I think I do not understand Fourier transform results properly. I am trying to improve my understanding. It would be very kind if someone can help by commenting/answering.
This is the following algorithm:
1> select [a,b] in X
2> compute grid points in [a,b] in X
3> compute frequency interval [f1, f2]
4> compute the grid in frequency interval [f1, f2]
5> compute pdf normal distribution for the points in step 2 i.e. compute $\frac{1}{\sqrt{2\pi}}e^{-x^2/2}$
6> now compute fft of the result in step 5 and plot the result.
This is matlab code for the process above:
N=100;
j=0:(N-1);
a=-5;
b=5;
dx = (b-a)/N;
x = a+j*dx;
dt = 2*pi/(N*dx);
f1 = -N/2*dt;
f2 = N/2*dt;
t= f1+ j*dt;
GX = normpdf(x,0,1);
fft_GX = real(fft(GX))';
When I plot fft_GX I do not get what I was expecting.
I was expecting the fft result of have a function form of $e^{-t^2/2}$.