weighted`: Wi·ai [m/s²]<br>• `overall`: aw [m/s²]<br>• `.plot()` | `ws.overall` |
| `running_rms` | `function` | **Running r.m.s. of a weighted signal (ISO 2631-1 Eqs. 2/3).**<br>• `signal` [m/s²], `fs` [Hz]<br>• `integration_time` τ [s] (Default: 1.0)<br>• `method`: 'linear' (Eq. 2) or 'exponential' (Eq. 3) (Default: 'linear') | `env = running_rms(aw_t, fs)`<br><br>• aw(t0) per sample [m/s²] |
| `mtvv` | `function` | **Maximum transient vibration value (Eq. 4).**<br>• `signal` [m/s²], `fs` [Hz]<br>• `integration_time` [s] (Default: 1.0) | `mtvv(aw_t, fs)`<br><br>• max aw(t0) [m/s²] |
| `vibration_dose_value` | `function` | **Vibration dose value VDV (Eq. 5).**<br>• `signal`: weighted acceleration [m/s²]<br>• `fs` [Hz] | `vdv = vibration_dose_value(aw_t, fs)`<br><br>• (∫aw⁴dt)^(1/4) [m/s^1.75] |
| `motion_sickness_dose_value` | `function` | **Motion sickness dose value MSDV (ISO 2631-1 clause 9).**<br>• `signal`: Wf-weighted acceleration [m/s²]<br>• `fs` [Hz] | `msdv = motion_sickness_dose_value(aw_t, fs)`<br><br>• (∫aw²dt)^(1/2) [m/s^1.5] |
| `crest_factor` | `function` | **Crest factor of a weighted signal (clause 6.2.1).**<br>• `signal` [m/s²] | `crest_factor(aw_t)`<br><br>• peak/r.m.s.; `HumanVibrationWarning` above 9 |
| `vibration_total_value` | `function` | **Vibration total value av/ahv (ISO 2631-1 Eq. 10).**<br>• `components`: axis-weighted r.m.s. [m/s²]<br>• `k`: per-axis factors (Default: None → 1, the ISO 5349-1 vector sum) | `vibration_total_value([0.3, 0.4, 0.5], k=[1.4, 1.4, 1.0])  # 0.86`<br><br>• √(Σ kj²awj²) [m/s²] |
| `daily_exposure` | `function` | **Daily exposure A(8) for one operation (ISO 5349-1 Eq. 2).**<br>• `total_value`: ahv or av [m/s²]<br>• `duration_s`: T [s] | `daily_exposure(3.0, 4*3600)  # 2.12`<br><br>• ahv √(T/8h) [m/s²] |
| `partial_exposure` | `function` | **Partial exposure Ai(8) of one operation (Eq. 2).**<br>• `total_value` [m/s²], `duration_s` [s] | `partial_exposure(3.0, 2*3600)  # 1.5` |
| `combine_partial_exposures` | `function` | **Combine partial exposures (Eq. 3).**<br>• `partials`: Ai(8) [m/s²] | `combine_partial_exposures([1.5, 2.1])  # 2.58`<br><br>• √(Σ Ai(8)²) [m/s²] |
| `hav_daily_exposure` | `function` | **A(8) for several operations (ISO 5349-1 Eq. 3).**<br>• `total_values`: ahvi [m/s²]<br>• `durations_s`: Ti [s] | `hav_daily_exposure([3.0, 5.0], [2*3600, 3600])  # 2.32` |
| `energy_equivalent_acceleration` | `function` | **Energy-equivalent magnitude aw,e (ISO 2631-1 Eq. B.3).**<br>• `magnitudes` [m/s²], `durations_s` [s] | `energy_equivalent_acceleration([0.5, 1.0], [3600, 1800])  # 0.707` |
| `hav_vwf_lifetime_years` | `function` | **Years to 10 % vibration-white-finger prevalence (ISO 5349-1 Eq. C.1).**<br>• `a8`: A(8) [m/s²] (> 0) | `hav_vwf_lifetime_years(2.5)  # 12.0`<br><br>• Dy = 31.8 A(8)^−1.06 [years] |
| `exposure_assessment` | `function` | **Assess a daily exposure (Directive 2002/44/EC Article 3).**<br>• `value`: A(8) [m/s²] or VDV [m/s^1.75]<br>• `kind`: 'hav'/'wbv'<br>• `metric`: 'a8' (Default) or 'vdv' (whole-body only) | `ea = exposure_assessment(3.0, kind='hav')`<br><br>• `ExposureAssessment` (zone 'action') |
| `ExposureAssessment` | `dataclass` | **Directive assessment.**<br>• `value`, `kind`, `metric`<br>• `action_value` (EAV), `limit_value` (ELV)<br>• `exceeds_action`, `exceeds_limit`<br>• `zone`: 'below action'/'action'/'limit' | `ea.zone, ea.exceeds_limit` |
| `daily_vibration_exposure` | `function` | **A(8) from several operations, assessed (ISO 5349 + Directive).**<br>• `total_values` [m/s²], `durations_s` [s]<br>• `kind`: 'hav'/'wbv'<br>• `labels` (Default: None → 'op 1', …) | `res = daily_vibration_exposure([3.0, 5.0], [2*3600, 3600], kind='hav')`<br><br>• `DailyVibrationExposure` (A(8) = 2.32) |
| `DailyVibrationExposure` | `dataclass` | **Assessed daily exposure.**<br>• `a8` [m/s²]<br>• `labels`, `total_values`, `durations_s`, `partials`<br>• `assessment`: `ExposureAssessment`<br>• `.plot()` | `res.a8, res.assessment.zone` |
| `HumanVibrationWarning` | `warning class` | **Human-vibration advisory.**<br>Emitted for a crest factor above 9 (ISO 2631-1 6.2.2) or other out-of-range measurement conditions | `warnings.simplefilter('error', HumanVibrationWarning)` |
| `REFERENCE_ACCELERATION` | `float` | **Vibration reference acceleration [m/s²].**<br>1e-6 (ISO 1683 acceleration level reference) | `REFERENCE_ACCELERATION  # 1e-06` |
| `REFERENCE_DURATION_S` | `float` | **Daily-exposure reference duration T0 [s].**<br>28800 (8 h) | `REFERENCE_DURATION_S  # 28800.0` |
| `HAV_EAV_A8` | `float` | **Hand-arm exposure action value [m/s²].**<br>2.5 (Directive 2002/44/EC) | `HAV_EAV_A8  # 2.5` |
| `HAV_ELV_A8` | `float` | **Hand-arm exposure limit value [m/s²].**<br>5.0 | `HAV_ELV_A8  # 5.0` |
| `WBV_EAV_A8` | `float` | **Whole-body A(8) action value [m/s²].**<br>0.5 | `WBV_EAV_A8  # 0.5` |
| `WBV_ELV_A8` | `float` | **Whole-body A(8) limit value [m/s²].**<br>1.15 | `WBV_ELV_A8  # 1.15` |
| `WBV_EAV_VDV` | `float` | **Whole-body VDV action value [m/s^1.75].**<br>9.1 | `WBV_EAV_VDV  # 9.1` |
| `WBV_ELV_VDV` | `float` | **Whole-body VDV limit value [m/s^1.75].**<br>21.0 | `WBV_ELV_VDV  # 21.0` |
| `PhonometryWarning` | `warning class` | **Base class for all phonometry warnings.**<br>Catch or silence every library advisory at once | `warnings.simplefilter('error', PhonometryWarning)` |
| `FilterBankWarning` | `warning class` | **Fractional-octave filter-bank advisory.**<br>Emitted for filter-bank processing pitfalls | `warnings.simplefilter('error', FilterBankWarning)` |
| `TonalityWarning` | `warning class` | **Tonality advisory.**<br>Emitted for biased tonality estimates (e.g. coarse FFT resolution) | `warnings.simplefilter('error', TonalityWarning)` |
| `STIWarning` | `warning class` | **STI/STIPA advisory.**<br>Emitted for suspect speech-intelligibility measurements or inputs | `warnings.simplefilter('error', STIWarning)` |
| `__version__` | `str` | **Package version string.**<br>(no parameters) | `phonometry.__version__  # '3.0.0'` |
| `.plot()` | `method` | **One-line canonical figure on every result object (soft matplotlib dependency).**<br>Available on `ZwickerLoudness`, `MooreGlasbergLoudness`, `MooreGlasbergTimeVaryingLoudness`, `EcmaLoudness`, `EcmaTonality`, `EcmaRoughness`, `STIResult`, `SIIResult`, `NCResult`, `RCResult`, `AgeThresholdResult`, `NiptsResult`, `HtlanResult`, `ImpulseProminenceResult`, `MultipleShockResult`, `ImpulseResponseResult`, `DecayCurve`, `RoomAcousticsResult`, `ReverberationResult`, `WeightedRatingResult`, `ImpactRatingResult`, `FacadeInsulationResult`, `LabAirborneInsulationResult`, `LabImpactInsulationResult`, `SoundPowerResult`, `ReverberationSoundPowerResult`, `SoundPowerIntensityResult`, `PrecisionSoundPowerResult`, `PrecisionIntensityResult`, `IntensityResult`, `UncertaintyResult`, `AbsorptionRatingResult`, `ScatteringResult`, `DiffusionResult`, `InsituAbsorptionResult`, `WeightingResponse`, `WeightedSpectrum` and `DailyVibrationExposure`.<br>• `ax`: existing Axes, or None to build a fresh figure (Default: None)<br>• returns the Matplotlib `Axes` (an array of Axes for multi-panel figures); never calls `plt.show()`<br>• needs matplotlib (`pip install phonometry[plot]`) | `res.plot()`<br>`decay_curve(ir, fs).plot()` |

## Notes

- `zero_phase=True` (in `OctaveFilterBank.filter` and `spectrogram`) filters
  forward-backward (`sosfiltfilt`): no group delay, doubled effective
  attenuation, offline analysis only. Incompatible with `stateful=True`. The
  passband also narrows, lowering the measured broadband band level by ~0.2 to
  0.3 dB per band (a pure in-band tone is unaffected); prefer forward filtering
  when the absolute band SPL must match single-pass conventions, and reserve
  zero-phase for when the temporal envelope matters (e.g. reverberation decay).
- `mode='peak'` includes the filter's onset transient; a tone that starts
  abruptly can overshoot by ~1 dB. See [Calibration and dBFS](https://jmrplens.github.io/phonometry/guides/calibration/).
- `octave_filter()` caches filter bank designs internally (32 entries), so
  repeated calls with the same parameters skip the design phase. For explicit
  control use `OctaveFilterBank`.
- Deprecated aliases (kept for one cycle, warn on use, removal in 4.0):
  `octavefilter` → `octave_filter`, `getansifrequencies` →
  `nominal_frequencies`, `normalizedfreq` → `normalized_frequencies`,
  `calculate_sensitivity` → `sensitivity`, `coverage_factor` →
  `insulation_coverage_factor`, `expanded_uncertainty` →
  `insulation_expanded_uncertainty`, plus the renamed names
  `OCTAVE_BANDS_HZ` → `OCTAVE_BANDS`, `THIRD_OCTAVE_BANDS_HZ` →
  `THIRD_OCTAVE_BANDS`, `BASE_PLATE_BANDS_HZ` → `BASE_PLATE_BANDS` and
  `ExposureWarning` → `OccupationalExposureWarning`.

---


<!-- source: docs/theory.md | canonical: https://jmrplens.github.io/phonometry/reference/theory/ -->

# Theoretical Background

## Octave Band Frequencies (ANSI S1.11 / IEC 61260)

The mid-band frequencies (fm) and edges (f1, f2) use a base-10 ratio:

$$
G = 10^{0.3}
$$

**Mid-band:**

$$
f_m = 1000 \cdot G^{x/b}
$$

(for odd b)

**Band edges:**

$$
f_1 = f_m \cdot G^{-1/2b}, \quad f_2 = f_m \cdot G^{1/2b}
$$

## Frequency Resolution vs FFT Bin Spacing

`octave_filter` is a **time-domain fractional-octave filter bank**, not an FFT or
Welch spectrum estimator. Therefore, its result does not have a frequency
resolution in the `fs / nfft` sense.

For `fraction=3`, the output contains one scalar level per third-octave band.
The relevant frequency granularity is the standardized band definition: center
frequency, lower edge, and upper edge. Because fractional-octave bands are
logarithmically spaced, their absolute bandwidth in Hz grows with frequency
while their relative bandwidth remains approximately constant.

For example, with `fraction=3` and `limits=[12, 20000]`, the exact third-octave
band around 1 kHz is approximately:

| Nominal band | Lower edge | Center | Upper edge | Bandwidth |
| :--- | ---: | ---: | ---: | ---: |
| 1 kHz | 891.25 Hz | 1000.00 Hz | 1122.02 Hz | 230.77 Hz |

You can inspect the exact bands with:

```python
from phonometry import nominal_frequencies

fc, fl, fu, labels = nominal_frequencies(fraction=3, limits=[12, 20000])
for label, center, lower, upper in zip(labels, fc, fl, fu):
    print(label, center, lower, upper, upper - lower)
```

If you need narrowband FFT bins for tonal inspection, run Welch/FFT on the
original signal and use the phonometry band edges as masks:

```python
import numpy as np
from scipy import signal
from phonometry import octave_filter, nominal_frequencies

fs = 100_000
# any 1D pressure signal in Pa (synthesized here so the example runs)
pressure_signal_pa = 0.02 * np.random.default_rng(0).standard_normal(fs)
x = pressure_signal_pa

# Standardized third-octave levels from phonometry.
levels, centers = octave_filter(
    x,
    fs=fs,
    fraction=3,
    limits=[12, 20_000],
)

# Same standardized band definitions, including lower/upper edges.
fc, fl, fu, labels = nominal_frequencies(fraction=3, limits=[12, 20_000])

# Narrowband Welch estimate on the original signal.
nperseg = min(2**15, len(x))
freq_bins, psd = signal.welch(
    x,
    fs=fs,
    window="hann",
    nperseg=nperseg,
    noverlap=nperseg // 2,
    scaling="density",
)

# Example: list the Welch bins inside the third-octave band closest to 1 kHz.
band_index = int(np.argmin(np.abs(np.asarray(fc) - 1000.0)))
in_band = (freq_bins >= fl[band_index]) & (freq_bins <= fu[band_index])

print("Selected third-octave band:", labels[band_index])
print("Welch bin spacing:", freq_bins[1] - freq_bins[0], "Hz")
for f, pxx in zip(freq_bins[in_band], psd[in_band]):
    print(f, pxx)
```

This keeps the two concepts separate: phonometry gives standardized
fractional-octave levels, while Welch gives narrowband FFT bins. With
`fs=100000` and `nperseg=2**15`, the Welch bin spacing is about `3.05 Hz`.
Window choice and overlap affect leakage and averaging variance, but they do not
change the bin spacing of each FFT segment.

