I'd like to understand the difference between an analytic and numeric FT of a function, using the numpy.fft module (i.e. why they are not the same). Starting with the equation
,
the Fourier Transform of this can be shown to give an analytical FT (see e.g. Arfken, Weber and Harris p966, or sympy: fourier_transform(exp(-abs(x)), x, k) which is a factor of 2 * pi different):
.
Using python/numpy to calculate the FFT of sig = np.exp(-np.abs(x)) gives a numerical FT that can plotted against the analytic solution (ft_numeric ~= ft_analytic x wave in plot).
The analytical FT can be seen to be a bounding window function for the numerical FT, and can be turned in to the numerical FT by multiplying by a suitable cos function (ft_analytic x wave overlaps ft_numeric, see example code for form of function).
My question is why does this numpy FFT produce a modified (modulated by a cos wave) numerical FT in this case? Is this to do with how the FFT is defined, and how can I tell this from its description: numpy FFT implementation.
import numpy as np
import numpy.fft as fft
import pylab as plt
x = np.linspace(-10, 10, 2001)
dx = x[1] - x[0]
normalization = 1 / dx
k = 2 * np.pi * fft.fftshift(fft.fftfreq(x.shape[0], d=dx))
# Signal.
sig = np.exp(-np.abs(x))
# Both shifted.
ft_numeric = fft.fftshift(fft.fft(sig))
ft_analytic = 2 / (1 + k**2)
wave = np.cos(2 * np.pi * k / (k[2] - k[0]))
plt.figure(1)
plt.clf()
plt.title('signal')
plt.plot(x, sig)
plt.xlabel('x')
plt.figure(2)
plt.clf()
plt.title('FT')
plt.plot(k, ft_analytic.real, label='ft_analytic')
plt.plot(k, normalization * ft_numeric, label='ft_numeric')
plt.plot(k, normalization * ft_numeric * wave, label='ft_analytic x wave')
plt.xlim((-15, 15))
plt.xlabel('k')
plt.legend()
plt.show()
There seems to be a fundamental misunderstanding. There is no cos modulation. You are just plotting the real part of the signal of your numeric FFT and the magnitude of your analytic.
The real part is obviously mirrored around 0 as you are dealing with a real signal. Thus the cosine.