I am trying to get the Fourier transform of a simple Gaussian function centered at zero. When I FT the positive (or negative) half of the function, I get the expected Gaussian transform, but when I try to FT the symmetric (even) Gaussian, I get strange rapid oscillations in the real and imaginary parts that form a Gaussian envelope. Why am I seeing this? I see this with any symmetric function, by the way, not just a Gaussian. Ideally, the imaginary part should be zero since the function is even, but I see rapid oscillations. What's going on?
import numpy as np
import matplotlib.pyplot as plt
import scipy.fft as sfft
def gaussian(t, sigma):
return np.exp(-0.5 * (t**2 / sigma**2))
N = 10000
t = np.array([i for i in range(N)])
fx = [gaussian(t[i], 1000) for i in range(N)]
# t_rev = -t[::-1][:-1]
# t = np.array(list(t_rev) + list(t))
fx_rev = fx[::-1][:-1]
fx = list(fx_rev) + list(fx)
ft = np.fft.fftshift(sfft.fft(fx))
plt.plot(fx)
plt.show()
plt.plot(ft.real)
plt.plot(ft.imag)
plt.show()
