Discretized continuous Fourier transform with numpy

You can use the numpy FFT module for that, but have to do some extra work. First let's look at the Fourier integral and discretize it:Here k,m are integers and N the number of data points for f(t). Using this discretization we get

The sum in the last expression is exactly the Discrete Fourier Transformation (DFT) numpy uses (see section "Implementation details" of the numpy FFT module).With this knowledge we can write the following python script

import numpy as npimport matplotlib.pyplot as pl#Consider function f(t)=1/(t^2+1)#We want to compute the Fourier transform g(w)#Discretize time tt0=-100.dt=0.001t=np.arange(t0,-t0,dt)#Define functionf=1./(t**2+1.)#Compute Fourier transform by numpy's FFT functiong=np.fft.fft(f)#frequency normalization factor is 2*np.pi/dtw = np.fft.fftfreq(f.size)*2*np.pi/dt#In order to get a discretisation of the continuous Fourier transform#we need to multiply g by a phase factorg*=dt*np.exp(-complex(0,1)*w*t0)/(np.sqrt(2*np.pi))#Plot Resultpl.scatter(w,g,color="r")#For comparison we plot the analytical solutionpl.plot(w,np.exp(-np.abs(w))*np.sqrt(np.pi/2),color="g")pl.gca().set_xlim(-10,10)pl.show()pl.close()

The resulting plot shows that the script works