When `sigbands=True`, `octave_filter` can also return the time-domain waveform
filtered by each band. Applying Welch/FFT to one selected filtered waveform can
be useful as a diagnostic view of the content inside that filtered band, but it
does not recover FFT bins from the scalar band levels.

## Magnitude Responses |H(jw)|

The library implements standard classical filter prototypes:

**1. Butterworth:** Maximally flat passband.

$$
|H(j\omega)| = \frac{1}{\sqrt{1 + (\omega/\omega_c)^{2n}}}
$$

**2. Chebyshev I:** Equiripple in passband, steeper roll-off.

$$
|H(j\omega)| = \frac{1}{\sqrt{1 + \epsilon^2 T_n^2(\omega/\omega_c)}}
$$

**3. Chebyshev II:** Inverse Chebyshev, equiripple in stopband, flat passband.

$$
|H(j\omega)| = \frac{1}{\sqrt{1 + \frac{1}{\epsilon^2 T_n^2(\omega_{stop}/\omega)}}}
$$

**4. Elliptic:** Equiripple in both, maximum selectivity.

$$
|H(j\omega)| = \frac{1}{\sqrt{1 + \epsilon^2 R_n^2(\omega/\omega_c, L)}}
$$

**5. Bessel:** Maximally flat group delay (linear phase).

$$
H(s) = \frac{\theta_n(0)}{\theta_n(s/\omega_0)}
$$

(Where $\theta_n$ is the reverse Bessel polynomial)

### Band-edge placement

For every architecture the bank places the **−3 dB points on the band edges**.
Two cases need special handling:

- **Chebyshev II**: scipy's `Wn` is the *stopband* edge. phonometry maps the
  desired −3 dB edges to stopband edges analytically — the prototype transition
  ratio is $\cosh(\operatorname{acosh}(\sqrt{10^{A/10}-1})/N)$ — applying the
  lowpass→bandpass transform in the pre-warped bilinear domain so the mapping
  stays exact for decimated bands close to Nyquist.
- **Bessel**: designed with `norm="mag"`, which defines the −3 dB point exactly
  at `Wn` (the `phase` norm would shift the edges to roughly −10 dB).

## Filter Bank Design & Numerical Stability

To ensure **100% stability** across the entire audible spectrum (even at low
frequencies like 16 Hz with high sample rates), phonometry employs two
critical strategies:

```mermaid
flowchart LR
    X["Input signal\nfs"] --> D{"Low band?"}
    D -- "yes" --> R["Decimate\nresample_poly (1/M)"] --> S1["SOS band filter\nat fs/M"]
    D -- "no" --> S2["SOS band filter\nat fs"]
    S1 --> L["Band level (RMS/peak)"]
    S2 --> L
    S1 -- "sigbands=True" --> U["Interpolate back\nresample_poly (M/1)"] --> Y["Band signal\nat fs"]
    S2 -- "sigbands=True" --> Y
```

1. **Second-Order Sections (SOS):** All filters are implemented as a series of
   cascaded biquads. This avoids the catastrophic numerical precision loss
   associated with high-order transfer functions (coefficients a, b).
2. **Multi-rate Decimation:** For low-frequency bands, the signal is
   automatically downsampled (decimated) before filtering and upsampled
   afterwards. This keeps the digital pole locations far from the unit circle
   boundary, preventing oscillation and noise. Chebyshev II banks reserve extra
   decimation headroom so their stopband edges stay below the decimated Nyquist.

## Weighting Curves (IEC 61672-1)

The A-weighting transfer function:

$$
R_A(f) = \frac{12194^2 \cdot f^4}{(f^2 + 20.6^2)\sqrt{(f^2 + 107.7^2)(f^2 + 737.9^2)}(f^2 + 12194^2)}
$$

$$
A(f) = 20 \log_{10}(R_A(f)) + 2.00
$$

The digital filter is obtained from the analog poles/zeros via the bilinear
transform. Because the bilinear transform compresses frequencies near Nyquist,
the default `high_accuracy` mode designs and runs the filter at an internally
oversampled rate (≥ 144 kHz) — see [Frequency Weighting](https://jmrplens.github.io/phonometry/guides/weighting/).

## Time Integration

Implemented as a first-order IIR exponential integrator:

$$
y[n] = \alpha \cdot x^2[n] + (1 - \alpha) \cdot y[n-1]
$$

$$
\alpha = 1 - e^{-1 / (f_s \cdot \tau)}
$$

Where `tau` is the time constant (e.g., 125 ms for Fast).

The default initial condition is `y[-1] = 0`. Use `initial_state='first'` to
start from the first input energy, or pass a scalar/array with the previous
mean-square output state. See [Why phonometry](https://jmrplens.github.io/phonometry/reference/why-phonometry/) for the
IEC 61672-1 tone-burst verification of this implementation.

## G-weighting (ISO 7196)

The G curve extends frequency weighting into the infrasound range. ISO 7196:1995 Table 1 (p. 2) defines it by four zeros at the origin and four complex-conjugate pole pairs, given as coordinates in Hz (multiplied by $2\pi$ to obtain rad/s):

$$
z_{1..4} = 0, \qquad
p = 2\pi \left\lbrace -0.707 \pm j0.707,\  -19.27 \pm j5.16,\  -14.11 \pm j14.11,\  -5.16 \pm j19.27 \right\rbrace \ \text{Hz}
$$

The gain $k$ is chosen so that the response is exactly **0 dB at 10 Hz** (clause 4):

$$
k = \left| \frac{\prod_i (j\omega_{10} - p_i)}{\prod_i (j\omega_{10} - z_i)} \right|, \qquad \omega_{10} = 2\pi \cdot 10 \ \text{rad/s}
$$

The four zeros against eight poles shape the characteristic response: a rise of approximately **+12 dB/octave between 1 Hz and 20 Hz**, with roll-offs of approximately **24 dB/octave** below 1 Hz and above 20 Hz. Infrasound needs its own curve because near the hearing threshold the perceived loudness of very-low-frequency tones grows much more steeply with sound pressure level than at mid frequencies — a small dB increase above threshold produces a large loudness jump — so the A curve (anchored at 1 kHz) grossly misrepresents infrasonic annoyance.

Since G acts on 0.25 Hz – 315 Hz, the plain bilinear transform is already exact there and the internal oversampling used for the A/C designs (whose action extends to 16 kHz) is not applied.

See the [Frequency Weighting guide](https://jmrplens.github.io/phonometry/guides/weighting/) for usage.

## Equal-loudness contours (ISO 226:2023)

A tone has a *loudness level* of $L_N$ phon when it is judged equally loud as a 1 kHz pure tone at $L_N$ dB SPL. ISO 226:2023 Formula (1) (clause 4.1, p. 2) gives the SPL of a pure tone at frequency $f$ that reaches loudness level $L_N$:

$$
L_f = \frac{10}{\alpha_f} \log_{10}\left[ \left(4 \cdot 10^{-10}\right)^{0.3 - \alpha_f} \left( 10^{\ 0.03 L_N} - 10^{\ 0.072} \right) + 10^{\ \alpha_f (T_f + L_U)/10} \right] - L_U
$$

Formula (2) (clause 4.2) inverts it, returning the loudness level of a tone at SPL $L_f$:

$$
L_N = \frac{100}{3} \log_{10}\left[ \frac{10^{\ \alpha_f (L_f + L_U)/10} - 10^{\ \alpha_f (T_f + L_U)/10}}{\left(4 \cdot 10^{-10}\right)^{0.3 - \alpha_f}} + 10^{\ 0.072} \right]
$$

The three parameters come from Table 1 (p. 4), tabulated at the 29 preferred third-octave frequencies of ISO 266 from 20 Hz to 12.5 kHz:

- $\alpha_f$ — exponent for loudness perception at frequency $f$,
- $L_U$ — magnitude of the linear transfer function, normalized at 1 kHz ($L_U = 0$ at 1 kHz),
- $T_f$ — threshold of hearing at $f$, in dB.

The standard specifies **no interpolation** between the tabulated frequencies. Formula (1) is specified for **20 phon to 90 phon** between 20 Hz and 4 kHz, and only up to **80 phon between 5 kHz and 12.5 kHz** — above 80 phon the contour therefore stops at 4 kHz. Values outside these limits from Formula (2) are extrapolations the standard labels as informative only.

See the [Psychoacoustics guide](https://jmrplens.github.io/phonometry/guides/psychoacoustics/) for usage.

## Tone prominence: TNR and PR (ECMA-418-1)

Both methods operate on a Hann-windowed, RMS-averaged power spectrum (clauses 11.1 / 12.1) and use the clause 10 critical-band model. The critical bandwidth centred on a tone at $f$ is (Formula 2):

$$
\Delta f_c = 25.0 + 75.0 \left(1.0 + 1.4 \left(\tfrac{f}{1000}\right)^2\right)^{0.69} \ \text{Hz}
$$

Band edges are placed **arithmetically** for $f \le 500$ Hz (Formulae 4–5): $f_{1,2} = f \mp \Delta f_c / 2$, and **geometrically** above (Formulae 7–8): $f_1 = -\Delta f_c/2 + \sqrt{\Delta f_c^2 + 4 f^2}/2$, $f_2 = f_1 + \Delta f_c$.

**TNR** (clause 11). The tone band spans the spectral minima on both sides of the peak within 15 % of $\Delta f_c$ (clause 11.2). The tone power subtracts the straight line connecting the band-edge bins (Formula 9): over $N$ tone-band bins, $P_t = \sum_k P_k - (P_{\text{lo}} + P_{\text{hi}})\ N/2$. The masking-noise power is the remaining critical-band power rescaled to the full critical bandwidth (Formula 10): $P_n = (P_{\text{band}} - P_t) \cdot \Delta f_c / \Delta f_{\text{band}}$, and $\mathrm{TNR} = 10\log_{10}(P_t/P_n)$ (Formula 11). The prominence criterion (Formulae 12–13) is

$$
\mathrm{TNR}_{\text{crit}} = \begin{cases} 8.0 + 8.33 \log_{10}(1000/f_t) \ \text{dB} & f_t < 1\ \text{kHz} \\ 8.0 \ \text{dB} & f_t \ge 1\ \text{kHz} \end{cases}
$$

**PR** (clause 12) compares the level of the critical band centred on the tone, $L_M$, with the mean power of the two **contiguous** critical bands $L_L$, $L_U$ (edges from the fitted Formulae 21–22 with Tables 2–3): $\mathrm{PR} = 10\log_{10} P_M - 10\log_{10}\left[(P_L + P_U)/2\right]$ (Formula 23). For $f_t \le 171.4$ Hz the lower band is truncated at 20 Hz and its power rescaled to a **100 Hz bandwidth** (Formula 24). The criterion (Formulae 25–26) is 9.0 dB at $f_t \ge 1$ kHz, rising as $9.0 + 10.0\log_{10}(1000/f_t)$ below. Tones are assessed within the 89.1 Hz – 11.2 kHz range of interest (clauses 11.5 / 12.6).

See the [Prominent Discrete Tones guide](https://jmrplens.github.io/phonometry/guides/tone-prominence/) for usage.

## Event and dose metrics

**Sound exposure level** (SEL; LAE with A-weighting, IEC 61672-1:2013) normalizes the energy of a discrete event (aircraft flyover, train pass) to a 1 s reference duration:

$$
\mathrm{SEL} = L_{eq,T} + 10 \log_{10}\left(\frac{T}{T_0}\right), \qquad T_0 = 1\ \text{s}
$$

**Sound exposure** $E$ (IEC 61252, 3.1) is the time integral of the squared A-weighted sound pressure, expressed in pascal-squared hours:

$$
E = \int_0^T p_A^2(t)\ dt = \overline{p_A^2} \cdot T \quad [\text{Pa}^2\text{h}]
$$

When the recording is a representative sample of a longer shift, $E$ scales the measured mean square by the actual exposure duration. The **normalized 8 h level** (IEC 61252, 3.3) converts exposure to the steady level that carries the same energy over a nominal working day:

$$
L_{EX,8h} = 10 \log_{10}\left(\frac{E}{8\ \text{h} \cdot p_0^2}\right), \qquad p_0 = 20\ \mu\text{Pa}
$$

It is identical to $L_{EP,d}$ of Directive 86/188/EEC and $L_{EX,8h}$ of ISO 1999 (IEC 61252, 3.3 NOTES 5–6). The anchor of IEC 61252 (3.3 NOTE 4): an exposure of **3.2 Pa²h corresponds to $L_{EX,8h}$ of exactly 90 dB**.

**LCpeak** (IEC 61672-1:2013, subclause 5.13) is the absolute maximum of the C-weighted sound pressure expressed in dB, $L_{Cpeak} = 20\log_{10}(\max|p_C(t)|/p_0)$ — the quantity behind the 135/137/140 dB(C) occupational action limits. The implementation is verified against the one-cycle and half-cycle reference responses of Table 5.

