+2566

.github/images/synchronous_average.svg

Reviewed

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1036 + <text style="font-size: 8px; font-family: sans-serif; text-anchor: start" x="266.220625" y="51.5975" transform="rotate(-0 266.220625 51.5975)">Average of N = 40 periods</text> 1037 1037 + </g> 1038 1038 + <g id="line2d_34"> 1039 1039 + <path d="M 243.820625 60.798125 1040 1040 + L 251.820625 60.798125 1041 1041 + L 259.820625 60.798125 1042 1042 + " style="fill: none; stroke-dasharray: 4.44,1.92; stroke-dashoffset: 0; stroke: #d62728; stroke-width: 1.2"/> 1043 1043 + </g> 1044 1044 + <g id="text_21"> 1045 1045 + <text style="font-size: 8px; font-family: sans-serif; text-anchor: start" x="266.220625" y="63.598125" transform="rotate(-0 266.220625 63.598125)">True periodic waveform</text> 1046 1046 + </g> 1047 1047 + </g> 1048 1048 + </g> 1049 1049 + <g id="axes_2"> 1050 1050 + <g id="patch_8"> 1051 1051 + <path d="M 426.942969 285.715781 1052 1052 + L 763.220625 285.715781 1053 1053 + L 763.220625 26.318125 1054 1054 + L 426.942969 26.318125 1055 1055 + z 1056 1056 + " style="fill: #ffffff"/> 1057 1057 + </g> 1058 1058 + <g id="matplotlib.axis_3"> 1059 1059 + <g id="xtick_8"> 1060 1060 + <g id="line2d_35"> 1061 1061 + <path d="M 426.942969 285.715781 1062 1062 + L 426.942969 26.318125 1063 1063 + " clip-path="url(#p422532d28b)" style="fill: none; stroke-dasharray: 2.96,1.28; stroke-dashoffset: 0; stroke: #e0e0e0; stroke-opacity: 0.5; stroke-width: 0.8"/> 1064 1064 + </g> 1065 1065 + <g id="line2d_36"> 1066 1066 + <g> 1067 1067 + <use xlink:href="#m3d27035b55" x="426.942969" y="285.715781" style="stroke: #000000; stroke-width: 0.8"/> 1068 1068 + </g> 1069 1069 + </g> 1070 1070 + <g id="text_22"> 1071 1071 + <text style="font-size: 10px; font-family: sans-serif; text-anchor: middle" x="426.942969" y="300.313437" transform="rotate(-0 426.942969 300.313437)">31.00</text> 1072 1072 + </g> 1073 1073 + </g> 1074 1074 + <g id="xtick_9"> 1075 1075 + <g id="line2d_37"> 1076 1076 + <path d="M 468.977676 285.715781 1077 1077 + L 468.977676 26.318125 1078 1078 + " clip-path="url(#p422532d28b)" style="fill: none; stroke-dasharray: 2.96,1.28; stroke-dashoffset: 0; stroke: #e0e0e0; stroke-opacity: 0.5; stroke-width: 0.8"/> 1079 1079 + </g> 1080 1080 + <g id="line2d_38"> 1081 1081 + <g> 1082 1082 + <use xlink:href="#m3d27035b55" x="468.977676" y="285.715781" style="stroke: #000000; stroke-width: 0.8"/> 1083 1083 + </g> 1084 1084 + </g> 1085 1085 + <g id="text_23"> 1086 1086 + <text style="font-size: 10px; font-family: sans-serif; text-anchor: middle" x="468.977676" y="300.313437" transform="rotate(-0 468.977676 300.313437)">31.25</text> 1087 1087 + </g> 1088 1088 + </g> 1089 1089 + <g id="xtick_10"> 1090 1090 + <g id="line2d_39"> 1091 1091 + <path d="M 511.012383 285.715781 1092 1092 + L 511.012383 26.318125 1093 1093 + " clip-path="url(#p422532d28b)" style="fill: none; stroke-dasharray: 2.96,1.28; stroke-dashoffset: 0; stroke: #e0e0e0; stroke-opacity: 0.5; stroke-width: 0.8"/> 1094 1094 + </g> 1095 1095 + <g id="line2d_40"> 1096 1096 + <g> 1097 1097 + <use xlink:href="#m3d27035b55" x="511.012383" y="285.715781" style="stroke: #000000; stroke-width: 0.8"/> 1098 1098 + </g> 1099 1099 + </g> 1100 1100 + <g id="text_24"> 1101 1101 + <text style="font-size: 10px; font-family: sans-serif; text-anchor: middle" x="511.012383" y="300.313437" transform="rotate(-0 511.012383 300.313437)">31.50</text> 1102 1102 + </g> 1103 1103 + </g> 1104 1104 + <g id="xtick_11"> 1105 1105 + <g id="line2d_41"> 1106 1106 + <path d="M 553.04709 285.715781 1107 1107 + L 553.04709 26.318125 1108 1108 + " clip-path="url(#p422532d28b)" style="fill: none; stroke-dasharray: 2.96,1.28; stroke-dashoffset: 0; stroke: #e0e0e0; stroke-opacity: 0.5; stroke-width: 0.8"/> 1109 1109 + </g> 1110 1110 + <g id="line2d_42"> 1111 1111 + <g> 1112 1112 + <use xlink:href="#m3d27035b55" x="553.04709" y="285.715781" style="stroke: #000000; stroke-width: 0.8"/> 1113 1113 + </g> 1114 1114 + </g> 1115 1115 + <g id="text_25"> 1116 1116 + <text style="font-size: 10px; font-family: sans-serif; text-anchor: middle" x="553.04709" y="300.313437" transform="rotate(-0 553.04709 300.313437)">31.75</text> 1117 1117 + </g> 1118 1118 + </g> 1119 1119 + <g id="xtick_12"> 1120 1120 + <g id="line2d_43"> 1121 1121 + <path d="M 595.081797 285.715781 1122 1122 + L 595.081797 26.318125 1123 1123 + " clip-path="url(#p422532d28b)" style="fill: none; stroke-dasharray: 2.96,1.28; stroke-dashoffset: 0; stroke: #e0e0e0; stroke-opacity: 0.5; stroke-width: 0.8"/> 1124 1124 + </g> 1125 1125 + <g id="line2d_44"> 1126 1126 + <g> 1127 1127 + <use xlink:href="#m3d27035b55" x="595.081797" y="285.715781" style="stroke: #000000; stroke-width: 0.8"/> 1128 1128 + </g> 1129 1129 + </g> 1130 1130 + <g id="text_26"> 1131 1131 + <text style="font-size: 10px; font-family: sans-serif; text-anchor: middle" x="595.081797" y="300.313437" transform="rotate(-0 595.081797 300.313437)">32.00</text> 1132 1132 + </g> 1133 1133 + </g> 1134 1134 + <g id="xtick_13"> 1135 1135 + <g id="line2d_45"> 1136 1136 + <path 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0.8"/> 1154 1154 + </g> 1155 1155 + <g id="line2d_48"> 1156 1156 + <g> 1157 1157 + <use xlink:href="#m3d27035b55" x="679.151211" y="285.715781" style="stroke: #000000; stroke-width: 0.8"/> 1158 1158 + </g> 1159 1159 + </g> 1160 1160 + <g id="text_28"> 1161 1161 + <text style="font-size: 10px; font-family: sans-serif; text-anchor: middle" x="679.151211" y="300.313437" transform="rotate(-0 679.151211 300.313437)">32.50</text> 1162 1162 + </g> 1163 1163 + </g> 1164 1164 + <g id="xtick_15"> 1165 1165 + <g id="line2d_49"> 1166 1166 + <path d="M 721.185918 285.715781 1167 1167 + L 721.185918 26.318125 1168 1168 + " clip-path="url(#p422532d28b)" style="fill: none; stroke-dasharray: 2.96,1.28; stroke-dashoffset: 0; stroke: #e0e0e0; stroke-opacity: 0.5; stroke-width: 0.8"/> 1169 1169 + </g> 1170 1170 + <g id="line2d_50"> 1171 1171 + <g> 1172 1172 + <use xlink:href="#m3d27035b55" x="721.185918" y="285.715781" style="stroke: #000000; stroke-width: 0.8"/> 1173 1173 + </g> 1174 1174 + </g> 1175 1175 + <g id="text_29"> 1176 1176 + <text style="font-size: 10px; font-family: sans-serif; text-anchor: middle" x="721.185918" y="300.313437" transform="rotate(-0 721.185918 300.313437)">32.75</text> 1177 1177 + </g> 1178 1178 + </g> 1179 1179 + <g id="xtick_16"> 1180 1180 + <g id="line2d_51"> 1181 1181 + <path d="M 763.220625 285.715781 1182 1182 + L 763.220625 26.318125 1183 1183 + " clip-path="url(#p422532d28b)" style="fill: none; stroke-dasharray: 2.96,1.28; stroke-dashoffset: 0; stroke: #e0e0e0; stroke-opacity: 0.5; stroke-width: 0.8"/> 1184 1184 + </g> 1185 1185 + <g id="line2d_52"> 1186 1186 + <g> 1187 1187 + <use xlink:href="#m3d27035b55" x="763.220625" y="285.715781" style="stroke: #000000; stroke-width: 0.8"/> 1188 1188 + </g> 1189 1189 + </g> 1190 1190 + <g id="text_30"> 1191 1191 + <text style="font-size: 10px; font-family: sans-serif; text-anchor: middle" x="763.220625" y="300.313437" transform="rotate(-0 763.220625 300.313437)">33.00</text> 1192 1192 + </g> 1193 1193 + 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clip-path="url(#p422532d28b)" style="fill: none; stroke-dasharray: 2.96,1.28; stroke-dashoffset: 0; stroke: #e0e0e0; stroke-opacity: 0.5; stroke-width: 0.8"/> 1234 1234 + </g> 1235 1235 + <g id="line2d_58"> 1236 1236 + <g> 1237 1237 + <use xlink:href="#mef640d9f3a" x="426.942969" y="186.897626" style="stroke: #000000; stroke-width: 0.8"/> 1238 1238 + </g> 1239 1239 + </g> 1240 1240 + <g id="text_34"> 1241 1241 + <text style="font-size: 10px; font-family: sans-serif; text-anchor: end" x="419.942969" y="190.696455" transform="rotate(-0 419.942969 190.696455)">0.4</text> 1242 1242 + </g> 1243 1243 + </g> 1244 1244 + <g id="ytick_11"> 1245 1245 + <g id="line2d_59"> 1246 1246 + <path d="M 426.942969 137.488549 1247 1247 + L 763.220625 137.488549 1248 1248 + " clip-path="url(#p422532d28b)" style="fill: none; stroke-dasharray: 2.96,1.28; stroke-dashoffset: 0; stroke: #e0e0e0; stroke-opacity: 0.5; stroke-width: 0.8"/> 1249 1249 + </g> 1250 1250 + <g id="line2d_60"> 1251 1251 + <g> 1252 1252 + 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+2566

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Reviewed

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style="font-size: 8.5px; font-family: sans-serif; fill: #ffffff">b</tspan> 982 982 + <tspan x="30.70459" y="-0.630625" style="font-size: 8.5px; font-family: sans-serif; fill: #ffffff">y</tspan> 983 983 + <tspan x="35.734863" y="-0.630625" style="font-size: 8.5px; font-family: sans-serif; fill: #ffffff"> </tspan> 984 984 + <tspan x="51.99057" y="-0.630625" style="font-size: 8.5px; font-family: sans-serif; fill: #ffffff"> </tspan> 985 985 + <tspan x="54.692474" y="-0.630625" style="font-size: 8.5px; font-family: sans-serif; fill: #ffffff">i</tspan> 986 986 + <tspan x="57.054047" y="-0.630625" style="font-size: 8.5px; font-family: sans-serif; fill: #ffffff">n</tspan> 987 987 + <tspan x="62.441254" y="-0.630625" style="font-size: 8.5px; font-family: sans-serif; fill: #ffffff"> </tspan> 988 988 + <tspan x="65.143158" y="-0.630625" style="font-size: 8.5px; font-family: sans-serif; fill: #ffffff">a</tspan> 989 989 + <tspan x="70.351898" y="-0.630625" style="font-size: 8.5px; font-family: sans-serif; fill: #ffffff">m</tspan> 990 990 + <tspan x="78.631927" y="-0.630625" style="font-size: 8.5px; font-family: sans-serif; fill: #ffffff">p</tspan> 991 991 + <tspan x="84.027435" y="-0.630625" style="font-size: 8.5px; font-family: sans-serif; fill: #ffffff">l</tspan> 992 992 + <tspan x="86.389008" y="-0.630625" style="font-size: 8.5px; font-family: sans-serif; fill: #ffffff">i</tspan> 993 993 + <tspan x="88.75058" y="-0.630625" style="font-size: 8.5px; font-family: sans-serif; fill: #ffffff">t</tspan> 994 994 + <tspan x="92.083344" y="-0.630625" style="font-size: 8.5px; font-family: sans-serif; fill: #ffffff">u</tspan> 995 995 + <tspan x="97.470551" y="-0.630625" style="font-size: 8.5px; font-family: sans-serif; fill: #ffffff">d</tspan> 996 996 + <tspan x="102.866058" y="-0.630625" style="font-size: 8.5px; font-family: sans-serif; fill: #ffffff">e</tspan> 997 997 + <tspan x="38.436768" y="-2.0525" style="font-size: 4.800299px; font-family: sans-serif; fill: #ffffff">√</tspan> 998 998 + <tspan x="44.569672" y="-0.265625" style="font-style: oblique; font-size: 8.5px; font-family: sans-serif; fill: #ffffff">N</tspan> 999 999 + </text> 1000 1000 + <rect x="43.507172" y="-10.46875" width="8.483398" height="0.53125"/> 1001 1001 + </g> 1002 1002 + </g> 1003 1003 + <g id="text_18"> 1004 1004 + <text style="font-weight: 700; font-size: 12px; font-family: sans-serif; text-anchor: middle; fill: #ffffff" x="211.081797" y="16.318125" transform="rotate(-0 211.081797 16.318125)">Periodic Waveform Extracted from Noise</text> 1005 1005 + </g> 1006 1006 + <g id="legend_1"> 1007 1007 + <g id="patch_7"> 1008 1008 + <path d="M 242.220625 68.72 1009 1009 + L 373.620625 68.72 1010 1010 + Q 375.220625 68.72 375.220625 67.12 1011 1011 + L 375.220625 31.918125 1012 1012 + Q 375.220625 30.318125 373.620625 30.318125 1013 1013 + L 242.220625 30.318125 1014 1014 + Q 240.620625 30.318125 240.620625 31.918125 1015 1015 + L 240.620625 67.12 1016 1016 + Q 240.620625 68.72 242.220625 68.72 1017 1017 + z 1018 1018 + " style="opacity: 0.8; stroke: #cccccc; stroke-linejoin: miter"/> 1019 1019 + </g> 1020 1020 + <g id="line2d_32"> 1021 1021 + <path d="M 243.820625 36.796875 1022 1022 + L 251.820625 36.796875 1023 1023 + L 259.820625 36.796875 1024 1024 + " style="fill: none; stroke: #555555; stroke-linecap: square"/> 1025 1025 + </g> 1026 1026 + <g id="text_19"> 1027 1027 + <text style="font-size: 8px; font-family: sans-serif; text-anchor: start; fill: #ffffff" x="266.220625" y="39.596875" transform="rotate(-0 266.220625 39.596875)">One noisy period</text> 1028 1028 + </g> 1029 1029 + <g id="line2d_33"> 1030 1030 + <path d="M 243.820625 48.7975 1031 1031 + L 251.820625 48.7975 1032 1032 + L 259.820625 48.7975 1033 1033 + " style="fill: none; stroke: #1f77b4; stroke-width: 1.8; stroke-linecap: square"/> 1034 1034 + </g> 1035 1035 + <g id="text_20"> 1036 1036 + <text style="font-size: 8px; font-family: sans-serif; text-anchor: start; fill: #ffffff" x="266.220625" y="51.5975" transform="rotate(-0 266.220625 51.5975)">Average of N = 40 periods</text> 1037 1037 + </g> 1038 1038 + <g id="line2d_34"> 1039 1039 + <path d="M 243.820625 60.798125 1040 1040 + L 251.820625 60.798125 1041 1041 + L 259.820625 60.798125 1042 1042 + " style="fill: none; stroke-dasharray: 4.44,1.92; stroke-dashoffset: 0; stroke: #d62728; stroke-width: 1.2"/> 1043 1043 + </g> 1044 1044 + <g id="text_21"> 1045 1045 + <text style="font-size: 8px; font-family: sans-serif; text-anchor: start; fill: #ffffff" x="266.220625" y="63.598125" transform="rotate(-0 266.220625 63.598125)">True periodic waveform</text> 1046 1046 + </g> 1047 1047 + </g> 1048 1048 + </g> 1049 1049 + <g id="axes_2"> 1050 1050 + <g id="patch_8"> 1051 1051 + <path d="M 426.942969 285.715781 1052 1052 + L 763.220625 285.715781 1053 1053 + L 763.220625 26.318125 1054 1054 + L 426.942969 26.318125 1055 1055 + z 1056 1056 + "/> 1057 1057 + </g> 1058 1058 + <g id="matplotlib.axis_3"> 1059 1059 + <g id="xtick_8"> 1060 1060 + <g id="line2d_35"> 1061 1061 + <path d="M 426.942969 285.715781 1062 1062 + L 426.942969 26.318125 1063 1063 + " clip-path="url(#p422532d28b)" style="fill: none; stroke-dasharray: 2.96,1.28; stroke-dashoffset: 0; stroke: #555555; stroke-opacity: 0.5; stroke-width: 0.8"/> 1064 1064 + </g> 1065 1065 + <g id="line2d_36"> 1066 1066 + <g> 1067 1067 + <use xlink:href="#m8d14ee1197" x="426.942969" y="285.715781" style="fill: #ffffff; stroke: #ffffff; stroke-width: 0.8"/> 1068 1068 + </g> 1069 1069 + </g> 1070 1070 + <g id="text_22"> 1071 1071 + <text style="font-size: 10px; font-family: sans-serif; text-anchor: middle; fill: #ffffff" x="426.942969" y="300.313437" transform="rotate(-0 426.942969 300.313437)">31.00</text> 1072 1072 + </g> 1073 1073 + </g> 1074 1074 + <g id="xtick_9"> 1075 1075 + <g id="line2d_37"> 1076 1076 + <path d="M 468.977676 285.715781 1077 1077 + L 468.977676 26.318125 1078 1078 + " clip-path="url(#p422532d28b)" style="fill: none; stroke-dasharray: 2.96,1.28; stroke-dashoffset: 0; stroke: #555555; stroke-opacity: 0.5; stroke-width: 0.8"/> 1079 1079 + </g> 1080 1080 + <g id="line2d_38"> 1081 1081 + <g> 1082 1082 + <use xlink:href="#m8d14ee1197" x="468.977676" y="285.715781" style="fill: #ffffff; stroke: #ffffff; stroke-width: 0.8"/> 1083 1083 + </g> 1084 1084 + </g> 1085 1085 + <g id="text_23"> 1086 1086 + <text style="font-size: 10px; font-family: sans-serif; text-anchor: middle; fill: #ffffff" x="468.977676" y="300.313437" transform="rotate(-0 468.977676 300.313437)">31.25</text> 1087 1087 + </g> 1088 1088 + </g> 1089 1089 + <g id="xtick_10"> 1090 1090 + <g id="line2d_39"> 1091 1091 + <path d="M 511.012383 285.715781 1092 1092 + L 511.012383 26.318125 1093 1093 + " clip-path="url(#p422532d28b)" style="fill: none; stroke-dasharray: 2.96,1.28; stroke-dashoffset: 0; stroke: #555555; stroke-opacity: 0.5; stroke-width: 0.8"/> 1094 1094 + </g> 1095 1095 + <g id="line2d_40"> 1096 1096 + <g> 1097 1097 + <use xlink:href="#m8d14ee1197" x="511.012383" y="285.715781" style="fill: #ffffff; stroke: #ffffff; stroke-width: 0.8"/> 1098 1098 + </g> 1099 1099 + </g> 1100 1100 + <g id="text_24"> 1101 1101 + <text style="font-size: 10px; font-family: sans-serif; text-anchor: middle; fill: #ffffff" x="511.012383" y="300.313437" transform="rotate(-0 511.012383 300.313437)">31.50</text> 1102 1102 + </g> 1103 1103 + </g> 1104 1104 + <g id="xtick_11"> 1105 1105 + <g id="line2d_41"> 1106 1106 + <path d="M 553.04709 285.715781 1107 1107 + L 553.04709 26.318125 1108 1108 + " clip-path="url(#p422532d28b)" style="fill: none; stroke-dasharray: 2.96,1.28; stroke-dashoffset: 0; stroke: #555555; stroke-opacity: 0.5; stroke-width: 0.8"/> 1109 1109 + </g> 1110 1110 + <g id="line2d_42"> 1111 1111 + <g> 1112 1112 + <use xlink:href="#m8d14ee1197" x="553.04709" y="285.715781" style="fill: #ffffff; stroke: #ffffff; stroke-width: 0.8"/> 1113 1113 + </g> 1114 1114 + </g> 1115 1115 + <g id="text_25"> 1116 1116 + <text style="font-size: 10px; font-family: sans-serif; text-anchor: middle; fill: #ffffff" x="553.04709" y="300.313437" transform="rotate(-0 553.04709 300.313437)">31.75</text> 1117 1117 + </g> 1118 1118 + </g> 1119 1119 + <g id="xtick_12"> 1120 1120 + <g id="line2d_43"> 1121 1121 + <path d="M 595.081797 285.715781 1122 1122 + L 595.081797 26.318125 1123 1123 + " clip-path="url(#p422532d28b)" style="fill: none; stroke-dasharray: 2.96,1.28; stroke-dashoffset: 0; stroke: #555555; stroke-opacity: 0.5; stroke-width: 0.8"/> 1124 1124 + </g> 1125 1125 + <g id="line2d_44"> 1126 1126 + <g> 1127 1127 + <use xlink:href="#m8d14ee1197" x="595.081797" y="285.715781" style="fill: #ffffff; stroke: #ffffff; stroke-width: 0.8"/> 1128 1128 + </g> 1129 1129 + </g> 1130 1130 + <g id="text_26"> 1131 1131 + <text style="font-size: 10px; font-family: sans-serif; text-anchor: middle; fill: #ffffff" x="595.081797" y="300.313437" transform="rotate(-0 595.081797 300.313437)">32.00</text> 1132 1132 + </g> 1133 1133 + </g> 1134 1134 + <g id="xtick_13"> 1135 1135 + <g id="line2d_45"> 1136 1136 + <path d="M 637.116504 285.715781 1137 1137 + L 637.116504 26.318125 1138 1138 + " clip-path="url(#p422532d28b)" style="fill: none; stroke-dasharray: 2.96,1.28; stroke-dashoffset: 0; stroke: #555555; stroke-opacity: 0.5; stroke-width: 0.8"/> 1139 1139 + </g> 1140 1140 + <g id="line2d_46"> 1141 1141 + <g> 1142 1142 + <use xlink:href="#m8d14ee1197" x="637.116504" y="285.715781" style="fill: #ffffff; stroke: #ffffff; stroke-width: 0.8"/> 1143 1143 + </g> 1144 1144 + </g> 1145 1145 + <g id="text_27"> 1146 1146 + <text style="font-size: 10px; font-family: sans-serif; text-anchor: middle; fill: #ffffff" x="637.116504" y="300.313437" transform="rotate(-0 637.116504 300.313437)">32.25</text> 1147 1147 + </g> 1148 1148 + </g> 1149 1149 + <g id="xtick_14"> 1150 1150 + <g id="line2d_47"> 1151 1151 + <path d="M 679.151211 285.715781 1152 1152 + L 679.151211 26.318125 1153 1153 + " clip-path="url(#p422532d28b)" style="fill: none; stroke-dasharray: 2.96,1.28; stroke-dashoffset: 0; stroke: #555555; stroke-opacity: 0.5; stroke-width: 0.8"/> 1154 1154 + </g> 1155 1155 + <g id="line2d_48"> 1156 1156 + <g> 1157 1157 + <use xlink:href="#m8d14ee1197" x="679.151211" y="285.715781" style="fill: #ffffff; stroke: #ffffff; stroke-width: 0.8"/> 1158 1158 + </g> 1159 1159 + </g> 1160 1160 + <g id="text_28"> 1161 1161 + <text style="font-size: 10px; font-family: sans-serif; text-anchor: middle; fill: #ffffff" x="679.151211" y="300.313437" transform="rotate(-0 679.151211 300.313437)">32.50</text> 1162 1162 + </g> 1163 1163 + </g> 1164 1164 + <g id="xtick_15"> 1165 1165 + <g id="line2d_49"> 1166 1166 + <path d="M 721.185918 285.715781 1167 1167 + L 721.185918 26.318125 1168 1168 + " clip-path="url(#p422532d28b)" style="fill: none; stroke-dasharray: 2.96,1.28; stroke-dashoffset: 0; stroke: #555555; stroke-opacity: 0.5; stroke-width: 0.8"/> 1169 1169 + </g> 1170 1170 + <g id="line2d_50"> 1171 1171 + <g> 1172 1172 + <use xlink:href="#m8d14ee1197" x="721.185918" y="285.715781" style="fill: #ffffff; stroke: #ffffff; stroke-width: 0.8"/> 1173 1173 + </g> 1174 1174 + </g> 1175 1175 + <g id="text_29"> 1176 1176 + <text style="font-size: 10px; font-family: sans-serif; text-anchor: middle; fill: #ffffff" x="721.185918" y="300.313437" transform="rotate(-0 721.185918 300.313437)">32.75</text> 1177 1177 + </g> 1178 1178 + </g> 1179 1179 + <g id="xtick_16"> 1180 1180 + <g id="line2d_51"> 1181 1181 + <path d="M 763.220625 285.715781 1182 1182 + L 763.220625 26.318125 1183 1183 + " clip-path="url(#p422532d28b)" style="fill: none; stroke-dasharray: 2.96,1.28; stroke-dashoffset: 0; stroke: #555555; stroke-opacity: 0.5; stroke-width: 0.8"/> 1184 1184 + </g> 1185 1185 + <g id="line2d_52"> 1186 1186 + <g> 1187 1187 + <use xlink:href="#m8d14ee1197" x="763.220625" y="285.715781" style="fill: #ffffff; stroke: #ffffff; stroke-width: 0.8"/> 1188 1188 + </g> 1189 1189 + </g> 1190 1190 + <g id="text_30"> 1191 1191 + <text style="font-size: 10px; font-family: sans-serif; text-anchor: middle; fill: #ffffff" x="763.220625" y="300.313437" transform="rotate(-0 763.220625 300.313437)">33.00</text> 1192 1192 + </g> 1193 1193 + </g> 1194 1194 + <g id="text_31"> 1195 1195 + <text style="font-size: 10px; font-family: sans-serif; text-anchor: middle; fill: #ffffff" x="595.081797" y="314.314219" transform="rotate(-0 595.081797 314.314219)">Frequency [orders]</text> 1196 1196 + </g> 1197 1197 + </g> 1198 1198 + <g id="matplotlib.axis_4"> 1199 1199 + <g id="ytick_8"> 1200 1200 + <g id="line2d_53"> 1201 1201 + <path d="M 426.942969 285.715781 1202 1202 + L 763.220625 285.715781 1203 1203 + " clip-path="url(#p422532d28b)" style="fill: none; stroke-dasharray: 2.96,1.28; stroke-dashoffset: 0; stroke: #555555; stroke-opacity: 0.5; stroke-width: 0.8"/> 1204 1204 + </g> 1205 1205 + <g id="line2d_54"> 1206 1206 + <g> 1207 1207 + <use xlink:href="#m5e07181663" x="426.942969" y="285.715781" style="fill: #ffffff; stroke: #ffffff; stroke-width: 0.8"/> 1208 1208 + </g> 1209 1209 + </g> 1210 1210 + <g id="text_32"> 1211 1211 + <text style="font-size: 10px; font-family: sans-serif; text-anchor: end; fill: #ffffff" x="419.942969" y="289.514609" transform="rotate(-0 419.942969 289.514609)">0.0</text> 1212 1212 + </g> 1213 1213 + </g> 1214 1214 + <g id="ytick_9"> 1215 1215 + <g id="line2d_55"> 1216 1216 + <path d="M 426.942969 236.306704 1217 1217 + L 763.220625 236.306704 1218 1218 + " clip-path="url(#p422532d28b)" style="fill: none; stroke-dasharray: 2.96,1.28; stroke-dashoffset: 0; stroke: #555555; stroke-opacity: 0.5; stroke-width: 0.8"/> 1219 1219 + </g> 1220 1220 + <g id="line2d_56"> 1221 1221 + <g> 1222 1222 + <use xlink:href="#m5e07181663" x="426.942969" y="236.306704" style="fill: #ffffff; stroke: #ffffff; stroke-width: 0.8"/> 1223 1223 + </g> 1224 1224 + </g> 1225 1225 + <g id="text_33"> 1226 1226 + <text style="font-size: 10px; font-family: sans-serif; text-anchor: end; fill: #ffffff" x="419.942969" y="240.105532" transform="rotate(-0 419.942969 240.105532)">0.2</text> 1227 1227 + </g> 1228 1228 + </g> 1229 1229 + <g id="ytick_10"> 1230 1230 + <g id="line2d_57"> 1231 1231 + <path d="M 426.942969 186.897626 1232 1232 + L 763.220625 186.897626 1233 1233 + " clip-path="url(#p422532d28b)" style="fill: none; stroke-dasharray: 2.96,1.28; stroke-dashoffset: 0; stroke: #555555; stroke-opacity: 0.5; stroke-width: 0.8"/> 1234 1234 + </g> 1235 1235 + <g id="line2d_58"> 1236 1236 + <g> 1237 1237 + <use xlink:href="#m5e07181663" x="426.942969" y="186.897626" style="fill: #ffffff; stroke: #ffffff; stroke-width: 0.8"/> 1238 1238 + </g> 1239 1239 + </g> 1240 1240 + <g id="text_34"> 1241 1241 + <text style="font-size: 10px; font-family: sans-serif; text-anchor: end; fill: #ffffff" x="419.942969" y="190.696455" transform="rotate(-0 419.942969 190.696455)">0.4</text> 1242 1242 + </g> 1243 1243 + </g> 1244 1244 + <g id="ytick_11"> 1245 1245 + <g id="line2d_59"> 1246 1246 + <path d="M 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+2559

