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# Psychoacoustic annoyance and fluctuation strength (Fastl & Zwicker)

How *annoying* a sound is depends on more than how loud it is. Fastl & Zwicker,
*Psychoacoustics: Facts and Models*, combine four psychoacoustic sensations —
**loudness**, **sharpness**, **roughness** and **fluctuation strength** — into a
single **psychoacoustic annoyance** `PA`, a scalar that grows with loudness and
is lifted further when the sound is sharp, rough or slowly fluctuating. This page
covers the exact `PA` model (Eqs 16.2–16.4), the **fluctuation strength** it
consumes — both the closed form for amplitude-modulated broadband noise
(Eq. 10.2) and the Osses et al. (2016) signal model — and the signal
convenience that derives all four sensations from a recording.

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/psychoacoustic_annoyance_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/psychoacoustic_annoyance.svg" alt="Psychoacoustic annoyance PA against percentile loudness N5 for three sensation profiles: a neutral baseline where PA equals N5, a sharp sound and a rough-and-fluctuating sound both lifted above the baseline, with the worked example PA = 30.82 marked" width="82%"></picture>

## 1. The four sensations

Psychoacoustic annoyance rests on four hearing sensations, each with its own
model and unit in the library:

- **Loudness** — the percentile loudness `N5` (the loudness exceeded 5 % of the
  time), in **sone**, from the ISO 532-1 Zwicker time-varying model
  (`loudness_zwicker`, see [Psychoacoustics](psychoacoustics.md)).
- **Sharpness** `S`, in **acum** — the spectral balance towards high
  frequencies, DIN 45692 (`sharpness_din`).
- **Roughness** `R`, in **asper** — the harshness of fast (~70 Hz) amplitude
  modulation, ECMA-418-2 Sottek model (`roughness_ecma`).
- **Fluctuation strength** `F`, in **vacil** — the sensation of slow (~4 Hz)
  loudness fluctuation (§3 below).

## 2. Psychoacoustic annoyance (Eqs 16.2–16.4)

The exact model (Fastl & Zwicker Eq. 16.2; origin Widmann 1992) scales `N5` by a
factor that grows with the sharpness weighting `wS` and the combined
roughness/fluctuation weighting `wFR`:

$$
PA = N_5\sqrt{1 + w_S^2 + w_{FR}^2}, \qquad
w_S = (S - 1.75)\,0.25\,\lg(N_5 + 10)\ \ (S > 1.75\ \mathrm{acum}), \qquad
w_{FR} = \frac{2.18}{N_5^{0.4}}\,(0.4\,F + 0.6\,R).
$$

`wS` is zero for `S ≤ 1.75 acum` (sharpness only adds annoyance above that
threshold); `wFR` weights roughness more heavily than fluctuation strength
(0.6 vs 0.4). `psychoacoustic_annoyance` takes the four quantities directly and
returns the annoyance together with the two intermediate weightings:

```python
import phonometry as ph

res = ph.psychoacoustic_annoyance(30.0, 2.0, 0.5, 0.3)   # N5, S, F, R
print(round(res.annoyance, 4))   # 30.8167
print(round(res.w_s, 4), round(res.w_fr, 4))   # 0.1001 0.2125
```

The figure above sweeps `PA` against `N5` for three profiles: a neutral baseline
(`S = 1.75 acum`, `F = R = 0`, so `PA = N5`), a sharp sound and a
rough-and-fluctuating sound — both lifted above the baseline — with the worked
example marked.

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

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

n5 = np.linspace(4.0, 60.0, 200)
profiles = [
    ("Baseline: S = 1.75 acum, F = R = 0", 1.75, 0.0, 0.0),
    ("Sharp: S = 3.5 acum", 3.5, 0.0, 0.0),
    ("Rough + fluctuating: F = 1.2 vacil, R = 0.7 asper", 2.0, 1.2, 0.7),
]
fig, ax = plt.subplots()
for label, s, f, r in profiles:
    pa = [ph.psychoacoustic_annoyance(v, s, f, r).annoyance for v in n5]
    ax.plot(n5, pa, label=label)

ex = ph.psychoacoustic_annoyance(30.0, 2.0, 0.5, 0.3)
ax.plot([30.0], [ex.annoyance], "o", label=f"Worked example (PA = {ex.annoyance:.2f})")
ax.set_xlabel("Percentile loudness N5 [sone]")
ax.set_ylabel("Psychoacoustic annoyance PA")
ax.legend()
plt.show()
```

</details>

### 2.1 From a signal (engineering estimate)

`psychoacoustic_annoyance_from_signal` is a convenience that derives all four
sensations from a calibrated pressure signal and combines them: `N5` from the
ISO 532-1 Zwicker time-varying loudness, `S` from DIN 45692 sharpness, `R` from
the ECMA-418-2 Sottek roughness and `F` from the fluctuation-strength signal
model.