See the [Levels guide](https://jmrplens.github.io/phonometry/guides/levels/) for usage and the [Calibration guide](https://jmrplens.github.io/phonometry/guides/calibration/) for absolute-scale setup.

## Environmental descriptors (ISO 1996-1)

The **day-evening-night level** $L_{den}$ (ISO 1996-1:2016, 3.6.4) is an energy average over the 24 h day with penalty weightings of **+5 dB for the evening** and **+10 dB for the night**:

$$
L_{den} = 10 \log_{10}\left\lbrace\frac{1}{24}\left[ t_d\ 10^{0.1 L_{day}} + t_e\ 10^{0.1 (L_{evening} + 5)} + t_n\ 10^{0.1 (L_{night} + 10)} \right]\right\rbrace
$$

with default period durations $(t_d, t_e, t_n) = (12, 4, 8)$ h — countries may define the periods differently (3.6.4 Note 1). The **day-night level** $L_{dn}$ (3.6.5) drops the evening period:

$$
L_{dn} = 10 \log_{10}\left\lbrace\frac{1}{24}\left[ t_d\ 10^{0.1 L_{day}} + t_n\ 10^{0.1 (L_{night} + 10)} \right]\right\rbrace, \qquad (t_d, t_n) = (15, 9)\ \text{h}
$$

Both are special cases of the **composite whole-day rating level** (6.5, generalizing Formulae 5–6), where each period $i$ contributes its rating level $L_i$ plus an adjustment $K_i$, weighted by its share of the day:

$$
L_R = 10 \log_{10}\left[ \sum_i \frac{h_i}{24}\ 10^{0.1 (L_i + K_i)} \right], \qquad \sum_i h_i = 24\ \text{h}
$$

The adjustments $K_i$ cover time-of-day penalties (ISO 1996-1 Table A.1: evening 5 dB, night 10 dB) as well as source-character adjustments — e.g. tonal penalties, which the ECMA-418-1 TNR/PR assessments can justify objectively.

See the [Levels guide](https://jmrplens.github.io/phonometry/guides/levels/) for usage.

## Impulsive-sound prominence (NT ACOU 112)

An impulse annoys beyond its energy, so environmental surveys after ISO 1996-2 penalize periods containing prominent impulsive sounds; NT ACOU 112:2002 makes that penalty objective. From the A-weighted, time-weighting-F level history of a single event, the onset rate (dB/s) and the level difference (dB) of the onset — which qualifies when steeper than 10 dB/s (clauses 4.5–4.7) — predict the perceived prominence (clause 7, Formula 1):

$$
P = 3 \lg(\text{onset rate}) + 2 \lg(\text{level difference}),
$$

designed to peak around 15 for very sudden, loud impulses. The adjustment to the measurement-period level takes the governing (highest-$P$) impulse (clause 8, Formula 2):

$$
K_I = 1.8\ (P - 5)\ \text{dB} \quad (P > 5;\ \text{else } K_I = 0),
$$

and the whole-day rating level combines the adjusted periods energetically (clause 8, Note 1):

$$
L_{Ar,T} = 10 \lg\Big[ \frac{1}{T} \sum_N \Delta t_N\ 10^{(L_{Aeq,N} + K_{I,N})/10} \Big].
$$

$K_I$ is exactly the kind of source-character adjustment that enters the ISO 1996-1 composite rating level above. The anchors $P(1000\ \text{dB/s}, 30\ \text{dB}) = 9 + 2\lg 30 = 11.95$ and $K_I(P{=}10) = 9.0$ dB are reproduced exactly.

See the [Impulse Prominence guide](impulse-prominence.md) for usage.

## Zwicker loudness (ISO 532-1)

The ear analyzes sound in **critical bands**: frequency regions within which energy is summed before loudness is formed. The **Bark scale** maps frequency to critical-band rate $z$, 0 to 24 Bark, and ISO 532-1:2017 samples the specific loudness $N'(z)$ at 0.1-Bark steps (240 values). The implementation is a clean-room port of the standard's normative reference program (Annex A.4) and proceeds in stages:

1. **One-third-octave levels** — 28 bands, 25 Hz to 12.5 kHz (the Annex A filterbank at 48 kHz, Tables A.1/A.2). For time-varying sounds the squared band outputs are smoothed by three cascaded low-passes with $\tau = 2/(3 f_c)$ ($f_c$ capped at 1 kHz) and sampled every 2 ms.
2. **Low-frequency grouping** — the 11 bands up to 250 Hz receive the equal-loudness corrections of Table A.3 and are summed into the first three critical bands (25–80, 100–160, 200–250 Hz).
3. **a0 transmission** — the outer/middle-ear transfer correction of Table A.4 (plus the diffuse-field difference of Table A.5 when `field='diffuse'`) yields the critical-band levels $L_E$.
4. **Core loudness** — each of the 20 critical bands is transformed with the threshold-in-quiet levels $L_{TQ}$ of Table A.6 (after the bandwidth adaptation DCB of Table A.7):

$$
N_c = 0.0635 \cdot 10^{0.025 L_{TQ}} \left[ \left( 1 - s + s \cdot 10^{(L_E - L_{TQ})/10} \right)^{0.25} - 1 \right] \ \text{sone/Bark}, \qquad s = 0.25
$$

(the reference program's form of Zwicker's loudness transformation; bands below threshold contribute zero).

5. **Slopes** — level-dependent upper masking slopes (steepness per specific-loudness range and critical band, Tables A.8/A.9) attach decaying flanks toward higher $z$; the total loudness is the area under the pattern:

$$
N = \int_0^{24} N'(z)\ dz \ \ \text{sone}
$$

For time-varying sounds a nonlinear temporal decay (time constants 5/15/75 ms, clause 6.3) and the duration-dependent weighting of the total loudness (3.5 ms and 70 ms low-passes weighted 0.47/0.53, clause 6.4) precede the 500 Hz loudness-vs-time output and the percentile values N5/N10 (clause 6.5).

**Sone and phon** are tied together by the 1 kHz anchor (1 sone = 40 phon; clause 5.6):

$$
N = 2^{(L_N - 40)/10} \ \text{sone} \qquad \Longleftrightarrow \qquad L_N = 40 + 10 \log_2 N \ \text{phon} \qquad (N \ge 1)
$$

below 1 sone the reference program uses $L_N = 40 (N + 0.0005)^{0.35}$, floored at 3 phon.

See the [Psychoacoustics guide](https://jmrplens.github.io/phonometry/guides/psychoacoustics/) for usage.

## Advanced loudness models & sound quality

ISO 532-1 is one of three loudness models; two newer families refine the auditory front-end and add the sound-quality metrics tonality and roughness.

### Moore-Glasberg loudness (ISO 532-2:2017, ISO 532-3:2023)

Instead of Zwicker's fixed critical bands, the Moore-Glasberg model forms a continuous **excitation pattern** on the ERB-number ("Cam") scale using level-dependent **rounded-exponential (roex)** auditory filters. As a function of the normalized frequency deviation $g = |f - f_c| / f_c$ from a filter centred at $f_c$, the filter weighting is

$$
W(g) = (1 + p\ g)\ e^{-p\ g}
$$

where the slope $p$ grows with the source level, broadening the lower skirt as level rises (ISO 532-2, Formulae 2–5); this reproduces the upward spread of masking. Passing the stimulus intensity through every filter gives the excitation $E(i)$, and a compressive law maps it to the **specific loudness** $N'(i)$ in sone/Cam (Formulae 7–9), of the mid-level form

$$
N'(i) = C \left[ \left( G\ \frac{E(i)}{E_0} + A \right)^{\alpha} - A^{\alpha} \right]
$$

with the calibration constant $C = 0.0617$ sone/Cam (ISO 532-2; $0.063$ in ISO 532-3). The total loudness is the area under the pattern,

$$
N = \int N'(i)\ di \ \ \text{sone}
$$

and a binaural-inhibition stage (Formulae 10–13) combines the ears so a diotic sound is louder than the same sound at one ear. The 1 kHz / 40 dB SPL anchor gives exactly 1 sone.

**ISO 532-3** makes this time-varying. A running spectrum from six parallel Hann-windowed FFTs (segment lengths 2–64 ms, each contributing its own frequency range, updated every $T_0 = 1$ ms) drives the same excitation and specific-loudness chain, integrated by two cascaded first-order smoothers with $\alpha = 1 - e^{-T_0 / \tau}$,

$$
S(t) = \alpha\ x(t) + (1 - \alpha)\ S(t - 1)
$$

using a fast time constant on the attack and a slower one on the release. This yields the **short-term loudness** $S'(t)$ (attack/release near 20–30 ms) and the **long-term loudness** $S''(t)$ (near 0.1–0.75 s); the peak long-term loudness $N_{\max} = \max_t S''(t)$ predicts the loudness of sounds up to about 5 s.

### Sottek Hearing Model (ECMA-418-2:2025)

ECMA-418-2 builds all three of its metrics on one auditory front-end (Clause 5): an outer/middle-ear filter, a bank of 53 overlapping gammatone-like band-pass filters spaced on the Bark_HMS scale ($z = 0.5$ to $26.5$), half-wave rectification, and a short-block RMS $\tilde{p}(l, z)$ per band $z$ and time block $l$. A compressive nonlinearity (Formula 23) turns the band RMS into the **specific basis loudness** $N'_{\mathrm{basis}}(l, z)$, whose calibration constant $c_N$ fixes a 1 kHz / 40 dB SPL tone at 1 sone_HMS. The loudness assembles the tonal and noise loudness (below) over bands and time (Formulae 113–117); it grows about $1.65\times$ per 10 dB, more slowly than Zwicker's factor of 2 — an intrinsic property of the Sottek summation.

### Tonality — autocorrelation of the band signal (ECMA-418-2)

A tonal component is periodic, so it survives in the **autocorrelation function** (ACF) of a band's rectified signal while broadband noise decorrelates. For each band the unbiased ACF of the block is

$$
\phi_z(m) = \frac{1}{M - m} \sum_{n=0}^{M - 1 - m} p_z(n)\ p_z(n + m)
$$

A windowed spectral estimate of $\phi_z$ separates a **tonal loudness** $N'_{\mathrm{tonal}}(l, z)$ from the **noise loudness** $N'_{\mathrm{noise}}(l, z)$ (Formulae 36–48). The specific tonality is the tonal loudness scaled by a smooth signal-to-noise gate $q(l)$ (Formulae 49–51),

$$
T'(l, z) = c_T\ q(l)\ N'_{\mathrm{tonal}}(l, z)
$$

and the single value $T$ (tu_HMS) is the gated time-average of the per-block maximum over bands (Formulae 61–64). The constant $c_T$ fixes the 1 kHz / 40 dB tone at 1 tu_HMS, and the band of the ACF peak gives the tonal frequency $f_{\mathrm{ton}}$.

### Roughness — envelope modulation (ECMA-418-2)

Roughness is the sensation of fast (roughly 20–300 Hz) amplitude modulation, strongest near 70 Hz. From each band's envelope $p_E(n)$ (Hilbert magnitude), a modulation spectrum is formed and weighted by a modulation-rate function peaking near 70 Hz and by the modulation depth; correlating the modulation across neighbouring bands and applying the specified temporal filtering yields the **specific roughness** $R'(l_{50}, z)$ and the time-dependent roughness

$$
R(l_{50}) = \sum_z R'(l_{50}, z) \ \ \text{asper}
$$

(Formulae 65–111). The single value $R$ is the 90th percentile of $R(l_{50})$ over time (Clause 7.1.10); the constant $c_R$ (Formula 104) calibrates the reference sound — a 1 kHz carrier 100 % amplitude-modulated at 70 Hz at 60 dB SPL — to 1 asper.

### Sharpness (DIN 45692)

Sharpness condenses the high-frequency emphasis of a sound into one number: the $g(z)$-weighted first moment of the ISO 532-1 stationary specific-loudness pattern (DIN 45692:2009, Equation 1):

$$
S = k\ \frac{\int_0^{24} N'(z)\ g(z)\ z\ dz}{\int_0^{24} N'(z)\ dz} \ \text{acum}, \qquad
g(z) = \begin{cases} 1 & z \le 15.8\ \text{Bark} \\ 0.15\ e^{0.42 (z - 15.8)} + 0.85 & z > 15.8\ \text{Bark} \end{cases}
$$

evaluated on the same 240-bin, 0.1-Bark grid. The constant $k$ is not hard-coded but derived from the calibration requirement (clause 6): a critical-band-wide narrowband noise 920–1080 Hz at 60 dB SPL scores exactly 1 acum — the derived $k = 0.108$ lands inside the normative window $0.105 \le k < 0.115$ (clause 5.2). The informative Annex B weightings are provided under the same 1-acum anchor: von Bismarck (knee at 15 Bark, $0.2\ e^{0.308(z-15)} + 0.8$) and Aures (loudness-dependent, $g(z) = 0.078\ (e^{0.171 z}/z)\ N/\ln(0.05 N + 1)$). The Table A.2 narrow-band targets are reproduced within the clause 6 tolerance (5 % or 0.05 acum): 0.38 acum at 250 Hz, 1.00 at 1 kHz, 1.78 at 2.5 kHz, 2.82 at 4 kHz.

See the [Psychoacoustics guide](https://jmrplens.github.io/phonometry/guides/psychoacoustics/) for usage.