.github/images/synchronous_average_es.svg

Reviewed

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<tspan x="66.500336" y="-0.630625" style="font-size: 8.5px; font-family: sans-serif">t</tspan> 988 988 + <tspan x="69.833099" y="-0.630625" style="font-size: 8.5px; font-family: sans-serif">u</tspan> 989 989 + <tspan x="75.220306" y="-0.630625" style="font-size: 8.5px; font-family: sans-serif">d</tspan> 990 990 + <tspan x="13.318604" y="-2.0525" style="font-size: 4.800299px; font-family: sans-serif">√</tspan> 991 991 + <tspan x="19.451508" y="-0.265625" style="font-style: oblique; font-size: 8.5px; font-family: sans-serif">N</tspan> 992 992 + </text> 993 993 + <rect x="18.389008" y="-10.46875" width="8.483398" height="0.53125"/> 994 994 + </g> 995 995 + </g> 996 996 + <g id="text_18"> 997 997 + <text style="font-weight: 700; font-size: 12px; font-family: sans-serif; text-anchor: middle" x="206.309922" y="16.798125" transform="rotate(-0 206.309922 16.798125)">Forma de onda periódica extraída del ruido</text> 998 998 + </g> 999 999 + <g id="legend_1"> 1000 1000 + <g id="patch_7"> 1001 1001 + <path d="M 200.08625 70.16 1002 1002 + L 368.84875 70.16 1003 1003 + Q 370.44875 70.16 370.44875 68.56 1004 1004 + L 370.44875 32.398125 1005 1005 + Q 370.44875 30.798125 368.84875 30.798125 1006 1006 + L 200.08625 30.798125 1007 1007 + Q 198.48625 30.798125 198.48625 32.398125 1008 1008 + L 198.48625 68.56 1009 1009 + Q 198.48625 70.16 200.08625 70.16 1010 1010 + z 1011 1011 + " style="fill: #ffffff; opacity: 0.8; stroke: #cccccc; stroke-linejoin: miter"/> 1012 1012 + </g> 1013 1013 + <g id="line2d_32"> 1014 1014 + <path d="M 201.68625 37.596875 1015 1015 + L 209.68625 37.596875 1016 1016 + L 217.68625 37.596875 1017 1017 + " style="fill: none; stroke: #e0e0e0; stroke-linecap: square"/> 1018 1018 + </g> 1019 1019 + <g id="text_19"> 1020 1020 + <text style="font-size: 8px; font-family: sans-serif; text-anchor: start" x="224.08625" y="40.396875" transform="rotate(-0 224.08625 40.396875)">Un período ruidoso</text> 1021 1021 + </g> 1022 1022 + <g id="line2d_33"> 1023 1023 + <path d="M 201.68625 49.9175 1024 1024 + L 209.68625 49.9175 1025 1025 + L 217.68625 49.9175 1026 1026 + " style="fill: none; stroke: #1f77b4; stroke-width: 1.8; stroke-linecap: square"/> 1027 1027 + </g> 1028 1028 + <g id="text_20"> 1029 1029 + <text style="font-size: 8px; font-family: sans-serif; text-anchor: start" x="224.08625" y="52.7175" transform="rotate(-0 224.08625 52.7175)">Promedio de N = 40 períodos</text> 1030 1030 + </g> 1031 1031 + <g id="line2d_34"> 1032 1032 + <path d="M 201.68625 62.238125 1033 1033 + L 209.68625 62.238125 1034 1034 + L 217.68625 62.238125 1035 1035 + " style="fill: none; stroke-dasharray: 4.44,1.92; stroke-dashoffset: 0; stroke: #d62728; stroke-width: 1.2"/> 1036 1036 + </g> 1037 1037 + <g id="text_21"> 1038 1038 + <text style="font-size: 8px; font-family: sans-serif; text-anchor: start" x="224.08625" y="65.038125" transform="rotate(-0 224.08625 65.038125)">Forma de onda periódica verdadera</text> 1039 1039 + </g> 1040 1040 + </g> 1041 1041 + </g> 1042 1042 + <g id="axes_2"> 1043 1043 + <g id="patch_8"> 1044 1044 + <path d="M 422.171094 286.195781 1045 1045 + L 758.44875 286.195781 1046 1046 + L 758.44875 26.798125 1047 1047 + L 422.171094 26.798125 1048 1048 + z 1049 1049 + " style="fill: #ffffff"/> 1050 1050 + </g> 1051 1051 + <g id="matplotlib.axis_3"> 1052 1052 + <g id="xtick_8"> 1053 1053 + <g id="line2d_35"> 1054 1054 + <path d="M 422.171094 286.195781 1055 1055 + L 422.171094 26.798125 1056 1056 + " clip-path="url(#p7743f044be)" style="fill: none; stroke-dasharray: 2.96,1.28; stroke-dashoffset: 0; stroke: #e0e0e0; stroke-opacity: 0.5; stroke-width: 0.8"/> 1057 1057 + </g> 1058 1058 + <g id="line2d_36"> 1059 1059 + <g> 1060 1060 + <use xlink:href="#m3d27035b55" x="422.171094" y="286.195781" style="stroke: #000000; stroke-width: 0.8"/> 1061 1061 + </g> 1062 1062 + </g> 1063 1063 + <g id="text_22"> 1064 1064 + <text style="font-size: 10px; font-family: sans-serif; text-anchor: middle" x="422.171094" y="300.793437" transform="rotate(-0 422.171094 300.793437)">31</text> 1065 1065 + </g> 1066 1066 + </g> 1067 1067 + <g id="xtick_9"> 1068 1068 + <g id="line2d_37"> 1069 1069 + <path d="M 464.205801 286.195781 1070 1070 + L 464.205801 26.798125 1071 1071 + " clip-path="url(#p7743f044be)" style="fill: none; stroke-dasharray: 2.96,1.28; stroke-dashoffset: 0; stroke: #e0e0e0; stroke-opacity: 0.5; stroke-width: 0.8"/> 1072 1072 + </g> 1073 1073 + <g id="line2d_38"> 1074 1074 + <g> 1075 1075 + <use xlink:href="#m3d27035b55" x="464.205801" y="286.195781" style="stroke: #000000; stroke-width: 0.8"/> 1076 1076 + </g> 1077 1077 + </g> 1078 1078 + <g id="text_23"> 1079 1079 + <text style="font-size: 10px; font-family: sans-serif; text-anchor: middle" x="464.205801" y="300.793437" transform="rotate(-0 464.205801 300.793437)">31,25</text> 1080 1080 + </g> 1081 1081 + </g> 1082 1082 + <g id="xtick_10"> 1083 1083 + <g id="line2d_39"> 1084 1084 + <path d="M 506.240508 286.195781 1085 1085 + L 506.240508 26.798125 1086 1086 + " clip-path="url(#p7743f044be)" style="fill: none; stroke-dasharray: 2.96,1.28; stroke-dashoffset: 0; stroke: #e0e0e0; stroke-opacity: 0.5; stroke-width: 0.8"/> 1087 1087 + </g> 1088 1088 + <g id="line2d_40"> 1089 1089 + <g> 1090 1090 + <use xlink:href="#m3d27035b55" x="506.240508" y="286.195781" style="stroke: #000000; stroke-width: 0.8"/> 1091 1091 + </g> 1092 1092 + </g> 1093 1093 + <g id="text_24"> 1094 1094 + <text style="font-size: 10px; font-family: sans-serif; text-anchor: middle" x="506.240508" y="300.793437" transform="rotate(-0 506.240508 300.793437)">31,5</text> 1095 1095 + </g> 1096 1096 + </g> 1097 1097 + <g id="xtick_11"> 1098 1098 + <g id="line2d_41"> 1099 1099 + <path d="M 548.275215 286.195781 1100 1100 + L 548.275215 26.798125 1101 1101 + " clip-path="url(#p7743f044be)" style="fill: none; stroke-dasharray: 2.96,1.28; stroke-dashoffset: 0; stroke: #e0e0e0; stroke-opacity: 0.5; stroke-width: 0.8"/> 1102 1102 + </g> 1103 1103 + <g id="line2d_42"> 1104 1104 + 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font-family: sans-serif; text-anchor: middle" x="590.309922" y="300.793437" transform="rotate(-0 590.309922 300.793437)">32</text> 1125 1125 + </g> 1126 1126 + </g> 1127 1127 + <g id="xtick_13"> 1128 1128 + <g id="line2d_45"> 1129 1129 + <path d="M 632.344629 286.195781 1130 1130 + L 632.344629 26.798125 1131 1131 + " clip-path="url(#p7743f044be)" style="fill: none; stroke-dasharray: 2.96,1.28; stroke-dashoffset: 0; stroke: #e0e0e0; stroke-opacity: 0.5; stroke-width: 0.8"/> 1132 1132 + </g> 1133 1133 + <g id="line2d_46"> 1134 1134 + <g> 1135 1135 + <use xlink:href="#m3d27035b55" x="632.344629" y="286.195781" style="stroke: #000000; stroke-width: 0.8"/> 1136 1136 + </g> 1137 1137 + </g> 1138 1138 + <g id="text_27"> 1139 1139 + <text style="font-size: 10px; font-family: sans-serif; text-anchor: middle" x="632.344629" y="300.793437" transform="rotate(-0 632.344629 300.793437)">32,25</text> 1140 1140 + </g> 1141 1141 + </g> 1142 1142 + <g id="xtick_14"> 1143 1143 + <g id="line2d_47"> 1144 1144 + <path d="M 674.379336 286.195781 1145 1145 + L 674.379336 26.798125 1146 1146 + " clip-path="url(#p7743f044be)" style="fill: none; stroke-dasharray: 2.96,1.28; stroke-dashoffset: 0; stroke: #e0e0e0; stroke-opacity: 0.5; stroke-width: 0.8"/> 1147 1147 + </g> 1148 1148 + <g id="line2d_48"> 1149 1149 + <g> 1150 1150 + <use xlink:href="#m3d27035b55" x="674.379336" y="286.195781" style="stroke: #000000; stroke-width: 0.8"/> 1151 1151 + </g> 1152 1152 + </g> 1153 1153 + <g id="text_28"> 1154 1154 + <text style="font-size: 10px; font-family: sans-serif; text-anchor: middle" x="674.379336" y="300.793437" transform="rotate(-0 674.379336 300.793437)">32,5</text> 1155 1155 + </g> 1156 1156 + </g> 1157 1157 + <g id="xtick_15"> 1158 1158 + <g id="line2d_49"> 1159 1159 + <path d="M 716.414043 286.195781 1160 1160 + L 716.414043 26.798125 1161 1161 + " clip-path="url(#p7743f044be)" style="fill: none; stroke-dasharray: 2.96,1.28; stroke-dashoffset: 0; stroke: #e0e0e0; stroke-opacity: 0.5; 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<use xlink:href="#mef640d9f3a" x="422.171094" y="88.559472" style="stroke: #000000; stroke-width: 0.8"/> 1261 1261 + </g> 1262 1262 + </g> 1263 1263 + <g id="text_36"> 1264 1264 + <text style="font-size: 10px; font-family: sans-serif; text-anchor: end" x="415.171094" y="92.3583" transform="rotate(-0 415.171094 92.3583)">0,8</text> 1265 1265 + </g> 1266 1266 + </g> 1267 1267 + <g id="ytick_13"> 1268 1268 + <g id="line2d_63"> 1269 1269 + <path d="M 422.171094 39.150394 1270 1270 + L 758.44875 39.150394 1271 1271 + " clip-path="url(#p7743f044be)" style="fill: none; stroke-dasharray: 2.96,1.28; stroke-dashoffset: 0; stroke: #e0e0e0; stroke-opacity: 0.5; stroke-width: 0.8"/> 1272 1272 + </g> 1273 1273 + <g id="line2d_64"> 1274 1274 + <g> 1275 1275 + <use xlink:href="#mef640d9f3a" x="422.171094" y="39.150394" style="stroke: #000000; stroke-width: 0.8"/> 1276 1276 + </g> 1277 1277 + </g> 1278 1278 + <g id="text_37"> 1279 1279 + <text style="font-size: 10px; font-family: sans-serif; 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+2559

.github/images/synchronous_average_es_dark.svg

Reviewed

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30.798125 1007 1007 + Q 198.48625 30.798125 198.48625 32.398125 1008 1008 + L 198.48625 68.56 1009 1009 + Q 198.48625 70.16 200.08625 70.16 1010 1010 + z 1011 1011 + " style="opacity: 0.8; stroke: #cccccc; stroke-linejoin: miter"/> 1012 1012 + </g> 1013 1013 + <g id="line2d_32"> 1014 1014 + <path d="M 201.68625 37.596875 1015 1015 + L 209.68625 37.596875 1016 1016 + L 217.68625 37.596875 1017 1017 + " style="fill: none; stroke: #555555; stroke-linecap: square"/> 1018 1018 + </g> 1019 1019 + <g id="text_19"> 1020 1020 + <text style="font-size: 8px; font-family: sans-serif; text-anchor: start; fill: #ffffff" x="224.08625" y="40.396875" transform="rotate(-0 224.08625 40.396875)">Un período ruidoso</text> 1021 1021 + </g> 1022 1022 + <g id="line2d_33"> 1023 1023 + <path d="M 201.68625 49.9175 1024 1024 + L 209.68625 49.9175 1025 1025 + L 217.68625 49.9175 1026 1026 + " style="fill: none; stroke: #1f77b4; stroke-width: 1.8; stroke-linecap: square"/> 1027 1027 + </g> 1028 1028 + <g id="text_20"> 1029 1029 + <text style="font-size: 8px; font-family: sans-serif; text-anchor: start; fill: #ffffff" x="224.08625" y="52.7175" transform="rotate(-0 224.08625 52.7175)">Promedio de N = 40 períodos</text> 1030 1030 + </g> 1031 1031 + <g id="line2d_34"> 1032 1032 + <path d="M 201.68625 62.238125 1033 1033 + L 209.68625 62.238125 1034 1034 + L 217.68625 62.238125 1035 1035 + " style="fill: none; stroke-dasharray: 4.44,1.92; stroke-dashoffset: 0; stroke: #d62728; stroke-width: 1.2"/> 1036 1036 + </g> 1037 1037 + <g id="text_21"> 1038 1038 + <text style="font-size: 8px; font-family: sans-serif; text-anchor: start; fill: #ffffff" x="224.08625" y="65.038125" transform="rotate(-0 224.08625 65.038125)">Forma de onda periódica verdadera</text> 1039 1039 + </g> 1040 1040 + </g> 1041 1041 + </g> 1042 1042 + <g id="axes_2"> 1043 1043 + <g id="patch_8"> 1044 1044 + <path d="M 422.171094 286.195781 1045 1045 + L 758.44875 286.195781 1046 1046 + L 758.44875 26.798125 1047 1047 + L 422.171094 26.798125 1048 1048 + z 1049 1049 + "/> 1050 1050 + </g> 1051 1051 + <g id="matplotlib.axis_3"> 1052 1052 + <g id="xtick_8"> 1053 1053 + <g id="line2d_35"> 1054 1054 + <path d="M 422.171094 286.195781 1055 1055 + L 422.171094 26.798125 1056 1056 + " clip-path="url(#p7743f044be)" style="fill: none; stroke-dasharray: 2.96,1.28; stroke-dashoffset: 0; stroke: #555555; stroke-opacity: 0.5; stroke-width: 0.8"/> 1057 1057 + </g> 1058 1058 + <g id="line2d_36"> 1059 1059 + <g> 1060 1060 + <use xlink:href="#m8d14ee1197" x="422.171094" y="286.195781" style="fill: #ffffff; stroke: #ffffff; stroke-width: 0.8"/> 1061 1061 + </g> 1062 1062 + </g> 1063 1063 + <g id="text_22"> 1064 1064 + <text style="font-size: 10px; font-family: sans-serif; text-anchor: middle; fill: #ffffff" x="422.171094" y="300.793437" transform="rotate(-0 422.171094 300.793437)">31</text> 1065 1065 + </g> 1066 1066 + </g> 1067 1067 + <g id="xtick_9"> 1068 1068 + <g id="line2d_37"> 1069 1069 + <path d="M 464.205801 286.195781 1070 1070 + L 464.205801 26.798125 1071 1071 + " clip-path="url(#p7743f044be)" style="fill: none; stroke-dasharray: 2.96,1.28; stroke-dashoffset: 0; stroke: #555555; stroke-opacity: 0.5; stroke-width: 0.8"/> 1072 1072 + </g> 1073 1073 + <g id="line2d_38"> 1074 1074 + <g> 1075 1075 + <use xlink:href="#m8d14ee1197" x="464.205801" y="286.195781" style="fill: #ffffff; stroke: #ffffff; stroke-width: 0.8"/> 1076 1076 + </g> 1077 1077 + </g> 1078 1078 + <g id="text_23"> 1079 1079 + <text style="font-size: 10px; font-family: sans-serif; text-anchor: middle; fill: #ffffff" x="464.205801" y="300.793437" transform="rotate(-0 464.205801 300.793437)">31,25</text> 1080 1080 + </g> 1081 1081 + </g> 1082 1082 + <g id="xtick_10"> 1083 1083 + <g id="line2d_39"> 1084 1084 + <path d="M 506.240508 286.195781 1085 1085 + L 506.240508 26.798125 1086 1086 + " clip-path="url(#p7743f044be)" style="fill: none; stroke-dasharray: 2.96,1.28; stroke-dashoffset: 0; stroke: #555555; stroke-opacity: 0.5; stroke-width: 0.8"/> 1087 1087 + </g> 1088 1088 + <g id="line2d_40"> 1089 1089 + <g> 1090 1090 + <use xlink:href="#m8d14ee1197" x="506.240508" y="286.195781" style="fill: #ffffff; stroke: #ffffff; stroke-width: 0.8"/> 1091 1091 + </g> 1092 1092 + </g> 1093 1093 + <g id="text_24"> 1094 1094 + <text style="font-size: 10px; font-family: sans-serif; text-anchor: middle; fill: #ffffff" x="506.240508" y="300.793437" transform="rotate(-0 506.240508 300.793437)">31,5</text> 1095 1095 + </g> 1096 1096 + </g> 1097 1097 + <g id="xtick_11"> 1098 1098 + <g id="line2d_41"> 1099 1099 + <path d="M 548.275215 286.195781 1100 1100 + L 548.275215 26.798125 1101 1101 + " clip-path="url(#p7743f044be)" style="fill: none; stroke-dasharray: 2.96,1.28; stroke-dashoffset: 0; stroke: #555555; stroke-opacity: 0.5; stroke-width: 0.8"/> 1102 1102 + </g> 1103 1103 + <g id="line2d_42"> 1104 1104 + <g> 1105 1105 + <use xlink:href="#m8d14ee1197" x="548.275215" y="286.195781" style="fill: #ffffff; stroke: #ffffff; stroke-width: 0.8"/> 1106 1106 + </g> 1107 1107 + </g> 1108 1108 + <g id="text_25"> 1109 1109 + <text style="font-size: 10px; font-family: sans-serif; text-anchor: middle; fill: #ffffff" x="548.275215" y="300.793437" transform="rotate(-0 548.275215 300.793437)">31,75</text> 1110 1110 + </g> 1111 1111 + </g> 1112 1112 + <g id="xtick_12"> 1113 1113 + <g id="line2d_43"> 1114 1114 + <path d="M 590.309922 286.195781 1115 1115 + L 590.309922 26.798125 1116 1116 + " clip-path="url(#p7743f044be)" style="fill: none; stroke-dasharray: 2.96,1.28; stroke-dashoffset: 0; stroke: #555555; stroke-opacity: 0.5; stroke-width: 0.8"/> 1117 1117 + </g> 1118 1118 + <g id="line2d_44"> 1119 1119 + <g> 1120 1120 + <use xlink:href="#m8d14ee1197" x="590.309922" y="286.195781" style="fill: #ffffff; stroke: #ffffff; stroke-width: 0.8"/> 1121 1121 + </g> 1122 1122 + </g> 1123 1123 + <g id="text_26"> 1124 1124 + <text style="font-size: 10px; font-family: sans-serif; text-anchor: middle; fill: #ffffff" x="590.309922" y="300.793437" transform="rotate(-0 590.309922 300.793437)">32</text> 1125 1125 + </g> 1126 1126 + </g> 1127 1127 + <g id="xtick_13"> 1128 1128 + <g id="line2d_45"> 1129 1129 + <path d="M 632.344629 286.195781 1130 1130 + L 632.344629 26.798125 1131 1131 + " clip-path="url(#p7743f044be)" style="fill: none; stroke-dasharray: 2.96,1.28; stroke-dashoffset: 0; stroke: #555555; stroke-opacity: 0.5; stroke-width: 0.8"/> 1132 1132 + </g> 1133 1133 + <g id="line2d_46"> 1134 1134 + <g> 1135 1135 + <use xlink:href="#m8d14ee1197" x="632.344629" y="286.195781" style="fill: #ffffff; stroke: #ffffff; stroke-width: 0.8"/> 1136 1136 + </g> 1137 1137 + </g> 1138 1138 + <g id="text_27"> 1139 1139 + <text style="font-size: 10px; font-family: sans-serif; text-anchor: middle; fill: #ffffff" x="632.344629" y="300.793437" transform="rotate(-0 632.344629 300.793437)">32,25</text> 1140 1140 + </g> 1141 1141 + </g> 1142 1142 + <g id="xtick_14"> 1143 1143 + <g id="line2d_47"> 1144 1144 + <path d="M 674.379336 286.195781 1145 1145 + L 674.379336 26.798125 1146 1146 + " clip-path="url(#p7743f044be)" style="fill: none; stroke-dasharray: 2.96,1.28; stroke-dashoffset: 0; stroke: #555555; stroke-opacity: 0.5; stroke-width: 0.8"/> 1147 1147 + </g> 1148 1148 + <g id="line2d_48"> 1149 1149 + <g> 1150 1150 + <use xlink:href="#m8d14ee1197" x="674.379336" y="286.195781" style="fill: #ffffff; stroke: #ffffff; stroke-width: 0.8"/> 1151 1151 + </g> 1152 1152 + </g> 1153 1153 + <g id="text_28"> 1154 1154 + <text style="font-size: 10px; font-family: sans-serif; text-anchor: middle; fill: #ffffff" x="674.379336" y="300.793437" transform="rotate(-0 674.379336 300.793437)">32,5</text> 1155 1155 + </g> 1156 1156 + </g> 1157 1157 + <g id="xtick_15"> 1158 1158 + <g id="line2d_49"> 1159 1159 + <path d="M 716.414043 286.195781 1160 1160 + L 716.414043 26.798125 1161 1161 + " clip-path="url(#p7743f044be)" style="fill: none; stroke-dasharray: 2.96,1.28; stroke-dashoffset: 0; stroke: #555555; stroke-opacity: 0.5; stroke-width: 0.8"/> 1162 1162 + </g> 1163 1163 + <g id="line2d_50"> 1164 1164 + <g> 1165 1165 + <use xlink:href="#m8d14ee1197" x="716.414043" y="286.195781" style="fill: #ffffff; stroke: #ffffff; stroke-width: 0.8"/> 1166 1166 + </g> 1167 1167 + </g> 1168 1168 + <g id="text_29"> 1169 1169 + <text style="font-size: 10px; font-family: sans-serif; text-anchor: middle; fill: #ffffff" x="716.414043" y="300.793437" transform="rotate(-0 716.414043 300.793437)">32,75</text> 1170 1170 + </g> 1171 1171 + </g> 1172 1172 + <g id="xtick_16"> 1173 1173 + <g id="line2d_51"> 1174 1174 + <path d="M 758.44875 286.195781 1175 1175 + L 758.44875 26.798125 1176 1176 + " clip-path="url(#p7743f044be)" style="fill: none; stroke-dasharray: 2.96,1.28; stroke-dashoffset: 0; stroke: #555555; stroke-opacity: 0.5; stroke-width: 0.8"/> 1177 1177 + </g> 1178 1178 + <g id="line2d_52"> 1179 1179 + <g> 1180 1180 + <use xlink:href="#m8d14ee1197" x="758.44875" y="286.195781" style="fill: #ffffff; stroke: #ffffff; stroke-width: 0.8"/> 1181 1181 + </g> 1182 1182 + </g> 1183 1183 + <g id="text_30"> 1184 1184 + <text style="font-size: 10px; font-family: sans-serif; text-anchor: middle; fill: #ffffff" x="758.44875" y="300.793437" transform="rotate(-0 758.44875 300.793437)">33</text> 1185 1185 + </g> 1186 1186 + </g> 1187 1187 + <g id="text_31"> 1188 1188 + <text style="font-size: 10px; font-family: sans-serif; text-anchor: middle; fill: #ffffff" x="590.309922" y="315.194219" transform="rotate(-0 590.309922 315.194219)">Frecuencia [órdenes]</text> 1189 1189 + </g> 1190 1190 + </g> 1191 1191 + <g id="matplotlib.axis_4"> 1192 1192 + <g id="ytick_8"> 1193 1193 + <g id="line2d_53"> 1194 1194 + <path d="M 422.171094 286.195781 1195 1195 + L 758.44875 286.195781 1196 1196 + " clip-path="url(#p7743f044be)" style="fill: none; stroke-dasharray: 2.96,1.28; stroke-dashoffset: 0; stroke: #555555; stroke-opacity: 0.5; stroke-width: 0.8"/> 1197 1197 + </g> 1198 1198 + <g id="line2d_54"> 1199 1199 + <g> 1200 1200 + <use xlink:href="#m5e07181663" x="422.171094" y="286.195781" style="fill: #ffffff; stroke: #ffffff; stroke-width: 0.8"/> 1201 1201 + </g> 1202 1202 + </g> 1203 1203 + <g id="text_32"> 1204 1204 + <text style="font-size: 10px; font-family: sans-serif; text-anchor: end; fill: #ffffff" x="415.171094" y="289.994609" transform="rotate(-0 415.171094 289.994609)">0</text> 1205 1205 + </g> 1206 1206 + </g> 1207 1207 + <g id="ytick_9"> 1208 1208 + <g id="line2d_55"> 1209 1209 + <path d="M 422.171094 236.786704 1210 1210 + L 758.44875 236.786704 1211 1211 + " clip-path="url(#p7743f044be)" style="fill: none; stroke-dasharray: 2.96,1.28; stroke-dashoffset: 0; stroke: #555555; stroke-opacity: 0.5; stroke-width: 0.8"/> 1212 1212 + </g> 1213 1213 + <g id="line2d_56"> 1214 1214 + <g> 1215 1215 + <use xlink:href="#m5e07181663" x="422.171094" y="236.786704" style="fill: #ffffff; stroke: #ffffff; stroke-width: 0.8"/> 1216 1216 + </g> 1217 1217 + </g> 1218 1218 + <g id="text_33"> 1219 1219 + <text style="font-size: 10px; font-family: sans-serif; text-anchor: end; fill: #ffffff" x="415.171094" y="240.585532" transform="rotate(-0 415.171094 240.585532)">0,2</text> 1220 1220 + </g> 1221 1221 + </g> 1222 1222 + <g id="ytick_10"> 1223 1223 + <g id="line2d_57"> 1224 1224 + <path d="M 422.171094 187.377626 1225 1225 + L 758.44875 187.377626 1226 1226 + " clip-path="url(#p7743f044be)" style="fill: none; stroke-dasharray: 2.96,1.28; stroke-dashoffset: 0; stroke: #555555; stroke-opacity: 0.5; stroke-width: 0.8"/> 1227 1227 + </g> 1228 1228 + <g id="line2d_58"> 1229 1229 + <g> 1230 1230 + <use xlink:href="#m5e07181663" x="422.171094" y="187.377626" style="fill: #ffffff; stroke: #ffffff; stroke-width: 0.8"/> 1231 1231 + </g> 1232 1232 + </g> 1233 1233 + <g id="text_34"> 1234 1234 + <text style="font-size: 10px; font-family: sans-serif; text-anchor: end; fill: #ffffff" x="415.171094" y="191.176455" transform="rotate(-0 415.171094 191.176455)">0,4</text> 1235 1235 + </g> 1236 1236 + </g> 1237 1237 + <g id="ytick_11"> 1238 1238 + <g id="line2d_59"> 1239 1239 + <path d="M 422.171094 137.968549 1240 1240 + L 758.44875 137.968549 1241 1241 + " clip-path="url(#p7743f044be)" style="fill: none; stroke-dasharray: 2.96,1.28; stroke-dashoffset: 0; stroke: #555555; stroke-opacity: 0.5; stroke-width: 0.8"/> 1242 1242 + </g> 1243 1243 + <g id="line2d_60"> 1244 1244 + <g> 1245 1245 + <use xlink:href="#m5e07181663" x="422.171094" y="137.968549" style="fill: #ffffff; stroke: #ffffff; stroke-width: 0.8"/> 1246 1246 + </g> 1247 1247 + </g> 1248 1248 + <g id="text_35"> 1249 1249 + <text style="font-size: 10px; font-family: sans-serif; text-anchor: end; fill: #ffffff" x="415.171094" y="141.767377" transform="rotate(-0 415.171094 141.767377)">0,6</text> 1250 1250 + </g> 1251 1251 + </g> 1252 1252 + <g id="ytick_12"> 1253 1253 + <g id="line2d_61"> 1254 1254 + <path d="M 422.171094 88.559472 1255 1255 + L 758.44875 88.559472 1256 1256 + " clip-path="url(#p7743f044be)" style="fill: none; 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= 32 lo deja pasar</text> 2503 2503 + </g> 2504 2504 + <g id="text_40"> 2505 2505 + <text style="font-weight: 700; font-size: 12px; font-family: sans-serif; text-anchor: middle; fill: #ffffff" x="590.309922" y="16.798125" transform="rotate(-0 590.309922 16.798125)">Rechazo de un tono eligiendo N (McFadden 1987)</text> 2506 2506 + </g> 2507 2507 + <g id="legend_2"> 2508 2508 + <g id="patch_13"> 2509 2509 + <path d="M 626.65625 69.2 2510 2510 + L 752.84875 69.2 2511 2511 + Q 754.44875 69.2 754.44875 67.6 2512 2512 + L 754.44875 32.398125 2513 2513 + Q 754.44875 30.798125 752.84875 30.798125 2514 2514 + L 626.65625 30.798125 2515 2515 + Q 625.05625 30.798125 625.05625 32.398125 2516 2516 + L 625.05625 67.6 2517 2517 + Q 625.05625 69.2 626.65625 69.2 2518 2518 + z 2519 2519 + " style="opacity: 0.8; stroke: #cccccc; stroke-linejoin: miter"/> 2520 2520 + </g> 2521 2521 + <g id="line2d_68"> 2522 2522 + <path d="M 628.25625 37.276875 2523 2523 + L 636.25625 37.276875 2524 2524 + L 644.25625 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+22 -3

