DFT for somthing similar to convolution

[menahemkrief]

Hi,

I have the following problem:
Let x,y be finite real valued sequences defined on 0...N-1 and let g be a non negative integer .


define

also on 0..N-1.


In addition, the DFT of y is known in closed form.
Is there a way to write z as some cyclic convolution, so that with the help of the convolution theorem z can be calculated in NLOG N istead of N^2?


I tried following the convolution therem proff but i get stuck:

The problem is that the second sum depends on k so the double sum doesn't factor to the product of DFTs.


what am I missing?


thank you

 

[MHanif]

Hi there,
I am not sure about the notations used in the above mentioned proof. If you want to use DFT to find linear convolution of x and y, then take the DFT for x and y of 2N-1 samples (by zero padding). This will give a 2N-1 size DFT of z which can be used to find the sequence z.
Hope this helps.
MHanif