## Modulation transfer and STI (IEC 60268-16)

Speech intelligibility rides on the slow intensity modulations of the speech envelope. The **modulation transfer function** $m(F)$ of a transmission channel is the ratio of received to emitted modulation depth of the octave-band intensity envelope at modulation frequency $F$; the full STI evaluates it at the 14 one-third-octave modulation frequencies 0.63–12.5 Hz in the seven octave bands 125 Hz – 8 kHz (A.2.2). From a measured impulse response the **Schroeder closed form** gives it directly (indirect method):

$$
m_k(f_m) = \frac{\left| \int_0^{\infty} h_k^2(t)\ e^{-j 2 \pi f_m t}\ dt \right|}{\int_0^{\infty} h_k^2(t)\ dt}
$$

Steady background noise multiplies each band's $m$ by the intensity ratio (the noise term):

$$
m'_k = m_k \cdot \frac{I_k}{I_k + I_{n,k}} = \frac{m_k}{1 + 10^{-\mathrm{SNR}_k/10}}
$$

and when absolute band levels are known the full correction $m'_k = m_k I_k / (I_k + I_{am,k} + I_{rt,k} + I_{n,k})$ adds the auditory masking intensity $I_{am,k}$ (from the next lower octave band, Table A.2) and the absolute reception threshold $I_{rt,k}$ (Table A.3). Each corrected $m$ maps to an **effective SNR**, clipped to the ±15 dB range where intelligibility actually varies, then to a transmission index (A.5.4/A.5.5):

$$
\mathrm{SNR}_{\mathrm{eff}} = 10 \log_{10} \frac{m}{1 - m}\ \text{dB}, \qquad \mathrm{TI} = \frac{\mathrm{SNR}_{\mathrm{eff}} + 15}{30}
$$

The band MTI is the mean TI over the modulation frequencies, and the STI weights the bands with the male factors $\alpha_k$, $\beta_k$ of Ed. 5 Table A.1 (A.5.6):

$$
\mathrm{STI} = \sum_{k=1}^{7} \alpha_k\ \mathrm{MTI}_k - \sum_{k=1}^{6} \beta_k \sqrt{\mathrm{MTI}_k\ \mathrm{MTI}_{k+1}}
$$

truncated to 1.0. STIPA (Annex B) samples the same physics with just two modulation frequencies per band (Table B.1) on a test signal with source modulation index 0.55; the received depths are measured by sine/cosine correlation of the ~100 Hz low-passed intensity envelopes $I_k(t)$ over an integer number of modulation periods:

$$
m_{dr} = \frac{2 \sqrt{\left( \sum_t I_k(t) \sin 2 \pi f_m t \right)^2 + \left( \sum_t I_k(t) \cos 2 \pi f_m t \right)^2}}{\sum_t I_k(t)}, \qquad m = \frac{m_{dr}}{0.55}
$$

See the [Speech Transmission Index guide](https://jmrplens.github.io/phonometry/guides/speech-transmission/) for usage.

## Speech Intelligibility Index (ANSI S3.5)

Where the STI characterizes a transmission channel, the SII (ANSI S3.5-1997) predicts intelligibility from what the listener can actually hear: 18 one-third-octave bands 160 Hz – 8 kHz, each contributing its band importance $I_i$ (Table 3, $\sum I_i = 1$, peaking near 2 kHz). All inputs are equivalent spectrum levels (clauses 3.11/3.55). Speech masks itself upward: each band's masking spectrum $Z_i$ (clause 5.4) accumulates the lower bands along slopes $C_i = -80 + 0.6\,(B_i + 10 \lg f_i - 6.353)$ dB, and the disturbance is the energetic sum of masking and hearing floor, $D_i = 10 \lg(10^{0.1 Z_i} + 10^{0.1 X'_i})$ with $X'_i = X_i + T'_i$ the reference internal noise spectrum plus the listener's hearing-threshold shift (clauses 5.5/5.6). The band audibility clips the speech-to-disturbance margin into $[0, 1]$ (clause 5.8), a level-distortion factor discounts overly loud presentation (clause 5.7), and the index sums (clause 6):

$$
A_i = \operatorname{clip}\Big( \frac{E'_i - D_i + 15}{30},\ 0,\ 1 \Big), \qquad
L_i = \operatorname{clip}\Big( 1 - \frac{E'_i - U_i - 10}{160},\ 0,\ 1 \Big), \qquad
\mathrm{SII} = \sum_{i=1}^{18} I_i\ L_i\ A_i .
$$

The Table 3 standard speech spectra for the normal, raised, loud and shout vocal efforts are built in (25.01 / 33.86 / 42.16 / 51.31 dB at 1 kHz); $U_i$ in the level-distortion factor is always the normal-effort spectrum. The anchor values: the normal-effort spectrum in quiet with normal hearing scores SII ≈ 0.996, the masking-spectrum reference values are matched to $10^{-4}$, and the vocal-effort spectra are cross-verified against the Google and CRAN reference implementations.

See the [Speech Intelligibility guide](https://jmrplens.github.io/phonometry/guides/speech-intelligibility/) for usage.

## Hearing thresholds and presbycusis (ISO 389-7, ISO 7029)

ISO 389-7:2006 Table 1 fixes the reference threshold of hearing of otologically normal young adults — the free-field and diffuse-field SPL corresponding to 0 dB HL at the 11 audiometric frequencies 125 Hz – 8 kHz (22.1 dB at 125 Hz for both fields, 2.4/0.8 dB free/diffuse at 1 kHz, diverging at high frequency to 12.6 vs 6.8 dB at 8 kHz). ISO 7029:2017 describes how that threshold shifts statistically with age: the median deviation from age 18 is (clause 4.2, Table 1)

$$
\Delta H_{md} = a\ (Y - 18)^b \ \text{dB},
$$

and any fractile follows a two-sided Gaussian model (clause 4.4), $\Delta H_Q = \Delta H_{md} + z(Q)\ s$, using the upper spread $s_u$ for $z \ge 0$ (worse than median) and the lower spread $s_l$ otherwise — each a degree-5 polynomial in $Y - 18$ per sex and frequency (clause 4.3, Tables 2–5). At age 18 every deviation is zero by construction. The formulae are established to 80 years at and below 2 kHz and to 70 years above; beyond that the evaluation is an extrapolation. Anchors: at 60 years the medians evaluate to 7.85 dB (male, 1 kHz), 20.21 dB (male, 4 kHz) and 15.32 dB (female, 4 kHz) — the Table 1 formula to $10^{-3}$.

See the [Hearing Threshold guide](hearing-threshold.md) for usage.

## Noise-induced hearing loss (ISO 1999)

ISO 1999:2013 predicts the permanent threshold shift a noise-exposed population accrues. The median noise-induced shift (NIPTS) for 10–40 years of exposure is (clause 6.3.1, Formula 2, Table 1):

$$
N_{50} = \big[ u + v \lg(t/t_0) \big]\ (L_{EX,8h} - L_0)^2, \qquad t_0 = 1\ \text{yr},
$$

quadratic in the excess over the frequency-dependent onset level $L_0$ (75 dB at 4 kHz — the most sensitive band — up to 93 dB at 500 Hz) and zero below it; under 10 years it scales as $\lg(t+1)/\lg 11$ (Formula 3). Fractiles add the spread, $N_Q = N_{50} + z\ d_{u,l}$ with $d = (X + Y \lg t)(L_{EX,8h} - L_0)^2$ (clause 6.3.2, Formulae 4–7, Tables 2/3), clamped at zero; the convention counts the fraction of the population with the *smaller* shift, so $Q = 0.9$ is the most-susceptible decile (reliable range 0.05–0.95). The hearing threshold level associated with age and noise (HTLAN) combines NIPTS with the ISO 7029 age component at the same fractile through the compressed sum (clause 6.1, Formula 1):

$$
H' = H + N - \frac{H\ N}{120}.
$$

The Annex D worked examples (Tables D.1–D.4; e.g. 100 dB / 40 yr at 3 kHz: 29/38/60 dB at the 0.10/0.50/0.90 fractiles) are reproduced exactly at the standard's integer rounding, and the Formula 2 hand value at 4 kHz / 20 yr / 90 dB is $N_{50} = 12.94$ dB.

See the [Noise-Induced Hearing Loss guide](noise-induced-hearing-loss.md) for usage.

## Sound intensity (IEC 61043)

Sound intensity is the time-averaged acoustic power flux $I = \overline{p u}$. The particle velocity follows from **Euler's equation** (linearized conservation of momentum):

$$
\rho_0 \frac{\partial u}{\partial t} = -\frac{\partial p}{\partial r}
$$

A p-p probe approximates the pressure gradient by the **finite difference** of two microphones a spacer distance $\Delta r$ apart (IEC 61043:1994, definition 3.2):

$$
p = \frac{p_1 + p_2}{2}, \qquad u = -\frac{1}{\rho_0 \Delta r} \int (p_2 - p_1)\ dt, \qquad I = \overline{p\ u}
$$

For stationary signals the same estimator has an exact frequency-domain form through the imaginary part of the one-sided **cross spectrum** $G_{12}$ of the two pressures — the implementation estimates it with Welch-averaged, Hann-windowed segments:

$$
I(f) = -\ \frac{\mathrm{Im}\lbrace G_{12}(f)\rbrace}{2 \pi f\ \rho_0\ \Delta r}
$$

The finite difference underestimates the true plane-wave intensity by the factor

$$
\frac{\sin(k \Delta r)}{k \Delta r}, \qquad k = \frac{2 \pi f}{c}
$$

— IEC 61043 clause 7.3 specifies the probe intensity response with exactly this argument and Table 3 tabulates it (e.g. −10.5 dB at 6.3 kHz for a 25 mm spacer). Below $f = 0.1 c / \Delta r$ (i.e. $k \Delta r$ under 0.63) the bias stays within about 0.3 dB; `bias_correction` provides the reciprocal factor per band and `max_valid_frequency` the bound.

The **pressure-intensity index** $\delta_{pI} = L_p - L_I$ measures how reactive the field is: in a free plane progressive wave it equals $10 \log_{10}(\rho_0 c / 400) = 0.14$ dB, while large values flag reactive or noisy fields in which the inter-channel phase error dominates. ISO 9614-1:1993 Annex A generalizes it over a measurement surface as the indicator F2 (with F3 for negative partial power and F4 for field non-uniformity), and the instrument's **dynamic capability** $L_d = \delta_{pI0} - K$ (pressure-residual intensity index minus the bias error factor: 10 dB for grades 1/2, 7 dB for grade 3) must exceed F2 for the measurement to be valid (criterion 1).

See the [Sound Intensity guide](https://jmrplens.github.io/phonometry/guides/intensity/) for usage.

## Room noise criteria (ANSI S12.2)

ANSI/ASA S12.2-2019 rates steady background noise in rooms against families of octave-band curves (16 Hz – 8 kHz). The **NC rating** uses the tangency method on the Table 1 curves (NC-15 to NC-70): each measured band is interpolated against the tabulated curve values, the rating is the highest per-band index and the band that sets it is the governing band — the interpolation makes the rating continuous (an NC-42.5 is reported as such, not snapped to a curve). The **RC Mark II** contour (Annex D) is a pure −5 dB/octave line keyed to its 1000 Hz value with a low-frequency floor of $\max(\mathrm{RC} + 25,\ 55)$ dB at 16/31.5 Hz; the rating is the arithmetic mean of the 500/1000/2000 Hz levels rounded to an integer (clause D.4), and the spectral-quality tag compares the spectrum with the reference contour (clause D.3): rumble "R" when any band at or below 500 Hz exceeds it by more than 5 dB, hiss "H" when any band at or above 1 kHz exceeds it by more than 3 dB (both together "RH"), else neutral "N" — reported as e.g. RC-35(N). The generated RC contours reproduce Table D.1 digit for digit, and feeding any Table 1 NC curve back returns its own rating. NCB, RNC (Annex A) and the QAI (clause D.5) are deliberately out of scope.

See the [Room Noise guide](room-noise.md) for usage.

## Room and building acoustics (ISO 18233, ISO 3382, ISO 16283, ISO 10140, EN 12354, ISO 12999, ISO 717, ISO 354)

### Deterministic-excitation impulse response (ISO 18233)

A room/transmission path is modelled as **linear time-invariant**, so its impulse response $h(t)$ carries everything. ISO 18233 replaces the classical noise-burst decay with a deterministic excitation that is **deconvolved** into $h(t)$, gaining 20–30 dB of effective signal-to-noise ratio. The exponential sine sweep (ESS, Annex B) has instantaneous frequency $f(t) = f_1 (f_2/f_1)^{t/T}$, so its phase is the closed-form integral of $2 \pi f(t)$:

$$
\varphi(t) = \frac{2 \pi f_1 T}{\ln(f_2/f_1)} \left[ \left( \frac{f_2}{f_1} \right)^{t/T} - 1 \right] .
$$

A constant time-per-octave makes the ESS spectrum pink (−3 dB/octave). Deconvolution is done by **linear** (non-circular, zero-padded) spectral division $H = Y\ \overline{X} / (|X|^2 + \varepsilon)$, the Tikhonov term $\varepsilon$ (a fraction of $\max |X|^2$) preventing noise blow-up at the band edges. Since a low-to-high sweep places harmonic-distortion products at negative arrival times, they fall in the wrapped tail and are removed by keeping the causal part (Farina). The MLS method (Annex A) instead exploits that the circular autocorrelation of a maximum-length sequence of length $2^N-1$ is a periodic delta, so $h = \operatorname{xcorr}_{\text{circ}}(\text{recorded}, \text{mls}) / 2^N$; synchronous averaging of $n$ periods adds $10 \log_{10} n$ dB.

### Schroeder backward integration (ISO 3382-1, 5.3.3)

The band decay curve is the **backward-integrated** squared IR (Schroeder):

$$
E(t) = \int_t^{\infty} p^2(\tau)\ d\tau = \int_0^{\infty} p^2\ d\tau - \int_0^t p^2\ d\tau , \qquad L(t) = 10 \log_{10} \frac{E(t)}{E(0)}\ \text{dB},
$$

i.e. a reversed cumulative sum in discrete time. Backward integration cancels the random fluctuation of a single squared IR: for a purely exponential energy decay $p^2(t) = e^{-a t}$ it gives $E(t) = e^{-a t}/a$, an exactly straight line $L(t) = -(10 a / \ln 10)\ t$. Background noise flattens $E(t)$, so integration is truncated at the crossing $t_1$ of the fitted decay line with the noise level and the missing tail is compensated by an exponential with the fitted rate; without that term the finite integral systematically **underestimates** $T$.