CHANGELOG.md

Reviewed

··· 8 8 ## [Unreleased] 9 9 10 10 ### Added 11 11 + - `time_synchronous_average`: extraction of a periodic waveform of known 12 12 + period from asynchronous noise by time domain averaging (P. D. McFadden, 13 13 + "A revised model for the extraction of periodic waveforms by time domain 14 14 + averaging", Mechanical Systems and Signal Processing 1(1), 1987, 83-95). 15 15 + Ensemble-averaging N successive periods (Eq. 5) reinforces every component 16 16 + synchronous with the period and suppresses the rest, with the residual 17 17 + asynchronous-noise standard deviation falling as 1/sqrt(N). The frozen 18 18 + `SynchronousAverageResult` carries the averaged one-period waveform, the 19 19 + residual and its RMS, the noise reduction in dB (10*log10(N)) with the 20 20 + sqrt(N) amplitude gain, the comb-filter response over the first harmonics, 21 21 + and a `.plot()` (EN/ES). When the sample rate does not divide the period 22 22 + the blocks are aligned to a common integer grid by the band-limited 23 23 + fractional delay of `fractional_delay`; an integer number of samples per 24 24 + period is recovered to machine precision. `comb_filter_response` evaluates 25 25 + the closed-form comb magnitude (Eq. 8/9), a Dirichlet kernel with unit 26 26 + teeth at the harmonics k/T and nodes at j/(N*T), so an interfering order is 27 27 + best rejected by choosing N to place a node on it (McFadden's 32.05-order 28 28 + example: N = 20 beats the power-of-two N = 32). 11 29 - `miso_coherence`: multiple and partial coherence of a multiple-input, 12 30 single-output system (Bendat & Piersol, Random Data 4e, Chapter 7). From the 13 13 - Welch cross-spectral matrix of two or three partially correlated inputs and 31 31 + Welch cross-spectral matrix of several partially correlated inputs and 14 32 one output it reports the ordinary coherence of each input (Eq. 7.109), the 15 33 multiple coherence explained by all inputs jointly (Eq. 7.35), and the 16 34 partial coherences (Eq. 7.87) obtained by the Gaussian-elimination ··· 1092 1110 cross-reference). 1093 1111 1094 1112 ### Changed 1095 1095 - - Raised the `scipy` floor in `pyproject.toml` to the current release 1096 1096 - (`scipy>=1.18.0`), matching the pin `requirements.txt` already carried. The 1113 1113 + - Raised the `scipy` floor in `pyproject.toml` to `scipy>=1.18.0`, the minimum 1114 1114 + supported scipy version, matching the floor `requirements.txt` already 1115 1115 + carried. The 1097 1116 `numpy` floor stays at `>=2.4.4` because the optional `perf` extra pulls 1098 1117 numba, which still requires NumPy 2.4 or older; the pure-Python matrix and the 1099 1118 figure stack already run on the latest NumPy through their own pins.

+14 -1

docs/CONFORMANCE.md

Reviewed

··· 15 15 16 16 ## Numerical conformance report 17 17 18 18 - &#9989; **405/405 conformance checks pass** across 51 domains and 267 standards - filters class 1 - weightings within IEC 61672-1 class 1. 18 18 + &#9989; **410/410 conformance checks pass** across 52 domains and 272 standards - filters class 1 - weightings within IEC 61672-1 class 1. 19 19 20 20 _{Each row pins a standard clause to its expected normative value and the value the library computes. Every section below is collapsible and stays collapsed while all of its rows pass; a section with any failing row opens automatically.} 21 21 ··· 587 587 | Havelock 2008 Ch. 27 Fig. 21 + Mercator series of ln(1+a*e^{-j*theta}) | Power-cepstrum height at the echo delay = reflection coefficient a | 0.4 (+/-0) | 0.4 | 0 | &#9989; | 588 588 | Havelock 2008 Ch. 87 Eq. (14): complex cepstrum, series term n = 2 | Second rahmonic of a reflection a = 0.4 equals -a^2/2 | -0.08 (+/-0) | -0.08 | 0 | &#9989; | 589 589 | Bendat & Piersol, Random Data 4e Sec. 13.3 (Fig. 13.11) | Envelope-spectrum line of an AM tone (A0 = 2, m = 0.35) at fm | 0.7 (+/-0.002) | 0.7 | 0 | &#9989; | 590 590 + 591 591 + </details> 592 592 + 593 593 + <details> 594 594 + <summary>&#9989; <b>Time synchronous averaging (McFadden 1987)</b>: 100% (5/5)</summary> 595 595 + 596 596 + | Standard | Quantity | Expected (norm) | Computed | &#916; | Status | 597 597 + |:---|:---|:---|:---|:---|:---:| 598 598 + | McFadden 1987 Eq. 8 / Eq. 9: comb filter \|C(f)\| at a harmonic k/T | Comb-filter tooth height at a harmonic equals unity (any N) | 1 (+/-0) | 1 | 0 | &#9989; | 599 599 + | McFadden 1987 Eq. 8: comb filter one quarter-order from a tooth, N = 2 | Comb-filter magnitude = 1/sqrt(2) at order 0.25 | 0.70710678 (+/-0) | 0.70710678 | 0 | &#9989; | 600 600 + | McFadden 1987 Sec. 4 (Fig. 5): node selection, tone at 32.05 orders | N = 20 places a comb node on 32.05 orders (\|C\| = 0), not the power-of-2 N = 32 | 0 (+/-0.0000000001) | 0 | 0 | &#9989; | 601 601 + | McFadden 1987 Eq. 5: exact recovery, integer samples per period | Noiseless periodic waveform (M = 256) recovered to machine precision | 0 (+/-0.0000000001) | 0 | 0 | &#9989; | 602 602 + | McFadden 1987 Sec. 1: asynchronous-noise variance reduced by 1/N | Residual noise std of the average falls as sigma/sqrt(N), N = 64 | 0.125 (+/-15%) | 0.12414 | -0.001 | &#9989; | 590 603 591 604 </details> 592 605

+2 -1

docs/README.md

Reviewed

··· 66 66 - [Calibrated spectral analysis](spectral-analysis.md): the Bendat & Piersol Welch estimators with their statistical quality: PSD and cross-spectral density with the effective number of averages, normalized random errors and chi-square confidence intervals, the coherent output spectrum with the spectral SNR, constant-power 1/n-octave smoothing, colored-noise generators with an exact power-law slope, and the Harris window figures of merit for choosing the taper 67 67 - [Time-frequency analysis](time-frequency.md): the calibrated STFT spectrogram in absolute dB SPL with the exact Welch-module scaling and the time-versus-frequency resolution trade-off, and the zoom FFT that resolves tones closer than a practical FFT bin 68 68 - [2D FDTD wave simulation](fdtd-simulation.md): the deterministic staggered-grid pressure-velocity FDTD solver (Attenborough & Van Renterghem 2021, chapter 4) with Gaussian, tone and arbitrary-signal sources, pressure probes, rasterised obstacles, rigid/impedance/absorbing boundaries, and a result object with probe histories and field snapshots 69 69 - - [Multiple and partial coherence](miso-coherence.md): the Bendat & Piersol multiple-input/output coherence functions for two or three correlated sources and one output, with the Gaussian-elimination conditioning that separates a genuine cause from a source that merely correlates with it, and the partial coherent output spectra that say which source dominates each band 69 69 + - [Multiple and partial coherence](miso-coherence.md): the Bendat & Piersol multiple-input/output coherence functions for several correlated sources and one output, with the Gaussian-elimination conditioning that separates a genuine cause from a source that merely correlates with it, and the partial coherent output spectra that say which source dominates each band 70 70 - [Correlation, time delay and envelope](correlation-delay.md): auto- and cross-correlation with the Bendat & Piersol normalizations and random errors, time-delay estimation by the direct correlator, the cross-spectrum phase slope and the Knapp & Carter GCC (Roth, SCOT, PHAT, maximum likelihood) with the Eq. 8.129 peak-location uncertainty, sub-sample impulse-response delay and alignment, and the Hilbert envelope with instantaneous phase and frequency 71 71 - [Test signals and sample-rate tools](test-signals.md): IEC 60268-1 tone bursts with exact gating (zero-crossing start, integral full periods, repetitive trains), polyphase resampling behind an explicit anti-alias specification whose designed Kaiser filter travels with the result, and band-limited fractional delay with a linear or circular boundary 72 72 - [Cepstrum, echoes and the envelope spectrum](cepstrum-echoes.md): the power, real and complex cepstrum with quefrency analysis, echo detection with the reflection coefficient read off the cepstral peak, lowpass/highpass liftering of a log spectrum, the homomorphic round trip of the complex cepstrum, and the envelope spectrum that turns amplitude modulations into discrete lines 73 73 + - [Time synchronous averaging](synchronous-averaging.md): extraction of a periodic waveform of known period by time domain averaging (McFadden 1987), the comb filter that describes the operation in the frequency domain with unit teeth at the harmonics and nodes between them, the square-root noise-reduction law, and the choice of the number of averages that places a comb node on an interfering order 73 74 - [System measurement: Golay, shaped sweeps, inversion](system-measurement.md): complementary Golay pairs whose periodic autocorrelations sum to an exact delta and deconvolve a noiseless system to machine precision, sweeps synthesized to follow an arbitrary target magnitude spectrum by group-delay shaping (Mueller & Massarani) with a near-ideal crest factor, and the regularized spectral inversion of a measured response with Kirkeby frequency-dependent regularization, achieved flatness and a capped out-of-band gain 74 75 75 76 ## Reference

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··· 625 625 | `cross_spectral_density` | `function` | **Welch cross-spectral density with magnitude/phase errors (Eqs. 9.33/9.52).**<br>• `x`, `y`, `fs`<br>• `window`, `nperseg`, `overlap`, `scaling` as above | `res = cross_spectral_density(x, y, fs)`<br><br>• `CrossSpectralDensityResult` | 626 626 | `coherent_output_spectrum` | `function` | **Gvv = γ²·Gyy, noise remainder and spectral SNR (Eqs. 9.55–9.56, 9.73).**<br>• `x` (input), `y` (output), `fs`<br>• `window`, `nperseg`, `overlap`, `scaling` as above | `res = coherent_output_spectrum(x, y, fs)`<br><br>• `CoherentOutputSpectrumResult` | 627 627 | `fractional_octave_smoothing` | `function` | **Constant-power 1/n-octave smoothing of a spectrum.**<br>• `frequencies`, `values`<br>• `fraction`: the n of 1/n octave (Default: 3)<br>• `domain`: 'power'/'amplitude'/'db' (Default: 'power') | `s = fractional_octave_smoothing(f, psd, 3.0)`<br><br>• smoothed array, same domain; flat spectra unchanged | 628 628 - | `miso_coherence` | `function` | **Multiple & partial coherence of a MISO system (Bendat & Piersol Ch. 7).**<br>• `inputs`: 2 or 3 input records (sequence of 1-D arrays or a (q, n) array), `output`, `fs`<br>• `order`: conditioning order (Default: 0..q−1)<br>• `window`, `nperseg`, `overlap`, `scaling` as above | `res = miso_coherence([x1, x2], y, fs)`<br><br>• `MISOCoherenceResult` | 628 628 + | `miso_coherence` | `function` | **Multiple & partial coherence of a MISO system (Bendat & Piersol Ch. 7).**<br>• `inputs`: q input records, q ≥ 2 (sequence of 1-D arrays or a (q, n) array), `output`, `fs`<br>• `order`: conditioning order (Default: 0..q−1)<br>• `window`, `nperseg`, `overlap`, `scaling` as above | `res = miso_coherence([x1, x2], y, fs)`<br><br>• `MISOCoherenceResult` | 629 629 | `MISOCoherenceResult` | `dataclass` | **Multiple/partial coherence of q correlated inputs and one output.**<br>• `ordinary_coherence` (q×F), `multiple_coherence` γ²y:x, `partial_coherence` (q×F, conditioned per `order`)<br>• `coherent_output_spectra` Gvi (q×F, Σ Gvi + `noise_psd` = `output_psd`)<br>• `multiple_coherence_random_error` (Eq. 9.98), `coherent_output_random_error` (Eq. 9.100)<br>• `.dominant_input()`: per-band index of the strongest source<br>• `.plot()`: coherent output spectra + coherences | `res.multiple_coherence, res.dominant_input()` | 630 630 | `resolution_bias_error` | `function` | **First-order resolution bias at a resonance peak, εb ≈ −(Be/Br)²/3 (Eq. 8.141).**<br>• `resolution_bandwidth` Be [Hz]<br>• `half_power_bandwidth` Br [Hz] | `resolution_bias_error(1.0, 4.0) # -1/48` | 631 631 | `noise_signal` | `function` | **Colored Gaussian noise with an exact power-law PSD slope.**<br>• `fs`, `seconds` (Default: 1.0)<br>• `color`: 'white'/'pink'/'red'/'blue'/'violet' (0/−3.01/−6.02/+3.01/+6.02 dB/oct)<br>• `rms` (Default: 1.0)<br>• `seed`: bit-reproducible per seed | `pink = noise_signal(48000, 10.0, color='pink', seed=7)` | ··· 657 657 | `EnvelopeResult` | `dataclass` | **Envelope analysis.**<br>• `times` [s], `envelope`, `phase` [rad, unwrapped], `instantaneous_frequency` [Hz]<br>• `fs` (output rate), `signal`, `signal_fs`, `decimation_factor`, `antialias`<br>• `.plot()`: signal + envelope, instantaneous frequency | `res.envelope, res.instantaneous_frequency` | 658 658 | `envelope_spectrum` | `function` | **Amplitude spectrum of the envelope: modulations as lines (B&P 13.3).**<br>• `x`, `fs`<br>• `kind`: 'magnitude' (Default) / 'squared' (square-law detector, Fig. 13.11)<br>• `window` (Default: 'hann', coherent-gain corrected), `nfft`<br>• `remove_dc` (Default: True; mean kept in `mean_level`)<br>AM tone A0, m: line A0·m at fm ('magnitude') | `res = envelope_spectrum(x, fs)`<br><br>• `EnvelopeSpectrumResult` | 659 659 | `EnvelopeSpectrumResult` | `dataclass` | **Envelope spectrum.**<br>• `frequencies` [Hz], `amplitude` (line heights), `mean_level`, `kind`<br>• `times`, `envelope` (detector output), `window`, `remove_dc`, `fs`, `nfft`<br>• `.plot()`: envelope + spectrum | `res.amplitude, res.mean_level` | 660 660 + | `time_synchronous_average` | `function` | **Extract a periodic waveform of known period by time domain averaging (McFadden 1987 Eq. 5).**<br>• `x`, `fs`, `period` [s] (one revolution)<br>• `n_averages`: whole periods to average (Default: as many as the record holds; choose N so N·q is integer to place a comb node on an interfering order q)<br>• `n_harmonics`: comb-response span (Default: 8)<br>Non-integer fs·period aligned by band-limited fractional delay | `res = time_synchronous_average(x, fs, 1/32)`<br><br>• `SynchronousAverageResult` | 661 661 + | `SynchronousAverageResult` | `dataclass` | **Time synchronous average.**<br>• `period_waveform`, `times` [s], `residual`, `residual_rms`<br>• `n_averages`, `samples_per_period`, `period`, `fs`, `interpolated`<br>• `noise_reduction_db` = 10·log₁₀N, `amplitude_snr_gain` = √N<br>• `comb_frequencies` [Hz], `comb_response` (\|C(f)\|)<br>• `.plot()`: averaged waveform + comb filter | `res.period_waveform, res.noise_reduction_db` | 662 662 + | `comb_filter_response` | `function` | **Magnitude of the N-period synchronous-averaging comb filter (McFadden 1987 Eq. 8).**<br>• `frequencies` [Hz], `period` [s], `n_averages`<br>\|C(f)\| = \|sin(N·π·f·T)/(N·sin(π·f·T))\|: unit teeth at harmonics k/T, nodes at j/(N·T) | `c = comb_filter_response(freqs, 1/32, 20)` | 660 663 | `regularized_inverse_filter` | `function` | **Kirkeby frequency-dependent regularized inversion (Kirkeby & Nelson 1999 Eq. (17)).**<br>• `response`: measured IR (array or `ImpulseResponseResult`; its `fs` rides along)<br>• `fs`: Sample rate [Hz]<br>• `f_range`: (f1, f2) equalized to unity (Required)<br>• `regularization_inside` (Default: 1e-6) / `regularization_outside` (Default: 1.0), fractions of max\|H\|²<br>• `transition_octaves` (Default: 1/3), `n_fft`, `delay` (Default: n_fft//2) | `inv = regularized_inverse_filter(ir, f_range=(100, 10000))`<br><br>• `InverseFilterResult` | 661 664 | `InverseFilterResult` | `dataclass` | **Regularized inverse filter.**<br>• `inverse` (time domain), `spectrum` (with modeling delay), `response_spectrum`, `regularization` = ε(f), `frequencies`, `f_range`, `delay`, `fs`<br>• `flatness_db`: worst in-band deviation of \|H·H_inv\| from 0 dB<br>• `max_gain_db`: out-of-band boost (≤ the 1/(2√ε) cap)<br>• `.apply(x)`: equalize, delay removed; `.plot()` | `flat = inv.apply(recording)` | 662 665 | `cepstrum` | `function` | **Power/real/complex cepstrum (Havelock Chs. 27/87).**<br>• `x`, `fs`<br>• `kind`: 'power' (Default, IDFT of ln\|X\|²) / 'real' (ln\|X\|) / 'complex' (invertible, phase unwrapped)<br>• `nfft`: even, ≥ record (Default: record length)<br>Echo a at t0: rahmonics (−1)^(n+1)·aⁿ/n at n·t0 | `res = cepstrum(x, fs, kind='complex')`<br><br>• `CepstrumResult` |

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docs/miso-coherence.md

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··· 9 9 Piersol, *Random Data* (4th ed., 2010, Chapter 7), resolve this for a 10 10 multiple-input/single-output (MISO) system with the **multiple** and 11 11 **partial** coherence functions. `miso_coherence` computes them from the same 12 12 - Welch cross-spectral core as the rest of `phonometry.metrology`, for two or 13 13 - three inputs and one output. 12 12 + Welch cross-spectral core as the rest of `phonometry.metrology`, for several 13 13 + correlated inputs and one output. 14 14 15 15 <picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/miso_coherence_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/miso_coherence.svg" alt="Two-panel figure. Top: the measured output autospectrum in dB with the coherent output contribution of two inputs shaded underneath; input 1 fills the low band and input 2 the high band, with the residual noise far below. Bottom: for the correlated second input, its ordinary coherence sits around 0.3 across the low band even though it drives no low-frequency path, while its partial coherence collapses to zero there once the first input is conditioned out; the multiple coherence stays near one except at the crossover null" width="82%"></picture> 16 16

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docs/spectral-analysis.md

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··· 407 407 detrend-off calibration), so a PSD, a coherence and an H1 computed with the 408 408 same segment length are mutually consistent bin by bin. The same cross-spectral 409 409 matrix underlies [multiple and partial coherence](miso-coherence.md), which 410 410 - extends the ordinary coherence to two or three correlated inputs and one 410 410 + extends the ordinary coherence to several correlated inputs and one 411 411 output. 412 412 413 413 ## References

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docs/synchronous-averaging.md

