倍頻程定義

在音頻分析領域，經常要分析音頻信號的頻譜，這時最常用的是倍頻程功率譜和1/3倍頻程功率譜。
所謂倍頻程功率譜，是將音頻分為一個個的頻段，然后分別計算每個頻段內的功率譜。相鄰頻段的寬度為二比一的關系。
1/3倍頻程是將倍頻程再細分為三段。 1/n倍頻程是將倍頻程再細分為n段。

倍頻程介紹參考鏈接

代碼實現

所需模塊

import numpy as np import matplotlib.pyplot as plt from scipy import signal as sg

實現代碼

首先實現1/96倍頻程中心頻率

order_96 = np.array([2.92644400e+00, 2.94806600e+00, 2.96984800e+00, 2.99179100e+00,3.01389700e+00, 3.03616500e+00, 3.05859800e+00, 3.08119700e+00,3.10396300e+00, 3.12689700e+00, 3.15000000e+00, 3.17362400e+00,3.19742400e+00, 3.22140400e+00, 3.24556300e+00, 3.26990300e+00,3.29442600e+00, 3.31913300e+00, 3.34402500e+00, 3.36910300e+00,3.39437000e+00, 3.41982600e+00, 3.44547300e+00, 3.47131300e+00,3.49734600e+00, 3.52357500e+00, 3.55000000e+00, 3.57657900e+00,3.60335700e+00, 3.63033600e+00, 3.65751600e+00, 3.68490000e+00,3.71248900e+00, 3.74028500e+00, 3.76828900e+00, 3.79650200e+00,3.82492700e+00, 3.85356400e+00, 3.88241600e+00, 3.91148400e+00,3.94076900e+00, 3.97027400e+00, 4.00000000e+00, 4.02955400e+00,4.05932700e+00, 4.08932000e+00, 4.11953400e+00, 4.14997200e+00,4.18063400e+00, 4.21152400e+00, 4.24264000e+00, 4.27398800e+00,4.30556700e+00, 4.33737900e+00, 4.36942600e+00, 4.40171000e+00,4.43423200e+00, 4.46699500e+00, 4.50000000e+00, 4.52973000e+00,4.55965700e+00, 4.58978200e+00, 4.62010500e+00, 4.65063000e+00,4.68135500e+00, 4.71228400e+00, 4.74341600e+00, 4.77475500e+00,4.80630100e+00, 4.83805500e+00, 4.87001900e+00, 4.90219400e+00,4.93458100e+00, 4.96718300e+00, 5.00000000e+00, 5.03554100e+00,5.07133400e+00, 5.10738200e+00, 5.14368700e+00, 5.18024900e+00,5.21707100e+00, 5.25415500e+00, 5.29150200e+00, 5.32911500e+00,5.36699600e+00, 5.40514600e+00, 5.44356600e+00, 5.48226000e+00,5.52122900e+00, 5.56047500e+00, 5.60000000e+00, 5.64137600e+00,5.68305800e+00, 5.72504800e+00, 5.76734800e+00, 5.80996100e+00,5.85288800e+00, 5.89613300e+00, 5.93969700e+00, 5.98358300e+00,6.02779300e+00, 6.07233000e+00, 6.11719700e+00, 6.16239400e+00,6.20792500e+00, 6.25379300e+00, 6.30000000e+00, 6.34724800e+00,6.39484900e+00, 6.44280700e+00, 6.49112600e+00, 6.53980600e+00,6.58885100e+00, 6.63826500e+00, 6.68804900e+00, 6.73820600e+00,6.78874000e+00, 6.83965300e+00, 6.89094700e+00, 6.94262600e+00,6.99469200e+00, 7.04715000e+00, 7.10000000e+00, 7.15315800e+00,7.20671400e+00, 7.26067100e+00, 7.31503200e+00, 7.36980100e+00,7.42497900e+00, 7.48057000e+00, 7.53657700e+00, 7.59300400e+00,7.64985400e+00, 7.70712900e+00, 7.76483200e+00, 7.82296800e+00,7.88153900e+00, 7.94054800e+00, 8.00000000e+00, 8.05910900e+00,8.11865400e+00, 8.17863900e+00, 8.23906900e+00, 8.29994400e+00,8.36126900e+00, 8.42304700e+00, 8.48528100e+00, 8.54797600e+00,8.61113400e+00, 8.67475800e+00, 8.73885200e+00, 8.80342000e+00,8.86846400e+00, 8.93399000e+00, 9.00000000e+00, 9.05946100e+00,9.11931400e+00, 9.17956400e+00, 9.24021100e+00, 