Convolution of fourier transform with gaussian function - fft

Question

I am new to Fourier transform so I hope someone could help!
I want to calculate the wave vector of the Fourier transform, which is why I am doing this test programme.
Basically I have vector u. I apply a Fourier transform to make a vector uf. Then I apply a convolution with a Gaussian function. The code in Scilab looks like this
u = [0.1 -0.1 -0.1 -0.1 0.1 0.1 -0.1 0.1 0.1 -0.1];
sizeu = size(u,'*');
// wave vector
for i=1:sizeu
q(i) = (i-1) * 2 * %pi / sizeu;
end
// forward transform
uf = fft(u);
// convolution with a Gaussian function
for i=2:sizeu
vf(i) = uf(i)*exp(-q(i)*q(i));
end
// backward transform
v = ifft(vf);
So I believe the result, i.e. the vector v should be a real vector. But it is actually a complex vector. I think the error is in the wave vector q but I don't know the correct answer.

Answer 1

Ok, I found it:
When scilab performs fft, the data in freq-space are swapped. Sometimes, there's no problem, because if you use ifft after fft, the ifft function will undo the swap. But, if isn't your objective to use ifft after the fft, or you have to make the product of an fft-transformed function with some non-swapped vector, you'll have troubles, because the frequency components of the functions aren't matched. The solution is to swap the gaussian function to match its values to the test function, using fftshift. Do not undo the swap of the test function to match to gaussian, because when you perform ifft, this function will undo the swap. So, here's the correct code:
u = [0.1 -0.1 -0.1 -0.1 0.1 0.1 -0.1 0.1 0.1 -0.1];
sizeu = size(u,'*');
// wave vector
q = (0:sizeu-1) * 2 * %pi / sizeu;
// forward transform
uf = fft(u);
// convolution with a Gaussian function
vf(2:$) = uf(2:$).*fftshift(exp(-q(2:$).*q(2:$)));
// backward transform
v = ifft(vf);
I cleaned some unnecessary fors, and didn't understand why you do the convolution from i=2 up to $ (last number of vector), and not from i=1. If going from i=1 up to $, remove (2:$) terms.