```python
import phonometry as ph

res = ph.psychoacoustic_annoyance_from_signal(x, fs, field="free")
print(res.annoyance, res.n5, res.sharpness, res.roughness, res.fluctuation_strength)
```

> **Model-mixing caveat.** This composite mixes model families — Zwicker-family
> `N5` and `S`, Sottek `R` and Osses `F`. The `PA` model was calibrated with
> Zwicker-family sensations, so treat the signal convenience as an *engineering
> estimate*. For exact, reproducible results, compute the four quantities with
> whichever models you trust and pass them to `psychoacoustic_annoyance`
> directly (the four-argument function is the exact model).

## 3. Fluctuation strength

**Fluctuation strength** `F` (vacil) quantifies the perception of *slow* loudness
fluctuation. Like roughness it is a band-pass sensation of the modulation
frequency, but it peaks about an order of magnitude lower — at `fmod ≈ 4 Hz`
rather than the ~70 Hz roughness peak. By definition, a 1 kHz tone at 60 dB,
100 % amplitude-modulated at 4 Hz, produces `1 vacil`.

<picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/fluctuation_strength_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/fluctuation_strength.svg" alt="Fluctuation strength versus modulation frequency on a log axis: the closed-form AM broadband-noise curve and the AM-tone signal-model sweep both show a band-pass characteristic peaking at 4 Hz" width="82%"></picture>

### 3.1 Closed form for AM broadband noise (Eq. 10.2)

For sinusoidally amplitude-modulated broadband noise, Fastl & Zwicker give a
closed form (Eq. 10.2) in modulation factor `m`, level `L` and modulation
frequency `fmod`:

$$
F = \frac{5.8\,(1.25\,m - 0.25)\,[0.05\,(L/\mathrm{dB}) - 1]}
{(f_{mod}/5\,\mathrm{Hz})^2 + (4\,\mathrm{Hz}/f_{mod}) + 1.5}\ \ \mathrm{vacil}.
$$

The denominator is the `4 Hz` band-pass: it bottoms out near `fmod ≈ 3.7 Hz` and
rises on either side. The result is clamped at `0` (the sensation vanishes below
~20 dB or `m < 0.2`). This exact form is the value to quote for AM broadband
noise.

```python
import phonometry as ph

print(round(ph.fluctuation_strength_am_noise(60.0, 1.0, 4.0), 4))   # 3.6943 vacil
```

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

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

# Exact closed form (Eq. 10.2), AM broadband noise at 60 dB, 100 % modulation:
fmod = np.logspace(np.log10(0.5), np.log10(32.0), 240)
f_bbn = [ph.fluctuation_strength_am_noise(60.0, 1.0, fm) for fm in fmod]

# Osses 2016 signal model on a 1 kHz / 70 dB AM tone over the same sweep:
fs = 48000
t = np.arange(int(2.0 * fs)) / fs
carrier = np.sin(2 * np.pi * 1000 * t)
fm_tone = [1.0, 2.0, 4.0, 8.0, 16.0, 32.0]
f_tone = []
for fm in fm_tone:
    am = (1.0 + np.sin(2 * np.pi * fm * t)) * carrier
    am = am / np.sqrt(np.mean(am ** 2)) * 2e-5 * 10 ** (70 / 20)
    f_tone.append(ph.fluctuation_strength(am, float(fs)).fluctuation_strength)

fig, ax = plt.subplots()
ax.semilogx(fmod, f_bbn, label="AM broadband noise (closed form)")
ax2 = ax.twinx()
ax2.plot(fm_tone, f_tone, "s--", color="tab:green", label="AM tone (signal model)")
ax.axvline(4.0, ls="--", color="0.4")
ax.set_xlabel("Modulation frequency f_mod [Hz]")
ax.set_ylabel("Fluctuation strength F [vacil]")
h1, l1 = ax.get_legend_handles_labels()
h2, l2 = ax2.get_legend_handles_labels()
ax.legend(h1 + h2, l1 + l2, loc="upper right")
plt.show()
```

</details>

### 3.2 The Osses 2016 signal model

`fluctuation_strength` implements the Osses et al. (2016) signal model: it builds
an excitation pattern over 47 auditory filters, extracts the `~4 Hz` envelope
modulation per band, weights and combines it, and returns the overall `F` (vacil)
plus the specific fluctuation strength over the Bark axis and the time-dependent
trace.

```python
import phonometry as ph

res = ph.fluctuation_strength(x, fs)
print(res.fluctuation_strength)   # vacil
res.plot()   # specific fluctuation strength F′(z) over the Bark axis (needs matplotlib)
```

> **No numeric standard.** Fluctuation strength has no ISO/IEC standard. This is
> a clean-room implementation from the Osses 2016 paper, calibrated so that the
> `1 kHz / 60 dB / m = 1 / 4 Hz` AM tone reads `1.00 vacil` by construction, and
> cross-checked against the Osses 2016 Table 1 literature values and the open
> SQAT reference (used only as a numeric oracle). Over the `70 dB` AM-tone sweep
> `fmod ∈ {1, 2, 4, 8, 16, 32} Hz` it gives `[0.42, 0.79, 1.09, 1.05, 0.19,
> 0.10]` vacil against the literature `[0.39, 0.84, 1.25, 1.30, 0.36, 0.06]`
> (Pearson `r = 0.98`, correct `4 Hz` peak, within ~2×). FM-tone accuracy is
> explicitly not pursued. For **AM broadband noise** the signal model overshoots
> the absolute level (it spreads the modulated energy across bands) — quote the
> closed form `fluctuation_strength_am_noise` (§3.1) for that stimulus.

---

**Standards.** Fastl & Zwicker (2006), *Psychoacoustics: Facts and Models*
(Springer): psychoacoustic annoyance `PA = N5·√(1 + wS² + wFR²)` with the
sharpness weighting `wS` and roughness/fluctuation weighting `wFR`
(Eqs 16.2–16.4; origin Widmann 1992), and the closed form for the fluctuation
strength of amplitude-modulated broadband noise (Eq. 10.2). The fluctuation-
strength signal model follows Osses García & Kohlrausch (2016), *Modelling the
sensation of fluctuation strength* (ICA 2016) — clean-room, with no numeric
standard: calibrated to `1 vacil` at the reference AM tone and cross-checked
against the paper's Table 1 literature trends and the open SQAT reference oracle.
The four sensations reuse the library's ISO 532-1 loudness, DIN 45692 sharpness
and ECMA-418-2 roughness (see [Psychoacoustics](psychoacoustics.md)).