### Regression windows and validity (ISO 3382-2, Clause 6, Annex B/C)

Reverberation time is a least-squares fit $L = a + b t$ over a window, extrapolated to 60 dB via $T = -60/b$ (Annex C): **EDT** on 0 to −10 dB, **T20** on −5 to −25 dB, **T30** on −5 to −35 dB. A single-slope decay gives EDT = T20 = T30; a fast early / slow late double slope gives EDT < T30. Validity uses the dynamic-range rule of 5.3.3 — the noise must sit at least 25 dB below the IR peak for EDT (evaluation span + 15 dB), tightened to 46 dB for T20 and 54 dB for T30 so the tail-compensation bias of a flagged-valid value stays within the 5 % JND — and the **curvature** $C = 100\ (T_{30}/T_{20} - 1)$ % (Annex B) flags a non-straight decay above 10 %.

### Clarity, definition and centre time (ISO 3382-1, Annex A)

Splitting the energy at an early/late boundary $t_e$ gives the early-to-late index and the definition ratio:

$$
C_{te} = 10 \log_{10} \frac{\int_0^{t_e} p^2\ dt}{\int_{t_e}^{\infty} p^2\ dt}\ \text{dB}, \qquad D_{50} = \frac{\int_0^{0.05} p^2\ dt}{\int_0^{\infty} p^2\ dt}, \qquad C_{50} = 10 \log_{10} \frac{D_{50}}{1 - D_{50}},
$$

with $t_e = 50$ ms (C50, speech) or 80 ms (C80, music), and the **centre time** $T_s = \int_0^{\infty} t\ p^2\ dt / \int_0^{\infty} p^2\ dt$. For a pure exponential decay these have closed forms $C_{te} = 10 \log_{10}(e^{a t_e} - 1)$ and $T_s = 1/a$; at $T = 1$ s ($a = 13.8155$) they evaluate to C80 = 3.05 dB, C50 = −0.02 dB, D50 = 0.499 and Ts = 72.4 ms — the values the implementation reproduces. Table A.1 JNDs (EDT 5 %, C80 1 dB, D50 0.05, Ts 10 ms) bound how finely each is worth reporting.

### Open-plan spatial decay (ISO 3382-3, Clause 6)

The spatial decay rate of A-weighted speech is the ordinary least-squares slope of $L_{p,A,S}$ against $\lg(r/r_0)$ ($r_0 = 1$ m) over the 2–16 m positions, rescaled to a per-doubling figure, and the nominal level is read off the same line at 4 m:

$$
L = a + b\ \lg(r/r_0), \qquad D_{2,S} = -\lg(2)\ b, \qquad L_{p,A,S,4\text{m}} = a + b\ \lg(4/r_0).
$$

The distraction distance rD and privacy distance rP are the distances where a **linear** (not logarithmic) regression of STI against distance crosses 0.50 and 0.20; a non-negative fitted slope (STI not falling with distance) makes them undefined, realising the standard's "can prove impossible to determine" note.

### Field insulation and weighted rating (ISO 16283-1, ISO 717-1)

Per one-third-octave band the level difference $D = L_1 - L_2$ (energy-averaged over microphone positions, $L = 10 \log_{10}[(1/n) \sum_i 10^{L_i/10}]$) is normalised two ways: the standardized level difference $D_{nT} = D + 10 \log_{10}(T/T_0)$ with $T_0 = 0.5$ s (so $D_{nT} = D$ when $T = T_0$), and the apparent sound reduction index $R' = D + 10 \log_{10}(S/A)$ with the Sabine absorption area $A = 0.16\ V / T$, hence $R' = D + 10 \log_{10}[S T / (0.16\ V)]$.

The single-number rating (ISO 717-1, Clause 4.4) shifts the Table 3 **reference curve** in 1 dB steps toward the measured curve until the sum of *unfavourable* deviations $\sum_i \max(0, \text{ref}_i + k - \text{meas}_i)$ is maximal but $\le$ 32.0 dB (16 thirds) or 10.0 dB (5 octaves); the rating $R_w$ is the shifted reference at 500 Hz. The **spectrum adaptation terms** are $C = X_{A1} - X_w$ and $C_{tr} = X_{A2} - X_w$ with $X_{Aj} = -10 \log_{10} \sum_i 10^{(L_{ij} - X_i)/10}$ (Table 4 spectra No. 1 pink noise, No. 2 urban traffic), each rounded to an integer. The ISO 717-1 Annex C worked example ($R_w = 30$, $C = -2$, $C_{tr} = -3$, unfavourable sum 31.8 dB) is reproduced exactly.

### Impact insulation and absorption (ISO 16283-2, ISO 717-2, ISO 354)

Impact insulation swaps the airborne source for a standardized **tapping
machine** and rates the receiving-room level, so the sign conventions flip. The
standardized and normalized impact levels are $L'_{nT} = L_i - 10 \log_{10}(T/T_0)$
(the reverberation term is *subtracted*, opposite to $D_{nT}$) and
$L'_n = L_i + 10 \log_{10}(A/A_0)$ with $A_0 = 10$ m² and $A = 0.16\ V/T$. The
ISO 717-2 rating shifts the Table 3 reference curve until $\sum_i \max(0, \text{meas}_i - (\text{ref}_i + k))$
is maximal but $\le$ 32.0 dB (16 thirds) or 10.0 dB (5 octaves) — the
*unfavourable* deviation now counts where the **measurement exceeds** the
reference (impact noise is worse when louder), the mirror image of ISO 717-1.
The rating is the shifted reference at 500 Hz, reduced by a further 5 dB for
octave bands, and the adaptation term is $C_I = L_{n,\text{sum}} - 15 - L_{n,w}$
with the energetic sum $L_{n,\text{sum}} = 10 \log_{10} \sum_i 10^{L_i/10}$ over
100–2500 Hz (thirds) or 125–2000 Hz (octaves). The ISO 717-2 Annex C examples
are reproduced exactly (thirds $L_{n,w} = 79$, $C_I = -11$; octaves $54$, $0$),
via the same monotone shift search as ISO 717-1 run on the negated curves.

Sound absorption (ISO 354) measures the equivalent absorption area from
Sabine's relation applied to a reverberation room empty and with the specimen:
$A = 55.3\ V/(c\ T) - 4 V m$ (the $4 V m$ term is the air absorption, $m$ the
power attenuation coefficient in 1/m), so the specimen area is
$A_T = A_2 - A_1$ and its coefficient $\alpha_s = A_T/S$. With the speed of
sound from Eq. (6), $c = 331 + 0.6\ t$ (°C), and $m$ converted from an
ISO 9613-1 attenuation coefficient by $m = \alpha / (10 \lg e)$. Because
diffraction and edge scattering intercept more than the flat sample area,
$\alpha_s$ is left unclamped and may exceed 1.0 (Clause 3.7 NOTE 2).

### Laboratory vs field normalization (ISO 10140, ISO 16283)

The field indices carry a prime because they include flanking transmission
around the partition; the laboratory indices do not, because a qualified
facility suppresses it. The algebra is otherwise identical, differing only in
which quantity is normalised. The airborne pair is the direct laboratory sound
reduction index $R = L_1 - L_2 + 10 \log_{10}(S/A)$ (ISO 10140-2) versus the
apparent field index $R' = L_1 - L_2 + 10 \log_{10}(S/A)$ (ISO 16283-1), the
same closed form evaluated with the facility's known $A$ or the room's measured
$A = 0.16\ V/T$. The impact pair is the normalized laboratory level
$L_n = L_i + 10 \log_{10}(A/A_0)$ (ISO 10140-3) versus the field $L'_n$
(ISO 16283-2), both referenced to $A_0 = 10$ m². Before either is formed the
receiving-room level is corrected for background noise by the energy
subtraction $L = 10 \log_{10}(10^{L_{sb}/10} - 10^{L_b/10})$ for a 6–15 dB
signal-to-background margin, capped at a fixed $1.3$ dB (the limit of
measurement) at or below 6 dB and omitted at or above 15 dB (ISO 10140-4,
Clause 4.3) — the laboratory analogue of the 6/10 dB rule of ISO 16283-1. The
façade extension (ISO 16283-3) replaces the source-room level by the level 2 m
in front of the façade, $D_{2m} = L_{1,2m} - L_2$, and adds a fixed
angle-of-incidence correction to the element sound reduction index, $-1.5$ dB
for the 45° loudspeaker method ($R'_{45°}$) and $-3$ dB for the all-angle
road-traffic method ($R'_{tr,s}$); all three carry the ISO 717-1 airborne
single number.

### Flanking transmission prediction (EN 12354-1/2)

The apparent field index is the energetic sum of the direct path $Dd$ and, for
each flanking element $F=f$ across its junction with the separating element, the
three paths $Ff$, $Df$ and $Fd$ (EN 12354-1, simplified single-number model,
Formula 26):

$$
R'_w = -10 \log_{10}\Big[ 10^{-R_{Dd,w}/10}
       + \sum 10^{-R_{Ff,w}/10} + \sum 10^{-R_{Df,w}/10}
       + \sum 10^{-R_{Fd,w}/10} \Big].
$$

The direct path is $R_{Dd,w} = R_{s,w} + \Delta R_{Dd,w}$ (Formula 27), the
separating-element laboratory index plus any lining improvement. Each flanking
path (Formula 28a) is

$$
R_{ij,w} = \frac{R_{i,w} + R_{j,w}}{2} + \Delta R_{ij,w} + K_{ij}
         + 10 \log_{10}\frac{S_s}{l_0\ l_f},
$$

with $R_{i,w}$, $R_{j,w}$ the laboratory indices of the two elements meeting at
the junction ($i$ source side, $j$ receiving side), $\Delta R_{ij,w}$ the
combined lining improvement, $S_s$ the separating-element area, $l_f$ the
junction coupling length and $l_0 = 1$ m the reference coupling length. $K_{ij}$
is the junction **vibration reduction index** (Annex E), an empirical function of
the mass ratio $M = \log_{10}(m'_{\perp,i}/m'_i)$ — for a rigid cross-junction
$K_{13} = 8.7 + 17.1 M + 5.7 M^2$ (through) and $K_{12} = 8.7 + 5.7 M^2$
(corner), read at 500 Hz — floored at $K_{ij,\min} = 10 \log_{10}[l_f\ l_0
(1/S_i + 1/S_j)]$ (Formula 29). Two linings combine as $\max(a,b) + \min(a,b)/2$
(Formulas 30/31). The impact counterpart (EN 12354-2, Formula 21) is the direct
subtraction $L'_{n,w} = L_{n,w,eq} - \Delta L_w + K$, with the bare-floor
equivalent level $L_{n,w,eq} = 164 - 35 \log_{10}(m'/m'_0)$ (Annex B), the
covering improvement $\Delta L_w$ (ISO 717-2) and the flanking correction $K$
from Table 1. The EN 12354-1 Annex H.3 ($R'_w = 52$ dB) and EN 12354-2 Annex E.3
($L'_{n,w} = 45$ dB) worked examples are reproduced exactly; the simplified
model is stated to have about a 2 dB standard deviation (Clause 5).

### Absorption in enclosed spaces (EN 12354-6)

EN 12354-6:2003 predicts the equivalent absorption area of a room from its
parts (the normative Clause 4 model). The total (Formula 1) sums the surfaces,
the objects and the air:

$$
A = \sum_i \alpha_{s,i}\ S_i + \sum_j A_{obj,j} + \sum_k \alpha_{s,k}\ S_k + A_{air},
\qquad A_{air} = 4\ m\ V\ (1 - \psi),
$$

with $m$ the power attenuation coefficient of air (Formula 2; Table 1
tabulates it for six temperature/humidity climates over the octave bands
125 Hz – 8 kHz), $\psi = \sum V_{obj} / V$ the volume fraction occupied by
objects (Formula 3), and a hard irregular object approximated by
$A_{obj} = V_{obj}^{2/3}$ (Formula 4). The reverberation time follows from
Sabine applied to the free volume (clause 4.4, Formula 5):

$$
T = \frac{55.3}{c_0}\ \frac{V\ (1 - \psi)}{A},
$$

with $c_0 = 345.6$ m/s chosen so that $55.3/c_0$ is the familiar $0.16$
(clause 4.4 NOTE). The three Annex E worked cases are reproduced: the
bare 29.75 m³ room gives $A = 2.26$ m² and $T = 2.1$ s at 1 kHz, and adding
hard objects ($\psi \approx 0.072$) raises $A$ to 5.03 m² and drops $T$ to
0.9 s. The informative Annex D method for irregular spaces and unevenly
distributed absorption is out of scope.

See the [Enclosed-Space Absorption guide](enclosed-space-absorption.md) for usage.