9.30125900e+00,9.36271000e+00, 9.42456700e+00, 9.48683300e+00, 9.54951000e+00,9.61260100e+00, 9.67610900e+00, 9.74003800e+00, 9.80438700e+00,9.86916300e+00, 9.93436600e+00, 1.00000000e+01, 1.00710820e+01,1.01426690e+01, 1.02147650e+01, 1.02873740e+01, 1.03604980e+01,1.04341420e+01, 1.05083100e+01, 1.05830050e+01, 1.06582310e+01,1.07339920e+01, 1.08102910e+01, 1.08871330e+01, 1.09645200e+01,1.10424590e+01, 1.11209500e+01, 1.12000000e+01, 1.12771350e+01,1.13548000e+01, 1.14330020e+01, 1.15117410e+01, 1.15910220e+01,1.16708510e+01, 1.17512280e+01, 1.18321590e+01, 1.19136480e+01,1.19956980e+01, 1.20783130e+01, 1.21614960e+01, 1.22452530e+01,1.23295870e+01, 1.24145010e+01, 1.25000000e+01, 1.25944540e+01,1.26896210e+01, 1.27855070e+01, 1.28821180e+01, 1.29794600e+01,1.30775370e+01, 1.31763530e+01, 1.32759180e+01, 1.33762350e+01,1.34773090e+01, 1.35791480e+01, 1.36817560e+01, 1.37851390e+01,1.38893040e+01, 1.39942560e+01, 1.41000000e+01, 1.42118430e+01,1.43245740e+01, 1.44381990e+01, 1.45527260e+01, 1.46681600e+01,1.47845110e+01, 1.49017840e+01, 1.50199870e+01, 1.51391280e+01,1.52592130e+01, 1.53802530e+01, 1.55022510e+01, 1.56252170e+01,1.57491590e+01, 1.58740840e+01, 1.60000000e+01, 1.61126060e+01,1.62260060e+01, 1.63402020e+01, 1.64552020e+01, 1.65710120e+01,1.66876370e+01, 1.68050840e+01, 1.69233570e+01, 1.70424610e+01,1.71624050e+01, 1.72831920e+01, 1.74048290e+01, 1.75273230e+01,1.76506790e+01, 1.77749020e+01, 1.79000000e+01, 1.80245360e+01,1.81499390e+01, 1.82762130e+01, 1.84033660e+01, 1.85314040e+01,1.86603340e+01, 1.87901590e+01, 1.89208870e+01, 1.90525260e+01,1.91850810e+01, 1.93185580e+01, 1.94529630e+01, 1.95883030e+01,1.97245850e+01, 1.98618140e+01, 2.00000000e+01, 2.01421640e+01,2.02853370e+01, 2.04295290e+01, 2.05747470e+01, 2.07209970e+01,2.08682840e+01, 2.10166210e+01, 2.11660100e+01, 2.13164620e+01,2.14679830e+01, 2.16205830e+01, 2.17742650e+01, 2.19290410e+01,2.20849170e+01, 2.22418990e+01, 2.24000000e+01, 2.25542700e+01,2.27096000e+01, 2.28660030e+01, 2.30234810e+01, 2.31820450e+01,2.33417020e+01, 2.35024570e+01, 2.36643180e+01, 2.38272950e+01,2.39913960e+01, 2.41566260e+01, 2.43229920e+01, 2.44905050e+01,2.46591740e+01, 2.48290020e+01, 2.50000000e+01, 2.51777040e+01,2.53566720e+01, 2.55369130e+01, 2.57184330e+01, 2.59012450e+01,2.60853560e+01, 2.62707750e+01, 2.64575140e+01, 2.66455780e+01,2.68349800e+01, 2.70257280e+01, 2.72178330e+01, 2.74113010e+01,2.76061460e+01, 2.78023760e+01, 2.80000000e+01, 2.82068810e+01,2.84152910e+01, 2.86252400e+01, 2.88367400e+01, 2.90498030e+01,2.92644410e+01, 2.94806630e+01, 2.96984840e+01, 2.99179150e+01,3.01389680e+01, 3.03616520e+01, 3.05859810e+01, 3.08119700e+01,3.10396270e+01, 3.12689670e+01, 3.15000000e+01, 3.17362370e+01,3.19742430e+01, 3.22140350e+01, 3.24556270e+01, 3.26990280e+01,3.29442600e+01, 