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··· 1 1 + ← [Documentation index](README.md) 2 2 + 3 3 + # Time synchronous averaging (McFadden 1987) 4 4 + 5 5 + A rotating machine repeats its signature once per revolution. Buried in 6 6 + broadband noise and in the tones of every other shaft, that repetitive 7 7 + waveform is hard to read directly. **Time synchronous averaging** (TSA) 8 8 + recovers it: given the period `T` of one revolution, it slices the record 9 9 + into successive length-`T` blocks and averages them. Every component 10 10 + synchronous with `T` reinforces; everything asynchronous, noise and the 11 11 + harmonics of unrelated shafts, averages down. `time_synchronous_average` 12 12 + implements the model of P. D. McFadden, *A revised model for the extraction 13 13 + of periodic waveforms by time domain averaging* (Mechanical Systems and 14 14 + Signal Processing 1(1), 1987, 83-95). 15 15 + 16 16 + <picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/synchronous_average_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/synchronous_average.svg" alt="Two-panel figure. Left: one noisy period of a signal in faint grey swings far above and below a smooth curve; the average of forty periods, in blue, lies almost exactly on the dashed red true waveform, so the asynchronous noise has been removed. Right: the comb-filter magnitude across the orders between 31 and 33, with unit-height teeth at the integer orders; for N = 20 a deep node falls exactly on the interfering tone marked at 32.05 orders, whereas the power-of-two N = 32 leaves a side lobe there and lets the tone through" width="92%"></picture> 17 17 + 18 18 + <details> 19 19 + <summary>Show the code for this figure</summary> 20 20 + 21 21 + ```python 22 22 + import matplotlib.pyplot as plt 23 23 + import numpy as np 24 24 + from phonometry import ( 25 25 + comb_filter_response, 26 26 + noise_signal, 27 27 + time_synchronous_average, 28 28 + ) 29 29 + 30 30 + fs = 8192.0 31 31 + period = 1.0 / 32.0 # one revolution: 256 samples at this rate 32 32 + m, n_avg = 256, 40 33 33 + phase = np.arange((n_avg + 1) * m) / m 34 34 + periodic = ( 35 35 + np.cos(2.0 * np.pi * phase) 36 36 + + 0.5 * np.cos(2.0 * np.pi * 3.0 * phase + 0.4) 37 37 + - 0.3 * np.cos(2.0 * np.pi * 6.0 * phase) 38 38 + ) 39 39 + signal = periodic + noise_signal(fs, phase.size / fs, rms=0.9, seed=11) 40 40 + res = time_synchronous_average(signal, fs, period, n_averages=n_avg) 41 41 + 42 42 + fig, (ax0, ax1) = plt.subplots(1, 2, figsize=(11, 4.6)) 43 43 + t_ms = 1e3 * res.times 44 44 + ax0.plot(t_ms, signal[:m], color="#cccccc", label="One noisy period") 45 45 + ax0.plot(t_ms, res.period_waveform, color="#1f77b4", lw=1.8, 46 46 + label=f"Average of N = {n_avg} periods") 47 47 + ax0.plot(t_ms, periodic[:m], "--", color="#d62728", label="True waveform") 48 48 + ax0.set_xlabel("Time [ms]"); ax0.set_ylabel("Amplitude"); ax0.legend() 49 49 + 50 50 + orders = np.linspace(31.0, 33.0, 4000) 51 51 + freqs = orders / period 52 52 + ax1.plot(orders, comb_filter_response(freqs, period, 32), color="#2ca02c", 53 53 + label="N = 32 (power of two)") 54 54 + ax1.plot(orders, comb_filter_response(freqs, period, 20), color="#1f77b4", 55 55 + label="N = 20 (node on 32.05)") 56 56 + ax1.axvline(32.05, color="#d62728", ls=":", label="Interfering tone") 57 57 + ax1.set_xlabel("Frequency [orders]"); ax1.set_ylabel("Comb filter magnitude") 58 58 + ax1.set_ylim(0, 1.05); ax1.legend() 59 59 + plt.show() 60 60 + ``` 61 61 + 62 62 + </details> 63 63 + 64 64 + The `.plot()` method draws the averaged waveform and the comb filter in one 65 65 + call: 66 66 + 67 67 + ```python 68 68 + res.plot() # English labels; res.plot(language="es") for Spanish 69 69 + ``` 70 70 + 71 71 + ## 1. The average is a comb filter 72 72 + 73 73 + Averaging `N` successive periods (McFadden Eq. 5), 74 74 + `a(t) = (1/N) Σ y(t + n·T)`, is, in the frequency domain, the multiplication 75 75 + of the signal spectrum by a **comb filter** (Eq. 8). Its magnitude (Eq. 9) is 76 76 + the Dirichlet kernel `|C(f)| = |sin(N·π·f·T) / (N·sin(π·f·T))|`. 77 77 + 78 78 + The comb has a **tooth** of unit height at every harmonic `k/T` (the orders 79 79 + `f·T = 1, 2, 3, ...`), *independent of `N`*: components synchronous with the 80 80 + period pass untouched. Between the teeth it has **nodes** at `j/(N·T)` for 81 81 + every `j` that is not a multiple of `N`, where the response is exactly zero. 82 82 + `comb_filter_response` evaluates this closed form directly: 83 83 + 84 84 + ```python 85 85 + import numpy as np 86 86 + from phonometry import comb_filter_response 87 87 + 88 88 + period = 1.0 / 32.0 89 89 + comb_filter_response(np.array([16.0 / period]), period, 8) # 1.0 at a tooth 90 90 + comb_filter_response(np.array([0.25 / period]), period, 2) # 1/sqrt(2) 91 91 + comb_filter_response(np.array([0.5 / period]), period, 2) # 0.0 at a node 92 92 + ``` 93 93 + 94 94 + `SynchronousAverageResult` carries the response over the first few harmonics 95 95 + in `comb_frequencies` and `comb_response`, so the shape of the filter that 96 96 + the average applied is available alongside the recovered waveform. 97 97 + 98 98 + ## 2. Noise falls as the square root of the number of averages 99 99 + 100 100 + Asynchronous noise of variance `σ²` averaged over `N` periods has residual 101 101 + variance `σ²/N`: the residual standard deviation falls as `1/√N`, and the 102 102 + amplitude signal-to-noise ratio improves by `√N`. That is a power reduction 103 103 + of `10·log₁₀ N` dB, reported as `noise_reduction_db`, with the amplitude gain 104 104 + `√N` as `amplitude_snr_gain`: 105 105 + 106 106 + ```python 107 107 + res = time_synchronous_average(signal, fs, period, n_averages=100) 108 108 + res.noise_reduction_db # 20.0 dB = 10*log10(100) 109 109 + res.amplitude_snr_gain # 10.0 = sqrt(100) 110 110 + ``` 111 111 + 112 112 + This law is the ideal one: it holds when the noise is uncorrelated from one 113 113 + period to the next, so colored or synchronous noise that is correlated across 114 114 + periods need not follow it. The `residual` (input minus the periodic 115 115 + reconstruction over the analysed span) and its `residual_rms` therefore report 116 116 + the noise actually left once the synchronous component is removed. 117 117 + 118 118 + ## 3. Choosing N to reject an interfering order 119 119 + 120 120 + Because a tooth sits on *every* integer order, TSA passes the harmonics of 121 121 + the target shaft but also any tone that happens to fall on an integer order. 122 122 + A tone at a *non-harmonic* order `q = f·T` is only attenuated by the comb, 123 123 + not removed, and how much depends on where the nearest node lands. McFadden's 124 124 + revised-model result is that such an interferer is best rejected by choosing 125 125 + `N` so that a node falls exactly on it, i.e. the smallest `N` with `N·q` an 126 126 + integer, rather than by the habitual power-of-two number of averages. An exact 127 127 + node exists only when the order `q` is rational, so some finite `N` makes `N·q` 128 128 + an integer; for an irrational or merely estimated order, choose the `N` whose 129 129 + node falls nearest the interfering order. 130 130 + 131 131 + His own example is a tone at 32.05 orders. With `N = 20` the product 132 132 + `20 · 32.05 = 641` is an integer, so a comb node lands on the tone and rejects 133 133 + it by more than 100 dB. The common choice `N = 32` gives `32 · 32.05 = 1025.6`, 134 134 + which sits on a side lobe: the tone is barely touched. The figure above shows 135 135 + both combs around order 32; the end-to-end average confirms it: 136 136 + 137 137 + ```python 138 138 + # true 8th-order component plus a strong interferer at 32.05 orders 139 139 + phase = np.arange(41 * 256) / 256 140 140 + signal = np.cos(2 * np.pi * 8.0 * phase) + 0.7 * np.cos(2 * np.pi * 32.05 * phase) 141 141 + 142 142 + leak_20 = time_synchronous_average(signal, fs, period, n_averages=20) 143 143 + leak_32 = time_synchronous_average(signal, fs, period, n_averages=32) 144 144 + # leak_20.period_waveform matches the clean 8th-order tone; leak_32 does not 145 145 + ``` 146 146 + 147 147 + So a power-of-two number of averages, convenient as it is, is not in general 148 148 + the optimal choice: the interfering orders present in the machine should set 149 149 + `N`. 150 150 + 151 151 + ## 4. Non-integer samples per period 152 152 + 153 153 + When `fs·T` is an integer, the period boundaries fall on samples, the blocks 154 154 + are sliced directly, and a noiseless periodic signal is recovered to machine 155 155 + precision (`interpolated` is `False`). When `fs·T` is not an integer the 156 156 + boundaries fall between samples; each block is then aligned to a common 157 157 + integer grid by the band-limited fractional delay of 158 158 + [`fractional_delay`](test-signals.md), and the waveform is recovered within 159 159 + that interpolation error (`interpolated` is `True`): 160 160 + 161 161 + ```python 162 162 + fs = 8192.0 163 163 + period = 1.0 / 31.7 # fs * period is not an integer 164 164 + t = np.arange(int(40 * period * fs)) / fs 165 165 + signal = np.cos(2.0 * np.pi * t / period) # one cycle per revolution 166 166 + res = time_synchronous_average(signal, fs, period) 167 167 + res.interpolated # True: fractional-delay alignment 168 168 + res.samples_per_period # integer samples of one period 169 169 + ``` 170 170 + 171 171 + By default the average uses as many whole periods as the record holds; pass 172 172 + `n_averages` to fix the count (for the node-selection choice of §3), and 173 173 + `n_harmonics` to set how many harmonics of `1/T` the returned comb response 174 174 + spans. 175 175 + 176 176 + The band-limited alignment shares its kernel with the sub-sample 177 177 + impulse-response alignment of the [test-signals page](test-signals.md), and 178 178 + the recovered waveform, being exactly one period, can be tiled to reconstruct 179 179 + the synchronous part of the signal for subtraction or for order analysis.

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··· 58 58 - [Correlation, time delay and envelope (Bendat & Piersol / Knapp & Carter)](https://jmrplens.github.io/phonometry/guides/correlation-delay/) 59 59 - [Test signals and sample-rate tools (IEC 60268-1)](https://jmrplens.github.io/phonometry/guides/test-signals/) 60 60 - [Cepstrum, echoes and the envelope spectrum (Havelock / Bendat & Piersol)](https://jmrplens.github.io/phonometry/guides/cepstrum-echoes/) 61 61 + - [Time synchronous averaging (McFadden 1987)](https://jmrplens.github.io/phonometry/guides/synchronous-averaging/) 61 62 - [Swept-sine distortion and phase utilities (Farina / Novak)](https://jmrplens.github.io/phonometry/guides/swept-sine-distortion/) 62 63 - [Block Processing](https://jmrplens.github.io/phonometry/guides/block-processing/) 63 64 - [Multichannel and Performance](https://jmrplens.github.io/phonometry/guides/multichannel/) ··· 10054 10055 detrend-off calibration), so a PSD, a coherence and an H1 computed with the 10055 10056 same segment length are mutually consistent bin by bin. The same cross-spectral 10056 10057 matrix underlies [multiple and partial coherence](https://jmrplens.github.io/phonometry/guides/miso-coherence/), which 10057 10057 - extends the ordinary coherence to two or three correlated inputs and one 10058 10058 + extends the ordinary coherence to several correlated inputs and one 10058 10059 output. 10059 10060 10060 10061 ## References ··· 10102 10103 Piersol, *Random Data* (4th ed., 2010, Chapter 7), resolve this for a 10103 10104 multiple-input/single-output (MISO) system with the **multiple** and 10104 10105 **partial** coherence functions. `miso_coherence` computes them from the same 10105 10105 - Welch cross-spectral core as the rest of `phonometry.metrology`, for two or 10106 10106 - three inputs and one output. 10106 10106 + Welch cross-spectral core as the rest of `phonometry.metrology`, for several 10107 10107 + correlated inputs and one output. 10107 10108 10108 10109 <picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/miso_coherence_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/miso_coherence.svg" alt="Two-panel figure. Top: the measured output autospectrum in dB with the coherent output contribution of two inputs shaded underneath; input 1 fills the low band and input 2 the high band, with the residual noise far below. Bottom: for the correlated second input, its ordinary coherence sits around 0.3 across the low band even though it drives no low-frequency path, while its partial coherence collapses to zero there once the first input is conditioned out; the multiple coherence stays near one except at the crossover null" width="82%"></picture> 10109 10110 ··· 10967 10968 behind the minimum-phase folding) and Section 13.3 with Figure 13.11 10968 10969 (envelope detection followed by DC removal, the structure of the 10969 10970 envelope spectrum). 10971 10971 + 10972 10972 + --- 10973 10973 + 10974 10974 + 10975 10975 + <!-- source: docs/synchronous-averaging.md | canonical: https://jmrplens.github.io/phonometry/guides/synchronous-averaging/ --> 10976 10976 + 10977 10977 + # Time synchronous averaging (McFadden 1987) 10978 10978 + 10979 10979 + A rotating machine repeats its signature once per revolution. Buried in 10980 10980 + broadband noise and in the tones of every other shaft, that repetitive 10981 10981 + waveform is hard to read directly. **Time synchronous averaging** (TSA) 10982 10982 + recovers it: given the period `T` of one revolution, it slices the record 10983 10983 + into successive length-`T` blocks and averages them. Every component 10984 10984 + synchronous with `T` reinforces; everything asynchronous, noise and the 10985 10985 + harmonics of unrelated shafts, averages down. `time_synchronous_average` 10986 10986 + implements the model of P. D. McFadden, *A revised model for the extraction 10987 10987 + of periodic waveforms by time domain averaging* (Mechanical Systems and 10988 10988 + Signal Processing 1(1), 1987, 83-95). 10989 10989 + 10990 10990 + <picture><source media="(prefers-color-scheme: dark)" srcset="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/synchronous_average_dark.svg"><img src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/synchronous_average.svg" alt="Two-panel figure. Left: one noisy period of a signal in faint grey swings far above and below a smooth curve; the average of forty periods, in blue, lies almost exactly on the dashed red true waveform, so the asynchronous noise has been removed. Right: the comb-filter magnitude across the orders between 31 and 33, with unit-height teeth at the integer orders; for N = 20 a deep node falls exactly on the interfering tone marked at 32.05 orders, whereas the power-of-two N = 32 leaves a side lobe there and lets the tone through" width="92%"></picture> 10991 10991 + 10992 10992 + <details> 10993 10993 + <summary>Show the code for this figure</summary> 10994 10994 + 10995 10995 + ```python 10996 10996 + import matplotlib.pyplot as plt 10997 10997 + import numpy as np 10998 10998 + from phonometry import ( 10999 10999 + comb_filter_response, 11000 11000 + noise_signal, 11001 11001 + time_synchronous_average, 11002 11002 + ) 11003 11003 + 11004 11004 + fs = 8192.0 11005 11005 + period = 1.0 / 32.0 # one revolution: 256 samples at this rate 11006 11006 + m, n_avg = 256, 40 11007 11007 + phase = np.arange((n_avg + 1) * m) / m 11008 11008 + periodic = ( 11009 11009 + np.cos(2.0 * np.pi * phase) 11010 11010 + + 0.5 * np.cos(2.0 * np.pi * 3.0 * phase + 0.4) 11011 11011 + - 0.3 * np.cos(2.0 * np.pi * 6.0 * phase) 11012 11012 + ) 11013 11013 + signal = periodic + noise_signal(fs, phase.size / fs, rms=0.9, seed=11) 11014 11014 + res = time_synchronous_average(signal, fs, period, n_averages=n_avg) 11015 11015 + 11016 11016 + fig, (ax0, ax1) = plt.subplots(1, 2, figsize=(11, 4.6)) 11017 11017 + t_ms = 1e3 * res.times 11018 11018 + ax0.plot(t_ms, signal[:m], color="#cccccc", label="One noisy period") 11019 11019 + ax0.plot(t_ms, res.period_waveform, color="#1f77b4", lw=1.8, 11020 11020 + label=f"Average of N = {n_avg} periods") 11021 11021 + ax0.plot(t_ms, periodic[:m], "--", color="#d62728", label="True waveform") 11022 11022 + ax0.set_xlabel("Time [ms]"); ax0.set_ylabel("Amplitude"); ax0.legend() 11023 11023 + 11024 11024 + orders = np.linspace(31.0, 33.0, 4000) 11025 11025 + freqs = orders / period 11026 11026 + ax1.plot(orders, comb_filter_response(freqs, period, 32), color="#2ca02c", 11027 11027 + label="N = 32 (power of two)") 11028 11028 + ax1.plot(orders, comb_filter_response(freqs, period, 20), color="#1f77b4", 11029 11029 + label="N = 20 (node on 32.05)") 11030 11030 + ax1.axvline(32.05, color="#d62728", ls=":", label="Interfering tone") 11031 11031 + ax1.set_xlabel("Frequency [orders]"); ax1.set_ylabel("Comb filter magnitude") 11032 11032 + ax1.set_ylim(0, 1.05); ax1.legend() 11033 11033 + plt.show() 11034 11034 + ``` 11035 11035 + 11036 11036 + </details> 11037 11037 + 11038 11038 + The `.plot()` method draws the averaged waveform and the comb filter in one 11039 11039 + call: 11040 11040 + 11041 11041 + ```python 11042 11042 + res.plot() # English labels; res.plot(language="es") for Spanish 11043 11043 + ``` 11044 11044 + 11045 11045 + ## 1. The average is a comb filter 11046 11046 + 11047 11047 + Averaging `N` successive periods (McFadden Eq. 5), 11048 11048 + `a(t) = (1/N) Σ y(t + n·T)`, is, in the frequency domain, the multiplication 11049 11049 + of the signal spectrum by a **comb filter** (Eq. 8). Its magnitude (Eq. 9) is 11050 11050 + the Dirichlet kernel `|C(f)| = |sin(N·π·f·T) / (N·sin(π·f·T))|`. 11051 11051 + 11052 11052 + The comb has a **tooth** of unit height at every harmonic `k/T` (the orders 11053 11053 + `f·T = 1, 2, 3, ...`), *independent of `N`*: components synchronous with the 11054 11054 + period pass untouched. Between the teeth it has **nodes** at `j/(N·T)` for 11055 11055 + every `j` that is not a multiple of `N`, where the response is exactly zero. 11056 11056 + `comb_filter_response` evaluates this closed form directly: 11057 11057 + 11058 11058 + ```python 11059 11059 + import numpy as np 11060 11060 + from phonometry import comb_filter_response 11061 11061 + 11062 11062 + period = 1.0 / 32.0 11063 11063 + comb_filter_response(np.array([16.0 / period]), period, 8) # 1.0 at a tooth 11064 11064 + comb_filter_response(np.array([0.25 / period]), period, 2) # 1/sqrt(2) 11065 11065 + comb_filter_response(np.array([0.5 / period]), period, 2) # 0.0 at a node 11066 11066 + ``` 11067 11067 + 11068 11068 + `SynchronousAverageResult` carries the response over the first few harmonics 11069 11069 + in `comb_frequencies` and `comb_response`, so the shape of the filter that 11070 11070 + the average applied is available alongside the recovered waveform. 11071 11071 + 11072 11072 + ## 2. Noise falls as the square root of the number of averages 11073 11073 + 11074 11074 + Asynchronous noise of variance `σ²` averaged over `N` periods has residual 11075 11075 + variance `σ²/N`: the residual standard deviation falls as `1/√N`, and the 11076 11076 + amplitude signal-to-noise ratio improves by `√N`. That is a power reduction 11077 11077 + of `10·log₁₀ N` dB, reported as `noise_reduction_db`, with the amplitude gain 11078 11078 + `√N` as `amplitude_snr_gain`: 11079 11079 + 11080 11080 + ```python 11081 11081 + res = time_synchronous_average(signal, fs, period, n_averages=100) 11082 11082 + res.noise_reduction_db # 20.0 dB = 10*log10(100) 11083 11083 + res.amplitude_snr_gain # 10.0 = sqrt(100) 11084 11084 + ``` 11085 11085 + 11086 11086 + This law is the ideal one: it holds when the noise is uncorrelated from one 11087 11087 + period to the next, so colored or synchronous noise that is correlated across 11088 11088 + periods need not follow it. The `residual` (input minus the periodic 11089 11089 + reconstruction over the analysed span) and its `residual_rms` therefore report 11090 11090 + the noise actually left once the synchronous component is removed. 11091 11091 + 11092 11092 + ## 3. Choosing N to reject an interfering order 11093 11093 + 11094 11094 + Because a tooth sits on *every* integer order, TSA passes the harmonics of 11095 11095 + the target shaft but also any tone that happens to fall on an integer order. 11096 11096 + A tone at a *non-harmonic* order `q = f·T` is only attenuated by the comb, 11097 11097 + not removed, and how much depends on where the nearest node lands. McFadden's 11098 11098 + revised-model result is that such an interferer is best rejected by choosing 11099 11099 + `N` so that a node falls exactly on it, i.e. the smallest `N` with `N·q` an 11100 11100 + integer, rather than by the habitual power-of-two number of averages. An exact 11101 11101 + node exists only when the order `q` is rational, so some finite `N` makes `N·q` 11102 11102 + an integer; for an irrational or merely estimated order, choose the `N` whose 11103 11103 + node falls nearest the interfering order. 11104 11104 + 11105 11105 + His own example is a tone at 32.05 orders. With `N = 20` the product 11106 11106 + `20 · 32.05 = 641` is an integer, so a comb node lands on the tone and rejects 11107 11107 + it by more than 100 dB. The common choice `N = 32` gives `32 · 32.05 = 1025.6`, 11108 11108 + which sits on a side lobe: the tone is barely touched. The figure above shows 11109 11109 + both combs around order 32; the end-to-end average confirms it: 11110 11110 + 11111 11111 + ```python 11112 11112 + # true 8th-order component plus a strong interferer at 32.05 orders 11113 11113 + phase = np.arange(41 * 256) / 256 11114 11114 + signal = np.cos(2 * np.pi * 8.0 * phase) + 0.7 * np.cos(2 * np.pi * 32.05 * phase) 11115 11115 + 11116 11116 + leak_20 = time_synchronous_average(signal, fs, period, n_averages=20) 11117 11117 + leak_32 = time_synchronous_average(signal, fs, period, n_averages=32) 11118 11118 + # leak_20.period_waveform matches the clean 8th-order tone; leak_32 does not 11119 11119 + ``` 11120 11120 + 11121 11121 + So a power-of-two number of averages, convenient as it is, is not in general 11122 11122 + the optimal choice: the interfering orders present in the machine should set 11123 11123 + `N`. 11124 11124 + 11125 11125 + ## 4. Non-integer samples per period 11126 11126 + 11127 11127 + When `fs·T` is an integer, the period boundaries fall on samples, the blocks 11128 11128 + are sliced directly, and a noiseless periodic signal is recovered to machine 11129 11129 + precision (`interpolated` is `False`). When `fs·T` is not an integer the 11130 11130 + boundaries fall between samples; each block is then aligned to a common 11131 11131 + integer grid by the band-limited fractional delay of 11132 11132 + [`fractional_delay`](https://jmrplens.github.io/phonometry/guides/test-signals/), and the waveform is recovered within 11133 11133 + that interpolation error (`interpolated` is `True`): 11134 11134 + 11135 11135 + ```python 11136 11136 + fs = 8192.0 11137 11137 + period = 1.0 / 31.7 # fs * period is not an integer 11138 11138 + t = np.arange(int(40 * period * fs)) / fs 11139 11139 + signal = np.cos(2.0 * np.pi * t / period) # one cycle per revolution 11140 11140 + res = time_synchronous_average(signal, fs, period) 11141 11141 + res.interpolated # True: fractional-delay alignment 11142 11142 + res.samples_per_period # integer samples of one period 11143 11143 + ``` 11144 11144 + 11145 11145 + By default the average uses as many whole periods as the record holds; pass 11146 11146 + `n_averages` to fix the count (for the node-selection choice of §3), and 11147 11147 + `n_harmonics` to set how many harmonics of `1/T` the returned comb response 11148 11148 + spans. 11149 11149 + 11150 11150 + The band-limited alignment shares its kernel with the sub-sample 11151 11151 + impulse-response alignment of the [test-signals page](https://jmrplens.github.io/phonometry/guides/test-signals/), and 11152 11152 + the recovered waveform, being exactly one period, can be tiled to reconstruct 11153 11153 + the synchronous part of the signal for subtraction or for order analysis. 10970 11154 10971 11155 --- 10972 11156 ··· 12198 12382 | `cross_spectral_density` | `function` | **Welch cross-spectral density with magnitude/phase errors (Eqs. 9.33/9.52).**<br>• `x`, `y`, `fs`<br>• `window`, `nperseg`, `overlap`, `scaling` as above | `res = cross_spectral_density(x, y, fs)`<br><br>• `CrossSpectralDensityResult` | 12199 12383 | `coherent_output_spectrum` | `function` | **Gvv = γ²·Gyy, noise remainder and spectral SNR (Eqs. 9.55–9.56, 9.73).**<br>• `x` (input), `y` (output), `fs`<br>• `window`, `nperseg`, `overlap`, `scaling` as above | `res = coherent_output_spectrum(x, y, fs)`<br><br>• `CoherentOutputSpectrumResult` | 12200 12384 | `fractional_octave_smoothing` | `function` | **Constant-power 1/n-octave smoothing of a spectrum.**<br>• `frequencies`, `values`<br>• `fraction`: the n of 1/n octave (Default: 3)<br>• `domain`: 'power'/'amplitude'/'db' (Default: 'power') | `s = fractional_octave_smoothing(f, psd, 3.0)`<br><br>• smoothed array, same domain; flat spectra unchanged | 12201 12201 - | `miso_coherence` | `function` | **Multiple & partial coherence of a MISO system (Bendat & Piersol Ch. 7).**<br>• `inputs`: 2 or 3 input records (sequence of 1-D arrays or a (q, n) array), `output`, `fs`<br>• `order`: conditioning order (Default: 0..q−1)<br>• `window`, `nperseg`, `overlap`, `scaling` as above | `res = miso_coherence([x1, x2], y, fs)`<br><br>• `MISOCoherenceResult` | 12385 12385 + | `miso_coherence` | `function` | **Multiple & partial coherence of a MISO system (Bendat & Piersol Ch. 7).**<br>• `inputs`: q input records, q ≥ 2 (sequence of 1-D arrays or a (q, n) array), `output`, `fs`<br>• `order`: conditioning order (Default: 0..q−1)<br>• `window`, `nperseg`, `overlap`, `scaling` as above | `res = miso_coherence([x1, x2], y, fs)`<br><br>• `MISOCoherenceResult` | 12202 12386 | `MISOCoherenceResult` | `dataclass` | **Multiple/partial coherence of q correlated inputs and one output.**<br>• `ordinary_coherence` (q×F), `multiple_coherence` γ²y:x, `partial_coherence` (q×F, conditioned per `order`)<br>• `coherent_output_spectra` Gvi (q×F, Σ Gvi + `noise_psd` = `output_psd`)<br>• `multiple_coherence_random_error` (Eq. 9.98), `coherent_output_random_error` (Eq. 9.100)<br>• `.dominant_input()`: per-band index of the strongest source<br>• `.plot()`: coherent output spectra + coherences | `res.multiple_coherence, res.dominant_input()` | 12203 12387 | `resolution_bias_error` | `function` | **First-order resolution bias at a resonance peak, εb ≈ −(Be/Br)²/3 (Eq. 8.141).**<br>• `resolution_bandwidth` Be [Hz]<br>• `half_power_bandwidth` Br [Hz] | `resolution_bias_error(1.0, 4.0) # -1/48` | 12204 12388 | `noise_signal` | `function` | **Colored Gaussian noise with an exact power-law PSD slope.**<br>• `fs`, `seconds` (Default: 1.0)<br>• `color`: 'white'/'pink'/'red'/'blue'/'violet' (0/−3.01/−6.02/+3.01/+6.02 dB/oct)<br>• `rms` (Default: 1.0)<br>• `seed`: bit-reproducible per seed | `pink = noise_signal(48000, 10.0, color='pink', seed=7)` | ··· 12230 12414 | `EnvelopeResult` | `dataclass` | **Envelope analysis.**<br>• `times` [s], `envelope`, `phase` [rad, unwrapped], `instantaneous_frequency` [Hz]<br>• `fs` (output rate), `signal`, `signal_fs`, `decimation_factor`, `antialias`<br>• `.plot()`: signal + envelope, instantaneous frequency | `res.envelope, res.instantaneous_frequency` | 12231 12415 | `envelope_spectrum` | `function` | **Amplitude spectrum of the envelope: modulations as lines (B&P 13.3).**<br>• `x`, `fs`<br>• `kind`: 'magnitude' (Default) / 'squared' (square-law detector, Fig. 13.11)<br>• `window` (Default: 'hann', coherent-gain corrected), `nfft`<br>• `remove_dc` (Default: True; mean kept in `mean_level`)<br>AM tone A0, m: line A0·m at fm ('magnitude') | `res = envelope_spectrum(x, fs)`<br><br>• `EnvelopeSpectrumResult` | 12232 12416 | `EnvelopeSpectrumResult` | `dataclass` | **Envelope spectrum.**<br>• `frequencies` [Hz], `amplitude` (line heights), `mean_level`, `kind`<br>• `times`, `envelope` (detector output), `window`, `remove_dc`, `fs`, `nfft`<br>• `.plot()`: envelope + spectrum | `res.amplitude, res.mean_level` | 12417 12417 + | `time_synchronous_average` | `function` | **Extract a periodic waveform of known period by time domain averaging (McFadden 1987 Eq. 5).**<br>• `x`, `fs`, `period` [s] (one revolution)<br>• `n_averages`: whole periods to average (Default: as many as the record holds; choose N so N·q is integer to place a comb node on an interfering order q)<br>• `n_harmonics`: comb-response span (Default: 8)<br>Non-integer fs·period aligned by band-limited fractional delay | `res = time_synchronous_average(x, fs, 1/32)`<br><br>• `SynchronousAverageResult` | 12418 12418 + | `SynchronousAverageResult` | `dataclass` | **Time synchronous average.**<br>• `period_waveform`, `times` [s], `residual`, `residual_rms`<br>• `n_averages`, `samples_per_period`, `period`, `fs`, `interpolated`<br>• `noise_reduction_db` = 10·log₁₀N, `amplitude_snr_gain` = √N<br>• `comb_frequencies` [Hz], `comb_response` (\|C(f)\|)<br>• `.plot()`: averaged waveform + comb filter | `res.period_waveform, res.noise_reduction_db` | 12419 12419 + | `comb_filter_response` | `function` | **Magnitude of the N-period synchronous-averaging comb filter (McFadden 1987 Eq. 8).**<br>• `frequencies` [Hz], `period` [s], `n_averages`<br>\|C(f)\| = \|sin(N·π·f·T)/(N·sin(π·f·T))\|: unit teeth at harmonics k/T, nodes at j/(N·T) | `c = comb_filter_response(freqs, 1/32, 20)` | 12233 12420 | `regularized_inverse_filter` | `function` | **Kirkeby frequency-dependent regularized inversion (Kirkeby & Nelson 1999 Eq. (17)).**<br>• `response`: measured IR (array or `ImpulseResponseResult`; its `fs` rides along)<br>• `fs`: Sample rate [Hz]<br>• `f_range`: (f1, f2) equalized to unity (Required)<br>• `regularization_inside` (Default: 1e-6) / `regularization_outside` (Default: 1.0), fractions of max\|H\|²<br>• `transition_octaves` (Default: 1/3), `n_fft`, `delay` (Default: n_fft//2) | `inv = regularized_inverse_filter(ir, f_range=(100, 10000))`<br><br>• `InverseFilterResult` | 12234 12421 | `InverseFilterResult` | `dataclass` | **Regularized inverse filter.**<br>• `inverse` (time domain), `spectrum` (with modeling delay), `response_spectrum`, `regularization` = ε(f), `frequencies`, `f_range`, `delay`, `fs`<br>• `flatness_db`: worst in-band deviation of \|H·H_inv\| from 0 dB<br>• `max_gain_db`: out-of-band boost (≤ the 1/(2√ε) cap)<br>• `.apply(x)`: equalize, delay removed; `.plot()` | `flat = inv.apply(recording)` | 12235 12422 | `cepstrum` | `function` | **Power/real/complex cepstrum (Havelock Chs. 27/87).**<br>• `x`, `fs`<br>• `kind`: 'power' (Default, IDFT of ln\|X\|²) / 'real' (ln\|X\|) / 'complex' (invertible, phase unwrapped)<br>• `nfft`: even, ≥ record (Default: record length)<br>Echo a at t0: rahmonics (−1)^(n+1)·aⁿ/n at n·t0 | `res = cepstrum(x, fs, kind='complex')`<br><br>• `CepstrumResult` |