### Measurement uncertainty (ISO 12999-1)

ISO 12999-1 supplies the uncertainty of the quantities above from
inter-laboratory (ISO 5725) reproducibility and repeatability rather than a
GUM functional model. Three **measurement situations** fix the standard
uncertainty $u$: situation **A** (laboratory characterisation) uses the
reproducibility standard deviation $\sigma_R$; situation **B** (same location,
different teams) the in-situ $\sigma_{situ}$; situation **C** (same location,
operator and equipment, repeated) the repeatability $\sigma_r$. The per-band and
single-number values are tabulated for airborne $R$/$R'$/$D_n$/$D_{nT}$
(Tables 2/3), impact $L_n$/$L'_n$ (Table 4 bands, situations B/C only; Table 5
ratings adding a situation-A estimate) and the
covering reduction $\Delta L$ (Tables 6/7, situation A only). The expanded
uncertainty is $U = k\ u$ (Formula 2) with the coverage factor $k$ of Table 8
(at 95 %, $k = 1.96$ two-sided, $k = 1.65$ one-sided; a minimum $k = 1$ is
enforced). A two-sided interval $Y = y \pm U$ reports a value (Formula 3); a
one-sided factor declares conformity, $y - U > $ requirement for a lower limit
(Formula 5) or $y + U <$ requirement for an upper limit (Formula 4).
Uncorrelated components combine in quadrature $u_c = \sqrt{\sum u_i^2}$
(Formula C.2), $m$ independent measurements reduce $u$ to $u/\sqrt{m}$
(Formula A.7), and the uncorrelated single-number uncertainty is the
energy-weighted quadrature sum of the band uncertainties (Formula B.2).

See the [Room Acoustics](https://jmrplens.github.io/phonometry/guides/room-acoustics/) and
[Building Acoustics](https://jmrplens.github.io/phonometry/guides/building-acoustics/) guides for usage.

## Outdoor propagation and occupational exposure (ISO 9613-1/2, ISO 9612)

### Atmospheric absorption (ISO 9613-1)

Air is a lossy medium: a propagating tone loses energy to shear viscosity and
heat conduction (classical and rotational losses, growing as $f^2$) and to the
**vibrational relaxation** of the oxygen and nitrogen molecules, each an energy
reservoir that resonates near a humidity- and temperature-dependent relaxation
frequency. ISO 9613-1:1993, Eq. (5) gives the pure-tone attenuation coefficient
$\alpha$ in decibels per metre:

$$
\alpha = 8.686\ f^2 \Big[ 1.84\times10^{-11} \big(p_a/p_r\big)^{-1} \big(T/T_0\big)^{1/2}
       + \big(T/T_0\big)^{-5/2} \big( 0.01275\ \tfrac{e^{-2239.1/T}}{f_{rO} + f^2/f_{rO}}
       + 0.1068\ \tfrac{e^{-3352.0/T}}{f_{rN} + f^2/f_{rN}} \big) \Big],
$$

with the oxygen and nitrogen relaxation frequencies $f_{rO}$, $f_{rN}$ of
Eq. (3)/(4), the reference conditions $T_0 = 293.15$ K, $p_r = 101.325$ kPa
(Clause 4.2) and the molar water-vapour concentration $h$ from the relative
humidity (Annex B). At low frequency $\alpha \propto f^2$; near each relaxation
frequency the corresponding term peaks and rolls off, which is why $\alpha$ rises
by two decades from 50 Hz to 10 kHz and why raising the humidity sweeps a peak
across the band. The library reproduces Table 1 to under 0.4 % (the standard's
own printed precision), well inside its stated $\pm 10$ %; passing
`exact_midband=True` snaps each frequency onto the exact midbands
$f_m = 1000 \cdot 10^{k/10}$ (Note 5) used to compute that table. The same
$\alpha$ is the only route to the ISO 354 power attenuation coefficient
$m = \alpha/(10 \lg e)$, exposed as `air_attenuation_m`.

### Outdoor propagation, general method (ISO 9613-2)

ISO 9613-2:1996 predicts the octave-band level at a receiver **downwind** of a
point source (or the equivalent moderate temperature inversion) as
$L_{fT}(DW) = L_W + D_c - A$ (Eq. (3)), where $D_c$ is the directivity correction
and $A$ is the octave-band attenuation, a sum of independent physical mechanisms
(Eq. (4)):

$$
A = A_{div} + A_{atm} + A_{gr} + A_{bar} + A_{misc}.
$$

The library implements the four general terms of Clause 7; the informative
$A_{misc}$ (foliage, industrial sites, housing) and reflections are left to the
caller. **Geometrical divergence** is spherical spreading from a point source,
$A_{div} = 20 \log_{10}(d/d_0) + 11$ dB with $d_0 = 1$ m (Eq. (7)) — exactly
51 dB at 100 m, +6 dB per distance doubling. **Atmospheric absorption** is
$A_{atm} = \alpha\ d$ (Eq. (8)) with $\alpha$ the ISO 9613-1 coefficient above.
**Ground effect** $A_{gr} = A_s + A_r + A_m$ (Eq. (9)) sums a source, receiver and
middle region, each evaluated from the Table 3 functions $a'/b'/c'/d'$ and its
ground factor $G$ (0 hard, 1 porous); a negative $A_{gr}$ denotes a net gain from
the ground reflection. An alternative A-weighted-only form
$A_{gr} = 4.8 - (2 h_m/d)[17 + 300/d] \ge 0$ (Eq. (10)) is offered for porous
ground when only the A-weighted level matters, paired with the solid-angle index
$D_\Omega$ (Eq. (11)). **Screening** by a barrier is the diffraction insertion
loss

$$
D_z = 10 \log_{10}\big[ 3 + (C_2/\lambda)\ C_3\ z\ K_{met} \big] \quad\text{dB},
$$

(Eq. (14)) with $C_2 = 20$ (or 40 when ground reflections are handled by image
sources), $C_3 = 1$ for a single edge or Eq. (15) for a double edge, the
path-length difference $z = d_{ss} + d_{sr} - d$ (Eq. (16)/(17)), wavelength
$\lambda = 340/f$ and the meteorological factor $K_{met}$ (Eq. (18)); $D_z$ is
capped at 20 dB (single) or 25 dB (double). For a top-edge barrier the ground
effect of the screened path is folded into the screening term,
$A_{bar} = D_z - A_{gr} \ge 0$ (Eq. (12), Note 13); for a lateral (vertical-edge)
barrier $A_{bar} = D_z$ and the ground term is kept (Eq. (13)). The long-term
average level subtracts the meteorological correction $C_{met}$ (Eq. (6),
(21)/(22)). The method's stated accuracy is $\pm 1$ to $\pm 3$ dB for broadband
noise up to 1000 m (Table 5).

### Occupational noise exposure and uncertainty (ISO 9612)

ISO 9612:2009 is the engineering method (accuracy grade 2) for a worker's daily
noise exposure level $L_{EX,8h}$, normalised to a nominal 8 h day. Three
**measurement strategies** trade effort for representativeness. The *task-based*
method (Clause 9) splits the day into tasks, energy-averages $I \ge 3$ samples
per task (Eq. 7) and sums the task contributions
$L_{EX,8h,m} = L_{p,A,eqT,m} + 10 \log_{10}(T_m/T_0)$ energetically (Eq. 9/10).
The *job-based* method (Clause 10) energy-averages $N \ge 5$ random samples over a
homogeneous exposure group (Eq. 11) and normalises the effective-day duration
(Eq. 12); the *full-day* method (Clause 11) does the same arithmetic on whole-day
measurements (Eq. 13).

The **Annex C** uncertainty budget is normative. The combined standard
uncertainty is $u^2 = \sum c_i^2 u_i^2$ (C.1) and the expanded uncertainty is
$U = k\ u$ with $k = 1.65$ for a **one-sided** 95 % interval (Clause 14), so the
reported upper limit is $L_{EX,8h} + U$. The task and job methods differ in an
instructive way: the task noise-sampling uncertainty $u_{1a}$ divides the summed
squared deviations by $I(I-1)$ — the standard error of the mean (Eq. C.6) —
whereas the job/full-day sampling uncertainty $u_1$ is the plain sample standard
deviation with denominator $N-1$ (Eq. C.12), so the same spread contributes more
in the job method (fewer, coarser samples). The task budget (Eq. C.3) adds the
sensitivity coefficients $c_{1a}$ (Eq. C.4) and $c_{1b}$ (Eq. C.5) and an optional
task-duration uncertainty $u_{1b}$ (Eq. C.7); the job/full-day budget (Eq. C.9)
reads $c_1 u_1$ from Table C.4 as a function of $(N, u_1)$ and adds the instrument
uncertainty $u_2$ (Table C.5) and microphone-position uncertainty $u_3 = 1.0$ dB
in quadrature. Peak levels $L_{p,Cpeak}$ are reported without an uncertainty:
Annex C provides no method for them (Table C.5, Note 1). The three worked
examples of Annexes D (task, $L_{EX,8h} = 84.3$ dB, $U = 2.7$ dB), E (job,
$88.1$ dB, $3.8$ dB) and F (full-day, $90.1$ dB, $3.4$ dB) are reproduced to
the standard's printed precision — every intermediate of Annex E is digit-exact,
and its final level differs only by the standard's own pre-rounding of the
effective-day level (see the [Occupational Noise Exposure guide](https://jmrplens.github.io/phonometry/guides/occupational-exposure/)).

See the [Outdoor Propagation guide](https://jmrplens.github.io/phonometry/guides/outdoor-propagation/) and the
[Occupational Noise Exposure guide](https://jmrplens.github.io/phonometry/guides/occupational-exposure/) for usage.

## Sound power determination (ISO 3744/3745/3746, ISO 3741, ISO 9614-2/3)

The sound power level $L_W = 10 \log_{10}(P/P_0)$ ($P_0 = 1$ pW) is an
*emission* quantity: unlike a pressure level it does not depend on the receiver
distance or the room. Three families of methods recover it.

### Enveloping-surface pressure (ISO 3744/3746)

Over a reflecting plane the free-field relation is simply
$L_W = \bar{L}_p + 10 \log_{10}(S/S_0)$: the mean-square pressure averaged over
an enveloping surface of area $S$, multiplied by $S$, is the radiated power.
Two corrections restore that idealisation. Uncorrelated **background noise**
adds its mean square to the source's, so with the margin
$\Delta L_p = L_{ST} - L_{bg}$ the source-only level is recovered by subtracting
$K_1 = -10 \log_{10}(1 - 10^{-\Delta L_p/10})$ (from $p_{src}^2 = p_{ST}^2 (1 - 10^{-\Delta L_p/10})$).
The **reverberant field** of a non-anechoic room adds a near-uniform energy
density $4P/(A c)$ to the direct $P/(S c)$, so the surface level exceeds the
free-field value by their ratio, $K_2 = 10 \log_{10}(1 + 4 S/A)$, with $A$ the
room's equivalent absorption area. The surface area is the closed form of the
geometry: a hemisphere $S = 2 \pi r^2$ over one reflecting plane (halved and
quartered for two and three planes), a one-plane box $S = 4(ab + bc + ca)$ with
$a = 0.5\ l_1 + d$, $b = 0.5\ l_2 + d$, $c = l_3 + d$. ISO 3746 (survey) shares
the maths with looser criteria. The expanded uncertainty is
$U = 2 \sqrt{\sigma_{R0}^2 + \sigma_{omc}^2}$.

### Precision grade in anechoic rooms (ISO 3745)

ISO 3745:2012 is the grade-1 (precision) sibling: a qualified anechoic or
hemi-anechoic room removes the reverberant field, so there is no $K_2$ term and
the corrections become meteorological. The power level is
$L_W = \bar{L}_p + 10 \lg(S/S_0) + C_1 + C_2 + C_3$ (Eq. 14/15) over a full
sphere $S = 4 \pi r^2$ or hemisphere $S = 2 \pi r^2$, with the background
correction $K_{1i} = -10 \lg(1 - 10^{-0.1 \Delta L_{pi}})$ applied per
microphone position *before* the energy average (Eq. 11) — no correction is
needed above a 15 dB margin, and below 10 dB (250 Hz – 5 kHz) or 6 dB (edge
bands) the correction is clamped and the result flagged as an upper bound
(clause 9.4.2). The meteorological terms are
$C_1 = -10 \lg(p_s/p_{s0}) + 5 \lg[(273 + \theta)/\theta_0]$ and
$C_2 = -10 \lg(p_s/p_{s0}) + 15 \lg[(273 + \theta)/\theta_1]$ with
$\theta_0 = 314$ K, $\theta_1 = 296$ K — at the 23 °C / 101.325 kPa reference
$C_2 = 0$ exactly and $C_1 = -0.128$ dB — and
$C_3 = A_0 (1.0053 - 0.0012 A_0)^{1.6}$ with $A_0 = a(f)\ r$ restores the
ISO 9613-1 air absorption over the measurement radius. The Annex D/E
microphone arrays are built in as digit-exact coordinate tables (40 equal-area
positions; the mirror set 21–40 is added when the band-SPL spread exceeds
$N_M/2$, clause 9.3.2), and the same positions yield the directivity index
$DI_i = L_{pi} - \bar{L}_p$ (Eq. 21). The clause 10.5 uncertainty example,
$U = 2\sqrt{0.5^2 + 2.0^2} = 4.12$ dB, is reproduced, along with the Table 2/3
per-band $\sigma_{R0}$ values.

### Reverberation room (ISO 3741)

In a qualified diffuse field the steady energy density $w = 4P/(A c)$ ties the
power to the room absorption, giving $L_W = \bar{L}_p + 10 \log_{10}(A/A_0) - 6$
plus higher-order corrections, with $A = (55.26/c)(V/T_{60})$ and
$c = 20.05 \sqrt{273 + \theta}$. The **Waterhouse correction**
$10 \log_{10}(1 + S c/(8 V f))$ compensates the extra energy stored in the
boundary layer that interior microphones miss ($S c/(8 V f) = S \lambda/(8 V)$,
so it fades as frequency rises); the $4.34\ A/S$ term is the mean-free-path air
correction, and $C_1$, $C_2$ carry the result to the reference meteorological
conditions (23 °C, 101.325 kPa). The **comparison method** subtracts a
reference source of known power measured in the same room,
$L_W = L_{W(\text{RSS})} + (\bar{L}_p - \bar{L}_{p,\text{RSS}} + C_2)$, so the
absorption-area, Waterhouse and $C_1$ terms cancel and the room need not be
characterised.

