Cauchy Lorentzian simulation on FFT with oscillation

Question

Recently I do simulation on Lorentzian Function with FFT

Lorentzian Function is 2a/(x**2+a**2)

import numpy as np
from scipy import fft
import matplotlib.pyplot as plt 
a =1 
N = 500
x =np.linspace(-30,30,N)
lorentz = (2*a) * (1/(a**2 + x**2))
fourier = (fft.fft(lorentz))
fig, (ax1) = plt.subplots(nrows=1, ncols=1)
ax1.loglog(abs(fourier[0:int(N/2)]),basey=np.e)
ax1.grid(True)
plt.show()

According to contour integration, it's supposed to be exp(-|k|*a)

It seems correct

It's supposed to be linear,when I plot it on log scale. But there are some oscillation on it.

when I extend my points till x = np.linspace(-100,100,N) The oscillation seems postpone.

x =np.linspace(-300,300,N)

I could not figure out the reason of oscillation.

Answer 1

Welcome to scicomp! If I remember correctly, then in order to Fourier transform a function it has to be a periodical so that you can use the sine and cosine functions as base for it. In your case the peak will have a discontinuity in the derivative at the ends at x=30 or x=-30. The Fourier base is not well suited for discontinuities. If my hunch is correct, then you should be able to extend your domain further outward, so that the discontinuity becomes less pronounced and consequently your oscillations should subside. To get a feel for what is happening look up the tent function and its Fourier transformation, there you will see similar oscillations.