根據公式解析離散傅立葉

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)   # 未歸一化
Y = np.fft.fft(y)/len(y)   # fft computing and normalization 歸一化
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')  # 從波形獲取功率譜圖
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)