Why signal add noise cause signal undiscoverable after `fft` and `ifft`？

Question

As we all know, Fourier transform of the sinc(t) is rect(f).
Simple python script to discover signal after fft and ifft with random normal noise.

import matplotlib.pyplot as plt
import numpy as np
np.random.seed(1234)

peak=np.fft.fft(np.sinc(np.linspace(-4, 4, 200)))
peak_ifft=np.fft.ifft(peak)
n=200
noise=np.random.normal(loc=0, scale=np.sqrt(2)/2, size=(n, 2)).view(np.complex128)
signal=noise+np.reshape(np.asarray(peak),(-1,1))
plt.figure()
plt.subplot(2,2,1)
plt.plot(np.abs(signal))
plt.title('power vs freq')
plt.subplot(2,2,2)
signal_ifft=np.fft.ifft(signal)
plt.plot(signal_ifft)
plt.title('should appear sinc,but not')
plt.subplot(2,2,3)
plt.plot(peak_ifft)
plt.title('should be sinc,and yes')
plt.show()

Output as below:

As x(t)+y(t)<->X(f)+Y(f),np.fft.ifft(np.fft(signal)) go back to origin signal,why cannot np.fft.ifft(np.fft(signal+noise))?

Answer 1

You want to just do signal + noise, and you'll be fine for the noise part.

For the plot, you need the absolute of the fft, you're just plotting the real part.

import matplotlib.pyplot as plt
import numpy as np
np.random.seed(1234)
# make data
x = np.linspace(-4, 4, 200)
y = np.sinc(x)
# make fft and ifft
peak = np.fft.fft(y)
peak_ifft = np.fft.ifft(peak)
# make the frequency domain
freq = np.fft.fftfreq(len(x), np.diff(x).mean())
# make some noise!!!
n = 200
noise = np.random.normal(loc=0, scale=0.1, size=n)
# signal with noise, and the fft and ifft
signal = y + noise
peak2 = np.fft.fft(signal)
peak_ifft2 = np.fft.ifft(peak2)
# plotting
plt.figure()
plt.plot(x, y)
plt.plot(x, peak_ifft, ':')
plt.figure()
plt.plot(freq, np.abs(peak))
plt.xlim([-5,5])
plt.figure()
plt.plot(x, signal)
plt.plot(x, peak_ifft2, ':')
plt.figure()
plt.plot(freq, np.abs(peak2))
plt.xlim([-5.,5])