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llms.txt

Reviewed

··· 58 58 - [Correlation, time delay and envelope (Bendat & Piersol / Knapp & Carter)](https://jmrplens.github.io/phonometry/guides/correlation-delay/) 59 59 - [Test signals and sample-rate tools (IEC 60268-1)](https://jmrplens.github.io/phonometry/guides/test-signals/) 60 60 - [Cepstrum, echoes and the envelope spectrum (Havelock / Bendat & Piersol)](https://jmrplens.github.io/phonometry/guides/cepstrum-echoes/) 61 61 + - [Time synchronous averaging (McFadden 1987)](https://jmrplens.github.io/phonometry/guides/synchronous-averaging/) 61 62 - [Swept-sine distortion and phase utilities (Farina / Novak)](https://jmrplens.github.io/phonometry/guides/swept-sine-distortion/) 62 63 - [Block Processing](https://jmrplens.github.io/phonometry/guides/block-processing/) 63 64 - [Multichannel and Performance](https://jmrplens.github.io/phonometry/guides/multichannel/)

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scripts/api_taxonomy.py

Reviewed

··· 278 278 "phonometry.metrology.signals", 279 279 "phonometry.metrology.phase", 280 280 "phonometry.metrology.cepstrum", 281 281 + "phonometry.metrology.synchronous_average", 281 282 "phonometry.metrology.inversion", 282 283 ), 283 284 ),

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scripts/conformance_report.py

Reviewed

··· 4918 4918 4919 4919 4920 4920 # =========================================================================== 4921 4921 + # Time synchronous averaging (McFadden 1987) 4922 4922 + # =========================================================================== 4923 4923 + _TSA = "Time synchronous averaging (McFadden 1987)" 4924 4924 + 4925 4925 + 4926 4926 + @register( 4927 4927 + _TSA, 4928 4928 + r"McFadden 1987 Eq. 8 / Eq. 9: comb filter \|C(f)\| at a harmonic k/T", 4929 4929 + "Comb-filter tooth height at a harmonic equals unity (any N)", 4930 4930 + ) 4931 4931 + def _chk_tsa_comb_tooth() -> Outcome: 4932 4932 + period = 1.0 / 32.0 4933 4933 + value = float(ph.comb_filter_response(np.array([16.0 / period]), period, 8)[0]) 4934 4934 + return numeric(1.0, value, 1e-10, places=8) 4935 4935 + 4936 4936 + 4937 4937 + @register( 4938 4938 + _TSA, 4939 4939 + "McFadden 1987 Eq. 8: comb filter one quarter-order from a tooth, N = 2", 4940 4940 + "Comb-filter magnitude = 1/sqrt(2) at order 0.25", 4941 4941 + ) 4942 4942 + def _chk_tsa_comb_midbin() -> Outcome: 4943 4943 + period = 1.0 / 32.0 4944 4944 + value = float( 4945 4945 + ph.comb_filter_response(np.array([0.25 / period]), period, 2)[0] 4946 4946 + ) 4947 4947 + return numeric(1.0 / math.sqrt(2.0), value, 1e-10, places=8) 4948 4948 + 4949 4949 + 4950 4950 + @register( 4951 4951 + _TSA, 4952 4952 + "McFadden 1987 Sec. 4 (Fig. 5): node selection, tone at 32.05 orders", 4953 4953 + r"N = 20 places a comb node on 32.05 orders (\|C\| = 0), not the power-of-2 N = 32", 4954 4954 + ) 4955 4955 + def _chk_tsa_node_selection() -> Outcome: 4956 4956 + period = 1.0 / 32.0 4957 4957 + freq = np.array([32.05 / period]) 4958 4958 + c20 = float(ph.comb_filter_response(freq, period, 20)[0]) 4959 4959 + c32 = float(ph.comb_filter_response(freq, period, 32)[0]) 4960 4960 + if not c32 > 0.15: # sanity: the power-of-two choice does not reject it 4961 4961 + return numeric(0.0, c32, 0.0, places=8) 4962 4962 + return numeric(0.0, c20, 1e-10, places=10) 4963 4963 + 4964 4964 + 4965 4965 + @register( 4966 4966 + _TSA, 4967 4967 + "McFadden 1987 Eq. 5: exact recovery, integer samples per period", 4968 4968 + "Noiseless periodic waveform (M = 256) recovered to machine precision", 4969 4969 + ) 4970 4970 + def _chk_tsa_exact_recovery() -> Outcome: 4971 4971 + fs = 8192.0 4972 4972 + period = 1.0 / 32.0 4973 4973 + m = 256 4974 4974 + phase = np.arange(m) / m 4975 4975 + one = np.cos(2.0 * np.pi * phase) + 0.5 * np.cos( 4976 4976 + 2.0 * np.pi * 3.0 * phase + 0.4 4977 4977 + ) 4978 4978 + res = ph.time_synchronous_average(np.tile(one, 24), fs, period) 4979 4979 + err = float(np.max(np.abs(res.period_waveform - one))) 4980 4980 + return numeric(0.0, err, 1e-10, places=12) 4981 4981 + 4982 4982 + 4983 4983 + @register( 4984 4984 + _TSA, 4985 4985 + "McFadden 1987 Sec. 1: asynchronous-noise variance reduced by 1/N", 4986 4986 + "Residual noise std of the average falls as sigma/sqrt(N), N = 64", 4987 4987 + ) 4988 4988 + def _chk_tsa_sqrt_n_law() -> Outcome: 4989 4989 + fs = 8192.0 4990 4990 + period = 1.0 / 32.0 4991 4991 + m = 256 4992 4992 + n_avg = 64 4993 4993 + phase = np.arange(m) / m 4994 4994 + one = np.cos(2.0 * np.pi * phase) 4995 4995 + rng = np.random.default_rng(2024) 4996 4996 + noise = rng.standard_normal(n_avg * m) 4997 4997 + res = ph.time_synchronous_average( 4998 4998 + np.tile(one, n_avg) + noise, fs, period, n_averages=n_avg 4999 4999 + ) 5000 5000 + measured = float(np.std(res.period_waveform - one)) 5001 5001 + return numeric(1.0 / math.sqrt(n_avg), measured, 0.15, rel=True, places=5) 5002 5002 + 5003 5003 + 5004 5004 + # =========================================================================== 4921 5005 # Data qualification and Rice statistics (Bendat & Piersol Chs. 4, 5, 10) 4922 5006 # =========================================================================== 4923 5007 _RANDOM_DATA = "Data qualification and Rice statistics (Bendat & Piersol)"

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scripts/generate_graphs.py

Reviewed

··· 721 721 "appears as one line at exactly $f_m$": 722 722 "la portadora está en 1 kHz; su modulación de amplitud\n" 723 723 "aparece como una línea exactamente en $f_m$", 724 724 + # Time synchronous averaging (McFadden 1987) 725 725 + "Periodic Waveform Extracted from Noise": 726 726 + "Forma de onda periódica extraída del ruido", 727 727 + "One noisy period": "Un período ruidoso", 728 728 + "Average of N = 40 periods": "Promedio de N = 40 períodos", 729 729 + "True periodic waveform": "Forma de onda periódica verdadera", 730 730 + "averaging N periods lowers the asynchronous\n" 731 731 + "noise by $\\sqrt{N}$ in amplitude": 732 732 + "promediar N períodos reduce el ruido asíncrono\n" 733 733 + "en $\\sqrt{N}$ en amplitud", 734 734 + "Rejecting a Tone by Choosing N (McFadden 1987)": 735 735 + "Rechazo de un tono eligiendo N (McFadden 1987)", 736 736 + "N = 32 (power of two)": "N = 32 (potencia de dos)", 737 737 + "N = 20 (node on 32.05)": "N = 20 (nodo en 32,05)", 738 738 + "Interfering tone (32.05)": "Tono interferente (32,05)", 739 739 + "Frequency [orders]": "Frecuencia [órdenes]", 740 740 + "Comb filter magnitude": "Magnitud del filtro peine", 741 741 + "N = 20 puts a node on 32.05 orders and removes\n" 742 742 + "it; the power-of-two N = 32 lets it through": 743 743 + "N = 20 sitúa un nodo en 32,05 órdenes y lo\n" 744 744 + "elimina; la potencia de dos N = 32 lo deja pasar", 724 745 # Multiple-input coherence (Bendat & Piersol Ch. 7) 725 746 "Multiple-Input Coherence: Which Source Dominates Each Band " 726 747 "(Bendat & Piersol Ch. 7)": ··· 4527 4548 color=COLOR_FG) 4528 4549 plt.tight_layout() 4529 4550 save_figure(output_dir, "envelope_spectrum.svg") 4551 4551 + plt.close() 4552 4552 + 4553 4553 + 4554 4554 + def generate_synchronous_average(output_dir: str) -> None: 4555 4555 + """TSA: a periodic waveform pulled from noise, and the comb filter.""" 4556 4556 + print("Generating synchronous_average...") 4557 4557 + from phonometry import ( 4558 4558 + comb_filter_response, 4559 4559 + noise_signal, 4560 4560 + time_synchronous_average, 4561 4561 + ) 4562 4562 + 4563 4563 + fs = 8192.0 4564 4564 + period = 1.0 / 32.0 # one revolution: 256 samples at this rate 4565 4565 + m = 256 4566 4566 + n_avg = 40 4567 4567 + phase = np.arange((n_avg + 1) * m) / m 4568 4568 + periodic = ( 4569 4569 + np.cos(2.0 * np.pi * phase) 4570 4570 + + 0.5 * np.cos(2.0 * np.pi * 3.0 * phase + 0.4) 4571 4571 + - 0.3 * np.cos(2.0 * np.pi * 6.0 * phase) 4572 4572 + ) 4573 4573 + signal = periodic + noise_signal(fs, phase.size / fs, rms=0.9, seed=11) 4574 4574 + res = time_synchronous_average(signal, fs, period, n_averages=n_avg) 4575 4575 + true_one = periodic[:m] 4576 4576 + 4577 4577 + fig, (ax0, ax1) = plt.subplots(1, 2, figsize=(11, 4.6)) 4578 4578 + 4579 4579 + # Panel (a): one noisy period against the recovered average. 4580 4580 + t_ms = 1e3 * res.times 4581 4581 + ax0.plot(t_ms, signal[:m], color=COLOR_GRID, linewidth=1.0, 4582 4582 + label="One noisy period") 4583 4583 + ax0.plot(t_ms, res.period_waveform, color=COLOR_PRIMARY, linewidth=1.8, 4584 4584 + label=f"Average of N = {n_avg} periods") 4585 4585 + ax0.plot(t_ms, true_one, color=COLOR_SECONDARY, linestyle="--", 4586 4586 + linewidth=1.2, label="True periodic waveform") 4587 4587 + ax0.set_xlim(0.0, 1e3 * period) 4588 4588 + ax0.set_xlabel("Time [ms]") 4589 4589 + ax0.set_ylabel("Amplitude") 4590 4590 + ax0.set_title("Periodic Waveform Extracted from Noise", 4591 4591 + fontweight="bold", pad=10) 4592 4592 + ax0.grid(color=COLOR_GRID, linestyle="--", alpha=0.5) 4593 4593 + ax0.set_axisbelow(True) 4594 4594 + ax0.legend(loc="upper right", fontsize=8) 4595 4595 + ax0.text(0.02, 0.03, 4596 4596 + "averaging N periods lowers the asynchronous\n" 4597 4597 + "noise by $\\sqrt{N}$ in amplitude", 4598 4598 + transform=ax0.transAxes, va="bottom", ha="left", fontsize=8.5, 4599 4599 + color=COLOR_FG) 4600 4600 + 4601 4601 + # Panel (b): comb filter, node selection at 32.05 orders. 4602 4602 + orders = np.linspace(31.0, 33.0, 4000) 4603 4603 + freqs = orders / period 4604 4604 + c20 = comb_filter_response(freqs, period, 20) 4605 4605 + c32 = comb_filter_response(freqs, period, 32) 4606 4606 + ax1.plot(orders, c32, color=COLOR_TERTIARY, linewidth=1.2, 4607 4607 + label="N = 32 (power of two)") 4608 4608 + ax1.plot(orders, c20, color=COLOR_PRIMARY, linewidth=1.4, 4609 4609 + label="N = 20 (node on 32.05)") 4610 4610 + ax1.axvline(32.05, color=COLOR_SECONDARY, linestyle=":", linewidth=1.3, 4611 4611 + label="Interfering tone (32.05)") 4612 4612 + ax1.set_xlim(31.0, 33.0) 4613 4613 + ax1.set_ylim(0.0, 1.05) 4614 4614 + ax1.set_xlabel("Frequency [orders]") 4615 4615 + ax1.set_ylabel("Comb filter magnitude") 4616 4616 + ax1.set_title("Rejecting a Tone by Choosing N (McFadden 1987)", 4617 4617 + fontweight="bold", pad=10) 4618 4618 + ax1.grid(color=COLOR_GRID, linestyle="--", alpha=0.5) 4619 4619 + ax1.set_axisbelow(True) 4620 4620 + ax1.legend(loc="upper right", fontsize=8) 4621 4621 + ax1.text(0.02, 0.55, 4622 4622 + "N = 20 puts a node on 32.05 orders and removes\n" 4623 4623 + "it; the power-of-two N = 32 lets it through", 4624 4624 + transform=ax1.transAxes, va="top", ha="left", fontsize=8.5, 4625 4625 + color=COLOR_FG) 4626 4626 + 4627 4627 + plt.tight_layout() 4628 4628 + save_figure(output_dir, "synchronous_average.svg") 4530 4629 plt.close() 4531 4630 4532 4631 ··· 8543 8642 # Chs. 27/87) and the envelope spectrum of an AM tone (B&P 13.3). 8544 8643 generate_cepstrum_echo, 8545 8644 generate_envelope_spectrum, 8645 8645 + # Time synchronous averaging of a periodic waveform in noise (McFadden 1987). 8646 8646 + generate_synchronous_average, 8546 8647 # Multiple-input/output coherence (Bendat & Piersol Ch. 7). 8547 8648 generate_miso_coherence, 8548 8649 # Data qualification: reverse arrangement stationarity test and the Rice

+1

scripts/generate_llms.py

Reviewed

··· 49 49 ("correlation-delay.md", "guides/correlation-delay"), 50 50 ("test-signals.md", "guides/test-signals"), 51 51 ("cepstrum-echoes.md", "guides/cepstrum-echoes"), 52 52 + ("synchronous-averaging.md", "guides/synchronous-averaging"), 52 53 ("swept-sine-distortion.md", "guides/swept-sine-distortion"), 53 54 ("block-processing.md", "guides/block-processing"), 54 55 ("multichannel.md", "guides/multichannel"),

+1

site/astro.config.mjs

Reviewed

··· 361 361 'guides/miso-coherence', 362 362 'guides/time-frequency', 363 363 'guides/cepstrum-echoes', 364 364 + 'guides/synchronous-averaging', 364 365 'guides/correlation-delay', 365 366 'guides/test-signals', 366 367 'guides/system-measurement',

+3 -3

site/src/content/docs/es/guides/miso-coherence.mdx

Reviewed

··· 1 1 --- 2 2 title: "Coherencia múltiple y parcial" 3 3 - description: "Las funciones de coherencia de sistemas de entradas múltiples y salida única de Bendat y Piersol: coherencia ordinaria, múltiple y parcial de dos o tres fuentes correladas que excitan una respuesta, el condicionamiento por eliminación de Gauss que separa una causa real de una fuente que solo correla con ella, y los espectros de salida coherente parciales que indican qué fuente domina cada banda." 3 3 + description: "Las funciones de coherencia de sistemas de entradas múltiples y salida única de Bendat y Piersol: coherencia ordinaria, múltiple y parcial de varias fuentes correladas que excitan una respuesta, el condicionamiento por eliminación de Gauss que separa una causa real de una fuente que solo correla con ella, y los espectros de salida coherente parciales que indican qué fuente domina cada banda." 4 4 references: 5 5 - type: book 6 6 authors: ["Bendat, J. S.", "Piersol, A. G."] ··· 22 22 capítulo 7), lo resuelven para un sistema de entradas múltiples y salida única 23 23 (MISO) con las funciones de coherencia **múltiple** y **parcial**. 24 24 `miso_coherence` las calcula desde el mismo núcleo de espectros cruzados de 25 25 - Welch que el resto de `phonometry.metrology`, para dos o tres entradas y una 26 26 - salida. 25 25 + Welch que el resto de `phonometry.metrology`, para varias entradas correladas 26 26 + y una salida. 27 27 28 28 <ThemeImage src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/miso_coherence.svg" alt="Figura de dos paneles. Arriba: el autoespectro de salida medido en dB con la contribución de salida coherente de dos entradas sombreada debajo; la entrada 1 llena la banda baja hasta unos 400 Hz y la entrada 2 la banda alta por encima de unos 1,5 kHz, con el ruido residual muy por debajo. Abajo: para la segunda entrada correlada, su coherencia ordinaria ronda 0,3 en toda la banda baja aunque no excita ningún camino de baja frecuencia, mientras que su coherencia parcial se desploma a cero ahí una vez que se condiciona fuera la primera entrada; la coherencia múltiple se mantiene cerca de uno salvo en el nulo del cruce." width="88%" loading="eager" /> 29 29

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site/src/content/docs/es/guides/sections/core-signal-analysis.md

Reviewed

··· 80 80 el espectro de salida coherente con la SNR espectral, suavizado en 1/n de 81 81 octava y generadores de ruido de colores con pendiente exacta. 82 82 - [Coherencia múltiple y parcial](/phonometry/es/guides/miso-coherence/): las 83 83 - funciones de coherencia de entradas múltiples para dos o tres fuentes 83 83 + funciones de coherencia de entradas múltiples para varias fuentes 84 84 correladas y una salida, con el condicionamiento que distingue una causa 85 85 real de una fuente que solo correla con ella, y los espectros de salida 86 86 coherente parciales que indican qué fuente domina cada banda. ··· 94 94 cepstral, liftering paso bajo/paso alto de un espectro logarítmico, y el 95 95 espectro de la envolvente que convierte las modulaciones de amplitud en 96 96 líneas discretas. 97 97 + - [Promediado síncrono en el tiempo](/phonometry/es/guides/synchronous-averaging/): 98 98 + extracción de una forma de onda periódica de período conocido por promediado 99 99 + en el dominio del tiempo, el filtro peine que lo describe en el dominio de la 100 100 + frecuencia, la ley de reducción de ruido en raíz cuadrada, y la elección del 101 101 + número de promedios que sitúa un nodo del peine sobre un orden interferente 102 102 + (McFadden 1987). 97 103 - [Correlación, retardo y envolvente](/phonometry/es/guides/correlation-delay/): 98 104 estimaciones de correlación con los errores aleatorios de Bendat y 99 105 Piersol, estimación del retardo por correlación directa, pendiente de fase

+14 -2

site/src/content/docs/es/guides/sections/signals-spectra.md

Reviewed

··· 25 25 pendiente exacta en ley de potencias. 26 26 [La coherencia múltiple y parcial](/phonometry/es/guides/miso-coherence/) lleva 27 27 esa misma maquinaria del espectro cruzado a varias fuentes correladas a la vez: 28 28 - a partir de dos o tres entradas y una salida separa la coherencia que una 28 28 + a partir de varias entradas correladas y una salida separa la coherencia que una 29 29 fuente aporta de verdad de la parte que solo comparte con otra, y sus espectros 30 30 de salida coherente parciales indican qué fuente domina cada banda. 31 31 ··· 43 43 pico cepstral, el liftering separa un espectro logarítmico en envolvente 44 44 suave y estructura fina, y el espectro de la envolvente convierte las 45 45 modulaciones de amplitud en líneas discretas en la frecuencia de modulación. 46 46 + [El promediado síncrono en el tiempo](/phonometry/es/guides/synchronous-averaging/) 47 47 + extrae una forma de onda repetitiva de período conocido del ruido asíncrono 48 48 + promediando períodos sucesivos: el ruido residual cae como la raíz cuadrada 49 49 + del número de promedios, y elegir ese número para situar un nodo del filtro 50 50 + peine sobre un orden interferente lo rechaza mucho mejor que la potencia de 51 51 + dos habitual. 46 52 47 53 [Correlación, retardo y envolvente](/phonometry/es/guides/correlation-delay/) 48 54 es la mitad del dominio del tiempo. La autocorrelación y la correlación ··· 88 94 generadores de ruido de colores con pendiente exacta. 89 95 - [Coherencia múltiple y parcial](/phonometry/es/guides/miso-coherence/): 90 96 las funciones de coherencia de entradas múltiples de Bendat y Piersol para 91 91 - dos o tres fuentes correladas y una salida, con el condicionamiento por 97 97 + varias fuentes correladas y una salida, con el condicionamiento por 92 98 eliminación de Gauss que distingue una causa real de una fuente que solo 93 99 correla con ella, y los espectros de salida coherente parciales que indican 94 100 qué fuente domina cada banda. ··· 101 107 de ecos con el coeficiente de reflexión leído en el pico, liftering paso 102 108 bajo/paso alto, la ida y vuelta homomórfica y el espectro de la envolvente 103 109 de las modulaciones de amplitud. 110 110 + - [Promediado síncrono en el tiempo](/phonometry/es/guides/synchronous-averaging/): 111 111 + extracción de una forma de onda periódica de período conocido por promediado 112 112 + en el dominio del tiempo, el filtro peine que describe la operación en el 113 113 + dominio de la frecuencia, la ley de reducción de ruido en raíz cuadrada, y la 114 114 + elección del número de promedios que sitúa un nodo del peine sobre un orden 115 115 + interferente (McFadden 1987). 104 116 - [Correlación, retardo y envolvente](/phonometry/es/guides/correlation-delay/): 105 117 estimaciones de correlación con sus errores aleatorios, estimación del 106 118 retardo por correlación directa, pendiente de fase y ponderaciones GCC,

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site/src/content/docs/es/guides/spectral-analysis.mdx

Reviewed

··· 508 508 PSD, una coherencia y una H1 calculadas con la misma longitud de segmento 509 509 son mutuamente consistentes bin a bin. La misma matriz de espectros cruzados 510 510 sustenta la [coherencia múltiple y parcial](/phonometry/es/guides/miso-coherence/), 511 511 - que extiende la coherencia ordinaria a dos o tres entradas correladas y una 511 511 + que extiende la coherencia ordinaria a varias entradas correladas y una 512 512 salida.

