# Need help dealing with Gaussians and their Fourier transforms in Matlab

• Mar 6th 2012, 11:35 PM
aussiedave
Need help dealing with Gaussians and their Fourier transforms in Matlab
Hey all. I'm trying to show using Matlab that the product of variances of a Gaussian and its Fourier transform (which is also a Gaussian) is always 1. This is quite easy to prove analytically (see here: Fourier Transforms and Uncertainty) but I am trying to do it using Matlab.

The problem I'm having is with working on discrete grids in Matlab. It's doing my head in. Here is my code so far:

Code:

```% USER INPUTS min = -100; max = 100; N = 100000; x0 = 0; y0 = 0; sigma_x = 30; rho = 0; A1 = 1/(sigma_x*(2*pi)^0.5); x = linspace(min,max,N); I1 = A1*exp(-0.5*(x-x0).^2/sigma_x^2); % Calculate **** for I1 int = sum(I1)*(max-min)/N mean_x = dot(I1,x) mean_x2 = dot(I1,x.^2) var1 = (mean_x2 - mean_x^2)*(max-min)/N sd = ((mean_x2 - mean_x^2)*(max-min)/N)^0.5 % Calculate the 2D Fourier Transform F = abs(fftshift(fft(I1))); % Calculate **** for F int = sum(F)*(max-min)/N F = F; mean_x = dot(F,x) mean_x2 = dot(F,x.^2) var2 = (mean_x2 - mean_x^2)*(max-min)/N sd = ((mean_x2 - mean_x^2)*(max-min)/N)^0.5```
I should get var1*var2 = 1 but it's not even close. I also find that the Gaussian in the Fourier transform isn't normalised, whilst the original one was. Is that to be expected?

Some help here would be much appreciated!