根据公式解析离散傅立叶

import numpy as np import matplotlib.pyplot as plt def signal_xHz(A, fi, time_s, sample): print(fi * time_s * 2 * np.pi) print(sample* time_s) return A * np.sin(np.linspace(0, fi * time_s * 2 * np.pi , num=sample* time_s)) sig = signal_xHz(20000,1000,5,8000) 31415.926535897932 40000 plt.plot(sig[0:10]) [<matplotlib.lines.Line2D at 0x11bbba3d0>]

sig[0:10] array([ 0.00000000e+00, 1.41424133e+04, 2.00000000e+04, 1.41413025e+04, -1.57083560e+00, -1.41435240e+04, -1.99999999e+04, -1.41401917e+04, 3.14167118e+00, 1.41446346e+04]) type(sig) sig.astype('int16').tofile('sin.pcm') import os os.system('sox -t raw -c 1 -e signed-integer -b 16 -r 8000 sin.pcm sin.wav') os.system('play sin.wav') 0 print (sig.astype('int16')[0:100]) [ 0 14142 19999 14141 -1 -14143 -19999 -14140 3 14144 19999 14139 -4 -14145 -19999 -14137 6 14146 19999 14136 -7 -14147 -19999 -14135 9 14149 19999 14134 -10 -14150 -19999 -14133 12 14151 19999 14132 -14 -14152 -19999 -14131 15 14153 19999 14130 -17 -14154 -19999 -14129 18 14155 19999 14127 -20 -14156 -19999 -14126 21 14157 19999 14125 -23 -14159 -19999 -14124 25 14160 19999 14123 -26 -14161 -19999 -14122 28 14162 19999 14121 -29 -14163 -19999 -14120 31 14164 19999 14119 -32 -14165 -19999 -14117 34 14166 19999 14116 -36 -14167 -19999 -14115 37 14169 19999 14114]

$$a_k=\frac{2}{N}\sum _{i=0}^{N-1}{x}_{i}cos\frac{2ki\pi }{N}$$

ak=0 N=8000 k=1000 for i in np.arange(0,8000-1): ak+=sig[i]*np.cos(2*np.pi*k*i/N) print(ak/N) 784.6553321430675

$$c_k=\frac{1}{N}\sum _{i=0}^{N-1}{x}_i{e}^{-j\frac{2ki\pi }{N}}$$

ak=0 N=8000 k=1000 for i in np.arange(0,8000-1): ak+=sig[i]*np.exp(2*np.pi*k*i*(-1.j)/N) print(ak/N) (784.6553321424174-9957.788439085387j) y=sig[0:8000] YY = np.fft.fft(y) # 未归一化 Y = np.fft.fft(y)/len(y) # fft computing and normalization 归一化 Y1 = Y[range(int(len(y)/2))] print(Y[999:1001]) [ 18.92979144 -242.80867967j 783.61624233-9958.8275289j ] plt.plot(Y1) /Users/chenpeiwen/opt/anaconda3/lib/python3.7/site-packages/numpy/core/_asarray.py:85: ComplexWarning: Casting complex values to real discards the imaginary part return array(a, dtype, copy=False, order=order)

[<matplotlib.lines.Line2D at 0x11bd3d050>] import librosa import librosa.display y, sr = librosa.load('sin.wav') S = np.abs(librosa.stft(y)) plt.figure() librosa.display.specshow(S**2, sr=sr, y_axis='log') # 从波形获取功率谱图 plt.colorbar() plt.title('Power spectrogram') Text(0.5, 1.0, 'Power spectrogram')

mfccs = librosa.feature.mfcc(y=y, sr=sr, n_mfcc=40) print(mfccs.shape) # (40, 65) (40, 216)