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site/src/content/docs/es/guides/synchronous-averaging.mdx

Reviewed

··· 1 1 + --- 2 2 + title: "Promediado síncrono en el tiempo" 3 3 + description: "Extracción de una forma de onda repetitiva de período conocido del ruido asíncrono mediante promediado en el dominio del tiempo (McFadden 1987): el filtro peine que describe la operación en el dominio de la frecuencia, la ley de reducción de ruido en raíz cuadrada, la elección del número de promedios que sitúa un nodo del peine sobre un orden interferente, y la alineación de banda limitada usada cuando el período no cae en un número entero de muestras." 4 4 + references: 5 5 + - type: article 6 6 + authors: ["McFadden, P. D."] 7 7 + year: 1987 8 8 + title: "A revised model for the extraction of periodic waveforms by time domain averaging" 9 9 + journal: "Mechanical Systems and Signal Processing 1(1), 83-95" 10 10 + doi: "10.1016/0888-3270(87)90043-2" 11 11 + note: "El modelo de filtro peine del promediado síncrono (Ec. 8, magnitud Ec. 9), el modelo revisado de registro finito que produce un resultado exactamente periódico, y la observación de que un orden interferente no armónico se rechaza mejor eligiendo el número de promedios de modo que un nodo del peine caiga sobre él, no con la potencia de dos habitual." 12 12 + --- 13 13 + 14 14 + import ThemeImage from '../../../../components/ThemeImage.astro'; 15 15 + 16 16 + Una máquina rotativa repite su firma una vez por revolución. Enterrada en el 17 17 + ruido de banda ancha y en los tonos del resto de los ejes, esa forma de onda 18 18 + repetitiva es difícil de leer directamente. El **promediado síncrono en el 19 19 + tiempo** (TSA) la recupera: dado el período `T` de una revolución, trocea el 20 20 + registro en bloques sucesivos de longitud `T` y los promedia. Toda componente 21 21 + síncrona con `T` se refuerza; todo lo asíncrono, el ruido y los armónicos de 22 22 + ejes no relacionados, se promedia hasta desaparecer. `time_synchronous_average` 23 23 + implementa el modelo de P. D. McFadden, *A revised model for the extraction of 24 24 + periodic waveforms by time domain averaging* (1987). 25 25 + 26 26 + <ThemeImage src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/synchronous_average_es.svg" alt="Figura de dos paneles. Izquierda: un período ruidoso de una señal en gris tenue oscila muy por encima y por debajo de una curva suave; el promedio de cuarenta períodos, en azul, se sitúa casi exactamente sobre la forma de onda verdadera en rojo discontinuo, de modo que el ruido asíncrono se ha eliminado. Derecha: la magnitud del filtro peine entre los órdenes 31 y 33, con dientes de altura unidad en los órdenes enteros; para N = 20 un nodo profundo cae exactamente sobre el tono interferente marcado en 32,05 órdenes, mientras que la potencia de dos N = 32 deja allí un lóbulo lateral y deja pasar el tono." width="92%" loading="eager" /> 27 27 + 28 28 + <details> 29 29 + <summary>Ver el código de esta figura</summary> 30 30 + 31 31 + ```python 32 32 + import matplotlib.pyplot as plt 33 33 + import numpy as np 34 34 + from phonometry import ( 35 35 + comb_filter_response, 36 36 + noise_signal, 37 37 + time_synchronous_average, 38 38 + ) 39 39 + 40 40 + fs = 8192.0 41 41 + period = 1.0 / 32.0 # una revolución: 256 muestras a esta frecuencia 42 42 + m, n_avg = 256, 40 43 43 + phase = np.arange((n_avg + 1) * m) / m 44 44 + periodic = ( 45 45 + np.cos(2.0 * np.pi * phase) 46 46 + + 0.5 * np.cos(2.0 * np.pi * 3.0 * phase + 0.4) 47 47 + - 0.3 * np.cos(2.0 * np.pi * 6.0 * phase) 48 48 + ) 49 49 + signal = periodic + noise_signal(fs, phase.size / fs, rms=0.9, seed=11) 50 50 + res = time_synchronous_average(signal, fs, period, n_averages=n_avg) 51 51 + 52 52 + fig, (ax0, ax1) = plt.subplots(1, 2, figsize=(11, 4.6)) 53 53 + t_ms = 1e3 * res.times 54 54 + ax0.plot(t_ms, signal[:m], color="#cccccc", label="Un período ruidoso") 55 55 + ax0.plot(t_ms, res.period_waveform, color="#1f77b4", lw=1.8, 56 56 + label=f"Promedio de N = {n_avg} períodos") 57 57 + ax0.plot(t_ms, periodic[:m], "--", color="#d62728", label="Forma verdadera") 58 58 + ax0.set_xlabel("Tiempo [ms]"); ax0.set_ylabel("Amplitud"); ax0.legend() 59 59 + 60 60 + orders = np.linspace(31.0, 33.0, 4000) 61 61 + freqs = orders / period 62 62 + ax1.plot(orders, comb_filter_response(freqs, period, 32), color="#2ca02c", 63 63 + label="N = 32 (potencia de dos)") 64 64 + ax1.plot(orders, comb_filter_response(freqs, period, 20), color="#1f77b4", 65 65 + label="N = 20 (nodo en 32,05)") 66 66 + ax1.axvline(32.05, color="#d62728", ls=":", label="Tono interferente") 67 67 + ax1.set_xlabel("Frecuencia [órdenes]"); ax1.set_ylabel("Magnitud del filtro peine") 68 68 + ax1.set_ylim(0, 1.05); ax1.legend() 69 69 + plt.show() 70 70 + ``` 71 71 + 72 72 + </details> 73 73 + 74 74 + El método `.plot()` dibuja la forma de onda promediada y el filtro peine en una 75 75 + sola llamada: 76 76 + 77 77 + ```python 78 78 + res.plot() # etiquetas en inglés; res.plot(language="es") en español 79 79 + ``` 80 80 + 81 81 + ## 1. El promedio es un filtro peine 82 82 + 83 83 + Promediar `N` períodos sucesivos (McFadden Ec. 5), 84 84 + 85 85 + $$ 86 86 + a(t) = \frac{1}{N} \sum_{n=0}^{N-1} y(t + n\,T), 87 87 + $$ 88 88 + 89 89 + es, en el dominio de la frecuencia, la multiplicación del espectro de la señal 90 90 + por un **filtro peine** (Ec. 8). Su magnitud (Ec. 9) es el núcleo de Dirichlet 91 91 + 92 92 + $$ 93 93 + |C(f)| = \left| \frac{\sin(N\pi f T)}{N \sin(\pi f T)} \right| . 94 94 + $$ 95 95 + 96 96 + El peine tiene un **diente** de altura unidad en cada armónico `k/T` (los 97 97 + órdenes `f·T = 1, 2, 3, ...`), *independiente de `N`*: las componentes 98 98 + síncronas con el período pasan intactas. Entre los dientes tiene **nodos** en 99 99 + `j/(N·T)` para todo `j` que no sea múltiplo de `N`, donde la respuesta es 100 100 + exactamente cero. `comb_filter_response` evalúa esta forma cerrada 101 101 + directamente: 102 102 + 103 103 + ```python 104 104 + import numpy as np 105 105 + from phonometry import comb_filter_response 106 106 + 107 107 + period = 1.0 / 32.0 108 108 + comb_filter_response(np.array([16.0 / period]), period, 8) # 1.0 en un diente 109 109 + comb_filter_response(np.array([0.25 / period]), period, 2) # 1/sqrt(2) 110 110 + comb_filter_response(np.array([0.5 / period]), period, 2) # 0.0 en un nodo 111 111 + ``` 112 112 + 113 113 + `SynchronousAverageResult` lleva la respuesta sobre los primeros armónicos en 114 114 + `comb_frequencies` y `comb_response`, de modo que la forma del filtro que el 115 115 + promedio aplicó está disponible junto a la forma de onda recuperada. 116 116 + 117 117 + ## 2. El ruido cae como la raíz cuadrada del número de promedios 118 118 + 119 119 + El ruido asíncrono de varianza $\sigma^2$ promediado sobre `N` períodos tiene 120 120 + varianza residual $\sigma^2/N$: la desviación típica residual cae como 121 121 + $1/\sqrt{N}$, y la relación señal-ruido en amplitud mejora en $\sqrt{N}$. Eso 122 122 + es una reducción de potencia de $10\log_{10} N$ dB, reportada como 123 123 + `noise_reduction_db`, con la ganancia en amplitud $\sqrt{N}$ como 124 124 + `amplitude_snr_gain`: 125 125 + 126 126 + ```python 127 127 + res = time_synchronous_average(signal, fs, period, n_averages=100) 128 128 + res.noise_reduction_db # 20.0 dB = 10*log10(100) 129 129 + res.amplitude_snr_gain # 10.0 = sqrt(100) 130 130 + ``` 131 131 + 132 132 + Esta ley es la ideal: se cumple cuando el ruido no está correlado de un período 133 133 + al siguiente, de modo que el ruido de color o síncrono correlado entre períodos 134 134 + no tiene por qué seguirla. El `residual` (entrada menos la reconstrucción 135 135 + periódica sobre el tramo analizado) y su `residual_rms` reportan, por tanto, el 136 136 + ruido realmente restante una vez eliminada la componente síncrona. 137 137 + 138 138 + ## 3. Elegir N para rechazar un orden interferente 139 139 + 140 140 + Como hay un diente sobre *todos* los órdenes enteros, el TSA deja pasar los 141 141 + armónicos del eje objetivo pero también cualquier tono que caiga en un orden 142 142 + entero. Un tono en un orden *no armónico* `q = f·T` solo es atenuado por el 143 143 + peine, no eliminado, y cuánto depende de dónde caiga el nodo más cercano. El 144 144 + resultado del modelo revisado de McFadden es que tal interferente se rechaza 145 145 + mejor eligiendo `N` de modo que un nodo caiga exactamente sobre él, es decir, 146 146 + el menor `N` con `N·q` entero, en vez de con el número de promedios potencia 147 147 + de dos habitual. Un nodo exacto solo existe cuando el orden `q` es racional, de 148 148 + modo que algún `N` finito hace `N·q` entero; para un orden irracional o solo 149 149 + estimado, elige el `N` cuyo nodo caiga más cerca del orden interferente. 150 150 + 151 151 + Su propio ejemplo es un tono en 32,05 órdenes. Con `N = 20` el producto 152 152 + `20 · 32,05 = 641` es entero, así que un nodo del peine cae sobre el tono y lo 153 153 + rechaza en más de 100 dB. La elección común `N = 32` da 154 154 + `32 · 32,05 = 1025,6`, que se sitúa en un lóbulo lateral: el tono apenas se 155 155 + toca. La figura de arriba muestra ambos peines en torno al orden 32; el 156 156 + promedio de extremo a extremo lo confirma: 157 157 + 158 158 + ```python 159 159 + # 8.o orden verdadero más un interferente fuerte en 32,05 órdenes 160 160 + phase = np.arange(41 * 256) / 256 161 161 + signal = np.cos(2 * np.pi * 8.0 * phase) + 0.7 * np.cos(2 * np.pi * 32.05 * phase) 162 162 + 163 163 + leak_20 = time_synchronous_average(signal, fs, period, n_averages=20) 164 164 + leak_32 = time_synchronous_average(signal, fs, period, n_averages=32) 165 165 + # leak_20.period_waveform coincide con el 8.o orden limpio; leak_32 no 166 166 + ``` 167 167 + 168 168 + Así que un número de promedios potencia de dos, por conveniente que sea, no es 169 169 + en general la elección óptima: los órdenes interferentes presentes en la 170 170 + máquina deberían fijar `N`. 171 171 + 172 172 + ## 4. Muestras por período no enteras 173 173 + 174 174 + Cuando `fs·T` es entero, las fronteras de período caen sobre muestras, los 175 175 + bloques se trocean directamente y una señal periódica sin ruido se recupera 176 176 + con precisión de máquina (`interpolated` es `False`). Cuando `fs·T` no es 177 177 + entero las fronteras caen entre muestras; cada bloque se alinea entonces a una 178 178 + malla entera común mediante el retardo fraccionario de banda limitada de 179 179 + [`fractional_delay`](/phonometry/es/guides/test-signals/), y la forma de onda 180 180 + se recupera dentro de ese error de interpolación (`interpolated` es `True`): 181 181 + 182 182 + ```python 183 183 + fs = 8192.0 184 184 + period = 1.0 / 31.7 # fs * period no es entero 185 185 + t = np.arange(int(40 * period * fs)) / fs 186 186 + signal = np.cos(2.0 * np.pi * t / period) # un ciclo por revolución 187 187 + res = time_synchronous_average(signal, fs, period) 188 188 + res.interpolated # True: alineación por retardo fraccionario 189 189 + res.samples_per_period # muestras enteras de un período 190 190 + ``` 191 191 + 192 192 + Por defecto el promedio usa tantos períodos completos como contenga el 193 193 + registro; pasa `n_averages` para fijar el número (para la elección de nodo del 194 194 + §3), y `n_harmonics` para fijar cuántos armónicos de `1/T` abarca la respuesta 195 195 + del peine devuelta. 196 196 + 197 197 + La alineación de banda limitada comparte su núcleo con la alineación 198 198 + submuestral de respuestas al impulso de la 199 199 + [página de señales de prueba](/phonometry/es/guides/test-signals/), y la forma 200 200 + de onda recuperada, al ser exactamente un período, puede repetirse para 201 201 + reconstruir la parte síncrona de la señal, para restarla o para análisis de 202 202 + órdenes.

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site/src/content/docs/guides/miso-coherence.mdx

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··· 1 1 --- 2 2 title: "Multiple and partial coherence" 3 3 - description: "The Bendat & Piersol multiple-input/output coherence functions: ordinary, multiple and partial coherence of two or three correlated sources driving one response, the Gaussian-elimination conditioning that separates a genuine cause from a source that merely correlates with it, and the partial coherent output spectra that say which source dominates each band." 3 3 + description: "The Bendat & Piersol multiple-input/output coherence functions: ordinary, multiple and partial coherence of several correlated sources driving one response, the Gaussian-elimination conditioning that separates a genuine cause from a source that merely correlates with it, and the partial coherent output spectra that say which source dominates each band." 4 4 references: 5 5 - type: book 6 6 authors: ["Bendat, J. S.", "Piersol, A. G."] ··· 21 21 Piersol, *Random Data* (4th ed., 2010, Chapter 7), resolve this for a 22 22 multiple-input/single-output (MISO) system with the **multiple** and 23 23 **partial** coherence functions. `miso_coherence` computes them from the same 24 24 - Welch cross-spectral core as the rest of `phonometry.metrology`, for two or 25 25 - three inputs and one output. 24 24 + Welch cross-spectral core as the rest of `phonometry.metrology`, for several 25 25 + correlated inputs and one output. 26 26 27 27 <ThemeImage src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/miso_coherence.svg" alt="Two-panel figure. Top: the measured output autospectrum in dB with the coherent output contribution of two inputs shaded underneath; input 1 fills the low band up to about 400 Hz and input 2 fills the high band above about 1.5 kHz, with the residual noise far below. Bottom: for the correlated second input, its ordinary coherence sits around 0.3 across the low band even though it drives no low-frequency path, while its partial coherence collapses to zero there once the first input is conditioned out; the multiple coherence stays near one except at the crossover null." width="88%" loading="eager" /> 28 28

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site/src/content/docs/guides/sections/core-signal-analysis.md

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··· 73 73 output spectrum with the spectral SNR, 1/n-octave smoothing and 74 74 exact-slope colored-noise generators. 75 75 - [Multiple and partial coherence](/phonometry/guides/miso-coherence/): the 76 76 - multiple-input/output coherence functions for two or three correlated 76 76 + multiple-input/output coherence functions for multiple correlated 77 77 sources and one output, with the conditioning that tells a genuine cause 78 78 from a source that merely correlates with it, and the partial coherent 79 79 output spectra that say which source dominates each band. ··· 85 85 detection with the reflection coefficient read off the cepstral peak, 86 86 lowpass/highpass liftering of a log spectrum, and the envelope spectrum 87 87 that turns amplitude modulations into discrete lines. 88 88 + - [Time synchronous averaging](/phonometry/guides/synchronous-averaging/): 89 89 + extraction of a periodic waveform of known period by time domain averaging, 90 90 + the comb filter that describes it in the frequency domain, the square-root 91 91 + noise-reduction law, and the choice of the number of averages that places a 92 92 + comb node on an interfering order (McFadden 1987). 88 93 - [Correlation, time delay and envelope](/phonometry/guides/correlation-delay/): 89 94 correlation estimates with the Bendat & Piersol random errors, time-delay 90 95 estimation by direct correlation, cross-spectrum phase slope and the

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site/src/content/docs/guides/sections/signals-spectra.md

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··· 22 22 white, pink, red, blue and violet test signals with an exact power-law slope. 23 23 [Multiple and partial coherence](/phonometry/guides/miso-coherence/) carries 24 24 that same cross-spectral machinery to several correlated sources at once: from 25 25 - two or three inputs and one output it separates the coherence a source 25 25 + multiple inputs and one output it separates the coherence a source 26 26 genuinely contributes from the part it merely shares with another, and its 27 27 partial coherent output spectra say which source dominates each band. 28 28 ··· 39 39 a log spectrum into smooth envelope and fine structure, and the envelope 40 40 spectrum turns amplitude modulations into discrete lines at the modulation 41 41 frequency. 42 42 + [Time synchronous averaging](/phonometry/guides/synchronous-averaging/) 43 43 + extracts a repetitive waveform of known period from asynchronous noise by 44 44 + ensemble-averaging successive periods: the residual noise falls as the square 45 45 + root of the number of averages, and choosing that number to place a comb node 46 46 + on an interfering order rejects it far better than the habitual power of two. 42 47 43 48 [Correlation, time delay and envelope](/phonometry/guides/correlation-delay/) 44 49 is the time-domain half. Auto- and cross-correlation come with the ··· 78 83 spectrum and spectral SNR, 1/n-octave smoothing and exact-slope 79 84 colored-noise generators. 80 85 - [Multiple and partial coherence](/phonometry/guides/miso-coherence/): 81 81 - the Bendat & Piersol multiple-input/output coherence functions for two or 82 82 - three correlated sources and one output, with the Gaussian-elimination 86 86 + the Bendat & Piersol multiple-input/output coherence functions for 87 87 + multiple correlated sources and one output, with the Gaussian-elimination 83 88 conditioning that tells a genuine cause from a source that merely 84 89 correlates with it, and the partial coherent output spectra that say which 85 90 source dominates each band. ··· 92 97 with the reflection coefficient read off the peak, lowpass/highpass 93 98 liftering, the homomorphic round trip and the envelope spectrum of 94 99 amplitude modulations. 100 100 + - [Time synchronous averaging](/phonometry/guides/synchronous-averaging/): 101 101 + extraction of a periodic waveform of known period by time domain averaging, 102 102 + the comb filter that describes the operation in the frequency domain, the 103 103 + square-root noise-reduction law, and the choice of the number of averages 104 104 + that places a comb node on an interfering order (McFadden 1987). 95 105 - [Correlation, time delay and envelope](/phonometry/guides/correlation-delay/): 96 106 correlation estimates with their random errors, time-delay estimation by 97 107 direct correlation, phase slope and GCC weightings, sub-sample

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site/src/content/docs/guides/spectral-analysis.mdx

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··· 486 486 calibration), so a PSD, a coherence and an H1 computed with the same segment 487 487 length are mutually consistent bin by bin. The same cross-spectral matrix 488 488 underlies [multiple and partial coherence](/phonometry/guides/miso-coherence/), 489 489 - which extends the ordinary coherence to two or three correlated inputs and one 489 489 + which extends the ordinary coherence to several correlated inputs and one 490 490 output.

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site/src/content/docs/guides/synchronous-averaging.mdx

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··· 1 1 + --- 2 2 + title: "Time synchronous averaging" 3 3 + description: "Extract a repetitive waveform of known period from asynchronous noise by time domain averaging (McFadden 1987): the comb filter that describes the operation in the frequency domain, the square-root noise-reduction law, the choice of the number of averages that places a comb node on an interfering order, and the band-limited alignment used when the period does not fall on an integer number of samples." 4 4 + references: 5 5 + - type: article 6 6 + authors: ["McFadden, P. D."] 7 7 + year: 1987 8 8 + title: "A revised model for the extraction of periodic waveforms by time domain averaging" 9 9 + journal: "Mechanical Systems and Signal Processing 1(1), 83-95" 10 10 + doi: "10.1016/0888-3270(87)90043-2" 11 11 + note: "The comb-filter model of synchronous averaging (Eq. 8, magnitude Eq. 9), the revised finite-record model that yields an exactly periodic result, and the observation that a non-harmonic interfering order is best rejected by choosing the number of averages so that a comb node lands on it, not by the habitual power of two." 12 12 + --- 13 13 + 14 14 + import ThemeImage from '../../../components/ThemeImage.astro'; 15 15 + 16 16 + A rotating machine repeats its signature once per revolution. Buried in 17 17 + broadband noise and in the tones of every other shaft, that repetitive 18 18 + waveform is hard to read directly. **Time synchronous averaging** (TSA) 19 19 + recovers it: given the period `T` of one revolution, it slices the record 20 20 + into successive length-`T` blocks and averages them. Every component 21 21 + synchronous with `T` reinforces; everything asynchronous, noise and the 22 22 + harmonics of unrelated shafts, averages down. `time_synchronous_average` 23 23 + implements the model of P. D. McFadden, *A revised model for the extraction 24 24 + of periodic waveforms by time domain averaging* (1987). 25 25 + 26 26 + <ThemeImage src="https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/synchronous_average.svg" alt="Two-panel figure. Left: one noisy period of a signal in faint grey swings far above and below a smooth curve; the average of forty periods, in blue, lies almost exactly on the dashed red true waveform, so the asynchronous noise has been removed. Right: the comb-filter magnitude across the orders between 31 and 33, with unit-height teeth at the integer orders; for N = 20 a deep node falls exactly on the interfering tone marked at 32.05 orders, whereas the power-of-two N = 32 leaves a side lobe there and lets the tone through." width="92%" loading="eager" /> 27 27 + 28 28 + <details> 29 29 + <summary>Show the code for this figure</summary> 30 30 + 31 31 + ```python 32 32 + import matplotlib.pyplot as plt 33 33 + import numpy as np 34 34 + from phonometry import ( 35 35 + comb_filter_response, 36 36 + noise_signal, 37 37 + time_synchronous_average, 38 38 + ) 39 39 + 40 40 + fs = 8192.0 41 41 + period = 1.0 / 32.0 # one revolution: 256 samples at this rate 42 42 + m, n_avg = 256, 40 43 43 + phase = np.arange((n_avg + 1) * m) / m 44 44 + periodic = ( 45 45 + np.cos(2.0 * np.pi * phase) 46 46 + + 0.5 * np.cos(2.0 * np.pi * 3.0 * phase + 0.4) 47 47 + - 0.3 * np.cos(2.0 * np.pi * 6.0 * phase) 48 48 + ) 49 49 + signal = periodic + noise_signal(fs, phase.size / fs, rms=0.9, seed=11) 50 50 + res = time_synchronous_average(signal, fs, period, n_averages=n_avg) 51 51 + 52 52 + fig, (ax0, ax1) = plt.subplots(1, 2, figsize=(11, 4.6)) 53 53 + t_ms = 1e3 * res.times 54 54 + ax0.plot(t_ms, signal[:m], color="#cccccc", label="One noisy period") 55 55 + ax0.plot(t_ms, res.period_waveform, color="#1f77b4", lw=1.8, 56 56 + label=f"Average of N = {n_avg} periods") 57 57 + ax0.plot(t_ms, periodic[:m], "--", color="#d62728", label="True waveform") 58 58 + ax0.set_xlabel("Time [ms]"); ax0.set_ylabel("Amplitude"); ax0.legend() 59 59 + 60 60 + orders = np.linspace(31.0, 33.0, 4000) 61 61 + freqs = orders / period 62 62 + ax1.plot(orders, comb_filter_response(freqs, period, 32), color="#2ca02c", 63 63 + label="N = 32 (power of two)") 64 64 + ax1.plot(orders, comb_filter_response(freqs, period, 20), color="#1f77b4", 65 65 + label="N = 20 (node on 32.05)") 66 66 + ax1.axvline(32.05, color="#d62728", ls=":", label="Interfering tone") 67 67 + ax1.set_xlabel("Frequency [orders]"); ax1.set_ylabel("Comb filter magnitude") 68 68 + ax1.set_ylim(0, 1.05); ax1.legend() 69 69 + plt.show() 70 70 + ``` 71 71 + 72 72 + </details> 73 73 + 74 74 + The `.plot()` method draws the averaged waveform and the comb filter in one 75 75 + call: 76 76 + 77 77 + ```python 78 78 + res.plot() # English labels; res.plot(language="es") for Spanish 79 79 + ``` 80 80 + 81 81 + ## 1. The average is a comb filter 82 82 + 83 83 + Averaging `N` successive periods (McFadden Eq. 5), 84 84 + 85 85 + $$ 86 86 + a(t) = \frac{1}{N} \sum_{n=0}^{N-1} y(t + n\,T), 87 87 + $$ 88 88 + 89 89 + is, in the frequency domain, the multiplication of the signal spectrum by a 90 90 + **comb filter** (Eq. 8). Its magnitude (Eq. 9) is the Dirichlet kernel 91 91 + 92 92 + $$ 93 93 + |C(f)| = \left| \frac{\sin(N\pi f T)}{N \sin(\pi f T)} \right| . 94 94 + $$ 95 95 + 96 96 + The comb has a **tooth** of unit height at every harmonic `k/T` (the orders 97 97 + `f·T = 1, 2, 3, ...`), *independent of `N`*: components synchronous with the 98 98 + period pass untouched. Between the teeth it has **nodes** at `j/(N·T)` for 99 99 + every `j` that is not a multiple of `N`, where the response is exactly zero. 100 100 + `comb_filter_response` evaluates this closed form directly: 101 101 + 102 102 + ```python 103 103 + import numpy as np 104 104 + from phonometry import comb_filter_response 105 105 + 106 106 + period = 1.0 / 32.0 107 107 + comb_filter_response(np.array([16.0 / period]), period, 8) # 1.0 at a tooth 108 108 + comb_filter_response(np.array([0.25 / period]), period, 2) # 1/sqrt(2) 109 109 + comb_filter_response(np.array([0.5 / period]), period, 2) # 0.0 at a node 110 110 + ``` 111 111 + 112 112 + `SynchronousAverageResult` carries the response over the first few harmonics 113 113 + in `comb_frequencies` and `comb_response`, so the shape of the filter that 114 114 + the average applied is available alongside the recovered waveform. 115 115 + 116 116 + ## 2. Noise falls as the square root of the number of averages 117 117 + 118 118 + Asynchronous noise of variance $\sigma^2$ averaged over `N` periods has 119 119 + residual variance $\sigma^2/N$: the residual standard deviation falls as 120 120 + $1/\sqrt{N}$, and the amplitude signal-to-noise ratio improves by $\sqrt{N}$. 121 121 + That is a power reduction of $10\log_{10} N$ dB, reported as 122 122 + `noise_reduction_db`, with the amplitude gain $\sqrt{N}$ as 123 123 + `amplitude_snr_gain`: 124 124 + 125 125 + ```python 126 126 + res = time_synchronous_average(signal, fs, period, n_averages=100) 127 127 + res.noise_reduction_db # 20.0 dB = 10*log10(100) 128 128 + res.amplitude_snr_gain # 10.0 = sqrt(100) 129 129 + ``` 130 130 + 131 131 + This law is the ideal one: it holds when the noise is uncorrelated from one 132 132 + period to the next, so colored or synchronous noise that is correlated across 133 133 + periods need not follow it. The `residual` (input minus the periodic 134 134 + reconstruction over the analysed span) and its `residual_rms` therefore report 135 135 + the noise actually left once the synchronous component is removed. 136 136 + 137 137 + ## 3. Choosing N to reject an interfering order 138 138 + 139 139 + Because a tooth sits on *every* integer order, TSA passes the harmonics of 140 140 + the target shaft but also any tone that happens to fall on an integer order. 141 141 + A tone at a *non-harmonic* order `q = f·T` is only attenuated by the comb, 142 142 + not removed, and how much depends on where the nearest node lands. McFadden's 143 143 + revised-model result is that such an interferer is best rejected by choosing 144 144 + `N` so that a node falls exactly on it, i.e. the smallest `N` with `N·q` an 145 145 + integer, rather than by the habitual power-of-two number of averages. An exact 146 146 + node exists only when the order `q` is rational, so some finite `N` makes `N·q` 147 147 + an integer; for an irrational or merely estimated order, choose the `N` whose 148 148 + node falls nearest the interfering order. 149 149 + 150 150 + His own example is a tone at 32.05 orders. With `N = 20` the product 151 151 + `20 · 32.05 = 641` is an integer, so a comb node lands on the tone and rejects 152 152 + it by more than 100 dB. The common choice `N = 32` gives `32 · 32.05 = 1025.6`, 153 153 + which sits on a side lobe: the tone is barely touched. The figure above shows 154 154 + both combs around order 32; the end-to-end average confirms it: 155 155 + 156 156 + ```python 157 157 + # true 8th-order component plus a strong interferer at 32.05 orders 158 158 + phase = np.arange(41 * 256) / 256 159 159 + signal = np.cos(2 * np.pi * 8.0 * phase) + 0.7 * np.cos(2 * np.pi * 32.05 * phase) 160 160 + 161 161 + leak_20 = time_synchronous_average(signal, fs, period, n_averages=20) 162 162 + leak_32 = time_synchronous_average(signal, fs, period, n_averages=32) 163 163 + # leak_20.period_waveform matches the clean 8th-order tone; leak_32 does not 164 164 + ``` 165 165 + 166 166 + So a power-of-two number of averages, convenient as it is, is not in general 167 167 + the optimal choice: the interfering orders present in the machine should set 168 168 + `N`. 169 169 + 170 170 + ## 4. Non-integer samples per period 171 171 + 172 172 + When `fs·T` is an integer, the period boundaries fall on samples, the blocks 173 173 + are sliced directly, and a noiseless periodic signal is recovered to machine 174 174 + precision (`interpolated` is `False`). When `fs·T` is not an integer the 175 175 + boundaries fall between samples; each block is then aligned to a common 176 176 + integer grid by the band-limited fractional delay of 177 177 + [`fractional_delay`](/phonometry/guides/test-signals/), and the waveform is 178 178 + recovered within that interpolation error (`interpolated` is `True`): 179 179 + 180 180 + ```python 181 181 + fs = 8192.0 182 182 + period = 1.0 / 31.7 # fs * period is not an integer 183 183 + t = np.arange(int(40 * period * fs)) / fs 184 184 + signal = np.cos(2.0 * np.pi * t / period) # one cycle per revolution 185 185 + res = time_synchronous_average(signal, fs, period) 186 186 + res.interpolated # True: fractional-delay alignment 187 187 + res.samples_per_period # integer samples of one period 188 188 + ``` 189 189 + 190 190 + By default the average uses as many whole periods as the record holds; pass 191 191 + `n_averages` to fix the count (for the node-selection choice of §3), and 192 192 + `n_harmonics` to set how many harmonics of `1/T` the returned comb response 193 193 + spans. 194 194 + 195 195 + The band-limited alignment shares its kernel with the sub-sample 196 196 + impulse-response alignment of the 197 197 + [test-signals page](/phonometry/guides/test-signals/), and the recovered 198 198 + waveform, being exactly one period, can be tiled to reconstruct the 199 199 + synchronous part of the signal for subtraction or for order analysis.

