What was fftshift created for?

fft_x_axis_arrays_demonstration.py

#!/usr/bin/env python

import numpy as np

import matplotlib.pyplot as plt

N = 10_000
t = np.linspace(-10, 10, N)
signal = np.exp(-t**2)
signal_f = np.abs(np.fft.fft(signal))

fig, axs = plt.subplots(3,2)
axs[0,0].plot(signal_f)
axs[0,0].set_xlabel("index")
axs[0,0].set_title("No x given to plot")

dt = t[1]-t[0]
f = np.fft.fftfreq(N, dt)

axs[1,0].plot(f, signal_f)
axs[1,0].set_xlabel("f [Hz]")
axs[1,0].set_title("numpy's fftfreq (correct x axis array)")

f_shifted = np.fft.fftshift(f)

axs[0,1].plot(f_shifted, signal_f)
axs[0,1].set_title("The fftfreq array shifted with fftshift (wrong x axis array!)")

f_natural = np.linspace(-1/dt, 1/dt, N)
f_my = np.fft.fftshift(f_natural)

axs[1,1].plot(f_my, signal_f)
axs[1,1].set_title("fftshifting a manually created linspace (correct x axis array)")

signal_f_shifted = np.fft.fftshift(signal_f)
axs[2,1].plot(f_natural, signal_f_shifted)
axs[2,1].set_title("fftshifting the fft of the signal, with a linearly spaced frequencies as x axis array input (correct x axis array)")

plt.show()

TL;DR If you use Numpy and not Matlab, you should never need to use fftshift, as fftfreq is much more declarative and understandable. If you use Matlab, the only way to get the correct frequency array, is to use fftshift on the linspace created given dt and N - where f_my is created.