3.31913260e+01, 3.34402470e+01, 3.36910320e+01,3.39436990e+01, 3.41982610e+01, 3.44547350e+01, 3.47131310e+01,3.49734610e+01, 3.52357480e+01, 3.55000000e+01, 3.57657890e+01,3.60335730e+01, 3.63033560e+01, 3.65751610e+01, 3.68490030e+01,3.71248930e+01, 3.74028510e+01, 3.76828880e+01, 3.79650230e+01,3.82492680e+01, 3.85356410e+01, 3.88241620e+01, 3.91148410e+01,3.94076960e+01, 3.97027440e+01, 4.00000000e+01, 4.02955440e+01,4.05932730e+01, 4.08932000e+01, 4.11953430e+01, 4.14997180e+01,4.18063430e+01, 4.21152340e+01, 4.24264070e+01, 4.27398800e+01,4.30556680e+01, 4.33737870e+01, 4.36942600e+01, 4.40170970e+01,4.43423230e+01, 4.46699520e+01, 4.50000000e+01, 4.52973060e+01,4.55965730e+01, 4.58978200e+01, 4.62010540e+01, 4.65062940e+01,4.68135490e+01, 4.71228370e+01, 4.74341660e+01, 4.77475510e+01,4.80630070e+01, 4.83805500e+01, 4.87001880e+01, 4.90219380e+01,4.93458140e+01, 4.96718290e+01, 5.00000000e+01, 5.03554080e+01,5.07133450e+01, 5.10738260e+01, 5.14368670e+01, 5.18024900e+01,5.21707120e+01, 5.25415500e+01, 5.29150280e+01, 5.32911570e+01,5.36699600e+01, 5.40514560e+01, 5.44356650e+01, 5.48226010e+01,5.52122920e+01, 5.56047520e+01, 5.60000000e+01, 5.64137610e+01,5.68305820e+01, 5.72504810e+01, 5.76734810e+01, 5.80996060e+01,5.85288810e+01, 5.89613270e+01, 5.93969690e+01, 5.98358310e+01,6.02779350e+01, 6.07233050e+01, 6.11719630e+01, 6.16239400e+01,6.20792540e+01, 6.25379330e+01, 6.30000000e+01, 6.34724730e+01,6.39484860e+01, 6.44280700e+01, 6.49112550e+01, 6.53980560e+01,6.58885190e+01, 6.63826520e+01, 6.68804930e+01, 6.73820650e+01,6.78873980e+01, 6.83965230e+01, 6.89094700e+01, 6.94262620e+01,6.99469220e+01, 7.04714970e+01, 7.10000000e+01, 7.15315780e+01,7.20671460e+01, 7.26067120e+01, 7.31503220e+01, 7.36980060e+01,7.42497860e+01, 7.48057020e+01, 7.53657760e+01, 7.59300460e+01,7.64985350e+01, 7.70712810e+01, 7.76483230e+01, 7.82296830e+01,7.88153920e+01, 7.94054870e+01, 8.00000000e+01, 8.05910870e+01,8.11865460e+01, 8.17864000e+01, 8.23906860e+01, 8.29994350e+01,8.36126860e+01, 8.42304690e+01, 8.48528140e+01, 8.54797590e+01,8.61113360e+01, 8.67475740e+01, 8.73885190e+01, 8.80341950e+01,8.86846470e+01, 8.93399050e+01, 9.00000000e+01, 9.05946120e+01,9.11931460e+01, 9.17956390e+01, 9.24021070e+01, 9.30125890e+01,9.36270980e+01, 9.42456740e+01, 9.48683320e+01, 9.54951020e+01,9.61260150e+01, 9.67611010e+01, 9.74003750e+01, 9.80438770e+01,9.86916280e+01, 9.93436580e+01, 1.00000000e+02, 1.00710815e+02,1.01426689e+02, 1.02147652e+02, 1.02873734e+02, 1.03604980e+02,1.04341423e+02, 1.05083099e+02, 1.05830055e+02, 1.06582314e+02,1.07339920e+02, 1.08102913e+02, 1.08871330e+02, 1.09645203e+02,1.10424583e+02, 1.11209503e+02, 1.12000000e+02, 1.12771347e+02,1.13548012e+02, 1.14330017e+02, 1.15117416e+02, 1.15910233e+02,1.16708511e+02, 1.17512283e+02, 1.18321594e+02, 