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site/src/content/docs/reference/api/index.md

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··· 223 223 | [`metrology.signals`](/phonometry/reference/api/spectra/signals/) | Test signals and sample-rate utilities. | 224 224 | [`metrology.phase`](/phonometry/reference/api/spectra/phase/) | Phase utilities: minimum phase, group delay and excess phase. | 225 225 | [`metrology.cepstrum`](/phonometry/reference/api/spectra/cepstrum/) | Cepstral analysis: real/power/complex cepstrum, liftering and echo detection. | 226 226 + | [`metrology.synchronous_average`](/phonometry/reference/api/spectra/synchronous-average/) | Time synchronous averaging (TSA) of a periodic waveform in noise. | 226 227 | [`metrology.inversion`](/phonometry/reference/api/spectra/inversion/) | Regularized spectral inversion with frequency-dependent regularization. | 227 228 228 229 ## Wave simulation

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site/src/content/docs/reference/api/spectra/miso.md

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··· 91 91 92 92 | Name | Description | 93 93 | :--- | :--- | 94 94 - | `inputs` | The `q` input records (2 or 3), a sequence of equal-length 1-D arrays or a 2-D `(q, n)` array. | 94 94 + | `inputs` | The `q` input records (`q >= 2`), a sequence of equal-length 1-D arrays or a 2-D `(q, n)` array. | 95 95 | `output` | The output record, 1-D, same length as the inputs. | 96 96 | `fs` | Sample rate, in Hz. | 97 97 | `order` | Conditioning order as input indices (default `0..q-1`). | ··· 146 146 | Name | Description | 147 147 | :--- | :--- | 148 148 | `frequencies` | One-sided frequency axis, in Hz. | 149 149 - | `n_inputs` | Number of inputs `q` (2 or 3). | 149 149 + | `n_inputs` | Number of inputs `q` (`q >= 2`). | 150 150 | `order` | Conditioning order actually applied, as original input indices; `partial_coherence[order[k]]` is conditioned on the inputs `order[:k]`. | 151 151 | `ordinary_coherence` | `γ²iy(f) ∈ [0, 1]` per input (Eq. 7.109), shape `(q, F)`: each input against the output on its own. | 152 152 | `multiple_coherence` | `γ²y:x(f) ∈ [0, 1]` (Eq. 7.35): the fraction of output power explained by all inputs jointly. Equals the sum of the partial coherences (Eq. 7.116) and `1 - noise_psd/output_psd`. |

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site/src/content/docs/reference/api/spectra/synchronous-average.md

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··· 1 1 + --- 2 2 + title: "metrology.synchronous_average" 3 3 + description: "Public API of phonometry.metrology.synchronous_average (auto-generated)." 4 4 + sidebar: 5 5 + label: "synchronous_average" 6 6 + --- 7 7 + 8 8 + > Auto-generated from the source docstrings by `scripts/generate_api_docs.py` (`make api-docs`). Do not edit by hand. 9 9 + 10 10 + Time synchronous averaging (TSA) of a periodic waveform in noise. 11 11 + 12 12 + Time domain averaging extracts a repetitive signal of known period `T` 13 13 + from additive noise by ensemble-averaging successive length-`T` blocks, 14 14 + following P. D. McFadden, "A revised model for the extraction of periodic 15 15 + waveforms by time domain averaging", *Mechanical Systems and Signal 16 16 + Processing* 1(1) 1987, 83-95. Given a signal `y(t) = x(t) + e(t)` with 17 17 + `x` periodic in `T` and `e` asynchronous, the average 18 18 + 19 19 + `a(t) = (1/N) Σ_{n=0}^{N-1} y(t + n·T)` (McFadden Eq. 5) 20 20 + 21 21 + reinforces every component synchronous with `T` and suppresses the rest. 22 22 + 23 23 + **Two models, one implementation.** McFadden distinguishes the *existing* 24 24 + comb-filter model from the *revised* model. In the frequency domain the 25 25 + average is the multiplication of `Y(f)` by the comb filter (Eq. 8) 26 26 + 27 27 + `C(f) = (1/N)·sin(N·π·f·T) / sin(π·f·T)`, 28 28 + 29 29 + whose magnitude `|C(f)| = |sin(N·π·f·T) / (N·sin(π·f·T))|` is a Dirichlet 30 30 + kernel: unity at every harmonic `k/T` (the teeth, Eq. 9, of unit height 31 31 + regardless of `N`) and zero at the nodes `j/(N·T)` with `j` not a 32 32 + multiple of `N`. That model assumes knowledge of `y` over infinite time 33 33 + and produces a result that is not exactly periodic. McFadden's *revised* 34 34 + model applies a rectangular window of width `T` in the time domain and 35 35 + samples the transform in the frequency domain, so it needs only a finite 36 36 + block of the signal and yields a result that is exactly periodic and can be 37 37 + stored as a single period. The digital block average computed here, `N` 38 38 + consecutive periods of an integer number of samples reduced to one period, 39 39 + *is* that revised model: the returned `period_waveform`, repeated, 40 40 + is exactly periodic. 41 41 + 42 42 + **Noise reduction.** Asynchronous noise of variance `σ²` averaged over 43 43 + `N` periods has residual variance `σ²/N`: the residual standard 44 44 + deviation falls as `1/√N` and the amplitude signal-to-noise ratio 45 45 + improves by `√N` (a power reduction of `10·log₁₀ N` dB, reported as 46 46 + `noise_reduction_db`). 47 47 + 48 48 + **Choosing N (McFadden's revised-model correction).** Because a discrete 49 49 + interfering tone at a *non-harmonic* order `q = f·T` is only attenuated, 50 50 + not removed, its rejection is optimised by choosing `N` so that a comb 51 51 + node lands exactly on it, i.e. the smallest `N` with `N·q` an integer. 52 52 + McFadden's own example, a tone at 32.05 orders, is suppressed by more than 53 53 + 100 dB with `N = 20` (since `20·32.05 = 641`) yet only ~14 dB with the 54 54 + common power-of-two choice `N = 32` (`32·32.05 = 1025.6`). Thus the 55 55 + habit of taking a power-of-two number of averages is not, in general, 56 56 + optimal. 57 57 + 58 58 + **Non-integer samples per period.** When `fs·T` is not an integer the 59 59 + period boundaries fall between samples. Each block is then aligned to a 60 60 + common integer grid by the band-limited fractional delay of 61 61 + [`phonometry.metrology.signals.fractional_delay`](/phonometry/reference/api/spectra/signals/#fractional_delay) before averaging, so 62 62 + the periodic waveform is recovered within the interpolation error of that 63 63 + band-limited shift. An integer `fs·T` needs no interpolation and the 64 64 + waveform is recovered to machine precision. 65 65 + 66 66 + ## comb_filter_response 67 67 + 68 68 + ```python 69 69 + comb_filter_response( 70 70 + frequencies: NDArray[np.float64] | list[float], 71 71 + period: float, 72 72 + n_averages: int, 73 73 + ) -> NDArray[np.float64] 74 74 + ``` 75 75 + 76 76 + Magnitude of the N-period synchronous-averaging comb filter. 77 77 + 78 78 + The closed form of McFadden Eq. 8, `|C(f)| = |sin(N·π·f·T) / 79 79 + (N·sin(π·f·T))|`, a Dirichlet kernel with unit-height teeth at the 80 80 + harmonics `k/T` (Eq. 9) and nodes at `j/(N·T)` for `j` not a 81 81 + multiple of `N`. 82 82 + 83 83 + **Parameters** 84 84 + 85 85 + | Name | Description | 86 86 + | :--- | :--- | 87 87 + | `frequencies` | Frequencies at which to evaluate, in Hz. | 88 88 + | `period` | Repetition period `T`, in seconds. | 89 89 + | `n_averages` | Number of averaged periods `N` (at least 1). | 90 90 + 91 91 + **Returns:** The filter magnitude at each frequency (unitless, in [0, 1]). 92 92 + 93 93 + **Raises** 94 94 + 95 95 + | Exception | When | 96 96 + | :--- | :--- | 97 97 + | ValueError | If the parameters are invalid. | 98 98 + 99 99 + ## SynchronousAverageResult 100 100 + 101 101 + ```python 102 102 + SynchronousAverageResult( 103 103 + period_waveform: NDArray[np.float64], 104 104 + times: NDArray[np.float64], 105 105 + residual: NDArray[np.float64], 106 106 + n_averages: int, 107 107 + samples_per_period: int, 108 108 + period: float, 109 109 + fs: float, 110 110 + interpolated: bool, 111 111 + noise_reduction_db: float, 112 112 + residual_rms: float, 113 113 + comb_frequencies: NDArray[np.float64], 114 114 + comb_response: NDArray[np.float64], 115 115 + ) 116 116 + ``` 117 117 + 118 118 + Time synchronous average of a periodic waveform in noise. 119 119 + 120 120 + **Attributes** 121 121 + 122 122 + | Name | Description | 123 123 + | :--- | :--- | 124 124 + | `period_waveform` | The averaged periodic waveform, one period of `samples_per_period` samples. | 125 125 + | `times` | Time axis of `period_waveform`, in seconds, spanning one period `[0, T)`. | 126 126 + | `residual` | Input minus the periodic reconstruction, over the analysed span (`n_averages·samples_per_period` samples, aligned to the integer period grid): what is left after the synchronous component is removed. | 127 127 + | `n_averages` | Number of periods averaged, `N`. | 128 128 + | `samples_per_period` | Integer samples per period `M` after any alignment. | 129 129 + | `period` | Repetition period `T`, in seconds. | 130 130 + | `fs` | Sample rate, in Hz. | 131 131 + | `interpolated` | Whether band-limited fractional-delay alignment was applied (`True` when `fs·T` is not an integer). | 132 132 + | `noise_reduction_db` | Power reduction of asynchronous noise, `10·log₁₀ N` dB (amplitude SNR gain `√N`). | 133 133 + | `residual_rms` | Root-mean-square of `residual`. | 134 134 + | `comb_frequencies` | Frequency axis of the comb-filter response, in Hz (from DC over a whole number of harmonics of `1/T`). | 135 135 + | `comb_response` | Magnitude of the comb filter (McFadden Eq. 8) on `comb_frequencies`. | 136 136 + 137 137 + ### SynchronousAverageResult.amplitude_snr_gain 138 138 + 139 139 + *property* 140 140 + 141 141 + Amplitude signal-to-noise improvement `√N` from averaging. 142 142 + 143 143 + ### SynchronousAverageResult.plot() 144 144 + 145 145 + ```python 146 146 + SynchronousAverageResult.plot( 147 147 + ax: Axes | None = None, 148 148 + *, 149 149 + language: str = 'en', 150 150 + **kwargs: Any, 151 151 + ) -> Axes | NDArray[Any] 152 152 + ``` 153 153 + 154 154 + Plot the averaged waveform and the comb-filter magnitude. 155 155 + 156 156 + With `ax` given, only the averaged-waveform panel is drawn on it. 157 157 + 158 158 + **Parameters** 159 159 + 160 160 + | Name | Description | 161 161 + | :--- | :--- | 162 162 + | `language` | Label language, `"en"` (default) or `"es"`. | 163 163 + 164 164 + ## time_synchronous_average 165 165 + 166 166 + ```python 167 167 + time_synchronous_average( 168 168 + x: NDArray[np.float64] | list[float], 169 169 + fs: float, 170 170 + period: float, 171 171 + *, 172 172 + n_averages: int | None = None, 173 173 + n_harmonics: int = 8, 174 174 + ) -> SynchronousAverageResult 175 175 + ``` 176 176 + 177 177 + Extract a periodic waveform of known period by time domain averaging. 178 178 + 179 179 + Ensemble-averages `N` successive periods of the record (McFadden 180 180 + Eq. 5) to reinforce the component synchronous with `period` and 181 181 + suppress asynchronous noise, whose residual standard deviation falls as 182 182 + `1/√N`. When `fs·period` is an integer the periods are sliced 183 183 + directly and a noiseless periodic signal is recovered exactly; otherwise 184 184 + each period is aligned to a common integer grid by the band-limited 185 185 + fractional delay of [`fractional_delay`](/phonometry/reference/api/spectra/signals/#fractional_delay) 186 186 + and recovered within that interpolation error. 187 187 + 188 188 + **Parameters** 189 189 + 190 190 + | Name | Description | 191 191 + | :--- | :--- | 192 192 + | `x` | Signal, 1-D, containing the periodic component plus noise. | 193 193 + | `fs` | Sample rate, in Hz. | 194 194 + | `period` | Known repetition period `T`, in seconds (e.g. one revolution of a rotating machine). | 195 195 + | `n_averages` | Number of whole periods to average (default: as many as the record holds). Choosing `N` so that `N·q` is an integer places a comb node on an interfering tone at order `q` and maximises its rejection (McFadden's revised-model result). | 196 196 + | `n_harmonics` | Number of harmonics of `1/T` spanned by the returned comb-filter response (default 8). | 197 197 + 198 198 + **Returns:** A [`SynchronousAverageResult`](/phonometry/reference/api/spectra/synchronous-average/#synchronousaverageresult). 199 199 + 200 200 + **Raises** 201 201 + 202 202 + | Exception | When | 203 203 + | :--- | :--- | 204 204 + | ValueError | If the inputs or parameters are invalid. |

+1

site/src/generated/api-sidebar.mjs

Reviewed

··· 223 223 'reference/api/spectra/signals', 224 224 'reference/api/spectra/phase', 225 225 'reference/api/spectra/cepstrum', 226 226 + 'reference/api/spectra/synchronous-average', 226 227 'reference/api/spectra/inversion', 227 228 ], 228 229 },

+8

src/phonometry/__init__.py

Reviewed

··· 280 280 ParametricEQ, 281 281 parametric_eq, 282 282 ) 283 283 + from .metrology.synchronous_average import ( 284 284 + SynchronousAverageResult, 285 285 + comb_filter_response, 286 286 + time_synchronous_average, 287 287 + ) 283 288 from .metrology.phase import ( 284 289 PhaseDecompositionResult, 285 290 excess_phase, ··· 1405 1410 "align_impulse_responses", 1406 1411 "envelope", 1407 1412 "envelope_spectrum", 1413 1413 + "time_synchronous_average", 1414 1414 + "comb_filter_response", 1415 1415 + "SynchronousAverageResult", 1408 1416 "cepstrum", 1409 1417 "lifter", 1410 1418 "echo_detection",

+68

src/phonometry/_plot/metrology.py

Reviewed

··· 31 31 StationarityTestResult, 32 32 ) 33 33 from ..metrology.signals import ToneBurstResult 34 34 + from ..metrology.synchronous_average import SynchronousAverageResult 34 35 from ..metrology.spectra import ( 35 36 CoherentOutputSpectrumResult, 36 37 CrossSpectralDensityResult, ··· 210 211 "Cascade": "Cascada", 211 212 "Parametric EQ response (Audio EQ Cookbook)": 212 213 "Respuesta del EQ paramétrico (Audio EQ Cookbook)", 214 214 + "Time synchronous average (McFadden 1987)": 215 215 + "Promediado síncrono en el tiempo (McFadden 1987)", 216 216 + "Averaged periodic waveform (N = {n})": 217 217 + "Forma de onda periódica promediada (N = {n})", 218 218 + "Time [ms]": "Tiempo [ms]", 219 219 + "Frequency [orders]": "Frecuencia [órdenes]", 220 220 + r"Comb filter $|C(f)|$ (Eq. 8)": r"Filtro peine $|C(f)|$ (Ec. 8)", 221 221 + "Harmonics of $1/T$": "Armónicos de $1/T$", 213 222 } 214 223 215 224 ··· 1777 1786 format_frequency_axis(axf, fmin, fmax) 1778 1787 localize_axes(axf, language) 1779 1788 return axes 1789 1789 + 1790 1790 + 1791 1791 + def plot_synchronous_average( 1792 1792 + result: "SynchronousAverageResult", ax: Axes | None = None, *, 1793 1793 + language: str = "en", **kwargs: Any 1794 1794 + ) -> Axes | np.ndarray: 1795 1795 + """Averaged periodic waveform and the synchronous-averaging comb filter. 1796 1796 + 1797 1797 + With ``ax`` given, only the averaged-waveform panel is drawn on it. 1798 1798 + 1799 1799 + :param result: A 1800 1800 + :class:`~phonometry.metrology.synchronous_average.SynchronousAverageResult`. 1801 1801 + :param ax: Existing axes for the waveform panel, or ``None`` for a fresh 1802 1802 + two-panel (waveform + comb filter) figure. 1803 1803 + :param language: Label language, ``"en"`` (default) or ``"es"``. 1804 1804 + :param kwargs: Forwarded to the averaged-waveform line. 1805 1805 + :return: The waveform axes (``ax`` given) or the array of two axes. 1806 1806 + """ 1807 1807 + from .._i18n import localize_axes 1808 1808 + 1809 1809 + def _waveform(axw: Axes) -> None: 1810 1810 + kwargs.setdefault("color", _C_PRIMARY) 1811 1811 + kwargs.setdefault("lw", 1.6) 1812 1812 + axw.plot( 1813 1813 + 1e3 * result.times, result.period_waveform, 1814 1814 + label=_t("Averaged periodic waveform (N = {n})", language, 1815 1815 + n=result.n_averages), 1816 1816 + **kwargs, 1817 1817 + ) 1818 1818 + axw.set_xlabel(_t("Time [ms]", language)) 1819 1819 + axw.set_ylabel(_t("Amplitude", language)) 1820 1820 + axw.grid(True, alpha=0.3) 1821 1821 + axw.legend(loc=_LEGEND_UPPER_RIGHT, fontsize="small") 1822 1822 + 1823 1823 + if ax is not None: 1824 1824 + _waveform(ax) 1825 1825 + localize_axes(ax, language) 1826 1826 + return ax 1827 1827 + 1828 1828 + axes = _new_axes_column(2, figsize=(8.0, 6.4)) 1829 1829 + _waveform(axes[0]) 1830 1830 + axes[0].set_title(_t("Time synchronous average (McFadden 1987)", language)) 1831 1831 + 1832 1832 + orders = result.comb_frequencies * result.period 1833 1833 + axes[1].plot(orders, result.comb_response, color=_C_PRIMARY, lw=1.2) 1834 1834 + top = int(np.floor(orders[-1] + 1e-9)) 1835 1835 + for k in range(1, top + 1): 1836 1836 + axes[1].axvline( 1837 1837 + float(k), color=_C_REFERENCE, linestyle=":", lw=0.8, alpha=0.6, 1838 1838 + label=_t("Harmonics of $1/T$", language) if k == 1 else None, 1839 1839 + ) 1840 1840 + axes[1].set_xlabel(_t("Frequency [orders]", language)) 1841 1841 + axes[1].set_ylabel(_t(r"Comb filter $|C(f)|$ (Eq. 8)", language)) 1842 1842 + axes[1].set_ylim(0.0, 1.05) 1843 1843 + axes[1].grid(True, alpha=0.3) 1844 1844 + axes[1].legend(loc=_LEGEND_UPPER_RIGHT, fontsize="small") 1845 1845 + for axf in axes: 1846 1846 + localize_axes(axf, language) 1847 1847 + return axes

+8

src/phonometry/metrology/__init__.py

Reviewed

··· 80 80 resample_signal, 81 81 tone_burst, 82 82 ) 83 83 + from .synchronous_average import ( 84 84 + SynchronousAverageResult, 85 85 + comb_filter_response, 86 86 + time_synchronous_average, 87 87 + ) 83 88 from .spectra import ( 84 89 CoherentOutputSpectrumResult, 85 90 CrossSpectralDensityResult, ··· 137 142 "SpectralDensityResult", 138 143 "SpectrogramResult", 139 144 "StationarityTestResult", 145 145 + "SynchronousAverageResult", 140 146 "TrendTestResult", 141 147 "TimeDelayResult", 142 148 "TimeWeighting", ··· 195 201 "spectrogram", 196 202 "stationarity_test", 197 203 "time_delay", 204 204 + "time_synchronous_average", 205 205 + "comb_filter_response", 198 206 "time_weighting", 199 207 "tone_burst", 200 208 "trend_test",

+5 -5

src/phonometry/metrology/miso.py

Reviewed

··· 95 95 rows = [np.ascontiguousarray(row, dtype=np.float64) for row in inputs] 96 96 else: 97 97 rows = [np.asarray(x, dtype=np.float64) for x in inputs] 98 98 - if not 2 <= len(rows) <= 3: 99 99 - raise ValueError("'inputs' must hold 2 or 3 input records.") 98 98 + if len(rows) < 2: 99 99 + raise ValueError("'inputs' must hold at least two input records.") 100 100 xs = [_validate_signal(x, f"inputs[{i}]") for i, x in enumerate(rows)] 101 101 ya = _validate_signal(output, "output") 102 102 for i, x in enumerate(xs): ··· 324 324 :attr:`order`. 325 325 326 326 :ivar frequencies: One-sided frequency axis, in Hz. 327 327 - :ivar n_inputs: Number of inputs ``q`` (2 or 3). 327 327 + :ivar n_inputs: Number of inputs ``q`` (``q >= 2``). 328 328 :ivar order: Conditioning order actually applied, as original input 329 329 indices; ``partial_coherence[order[k]]`` is conditioned on the inputs 330 330 ``order[:k]``. ··· 437 437 Absent a physical basis, Bendat & Piersol (Section 7.2.4) recommend 438 438 ordering the inputs by descending ordinary coherence with the output. 439 439 440 440 - :param inputs: The ``q`` input records (2 or 3), a sequence of equal-length 441 441 - 1-D arrays or a 2-D ``(q, n)`` array. 440 440 + :param inputs: The ``q`` input records (``q >= 2``), a sequence of 441 441 + equal-length 1-D arrays or a 2-D ``(q, n)`` array. 442 442 :param output: The output record, 1-D, same length as the inputs. 443 443 :param fs: Sample rate, in Hz. 444 444 :param order: Conditioning order as input indices (default ``0..q-1``).