### Intensity scanning (ISO 9614-2)

Sound intensity is the net energy flux $\vec{I} = \overline{p\ \vec{u}}$, so by
the divergence theorem the power through a closed surface is
$P = \sum_i \langle I_{n,i} \rangle\ S_i$. A steady source *outside* the surface
contributes zero net flux (its energy enters and leaves), which is why
intensity rejects stationary background noise — but it can still drive a band's
$P$ negative, in which case that band is not determinable. Two normative field
indicators gate validity: the surface pressure-intensity indicator
$F_{pI} = [L_p] - L_W + 10 \log_{10}(S/S_0)$ (reactivity) and the
negative-partial-power indicator
$F_{+/-} = 10 \log_{10}(\sum_i \lvert P_i \rvert / \lvert \sum_i P_i \rvert)$
(recirculation), together with the probe's dynamic capability
$L_d = \delta_{pI0} - K$ ($K = 10$ dB grade 2, 7 dB grade 3), which must exceed
$F_{pI}$. A band earns the engineering grade when $L_d > F_{pI}$, $F_{+/-} \le 3$ dB
and the two repeated sweeps agree within the Table 2 limit.

### Precision intensity scanning (ISO 9614-3)

ISO 9614-3:2002 upgrades the scanning method to precision grade with a tighter
indicator machinery. The partial powers $P_i = I_{n,i} S_i$ (Eq. 5) sum as
before, but validity now rests on the signed and unsigned pressure-intensity
indicators $F_{pIn} = \bar{L}_p - L_{In}$ (Eqs. B.3/B.6 — the F2/F3 of
ISO 9614-1) and the normalized intensity non-uniformity $F_S$ (Eq. B.8),
through five acceptance criteria (Annex C): scan repeatability
$|L_{In}(1) - L_{In}(2)| \le s/2$ (C.1), dynamic capability
$L_d = \delta_{pI0} - K \ge F_{pIn}(\text{signed})$ with the precision
bias-error factor $K = 10$ dB (C.2),
$F_{pIn}(\text{signed}) - F_{pIn}(\text{unsigned}) \le 3$ dB (C.3),
$F_S \le 2$ (C.4) and the scan-density convergence
$0.83 \le F_S(1)/F_S(2) \le 1.2$ (C.5). Eq. 10 normalizes the result to the
reference meteorological conditions,
$L_{W0} = L_W - 15 \lg[(B/101325) \cdot 296.15/(273.15 + \theta)]$. Bands whose
net power is negative are not determinable (clause 9.2) and are flagged. A
uniform normal intensity recovers the power exactly (100 µW over 3.75 m² →
80.0 dB re 1 pW), independent of how the surface is segmented.

See the [Sound Power guide](https://jmrplens.github.io/phonometry/guides/sound-power/) for usage.

## Surface scattering and diffusion (ISO 17497-1, ISO 17497-2)

### Random-incidence scattering coefficient (ISO 17497-1)

A rough surface splits the reflected energy into a specular and a scattered
part; the scattering coefficient $s$ is the non-specular energy fraction.
ISO 17497-1:2004+A1:2014 measures it in a reverberation room with the test
sample on a turntable: four reverberation times — stationary and rotating,
each without and with the sample (Table 2) — give the random-incidence
absorption $\alpha_s$ (clause 8.1.1, Formula 1) and the *specular* absorption
$\alpha_{spec}$ (clause 8.1.2, Formula 4). Rotation decorrelates the scattered
reflections between decays, so they average out and register as extra
"absorption", and the scattering coefficient follows (clause 8.1.3,
Formula 5):

$$
s = \frac{\alpha_{spec} - \alpha_s}{1 - \alpha_s},
$$

each $\alpha$ being a two-condition Sabine difference
$55.3 (V/S) [1/(c_b T_b) - 1/(c_a T_a)] - 4 (V/S)(m_b - m_a)$ with
$c = 343.2 \sqrt{(273.15 + t)/293.15}$ (Formula 2) and $m$ from ISO 9613-1
via $m = \alpha_{dB}/(10 \lg e)$ (Formula 3). The base plate itself must
scatter little: Table 1 caps its coefficient (Formula 6) at 0.05–0.25 across
100 Hz – 5 kHz (clause 6.2). Negative $s$ is truncated to zero for
presentation (clause 8.3), but values above 1 near grazing bands are kept
(clause 6.3.2). The Annex A uncertainty chain ($u_\alpha$, Formulae A.3/A.4;
$u_s$, Formula A.5; $U = 2 u_s$) is implemented. Since the standard prints no
worked example, the oracle is a synthetic end-to-end chain
($V = 200$ m³, $S = 10$ m², $T = 8.0/6.0/7.5/5.0$ s → $s = 0.093$) plus the
Formula A.5 hand value $u_s = 0.0297$.

### Directional diffusion coefficient (ISO 17497-2)

ISO 17497-2:2012 measures, in the free field, how uniformly a surface spreads
its reflected polar response over $n$ microphones. The autocorrelation-based
coefficient (clause 8.1, Formula 5) is

$$
d_\theta = \frac{\left( \sum_i p_i \right)^2 - \sum_i p_i^2}{(n - 1) \sum_i p_i^2},
\qquad p_i = 10^{L_i/10},
$$

1 for a perfectly uniform response and tending to 0 for a single specular
lobe; Formula 6 is the area-weighted form with $N_i = A_i / A_{min}$ from the
Formula 8 solid-angle factors ($A_i = (4\pi/\Delta\phi) \sin^2(\Delta\theta/4)$
at the zenith). Normalizing against a flat reference reflector of the same
size removes edge diffraction (clause 8.2, Formula 7):
$d_{\theta,n} = (d_\theta - d_{\theta,r})/(1 - d_{\theta,r})$. The
random-incidence value averages the source angles with weights 1:3:3:3:3 for
0°, ±30°, ±60° (clause 8.4). Anchors: levels (70, 74, 68, 72) dB →
$d = 0.7367$; zenith area factor 1.5710.

See the [Surface Scattering guide](surface-scattering.md) for usage.

## In-situ road surface absorption (ISO 13472-1, ISO 13472-2)

ISO 13472-1:2002 (extended surface method) recovers the normal-incidence
absorption of a road surface in place, from one microphone above it: the
direct and reflected components of an impulse response are separated by the
**subtraction technique** and the **Adrienne window** (clause 6.4: a sharp
leading edge, a mandated 5 ms flat top and a Blackman-Harris trailing edge),
and

$$
\alpha(f) = 1 - \frac{1}{K_r^2} \left| \frac{H_r(f)}{H_i(f)} \right|^2,
\qquad K_r = \frac{d_s - d_m}{d_s + d_m} = \frac{2}{3}
$$

for the mandatory geometry $d_s = 1.25$ m, $d_m = 0.25$ m (clause 4.2,
Annex C) — $K_r$ is the spherical-spreading ratio between the direct and the
image path. Ratioing the road measurement against one on a highly reflective
reference surface cancels the entire electro-acoustic chain along with $K_r$
(Annex B). The 5 ms window bounds the sampled area (Annex A closed form:
radius ≈ 1.34 m for the standard geometry) and the valid range is
250 Hz – 4 kHz in one-third octaves. ISO 13472-2:2010 (spot method,
250–1600 Hz) instead couples a small impedance tube to the surface and defers
the mathematics to the ISO 10534-2 transfer-function method below (its
clauses 4/5.7/6.6) — the implementation reuses that module, adding the Part 2
geometry and validity limits ($f_u = 0.58\ c_0/d$; microphone spacing bounds
$0.45\ c_0/f_{max}$ and $0.05\ c_0/f_{min}$, clause 5.4) and the Annex A
subtractive correction for internal system losses.

See the [Surface Scattering guide](surface-scattering.md) for usage.

## Acoustic material characterisation (ISO 11654, ISO 9053-1/2, ISO 10534-1/2, ASTM E2611)

### Weighted sound absorption (ISO 11654)

ISO 11654:1997 condenses an ISO 354 third-octave absorption curve into a
single number. The practical coefficient $\alpha_p$ averages the three thirds
of each octave 250 Hz – 4 kHz and rounds to steps of 0.05 (clause 4.1). The
reference curve (0.80, 1.00, 1.00, 1.00, 0.90 at 250–4000 Hz) is then shifted
downward in 0.05 steps until the sum of unfavourable deviations — counted only
where the measurement falls *below* the shifted curve — is $\le 0.10$;
$\alpha_w$ is the shifted curve at 500 Hz (clause 4.2). A shape indicator
flags excess absorption $\ge 0.25$ above the shifted curve: L at 250 Hz, M at
500/1000 Hz, H at 2000/4000 Hz (clause 4.3), and the informative Annex B maps
$\alpha_w$ to the absorption classes A–E. Because every quantity is a multiple
of 0.05, the implementation does the whole grid arithmetic in integer
twentieths, making the shift search and class boundaries exact and
float-safe. The two Annex A worked examples are reproduced:
$\alpha_p = (0.35, 0.70, 0.65, 0.60, 0.55)$ → $\alpha_w = 0.60$, class C; and
raising 500 Hz to 1.00 keeps $\alpha_w = 0.60$ but adds the indicator, "0.60(M)".

### Airflow resistance (ISO 9053-1/2)

Airflow resistivity $\sigma = R\,A/d$ is the key transport parameter of a
porous absorber. ISO 9053-1:2018 (static method) drives a steady flow through
the specimen and fits $\Delta p = a\,u + b\,u^2$ through the origin
(clause 7.5); since $R_s = \Delta p / u = a + b\,u$, the linear coefficient is
the zero-velocity specific resistance, reported at the reference velocity
$u = 0.5$ mm/s. ISO 9053-2:2020 (alternating method) replaces the flowmeter
with a ~2 Hz piston and a microphone in a closed cavity (clause 8.7,
Formula 2):

$$
R = \kappa'\ \frac{p_s}{2 \pi f V}\ \frac{h_t}{h_s}\ 10^{(L_{ps} - L_{pt})/20}
$$

— only a level *difference* enters, so the sound-level device needs no
absolute calibration. The effective exponent $\kappa'$ (Annex A,
Formula A.7) corrects the adiabatic $\kappa$ for wall heat conduction through
the thermal boundary layer $b = \sqrt{2 c_0 l_h / \omega}$ (Formulae A.4/A.5).
The Annex A.3 worked example (100 mm closed cylinder at 2 Hz: $b = 1.83$ mm,
$\kappa' = 1.370 = 0.978\,\kappa$) is reproduced, and the validity guards of
Formula 3 (transfer ratio < 0.3) and Formula 4 (10 dB background margin) are
enforced.

### Impedance tube (ISO 10534-1, ISO 10534-2, ASTM E2611)

A tube below its cut-on frequency ($f d < 0.58\ c_0$ circular,
$< 0.50\ c_0$ rectangular; microphone-spacing limits $f s < 0.45\ c_0$ and
$f > c_0/(20 s)$; clauses 4.2–4.5) carries only plane waves, so the surface
reflection factor of a sample is fully observable. ISO 10534-2
(transfer-function method) compares the measured two-microphone transfer
function $H_{12}$ with the analytic incident and reflected ones
$H_I = e^{-j k_0 s}$, $H_R = e^{+j k_0 s}$ (Annex D) to give (clause 7,
Eq. 17):

$$
r = \frac{H_{12} - H_I}{H_R - H_{12}}\ e^{2 j k_0 x_1}, \qquad
\alpha = 1 - |r|^2, \qquad \frac{Z}{\rho c_0} = \frac{1 + r}{1 - r},
$$

with the complex wavenumber's attenuation lower bound
$k_0'' = 1.94 \times 10^{-2} \sqrt{f}/(c_0 d)$ (Eq. A.18). ISO 10534-1
(standing-wave-ratio method) is the closed-form classic:
$|r| = (s - 1)/(s + 1)$ from the max/min ratio $s = 10^{\Delta L/20}$ and the
phase from the first-minimum position (Eqs. 12–26) — an SWR of 3 gives exactly
$|r| = 0.5$ and $\alpha = 0.75$. ASTM E2611-19 adds transmission: four
microphones decompose the up- and downstream fields into the $A, B, C, D$
waves (Eqs. 17–20) and a two-load (or symmetric one-load) solve yields the
specimen's 2×2 **transfer matrix** $[p; u]_0 = T\,[p; u]_d$ (Eqs. 16/22–24),
from which the anechoic-backing normal-incidence transmission loss is
(Eqs. 25/26)

$$
TL = 20 \lg \frac{\left| T_{11} + T_{12}/\rho c + \rho c\ T_{21} + T_{22} \right|}{2},
$$

plus the hard-backed reflection
$R = (T_{11} - \rho c\,T_{21})/(T_{11} + \rho c\,T_{21})$ (Eq. 27), the
material wavenumber $\arccos(T_{11})/d$ (Eq. 29) and the characteristic
impedance $\sqrt{T_{12}/T_{21}}$ (Eq. 30). The three standards deliberately
keep their own sign ansatz and temperature units (ISO in kelvin, ASTM in
Celsius), and near-singular load solves raise a warning. Since neither
standard prints a numeric example, the oracles are physics identities: the
analytic air-layer matrix ($\det T = 1$, $T_{11} = T_{22}$, TL = 0 dB,
hard-backed $|R| = 1$), synthetic round-trips that recover a known $r$, and
two-load recovery of an asymmetric reciprocal specimen.

See the [Materials guide](materials.md) for usage.

