一般来说，我们知道，如果我们在频域中三个函数的傅里叶变换之间存在这种关系：

$$\mathfrak{F}\{\mathsf{P}(t)\} = \mathfrak{F}\{\mathsf{Z}(t)\}\mathfrak{F}\{\mathsf{Q}(t)\}$$

我们应该在时域中有这种关系：

$$\mathsf{P}(t) = \int \mathsf{Z}(\tau) \mathsf{Q}(t-\tau) d\tau$$

现在我在 Python 中有这个：

import numpy as np

time = np.genfromtxt('time-data.txt',delimiter=',').T

fft = np.genfromtxt('fft-data.txt',delimiter=',',dtype=np.complex_).T

Z_t = time[0]
Q_t = time[1]

Z_fft = fft[0]
Q_fft = fft[1]
P_fft = fft[2]

#Shows that indeed Z_fft is the Fourier transform of Z_t
assert (np.fft.irfft(Z_fft,n=len(Q_t)) == Z_t).all()

#Shows that indeed Q_fft is the Fourier transform of Q_t
assert (np.fft.irfft(Q_fft,n=len(Q_t)) == Q_t).all()

#Shows that indeed Z_fft = P_fft / Q_fft
assert (np.divide(P_fft,Q_fft) == Z_fft).all()

P_t = np.fft.irfft(P_fft,n=len(Q_t))

P_t_from_convolve = np.convolve(Z_t,Q_t,mode='same')

#Expect to see that P_t = P_t_convolve but it's not the case!
assert (P_t == P_t_from_convolve).all()

有什么理由$\int \mathsf{Z}(\tau) \mathsf{Q}(t-\tau) d\tau \neq \mathfrak{F^{-1}}\{\mathfrak{F} \{ \mathsf{P}(t) \} \}$?

这些是这里的数据文件：

• time-data.txt
• fft-data.txt