amplitude of numpy's fft results is to be multiplied by sampling period?

Question

I try to validate my understanding of Numpy\'s FFT with an example: the Fourier transform of exp(-pi*t^2) should be exp(-pi*f^2) when no scaling is

Answer 1

Be careful, you are not computing the continuous time Fourier transform, computers work with discrete data, so does Numpy, if you take a look to numpy.fft.fft documentation it says:

numpy.fft.fft(a, n=None, axis=-1)[source]

Compute the one-dimensional discrete Fourier Transform.

This function computes the one-dimensional n-point discrete Fourier Transform (DFT) with the efficient Fast Fourier Transform (FFT) algorithm

That means that your are computing the DFT which is defined by equation:

the continuous time Fourier transform is defined by:

And if you do the maths to look for the relationship between them:

As you can see there is a constant factor 1/N which is exactly your scale value dt (x[n] - x[n-1] where n is in [0,T] interval is equivalent to 1/N).

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Just a comment on your code, it is not a good practice to import everything from numpy import * instead use:

import numpy as np
import matplotlib.pyplot as plt

# create data
N = 4097
T = 100.0
t = np.linspace(-T/2,T/2,N)
f = np.exp(-np.pi*t**2)

# perform FT and multiply by dt
dt = t[1]-t[0]
ft = np.fft.fft(f) * dt      
freq = np.fft.fftfreq(N, dt)
freq = freq[:N/2+1]

# plot results
plt.plot(freq, np.abs(ft[:N/2+1]),'o')
plt.plot(freq, np.exp(-np.pi * freq**2),'r')
plt.legend(('numpy fft * dt', 'exact solution'), loc='upper right')
plt.xlabel('f')
plt.ylabel('amplitude')
plt.xlim([0, 1.4])
plt.show()