Generally, we know that if we have this relation between Fourier transforms of three functions in frequency domain as:
$$\mathfrak{F}\{\mathsf{P}(t)\} = \mathfrak{F}\{\mathsf{Z}(t)\}\mathfrak{F}\{\mathsf{Q}(t)\}$$
We should have this relationship in the time domain as:
$$\mathsf{P}(t) = \int \mathsf{Z}(\tau) \mathsf{Q}(t-\tau) d\tau$$
Now I have this in Python:
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()
Is there any reason that $\int \mathsf{Z}(\tau) \mathsf{Q}(t-\tau) d\tau \neq \mathfrak{F^{-1}}\{\mathfrak{F} \{ \mathsf{P}(t) \} \}$?
These are the data files here:
