傅立叶外推算法的python实现和缺点

在信号处理中有时会希望预测信号的走势，可以预测信号走势的算法有插值法外推，回归算法。

我们可以对已知信号进行傅立叶变换，得到信号的频谱，再进行反变换并扩展时间轴就是傅立叶外推算法，以下为python代码的实现：

#!/usr/bin/python3

import numpy as np
import pylab as pl
from scipy import signal
from numpy import fft

def fourierExtrapolation(x, n_predict):
n = len(x)
n_harm = 50                        # number of harmonics in model
t = np.arange(0, n)
p = np.polyfit(t, x, 1)            # find linear trend in x
x_notrend = x - p[0] * t        # detrended x
x_freqdom = fft.fft(x_notrend)    # detrended x in frequency domain
f = fft.fftfreq(n)                # frequencies
indexes = list(range(n))
# sort indexes by frequency, lower -> higher
indexes.sort(key = lambda i: np.absolute(f[i]))
t = np.arange(0, n + n_predict)
restored_sig = np.zeros(t.size)
for i in indexes[:1 + n_harm * 2]:
ampli = np.absolute(x_freqdom[i]) / n    # amplitude
phase = np.angle(x_freqdom[i])             # phase
restored_sig += ampli * np.cos(2 * np.pi * f[i] * t + phase)
return ((restored_sig + p[0] * t),x_freqdom/100)

def main():
t = np.linspace(0, 1, 500)
#y = list(signal.sawtooth((2 * np.pi * 5 * t ),0.5))
y = list(np.sin(2 * np.pi * 5 * t))
n_predict = 1000
extrapolation, frequency = fourierExtrapolation(y, n_predict)
pl.plot(np.arange(0, 500), y, 'r', label = 'triangle')
pl.plot(np.arange(0, 500), frequency, 'g', label = 'frequency')
pl.plot(np.arange(0, extrapolation.size), extrapolation, 'b', label = 'extrapolation')
pl.legend()
pl.show()

if __name__ == "__main__":
main()

缺点是，已知信号最好是cos的整数倍，不然很容易引入干扰。如图