+314

src/phonometry/metrology/synchronous_average.py

Reviewed

··· 1 1 + # Copyright (c) 2026. Jose M. Requena-Plens 2 2 + """ 3 3 + Time synchronous averaging (TSA) of a periodic waveform in noise. 4 4 + 5 5 + Time domain averaging extracts a repetitive signal of known period ``T`` 6 6 + from additive noise by ensemble-averaging successive length-``T`` blocks, 7 7 + following P. D. McFadden, "A revised model for the extraction of periodic 8 8 + waveforms by time domain averaging", *Mechanical Systems and Signal 9 9 + Processing* 1(1) 1987, 83-95. Given a signal ``y(t) = x(t) + e(t)`` with 10 10 + ``x`` periodic in ``T`` and ``e`` asynchronous, the average 11 11 + 12 12 + ``a(t) = (1/N) Σ_{n=0}^{N-1} y(t + n·T)`` (McFadden Eq. 5) 13 13 + 14 14 + reinforces every component synchronous with ``T`` and suppresses the rest. 15 15 + 16 16 + **Two models, one implementation.** McFadden distinguishes the *existing* 17 17 + comb-filter model from the *revised* model. In the frequency domain the 18 18 + average is the multiplication of ``Y(f)`` by the comb filter (Eq. 8) 19 19 + 20 20 + ``C(f) = (1/N)·sin(N·π·f·T) / sin(π·f·T)``, 21 21 + 22 22 + whose magnitude ``|C(f)| = |sin(N·π·f·T) / (N·sin(π·f·T))|`` is a Dirichlet 23 23 + kernel: unity at every harmonic ``k/T`` (the teeth, Eq. 9, of unit height 24 24 + regardless of ``N``) and zero at the nodes ``j/(N·T)`` with ``j`` not a 25 25 + multiple of ``N``. That model assumes knowledge of ``y`` over infinite time 26 26 + and produces a result that is not exactly periodic. McFadden's *revised* 27 27 + model applies a rectangular window of width ``T`` in the time domain and 28 28 + samples the transform in the frequency domain, so it needs only a finite 29 29 + block of the signal and yields a result that is exactly periodic and can be 30 30 + stored as a single period. The digital block average computed here, ``N`` 31 31 + consecutive periods of an integer number of samples reduced to one period, 32 32 + *is* that revised model: the returned :attr:`period_waveform`, repeated, 33 33 + is exactly periodic. 34 34 + 35 35 + **Noise reduction.** Asynchronous noise of variance ``σ²`` averaged over 36 36 + ``N`` periods has residual variance ``σ²/N``: the residual standard 37 37 + deviation falls as ``1/√N`` and the amplitude signal-to-noise ratio 38 38 + improves by ``√N`` (a power reduction of ``10·log₁₀ N`` dB, reported as 39 39 + :attr:`noise_reduction_db`). 40 40 + 41 41 + **Choosing N (McFadden's revised-model correction).** Because a discrete 42 42 + interfering tone at a *non-harmonic* order ``q = f·T`` is only attenuated, 43 43 + not removed, its rejection is optimised by choosing ``N`` so that a comb 44 44 + node lands exactly on it, i.e. the smallest ``N`` with ``N·q`` an integer. 45 45 + McFadden's own example, a tone at 32.05 orders, is suppressed by more than 46 46 + 100 dB with ``N = 20`` (since ``20·32.05 = 641``) yet only ~14 dB with the 47 47 + common power-of-two choice ``N = 32`` (``32·32.05 = 1025.6``). Thus the 48 48 + habit of taking a power-of-two number of averages is not, in general, 49 49 + optimal. 50 50 + 51 51 + **Non-integer samples per period.** When ``fs·T`` is not an integer the 52 52 + period boundaries fall between samples. Each block is then aligned to a 53 53 + common integer grid by the band-limited fractional delay of 54 54 + :func:`phonometry.metrology.signals.fractional_delay` before averaging, so 55 55 + the periodic waveform is recovered within the interpolation error of that 56 56 + band-limited shift. An integer ``fs·T`` needs no interpolation and the 57 57 + waveform is recovered to machine precision. 58 58 + """ 59 59 + 60 60 + from __future__ import annotations 61 61 + 62 62 + from dataclasses import dataclass 63 63 + from typing import TYPE_CHECKING, Any 64 64 + 65 65 + import numpy as np 66 66 + 67 67 + from .signals import _validate_1d_finite, fractional_delay 68 68 + from .spectra import _positive 69 69 + 70 70 + if TYPE_CHECKING: 71 71 + from matplotlib.axes import Axes 72 72 + from numpy.typing import NDArray 73 73 + 74 74 + __all__ = [ 75 75 + "SynchronousAverageResult", 76 76 + "comb_filter_response", 77 77 + "time_synchronous_average", 78 78 + ] 79 79 + 80 80 + #: Below this a fractional sample offset counts as integer-aligned, so no 81 81 + #: interpolation is applied and an integer-period record is exact. 82 82 + _ALIGN_TOL = 1e-9 83 83 + 84 84 + 85 85 + def comb_filter_response( 86 86 + frequencies: "NDArray[np.float64] | list[float]", 87 87 + period: float, 88 88 + n_averages: int, 89 89 + ) -> "NDArray[np.float64]": 90 90 + """Magnitude of the N-period synchronous-averaging comb filter. 91 91 + 92 92 + The closed form of McFadden Eq. 8, ``|C(f)| = |sin(N·π·f·T) / 93 93 + (N·sin(π·f·T))|``, a Dirichlet kernel with unit-height teeth at the 94 94 + harmonics ``k/T`` (Eq. 9) and nodes at ``j/(N·T)`` for ``j`` not a 95 95 + multiple of ``N``. 96 96 + 97 97 + :param frequencies: Frequencies at which to evaluate, in Hz. 98 98 + :param period: Repetition period ``T``, in seconds. 99 99 + :param n_averages: Number of averaged periods ``N`` (at least 1). 100 100 + :return: The filter magnitude at each frequency (unitless, in [0, 1]). 101 101 + :raises ValueError: If the parameters are invalid. 102 102 + """ 103 103 + period_v = _positive(period, "period") 104 104 + n = int(n_averages) 105 105 + if n < 1: 106 106 + raise ValueError("'n_averages' must be a positive integer.") 107 107 + freqs = np.asarray(frequencies, dtype=np.float64) 108 108 + if not np.all(np.isfinite(freqs)): 109 109 + raise ValueError("'frequencies' must be finite.") 110 110 + order = freqs * period_v 111 111 + lower = np.sin(np.pi * order) 112 112 + upper = np.sin(n * np.pi * order) 113 113 + with np.errstate(divide="ignore", invalid="ignore"): 114 114 + ratio = np.abs(upper / (n * lower)) 115 115 + # At the teeth (integer order) the 0/0 limit is exactly 1 (Eq. 9). 116 116 + at_tooth = np.abs(lower) < _ALIGN_TOL 117 117 + response = np.where(at_tooth, 1.0, ratio) 118 118 + return np.asarray(response, dtype=np.float64) 119 119 + 120 120 + 121 121 + @dataclass(frozen=True) 122 122 + class SynchronousAverageResult: 123 123 + """Time synchronous average of a periodic waveform in noise. 124 124 + 125 125 + :ivar period_waveform: The averaged periodic waveform, one period of 126 126 + :attr:`samples_per_period` samples. 127 127 + :ivar times: Time axis of :attr:`period_waveform`, in seconds, spanning 128 128 + one period ``[0, T)``. 129 129 + :ivar residual: Input minus the periodic reconstruction, over the 130 130 + analysed span (``n_averages·samples_per_period`` samples, aligned 131 131 + to the integer period grid): what is left after the synchronous 132 132 + component is removed. 133 133 + :ivar n_averages: Number of periods averaged, ``N``. 134 134 + :ivar samples_per_period: Integer samples per period ``M`` after any 135 135 + alignment. 136 136 + :ivar period: Repetition period ``T``, in seconds. 137 137 + :ivar fs: Sample rate, in Hz. 138 138 + :ivar interpolated: Whether band-limited fractional-delay alignment was 139 139 + applied (``True`` when ``fs·T`` is not an integer). 140 140 + :ivar noise_reduction_db: Power reduction of asynchronous noise, 141 141 + ``10·log₁₀ N`` dB (amplitude SNR gain ``√N``). 142 142 + :ivar residual_rms: Root-mean-square of :attr:`residual`. 143 143 + :ivar comb_frequencies: Frequency axis of the comb-filter response, in 144 144 + Hz (from DC over a whole number of harmonics of ``1/T``). 145 145 + :ivar comb_response: Magnitude of the comb filter (McFadden Eq. 8) on 146 146 + :attr:`comb_frequencies`. 147 147 + """ 148 148 + 149 149 + period_waveform: "NDArray[np.float64]" 150 150 + times: "NDArray[np.float64]" 151 151 + residual: "NDArray[np.float64]" 152 152 + n_averages: int 153 153 + samples_per_period: int 154 154 + period: float 155 155 + fs: float 156 156 + interpolated: bool 157 157 + noise_reduction_db: float 158 158 + residual_rms: float 159 159 + comb_frequencies: "NDArray[np.float64]" 160 160 + comb_response: "NDArray[np.float64]" 161 161 + 162 162 + @property 163 163 + def amplitude_snr_gain(self) -> float: 164 164 + """Amplitude signal-to-noise improvement ``√N`` from averaging.""" 165 165 + return float(np.sqrt(self.n_averages)) 166 166 + 167 167 + def plot( 168 168 + self, ax: "Axes | None" = None, *, language: str = "en", **kwargs: Any 169 169 + ) -> "Axes | NDArray[Any]": 170 170 + """Plot the averaged waveform and the comb-filter magnitude. 171 171 + 172 172 + With ``ax`` given, only the averaged-waveform panel is drawn on it. 173 173 + 174 174 + :param language: Label language, ``"en"`` (default) or ``"es"``. 175 175 + """ 176 176 + from .._i18n import check_language 177 177 + from .._plot.metrology import plot_synchronous_average 178 178 + 179 179 + check_language(language) 180 180 + return plot_synchronous_average(self, ax=ax, language=language, **kwargs) 181 181 + 182 182 + 183 183 + def _samples_per_period(fs: float, period: float) -> tuple[float, int]: 184 184 + """Exact and integer samples per period, with an integer-fit check.""" 185 185 + samples = fs * period 186 186 + rounded = int(round(samples)) 187 187 + if rounded < 2: 188 188 + raise ValueError( 189 189 + "'period' is too short for the sample rate: it must span at " 190 190 + "least 2 samples." 191 191 + ) 192 192 + return samples, rounded 193 193 + 194 194 + 195 195 + def _resolve_n_averages( 196 196 + n_averages: int | None, length: int, samples: float, m_int: int 197 197 + ) -> int: 198 198 + """Number of whole periods to average, validated against the record.""" 199 199 + available = int(np.floor((length - m_int) / samples)) + 1 200 200 + if available < 1: 201 201 + raise ValueError( 202 202 + "The record is shorter than one period; nothing to average." 203 203 + ) 204 204 + if n_averages is None: 205 205 + return available 206 206 + requested = int(n_averages) 207 207 + if requested < 1: 208 208 + raise ValueError("'n_averages' must be a positive integer.") 209 209 + if requested > available: 210 210 + raise ValueError( 211 211 + f"'n_averages' = {requested} exceeds the {available} whole " 212 212 + f"periods available in the record." 213 213 + ) 214 214 + return requested 215 215 + 216 216 + 217 217 + def _extract_period( 218 218 + x: "NDArray[np.float64]", start: float, m_int: int 219 219 + ) -> "NDArray[np.float64]": 220 220 + """One period of ``m_int`` samples starting at fractional ``start``. 221 221 + 222 222 + The block is aligned to the integer grid by a band-limited fractional 223 223 + delay; an integer ``start`` (within :data:`_ALIGN_TOL`) is sliced 224 224 + directly, so an integer-period record stays exact. 225 225 + """ 226 226 + i0 = int(round(start)) 227 227 + frac = start - i0 228 228 + if abs(frac) < _ALIGN_TOL: 229 229 + return x[i0 : i0 + m_int] 230 230 + # fractional_delay(x, d) yields x(j - d); advancing by frac (d = -frac) 231 231 + # gives x(j + frac), so sample i0 + m carries x(i0 + frac + m). 232 232 + shifted = fractional_delay(x, -frac) 233 233 + return np.asarray(shifted[i0 : i0 + m_int], dtype=np.float64) 234 234 + 235 235 + 236 236 + def _comb_grid( 237 237 + period: float, n_averages: int, n_harmonics: int 238 238 + ) -> tuple["NDArray[np.float64]", "NDArray[np.float64]"]: 239 239 + """Frequency axis (Hz) and comb-filter magnitude over the first teeth.""" 240 240 + points = max(256, 200 * n_harmonics) 241 241 + freqs = np.linspace(0.0, n_harmonics / period, points) 242 242 + response = comb_filter_response(freqs, period, n_averages) 243 243 + return freqs, response 244 244 + 245 245 + 246 246 + def time_synchronous_average( 247 247 + x: "NDArray[np.float64] | list[float]", 248 248 + fs: float, 249 249 + period: float, 250 250 + *, 251 251 + n_averages: int | None = None, 252 252 + n_harmonics: int = 8, 253 253 + ) -> SynchronousAverageResult: 254 254 + """Extract a periodic waveform of known period by time domain averaging. 255 255 + 256 256 + Ensemble-averages ``N`` successive periods of the record (McFadden 257 257 + Eq. 5) to reinforce the component synchronous with ``period`` and 258 258 + suppress asynchronous noise, whose residual standard deviation falls as 259 259 + ``1/√N``. When ``fs·period`` is an integer the periods are sliced 260 260 + directly and a noiseless periodic signal is recovered exactly; otherwise 261 261 + each period is aligned to a common integer grid by the band-limited 262 262 + fractional delay of :func:`~phonometry.metrology.signals.fractional_delay` 263 263 + and recovered within that interpolation error. 264 264 + 265 265 + :param x: Signal, 1-D, containing the periodic component plus noise. 266 266 + :param fs: Sample rate, in Hz. 267 267 + :param period: Known repetition period ``T``, in seconds (e.g. one 268 268 + revolution of a rotating machine). 269 269 + :param n_averages: Number of whole periods to average (default: as many 270 270 + as the record holds). Choosing ``N`` so that ``N·q`` is an integer 271 271 + places a comb node on an interfering tone at order ``q`` and 272 272 + maximises its rejection (McFadden's revised-model result). 273 273 + :param n_harmonics: Number of harmonics of ``1/T`` spanned by the 274 274 + returned comb-filter response (default 8). 275 275 + :return: A :class:`SynchronousAverageResult`. 276 276 + :raises ValueError: If the inputs or parameters are invalid. 277 277 + """ 278 278 + xa = _validate_1d_finite(x, "x") 279 279 + fs_v = _positive(fs, "fs") 280 280 + period_v = _positive(period, "period") 281 281 + n_harmonics_v = int(n_harmonics) 282 282 + if n_harmonics_v < 1: 283 283 + raise ValueError("'n_harmonics' must be a positive integer.") 284 284 + 285 285 + samples, m_int = _samples_per_period(fs_v, period_v) 286 286 + n_avg = _resolve_n_averages(n_averages, xa.size, samples, m_int) 287 287 + 288 288 + interpolated = abs(samples - round(samples)) >= _ALIGN_TOL 289 289 + blocks = np.empty((n_avg, m_int), dtype=np.float64) 290 290 + for n in range(n_avg): 291 291 + blocks[n] = _extract_period(xa, n * samples, m_int) 292 292 + 293 293 + period_waveform = np.asarray(blocks.mean(axis=0), dtype=np.float64) 294 294 + residual = np.asarray((blocks - period_waveform).ravel(), dtype=np.float64) 295 295 + residual_rms = float(np.sqrt(np.mean(residual * residual))) 296 296 + noise_reduction_db = 10.0 * float(np.log10(n_avg)) 297 297 + 298 298 + comb_freqs, comb_response = _comb_grid(period_v, n_avg, n_harmonics_v) 299 299 + times = np.arange(m_int, dtype=np.float64) * (period_v / m_int) 300 300 + 301 301 + return SynchronousAverageResult( 302 302 + period_waveform=period_waveform, 303 303 + times=times, 304 304 + residual=residual, 305 305 + n_averages=n_avg, 306 306 + samples_per_period=m_int, 307 307 + period=period_v, 308 308 + fs=fs_v, 309 309 + interpolated=bool(interpolated), 310 310 + noise_reduction_db=noise_reduction_db, 311 311 + residual_rms=residual_rms, 312 312 + comb_frequencies=comb_freqs, 313 313 + comb_response=comb_response, 314 314 + )

+48 -8

tests/metrology/test_miso.py

Reviewed

··· 184 184 assert float(np.median(np.abs(diff))) < 0.02 185 185 186 186 187 187 + def test_four_uncorrelated_inputs_general_q_identity() -> None: 188 188 + """Four mutually uncorrelated inputs (general q, not the old 2-3 cap). 189 189 + 190 190 + Oracle: Bendat & Piersol Eqs. 7.116/7.117. When the ``q`` inputs are 191 191 + mutually uncorrelated the conditioning removes nothing, so the partial 192 192 + coherence of each input equals its ordinary coherence (Eq. 7.117) and the 193 193 + multiple coherence equals the sum of the ordinary coherences (Eq. 7.116), 194 194 + bounded by one. The expected relations come from the standard, not from 195 195 + the code: four independent white records drive four distinct FIR paths 196 196 + into one output with additive measurement noise, so the inputs stay 197 197 + mutually uncorrelated by construction and both identities must hold. 198 198 + """ 199 199 + x0 = _white(50) 200 200 + x1 = _white(51) 201 201 + x2 = _white(52) 202 202 + x3 = _white(53) 203 203 + y = ( 204 204 + _fir(x0, [1.0, 0.4, -0.2]) 205 205 + + _fir(x1, [0.2, -0.6, 0.4]) 206 206 + + _fir(x2, [0.5, 0.3]) 207 207 + + _fir(x3, [-0.3, 0.5, 0.1]) 208 208 + + _white(54, rms=0.6) 209 209 + ) 210 210 + res = ph.miso_coherence([x0, x1, x2, x3], y, FS, nperseg=2048) 211 211 + 212 212 + # (a) general q >= 2 no longer raises, and (d) every array carries q = 4. 213 213 + assert res.n_inputs == 4 214 214 + assert res.ordinary_coherence.shape[0] == 4 215 215 + assert res.partial_coherence.shape[0] == 4 216 216 + assert res.coherent_output_spectra.shape[0] == 4 217 217 + assert res.multiple_coherence.shape == res.frequencies.shape 218 218 + 219 219 + band = _band(res.frequencies) 220 220 + # (b) per-input partial == ordinary (Eq. 7.117). The extra input widens the 221 221 + # Section 9.3 bias slightly over the two-input case, so allow a comfortable 222 222 + # few hundredths. 223 223 + for i in range(4): 224 224 + diff = np.abs( 225 225 + res.partial_coherence[i][band] - res.ordinary_coherence[i][band] 226 226 + ) 227 227 + assert float(np.median(diff)) < 0.03 228 228 + # (c) multiple == sum of ordinaries, clipped to one (Eq. 7.116). 229 229 + expected = np.minimum(res.ordinary_coherence[:, band].sum(axis=0), 1.0) 230 230 + diff_multiple = res.multiple_coherence[band] - expected 231 231 + assert float(np.median(np.abs(diff_multiple))) < 0.03 232 232 + 233 233 + 187 234 # --------------------------------------------------------------------------- 188 235 # Correlated inputs: partial coherence removes the shared path (the point) 189 236 # --------------------------------------------------------------------------- ··· 377 424 def test_rejects_single_input() -> None: 378 425 x1 = _white(140, n=1 << 14) 379 426 y = x1.copy() 380 380 - with pytest.raises(ValueError, match="2 or 3"): 427 427 + with pytest.raises(ValueError, match="at least two"): 381 428 ph.miso_coherence([x1], y, FS) 382 382 - 383 383 - 384 384 - def test_rejects_four_inputs() -> None: 385 385 - inputs = [_white(150 + i, n=1 << 14) for i in range(4)] 386 386 - y = _white(160, n=1 << 14) 387 387 - with pytest.raises(ValueError, match="2 or 3"): 388 388 - ph.miso_coherence(inputs, y, FS) 389 429 390 430 391 431 def test_rejects_mismatched_length() -> None:

+282

tests/metrology/test_synchronous_average.py

Reviewed

··· 1 1 + # Copyright (c) 2026. Jose M. Requena-Plens 2 2 + """Tests for time synchronous averaging (McFadden 1987). 3 3 + 4 4 + Clean-room oracles derived from P. D. McFadden, "A revised model for the 5 5 + extraction of periodic waveforms by time domain averaging", *Mechanical 6 6 + Systems and Signal Processing* 1(1) 1987, 83-95, with synthesised signals 7 7 + whose periodic part is known exactly: 8 8 + 9 9 + * the comb-filter closed form ``|C(f)| = |sin(N·π·f·T)/(N·sin(π·f·T))|`` 10 10 + (Eq. 8): unit teeth at the harmonics (Eq. 9), the mid-bin closed forms, 11 11 + and zeros at the nodes; 12 12 + * the revised-model correction, that a non-harmonic interfering tone is 13 13 + best rejected by placing a comb node on it (McFadden's 32.05-order 14 14 + example: ``N = 20`` beats the power-of-two ``N = 32``); 15 15 + * exact recovery of a noiseless periodic signal with an integer number of 16 16 + samples per period; 17 17 + * recovery within a band-limited interpolation bound for a non-integer 18 18 + number of samples per period; 19 19 + * the ``1/√N`` fall of the residual asynchronous-noise standard deviation. 20 20 + """ 21 21 + 22 22 + from __future__ import annotations 23 23 + 24 24 + import matplotlib 25 25 + 26 26 + matplotlib.use("Agg") 27 27 + 28 28 + import matplotlib.pyplot as plt # noqa: E402 29 29 + import numpy as np # noqa: E402 30 30 + import pytest # noqa: E402 31 31 + 32 32 + import phonometry as ph # noqa: E402 33 33 + from phonometry.metrology.synchronous_average import comb_filter_response 34 34 + 35 35 + FS = 8192.0 36 36 + #: One revolution spanning exactly 256 samples (32 revolutions per second). 37 37 + PERIOD = 1.0 / 32.0 38 38 + M = int(round(FS * PERIOD)) # 256 samples per period 39 39 + 40 40 + 41 41 + def _periodic(period: float, m: int, orders: tuple[float, ...]) -> np.ndarray: 42 42 + """A smooth signal periodic in ``period`` sampled on ``m`` points/period.""" 43 43 + phase = np.arange(m) * (period / m) / period 44 44 + out = np.zeros(m, dtype=np.float64) 45 45 + for k, order in enumerate(orders): 46 46 + out += (1.0 / (k + 1)) * np.cos(2.0 * np.pi * order * phase + 0.3 * k) 47 47 + return out 48 48 + 49 49 + 50 50 + def _repeat(one_period: np.ndarray, n: int) -> np.ndarray: 51 51 + return np.tile(one_period, n) 52 52 + 53 53 + 54 54 + # --------------------------------------------------------------------------- 55 55 + # Comb-filter closed form (McFadden Eq. 8 / Eq. 9) 56 56 + # --------------------------------------------------------------------------- 57 57 + 58 58 + 59 59 + def test_comb_unit_teeth_at_harmonics() -> None: 60 60 + """|C| = 1 at every harmonic k/T, independent of N (Eq. 9).""" 61 61 + harmonics = np.array([k / PERIOD for k in range(1, 9)]) 62 62 + for n in (1, 2, 4, 8, 20): 63 63 + response = comb_filter_response(harmonics, PERIOD, n) 64 64 + assert np.allclose(response, 1.0, atol=1e-9) 65 65 + 66 66 + 67 67 + def test_comb_midbin_closed_forms() -> None: 68 68 + """Closed-form values between the teeth.""" 69 69 + quarter = comb_filter_response(np.array([0.25 / PERIOD]), PERIOD, 2)[0] 70 70 + assert quarter == pytest.approx(1.0 / np.sqrt(2.0), abs=1e-12) 71 71 + half_n3 = comb_filter_response(np.array([0.5 / PERIOD]), PERIOD, 3)[0] 72 72 + assert half_n3 == pytest.approx(1.0 / 3.0, abs=1e-12) 73 73 + 74 74 + 75 75 + def test_comb_node_between_teeth() -> None: 76 76 + """|C| = 0 at the mid-node j/(N·T) with j not a multiple of N.""" 77 77 + node = comb_filter_response(np.array([0.5 / PERIOD]), PERIOD, 2)[0] 78 78 + assert node < 1e-12 79 79 + 80 80 + 81 81 + def test_comb_response_is_bounded() -> None: 82 82 + """The comb magnitude never exceeds unity on a dense grid.""" 83 83 + freqs = np.linspace(0.0, 8.0 / PERIOD, 4000) 84 84 + response = comb_filter_response(freqs, PERIOD, 7) 85 85 + assert float(np.max(response)) <= 1.0 + 1e-9 86 86 + 87 87 + 88 88 + # --------------------------------------------------------------------------- 89 89 + # McFadden's revised-model correction: choosing N to place a node 90 90 + # --------------------------------------------------------------------------- 91 91 + 92 92 + 93 93 + def test_mcfadden_node_selection_closed_form() -> None: 94 94 + """A 32.05-order tone: N=20 lands on a node, N=32 (power of 2) does not.""" 95 95 + freq = np.array([32.05 / PERIOD]) 96 96 + c20 = comb_filter_response(freq, PERIOD, 20)[0] 97 97 + c32 = comb_filter_response(freq, PERIOD, 32)[0] 98 98 + assert c20 < 1e-10 # 20 * 32.05 = 641 -> exact node 99 99 + assert c32 > 0.15 # 32 * 32.05 = 1025.6 -> passed with a side lobe 100 100 + assert c32 > 1e6 * max(c20, 1e-300) # >100 dB better rejection with N=20 101 101 + 102 102 + 103 103 + def test_mcfadden_node_selection_end_to_end() -> None: 104 104 + """The averaged waveform confirms the node: N=20 removes the interferer.""" 105 105 + n_span = 40 106 106 + phase = np.arange((n_span + 1) * M) / FS / PERIOD 107 107 + true = np.cos(2.0 * np.pi * 8.0 * phase) 108 108 + interferer = 0.7 * np.cos(2.0 * np.pi * 32.05 * phase + 0.4) 109 109 + signal = true + interferer 110 110 + true_one = np.cos(2.0 * np.pi * 8.0 * np.arange(M) / M) 111 111 + 112 112 + leak_20 = np.max( 113 113 + np.abs( 114 114 + ph.time_synchronous_average( 115 115 + signal, FS, PERIOD, n_averages=20 116 116 + ).period_waveform 117 117 + - true_one 118 118 + ) 119 119 + ) 120 120 + leak_32 = np.max( 121 121 + np.abs( 122 122 + ph.time_synchronous_average( 123 123 + signal, FS, PERIOD, n_averages=32 124 124 + ).period_waveform 125 125 + - true_one 126 126 + ) 127 127 + ) 128 128 + assert leak_20 < 1e-9 129 129 + assert leak_32 > 0.1 130 130 + 131 131 + 132 132 + # --------------------------------------------------------------------------- 133 133 + # Exact recovery (integer samples per period) 134 134 + # --------------------------------------------------------------------------- 135 135 + 136 136 + 137 137 + def test_exact_recovery_integer_period() -> None: 138 138 + """Noiseless periodic signal, integer M, recovered to machine precision.""" 139 139 + one = _periodic(PERIOD, M, (1.0, 3.0, 5.0)) 140 140 + signal = _repeat(one, 24) 141 141 + result = ph.time_synchronous_average(signal, FS, PERIOD) 142 142 + 143 143 + assert result.interpolated is False 144 144 + assert result.samples_per_period == M 145 145 + assert result.n_averages == 24 146 146 + assert np.max(np.abs(result.period_waveform - one)) < 1e-12 147 147 + assert result.residual_rms < 1e-12 148 148 + 149 149 + 150 150 + def test_times_span_one_period() -> None: 151 151 + one = _periodic(PERIOD, M, (2.0,)) 152 152 + result = ph.time_synchronous_average(_repeat(one, 10), FS, PERIOD) 153 153 + assert result.times.size == M 154 154 + assert result.times[0] == pytest.approx(0.0, abs=1e-15) 155 155 + assert result.times[-1] < PERIOD 156 156 + 157 157 + 158 158 + # --------------------------------------------------------------------------- 159 159 + # Non-integer samples per period: fractional-delay alignment 160 160 + # --------------------------------------------------------------------------- 161 161 + 162 162 + 163 163 + def test_noninteger_period_recovered_within_bound() -> None: 164 164 + """A non-integer M is aligned by band-limited fractional delay.""" 165 165 + period = 1.0 / 31.7 # FS * period is not an integer 166 166 + m_int = int(round(FS * period)) 167 167 + phase = np.arange(30 * m_int) / FS / period 168 168 + signal = np.cos(2.0 * np.pi * phase) + 0.4 * np.cos( 169 169 + 2.0 * np.pi * 2.0 * phase + 0.3 170 170 + ) 171 171 + result = ph.time_synchronous_average(signal, FS, period) 172 172 + 173 173 + assert result.interpolated is True 174 174 + reference = np.cos( 175 175 + 2.0 * np.pi * np.arange(m_int) * (period / m_int) / period 176 176 + ) + 0.4 * np.cos( 177 177 + 2.0 * np.pi * 2.0 * np.arange(m_int) * (period / m_int) / period + 0.3 178 178 + ) 179 179 + assert np.max(np.abs(result.period_waveform - reference)) < 0.05 180 180 + 181 181 + 182 182 + # --------------------------------------------------------------------------- 183 183 + # Noise reduction: 1/sqrt(N) law 184 184 + # --------------------------------------------------------------------------- 185 185 + 186 186 + 187 187 + def test_noise_reduction_sqrt_n_law() -> None: 188 188 + """Residual asynchronous-noise std falls as 1/sqrt(N) (statistical).""" 189 189 + one = _periodic(PERIOD, M, (1.0, 4.0)) 190 190 + rng = np.random.default_rng(2024) 191 191 + n_avg = 64 192 192 + sigma = 1.0 193 193 + signal = _repeat(one, n_avg) + rng.standard_normal(n_avg * M) * sigma 194 194 + result = ph.time_synchronous_average(signal, FS, PERIOD, n_averages=n_avg) 195 195 + 196 196 + residual_of_average = result.period_waveform - one 197 197 + measured = float(np.std(residual_of_average)) 198 198 + predicted = sigma / np.sqrt(n_avg) 199 199 + # 256 samples estimate the std of a zero-mean Gaussian to a relative 200 200 + # error ~ 1/sqrt(2*256) ~ 4.4 %; 15 % is a comfortable statistical band. 201 201 + assert measured == pytest.approx(predicted, rel=0.15) 202 202 + 203 203 + 204 204 + def test_noise_reduction_db_matches_n() -> None: 205 205 + one = _periodic(PERIOD, M, (1.0,)) 206 206 + result = ph.time_synchronous_average(_repeat(one, 100), FS, PERIOD) 207 207 + assert result.noise_reduction_db == pytest.approx(20.0, abs=1e-9) 208 208 + assert result.amplitude_snr_gain == pytest.approx(10.0, abs=1e-9) 209 209 + 210 210 + 211 211 + # --------------------------------------------------------------------------- 212 212 + # Validation and plotting 213 213 + # --------------------------------------------------------------------------- 214 214 + 215 215 + 216 216 + def test_default_n_averages_uses_whole_record() -> None: 217 217 + one = _periodic(PERIOD, M, (1.0,)) 218 218 + result = ph.time_synchronous_average(_repeat(one, 12), FS, PERIOD) 219 219 + assert result.n_averages == 12 220 220 + 221 221 + 222 222 + def test_requested_n_averages_over_available_raises() -> None: 223 223 + signal = _repeat(_periodic(PERIOD, M, (1.0,)), 5) 224 224 + with pytest.raises(ValueError, match="exceeds"): 225 225 + ph.time_synchronous_average(signal, FS, PERIOD, n_averages=6) 226 226 + 227 227 + 228 228 + @pytest.mark.parametrize( 229 229 + ("kwargs", "match"), 230 230 + [ 231 231 + ({"n_averages": 0}, "positive integer"), 232 232 + ({"n_harmonics": 0}, "positive integer"), 233 233 + ], 234 234 + ) 235 235 + def test_invalid_parameters_raise(kwargs: dict, match: str) -> None: 236 236 + signal = _repeat(_periodic(PERIOD, M, (1.0,)), 4) 237 237 + with pytest.raises(ValueError, match=match): 238 238 + ph.time_synchronous_average(signal, FS, PERIOD, **kwargs) 239 239 + 240 240 + 241 241 + def test_period_too_short_raises() -> None: 242 242 + signal = _repeat(_periodic(PERIOD, M, (1.0,)), 4) 243 243 + tiny_period = 1.0 / FS # spans a single sample 244 244 + with pytest.raises(ValueError, match="at least 2 samples"): 245 245 + ph.time_synchronous_average(signal, FS, tiny_period) 246 246 + 247 247 + 248 248 + def test_record_shorter_than_one_period_raises() -> None: 249 249 + short = np.zeros(M // 2, dtype=np.float64) 250 250 + with pytest.raises(ValueError, match="shorter than one period"): 251 251 + ph.time_synchronous_average(short, FS, PERIOD) 252 252 + 253 253 + 254 254 + def test_comb_filter_response_validation() -> None: 255 255 + one = np.array([1.0]) 256 256 + with pytest.raises(ValueError, match="positive integer"): 257 257 + comb_filter_response(one, PERIOD, 0) 258 258 + with pytest.raises(ValueError, match="positive"): 259 259 + comb_filter_response(one, -1.0, 2) 260 260 + 261 261 + 262 262 + def test_plot_returns_axes() -> None: 263 263 + one = _periodic(PERIOD, M, (1.0, 3.0)) 264 264 + result = ph.time_synchronous_average(_repeat(one, 8), FS, PERIOD) 265 265 + 266 266 + axes = result.plot() 267 267 + assert axes.shape == (2,) 268 268 + plt.close("all") 269 269 + 270 270 + _, ax = plt.subplots() 271 271 + returned = result.plot(ax=ax, language="es") 272 272 + assert returned is ax 273 273 + plt.close("all") 274 274 + 275 275 + 276 276 + def test_plot_rejects_unknown_language() -> None: 277 277 + result = ph.time_synchronous_average( 278 278 + _repeat(_periodic(PERIOD, M, (1.0,)), 4), FS, PERIOD 279 279 + ) 280 280 + with pytest.raises(ValueError): 281 281 + result.plot(language="fr") 282 282 + plt.close("all")