## Human vibration (ISO 8041-1, ISO 2631-1/2, ISO 5349-1/2, Directive 2002/44/EC)

Human response to vibration depends on frequency, axis and body part, so
acceleration is filtered by the frequency weightings of ISO 8041-1:2017 before
any metric. Each weighting is the analog cascade
$H(s) = H_h(s) H_l(s) H_t(s) H_s(s)$ (Formula 5): two-pole Butterworth
band-limiting high-pass and low-pass stages (Formulae 1/2), an
acceleration–velocity transition (Formula 3, carrying the only non-unity gain,
$K = 1.024$ for Wb) and an upward step (Formula 4), with the Table 3 corner
frequencies and Q factors; a corner at infinity collapses its stage to unity
(Table 3 NOTEs). Wk (vertical whole-body) and Wd (horizontal) of
ISO 2631-1, Wm (buildings, ISO 2631-2), Wb (rail, ISO 2631-4), Wc/We/Wj
(seat-back, rotational, head) and Wh (hand-arm, ISO 5349-1) plus Wf (motion
sickness) are all implemented from the exact cascade — the filter is applied
as the exact complex response via FFT (magnitude *and* phase), not a
bilinear-warped digital approximation — and the ISO 8041-1 Annex B design-goal
tables (B.1–B.9) are reproduced to 0.1 %.

The weighted metrics follow ISO 2631-1:1997: running rms with linear or
exponential integration (Eqs. 2/3), **MTVV** as its maximum (Eq. 4), the
fourth-power **VDV** $= (\int a_w^4\, dt)^{1/4}$ in m/s^1.75 (Eq. 5), the crest
factor with the basic method deemed adequate up to 9 (clause 6.2), and the
vibration total value $a_v = \sqrt{\sum_j k_j^2 a_{wj}^2}$ (Eq. 10). Hand-arm
exposure follows ISO 5349-1:2001: $a_{hv}$ (Eq. 1, all $k = 1$), daily
exposure $A(8) = a_{hv} \sqrt{T/T_0}$ with $T_0 = 8$ h (Eq. 2), partial
exposures combined in quadrature (ISO 5349-2:2001, Eqs. 1–3), and the Annex C
vascular-risk model $D_y = 31.8\ A(8)^{-1.06}$ for the years to 10 %
white-finger prevalence. The Directive 2002/44/EC action and limit values are
built in: hand-arm $A(8)$ 2.5/5.0 m/s², whole-body $A(8)$ 0.5/1.15 m/s² or
VDV 9.1/21.0 m/s^1.75 (Article 3). The ISO 5349-2 worked examples are
reproduced (E.2.1: 7.4 m/s² for 2.5 h → $A(8) = 4.1$ m/s²; E.3 forestry,
three tools → 3.6 m/s²), as are the ISO 5349-1 Table C.1 exposure-duration
rows.

### Multiple shocks (ISO 2631-5)

Repeated shocks damage the lumbar spine through peak compression rather than
average energy, so ISO 2631-5:2018 replaces the Wk weighting with the
seat-to-spine transfer function of clause 5.2 (Formula 1: one complex zero and
six complex pole pairs, unity at DC, resonance near 5 Hz —
$|H| \approx 1.54$ at 5 Hz) and accumulates the positive spinal-response peaks with a
sixth-power (Palmgren-Miner) dose (clause 5.3, Formulae 3/4):

$$
D_z = 1.07 \left( \sum_i A_{z,i}^6 \right)^{1/6}, \qquad
D_{zd} = D_z\ (t_d / t_m)^{1/6}.
$$

Annex C converts the daily dose to a compressive stress $S_d = m_z D_{zd}$
($m_z = 0.029/0.025$ MPa per m/s² for the 82 kg male / 64 kg female), tracks
the age-declining ultimate strength $S_u = 6.75 - S_{age}(b + i)$ and forms
the cumulative stress variable $R$ (Formulae C.3/C.4), mapped to an injury
probability by the Table C.1 Weibull law $\Pi = 1 - e^{-(R/\alpha)^\beta}$.
The spinal filter is evaluated analytically in the frequency domain and
validated against the Annex D 256 Hz digital-filter tabulation within the
clause 5.2 tolerance; the Annex C worked example (five 40 m/s² shocks per day
over 20 years) is reproduced: $D_{zd} = 55.97$ m/s², $R = 1.22$,
$\Pi = 0.37$. The Annex A finite-element spinal model (distributed by ISO as
separate software) is out of scope.

See the [Human Vibration guide](human-vibration.md) and the
[Multiple-Shock Vibration guide](multiple-shock-vibration.md) for usage.

## Measurement uncertainty (ISO/IEC Guide 98-3 — GUM and Supplement 1)

Domain budgets like ISO 12999-1 and ISO 9612 Annex C are instances of the
general framework of the GUM (ISO/IEC Guide 98-3:2008). Given a measurement
model $y = f(x_1, \ldots, x_N)$, the law of propagation of uncertainty
(clause 5) combines the input standard uncertainties through sensitivity
coefficients:

$$
u_c^2(y) = \sum_{i=1}^{N} \left( \frac{\partial f}{\partial x_i} \right)^2 u^2(x_i),
$$

generalized to $(c \odot u)^{\top} r\ (c \odot u)$ for correlated inputs. The
sensitivities are obtained by central differences on the user's model callable
(step scaled to $10^{-3}$ of each input uncertainty), so no hand-derived
partials are needed. Type B inputs enter through the clause 4.3 half-width
rules: rectangular $a/\sqrt{3}$ (4.3.7), triangular $a/\sqrt{6}$ (4.3.9),
U-shaped $a/\sqrt{2}$. The expanded uncertainty $U = k\,u_c$ takes $k$ from
the t-distribution at the Welch–Satterthwaite effective degrees of freedom
(Annex G.4):

$$
\nu_{\mathrm{eff}} = \frac{u_c^4}{\sum_i u_i^4 / \nu_i}.
$$

**Supplement 1** (ISO/IEC Guide 98-3-1:2008) propagates the full distributions
instead: $10^6$ Monte Carlo draws (clause 6.4) through the same model give
$u(y)$ and the probabilistically symmetric coverage interval from the
$\frac{1}{2}(1 \mp p)$ fractiles (clause 7.7) — the route when the model is
non-linear or the output visibly non-Gaussian. The Guides' own examples are
reproduced: the four-term additive model gives $u_c = 2.0$ and the Monte Carlo
95 % interval $\pm 3.88$ of Supplement 1 clause 9.2/Table 3 (four rectangular
inputs — the output is nearly trapezoidal, not Gaussian, so the interval is
narrower than $\pm 1.96\,u$), and the GUM Annex H.1 end-gauge example gives
$k = t_{0.99}(\nu_{\mathrm{eff}} = 16) = 2.92$ and $U_{99} = 93$ nm.

See the [GUM Uncertainty guide](gum-uncertainty.md) for usage.

---


<!-- source: docs/why-phonometry.md | canonical: https://jmrplens.github.io/phonometry/reference/why-phonometry/ -->

# Why phonometry

phonometry is a standards-based acoustic measurement toolkit. Its
differentiator is not the list of features but how they are built: every
metric is implemented from the governing standard's text, and the standard's
own reference values and acceptance limits are transcribed into the test
suite and enforced in CI. This page explains that approach with a concrete
case study — time weighting under **IEC 61672-1:2013** — and summarizes what
is conformance-tested today. The time-weighting analysis was originally
published in [issue #38](https://github.com/jmrplens/phonometry/issues/38).

## Design philosophy: a case study in time weighting

Standard time weighting is defined as a **continuous function of time** via the
differential equation

$$
\tau \frac{dy(t)}{dt} + y(t) = x^2(t)
$$

which corresponds to a stable first-order low-pass filter with a pole in the
left half-plane ($s = -1/\tau$).

| | phonometry | python-acoustics |
| :--- | :--- | :--- |
| Output | Continuous time-weighted envelope (one value per sample) | Stepped output (one value every τ seconds) |
| Input units | Raw sound pressure (Pa); squared internally | Energetic quantity (Pa²) expected as input |
| Filter | Stable exponential averaging (pole at $s=-1/\tau$) | Theoretically unstable design (pole in the right half-plane) stabilized by resetting the filter state every τ seconds |
| Behavior | True IEC time weighting | Closer to a block integrator ($L_{eq,\tau}$) |

A pole on the negative real axis corresponds to a decaying exponential impulse
response ($h(t) \propto e^{-t/\tau}$) — exactly what "exponential time
weighting" means: past events are forgotten exponentially. A pole on the
positive real axis grows without bound; block-resetting hides this but changes
the measurement's nature.

## Verification against IEC 61672-1 (tone bursts)

The rigorous test for time weighting is the **Tone Burst Response**
(IEC 61672-1, Table 4), using a 4 kHz sine burst referenced to the steady-state
level.

**phonometry results (FAST):**

| Burst duration | IEC target (dB) | phonometry (dB) | Error (dB) | Status |
| :--- | :--- | :--- | :--- | :--- |
| 200 ms | −1.0 | −0.98 | +0.02 | ✅ PASS |
| 50 ms | −4.8 | −4.82 | −0.02 | ✅ PASS |
| 10 ms | −11.1 | −11.14 | −0.04 | ✅ PASS |
| 1 ms | −20.9 | −20.99 | −0.09 | ✅ PASS |

**python-acoustics results (FAST, squared signal passed as required by that
library):**

| Burst duration | IEC target (dB) | python-acoustics (dB) | Error (dB) | Status |
| :--- | :--- | :--- | :--- | :--- |
| 200 ms | −1.0 | −0.97 | +0.03 | ✅ PASS |
| 50 ms | −4.8 | −3.93 | **+0.87** | ⚠️ FAIL |
| 10 ms | −11.1 | −10.90 | +0.20 | ✅ PASS |
| 1 ms | −20.9 | −20.90 | +0.00 | ✅ PASS |

phonometry maintains high precision across all test cases. The block-based
approach deviates significantly (> 0.8 dB) for the 50 ms burst because 125 ms
blocks cannot resolve short transient events accurately — the result depends on
how the burst aligns with the block boundaries.

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/tone_burst_iec_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/tone_burst_iec.svg" alt="Fast envelope responses to 200, 50 and 10 ms tone bursts peaking exactly at the IEC 61672-1 Table 4 reference values" width="80%"></picture>

*Measured Fast envelopes (blue) matching the Table 4 reference values
(dashed) within 0.1 dB for 200/50/10 ms bursts.*

## What this means in practice

- If you need **standard-compliant Fast/Slow/Impulse envelopes** (sound level
  meter behavior, one level per sample), use phonometry's
  [`time_weighting`](https://jmrplens.github.io/phonometry/guides/time-weighting/).
- If you need **block-averaged Leq per interval**, that is a different, equally
  valid metric — you can compute it with [`leq`](https://jmrplens.github.io/phonometry/guides/levels/) over consecutive
  slices.
- Both approaches are useful; they just answer different questions. The
  discrepancy reported in issue #38 comes from comparing a continuous envelope
  against a block integrator, not from an implementation error.

## Conformance testing across the library

The tone-burst case above is not an isolated check. For each standard the
library implements, the reference values and acceptance limits are transcribed
from the official text into the test suite, so any regression fails CI:

| Standard | What is verified | Test file |
| :--- | :--- | :--- |
| IEC 61672-1:2013 Table 3 | A/C/Z weighting at all 34 nominal frequencies, class 1 limits, at 48 and 96 kHz | `tests/test_iec_weighting_table3.py` |
| IEC 61672-1:2013 Table 4 | F/S tone-burst responses (1 s to 1 ms) and the LAE column for `sel()` | `tests/test_iec_compliance.py` |
| IEC 61672-1:2013 Table 5 | `lc_peak()` one-cycle/half-cycle peak responses, class 1 limits | `tests/test_levels.py` |
| IEC 61260-1:2014 Table 1 | Filter-bank class 1/2 acceptance limits via `verify_filter_class()` | `tests/test_compliance.py` |
| ISO 7196:1995 Table 2 | G weighting (infrasound) at every nominal response value, 0.25–315 Hz | `tests/test_g_weighting.py` |
| ISO 226:2023 Annex B | Equal-loudness contours, loudness levels and hearing threshold against the Annex B tables | `tests/test_loudness_contours.py` |
| ECMA-418-1:2024 | TNR/PR tone prominence: critical bandwidths, proximity spacing and prominence criteria against the worked examples in clauses 10–12 | `tests/test_tonality.py` |
| ISO 1996-1:2016 | `lden()`, `ldn()` and `composite_rating_level()` against hand-computed formula values | `tests/test_environmental.py` |
| IEC 60942:2017 Table 2 | Calibrator short-term stability limits (frequency-dependent, class 1) in `sensitivity()` | `tests/test_calibration_validation.py` |

The full numerical report — the expected value and the value the library
computes for every check, regenerated on every pull request — is published
as [CONFORMANCE.md](CONFORMANCE.md).

Beyond IEC 61252-style noise dose (`sound_exposure()`, `lex_8h()`), the same
standards-first mindset shows up in the numerics: filter banks place their
−3 dB points on the **ANSI S1.11 / IEC 61260-1** band edges for every
architecture (including Chebyshev II and Bessel, where scipy's raw
parametrization would not), and A/C weighting stays within class 1 tolerances
up to 16 kHz at common audio rates via internal oversampling (see
[Frequency Weighting](https://jmrplens.github.io/phonometry/guides/weighting/)).

## Where phonometry fits in the Python ecosystem

- **python-acoustics** was archived in February 2024 and is no longer
  maintained. Its comparison above reflects the last released code.
- **acoustic-toolbox**, the community successor to python-acoustics, depends
  on phonometry for its weighted level computations rather than reimplementing
  them.
- **MoSQITo** focuses on psychoacoustic sound-quality metrics (loudness,
  sharpness, roughness). It complements rather than overlaps phonometry:
  it does not cover sound level metrology (weighting filters, ballistics,
  Leq/SEL/Lden, calibration) and does not claim conformance testing against
  the standards' tolerance tables.

If your work needs numbers you can defend against a standard's tolerance
table — measurement reports, environmental assessments, instrument
cross-checks — that verification layer is what phonometry is for.

---