1.19136482e+02,1.19956978e+02, 1.20783127e+02, 1.21614967e+02, 1.22452530e+02,1.23295868e+02, 1.24145012e+02, 1.25000000e+02, 1.25888527e+02,1.26783363e+02, 1.27684563e+02, 1.28592163e+02, 1.29506226e+02,1.30426773e+02, 1.31353882e+02, 1.32287567e+02, 1.33227890e+02,1.34174896e+02, 1.35128647e+02, 1.36089157e+02, 1.37056503e+02,1.38030731e+02, 1.39011871e+02, 1.40000000e+02, 1.41173294e+02,1.42356415e+02, 1.43549454e+02, 1.44752487e+02, 1.45965607e+02,1.47188889e+02, 1.48422424e+02, 1.49666290e+02, 1.50920593e+02,1.52185410e+02, 1.53460815e+02, 1.54746918e+02, 1.56043793e+02,1.57351532e+02, 1.58670242e+02, 1.60000000e+02, 1.61182175e+02,1.62373093e+02, 1.63572800e+02, 1.64781372e+02, 1.65998871e+02,1.67225372e+02, 1.68460938e+02, 1.69705627e+02, 1.70959518e+02,1.72222672e+02, 1.73495148e+02, 1.74777039e+02, 1.76068390e+02,1.77369293e+02, 1.78679810e+02, 1.80000000e+02, 1.81189224e+02,1.82386292e+02, 1.83591278e+02, 1.84804214e+02, 1.86025177e+02,1.87254196e+02, 1.88491348e+02, 1.89736664e+02, 1.90990204e+02,1.92252029e+02, 1.93522202e+02, 1.94800751e+02, 1.96087753e+02,1.97383255e+02, 1.98687317e+02, 2.00000000e+02, 2.01421631e+02,2.02853378e+02, 2.04295303e+02, 2.05747467e+02, 2.07209961e+02,2.08682846e+02, 2.10166199e+02, 2.11660110e+02, 2.13164627e+02,2.14679840e+02, 2.16205826e+02, 2.17742661e+02, 2.19290405e+02,2.20849167e+02, 2.22419006e+02, 2.24000000e+02, 2.25542694e+02,2.27096024e+02, 2.28660034e+02, 2.30234833e+02, 2.31820465e+02,2.33417023e+02, 2.35024567e+02, 2.36643188e+02, 2.38272964e+02,2.39913956e+02, 2.41566254e+02, 2.43229935e+02, 2.44905060e+02,2.46591736e+02, 2.48290024e+02, 2.50000000e+02, 2.51777054e+02,2.53566727e+02, 2.55369125e+02, 2.57184326e+02, 2.59012451e+02,2.60853546e+02, 2.62707764e+02, 2.64575134e+02, 2.66455780e+02,2.68349792e+02, 2.70257294e+02, 2.72178314e+02, 2.74113007e+02,2.76061462e+02, 2.78023743e+02, 2.80000000e+02, 2.82068817e+02,2.84152893e+02, 2.86252380e+02, 2.88367401e+02, 2.90498047e+02,2.92644409e+02, 2.94806641e+02, 2.96984863e+02, 2.99179138e+02,3.01389679e+02, 3.03616516e+02, 3.05859802e+02, 3.08119690e+02,3.10396271e+02, 3.12689667e+02, 3.15000000e+02, 3.17362366e+02,3.19742432e+02, 3.22140350e+02, 3.24556274e+02, 3.26990295e+02,3.29442566e+02, 3.31913239e+02, 3.34402466e+02, 3.36910309e+02,3.39437012e+02, 3.41982635e+02, 3.44547333e+02, 3.47131287e+02,3.49734619e+02, 3.52357483e+02, 3.55000000e+02, 3.57657898e+02,3.60335724e+02, 3.63033569e+02, 3.65751617e+02, 3.68490021e+02,3.71248932e+02, 3.74028503e+02, 3.76828888e+02, 3.79650208e+02,3.82492676e+02, 3.85356415e+02, 3.88241608e+02, 3.91148407e+02,3.94076965e+02, 3.97027435e+02, 4.00000000e+02, 4.02955444e+02,4.05932709e+02, 4.08932007e+02, 4.11953430e+02, 4.14997192e+02,4.18063446e+02, 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定義倍頻程計算函數

