傅里叶频谱、能量谱密度、自相关函数

scipy.fftpack中实现的 FFT 变换为：
$$X_k=\sum _{n=0}^{N-1}x[n]e^{-i{\omega }_{0}kn}$$
逆变换为：
$$x[n]=\frac{1}{N}\sum _{k=0}^{N-1}{X}_k{e}^{i{\omega }_{0}kn}$$
对应的帕塞瓦尔等式为：
$$\begin{array}{rl}& \sum _{n=0}^{N-1}{\left|x[n]\right|}^2\\ & \\ =& \sum _{n=0}^{N-1}x[n]\overline{x[n]}\\ & \\ =& \frac{1}{N}\sum _{n=0}^{N-1}x[n]\overline{\sum _{k=0}^{N-1}{X}_k{e}^{i{\omega }_{0}kn}}\\ & \\ =& \frac{1}{N}\sum _{n=0}^{N-1}x[n]\sum _{k=0}^{N-1}\overline{X_k}{e}^{-i{\omega }_{0}kn}\\ & \\ =& \frac{1}{N}\sum _{k=0}^{N-1}\left[\sum _{n=0}^{N-1}x[n]e^{-i{\omega }_{0}kn}\right]\overline{X_k}\\ & \\ =& \frac{1}{N}\sum _{k=0}^{N-1}{X}_k\overline{X_k}\\ & \\ =& \frac{1}{N}\sum _{k=0}^{N-1}|X_k{|}^2\end{array}$$
故可以定义能量谱密度：
$$P_k=\frac{|X_k{|}^2}{N}$$

from scipy.fftpack import fft, fftshift, ifft
from scipy.fftpack import fftfreq
import numpy as np
import matplotlib.pyplot as plt
import warnings
warnings.filterwarnings("ignore")

fs = 2048  # 采样频率
N = fs  # 采样点数

num_fft = N

"""
生成原始信号序列

在原始信号中加上噪声
np.random.randn(t.size)
"""
t = np.arange(0, 1, 1/fs)
f0 = 100
f1 = 200
x = np.cos(2*np.pi*f0*t)*1 + np.cos(2*np.pi*f1*t)*3 + np.random.randn(t.size)*10
print('total energy in time domain:', np.sum(x**2))

plt.figure(figsize=(15, 12))
ax=plt.subplot(511)
ax.set_title('original signal')
plt.tight_layout()
plt.plot(x)

"""
FFT(Fast Fourier Transformation)快速傅里叶变换
"""
Y = fft(x, num_fft)
Y = np.abs(Y)

ax=plt.subplot(512)
ax.set_title('fft transform')
plt.plot(np.arange(num_fft//2)*fs/num_fft, Y[:num_fft//2])

"""
功率谱 power spectrum
直接平方
"""
ps = Y**2/num_fft
ax=plt.subplot(513)
ax.set_title('direct method')
plt.plot(np.arange(num_fft//2)*fs/num_fft, ps[:num_fft//2])
print('total energy in frequency domain:', np.sum(ps))

"""
相关功谱率 power spectrum using correlate
间接法
"""
cor_x = np.correlate(x, x, 'same')

ax=plt.subplot(514)
ax.set_title('cor_x')
plt.plot(cor_x[N//2:])
plt.tight_layout()

cor_X = fft(cor_x, num_fft)
ps_cor = np.abs(cor_X)/num_fft

ax=plt.subplot(515)
ax.set_title('indirect method')
plt.plot(np.arange(num_fft//2)*fs/num_fft, ps_cor[:num_fft//2])
plt.tight_layout()
print('total energy by correlate:', np.sum(ps_cor))