def n_octave(x, sample_rate, order=3, window='hanning', nperseg=8192):"""1/n倍頻程帶功率幅值計算函數:param x: 輸入數據，為一維數組，數據長度建議大于8192:param sample_rate: 數據采樣率:param order: 倍頻程階次，默認為3，支持1,3,12,24,48,96:param window: fft濾波窗函數:param nperseg: fft計算譜線數，一般為2的倍數:return: x為橫坐標頻率（Hz），y為縱坐標幅值"""# order_96 = np.load(r'96thOctave.npy')if order == 96:freqs = order_96elif order == 48:freqs = order_96[::2]elif order == 24:freqs = order_96[194::4]elif order == 12:freqs = order_96[194::8]elif order == 3:freqs = order_96[202::32]elif order == 1:freqs = order_96[234::96]else:raise ValueError(r'order mast be in (1,3,12,24,48,96)')f_start = 2 ** (-1 / 2 / order) * freqs # 平滑起始頻率f_stop = 2 ** (1 / 2 / order) * freqs # 平滑截止頻率x_out = freqsm = len(x_out)fx, mag = sg.welch(x, sample_rate, window, nperseg) # welch方法計算功率密度譜mag = mag * sample_rate / nperseg # 恢復功率密度譜幅值y_out = np.zeros(m)for i in range(m): # 按照倍頻程階次，計算響應曲線y均值index = np.where((fx >= f_start[i]) & (fx < f_stop[i])) # 取交集if len(index[0]) == 0 and i > 0: # fft計算結果點數不足時取附近值，會影響系統總能量導致結果不準確y_out[i] = y_out[i - 1]elif len(index[0]) == 0 and i == 0:y_out[0] = mag[0]else:data = np.take(mag, index)[0]y_out[i] = np.sum(data) # 倍頻程范圍內功率譜密度求和（連續函數求積分）即為倍頻程內能量值return x_out, y_out

代碼測試

if __name__ == '__main__':fs = 48000N = 48000nfft = 8192n = np.linspace(0, N - 1, N)t = n / fsy = np.sin(2 * np.pi * 800 * t) * np.sqrt(2)fx, mag = n_octave(y, fs, order=12, window='hanning', nperseg=nfft)print(max(mag))plt.semilogx(fx, mag, 'r')plt.show()

測試結果：

總結

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