Using sinc as a filter

I'm trying to use a cardinal sine as a lowpass filter for a cosine signal with a fundamental of 1k.

In frequency domain, what really happens is that I'm multiplying two impulses (centered at -1k and 1k) with a rectangular pulse of width equals 2 (convolution is multiplication in frecuency domain). The result would be a constant zero function. However using the above code in Matlab I'm getting again the sinc function as output. Any help would be appreciated (attached the full plot).

 D = 5;
 f0 = 1000;
 fm = 2*f0;    % nyquist reestriction

 t = (-D:1/fm:D);
 x = cos(1000*2*pi*t);
 h = sinc(t);

 x_p = filter(h,1,x);
 plot(x_p);

This questions was also asked on http://dsp.stackexchange.com/questions/15961/using-sinc-as-a-filter but is still unanswered. Using conv(h,x) I'm getting two cardinal sines, also tested with filter(x,1,h)

Answer 1

[Honglei Chen]

Hi Juan,

Your filter is as long as your signal, that means whatever you see is basically the transient while Fourier analysis gives the stabilized result. If you shorten your filter to a quarter of length, you will see that after the transient, the result is close to 0. (Note I changed your total points to even to make the bookkeeping easy but you can surely reproduce this with odd number of points)

 D = 5;
 f0 = 1000;
 fm = 2*f0;    % nyquist reestriction

 t = (-D:1/fm:D-1/fm);
 x = cos(1000*2*pi*t);
 h = sinc(t(3*numel(t)/8:5*numel(t)/8-1));
 x_p = filter(h,1,x);
 plot(x_p);

Two more minor comments:

1. Your signal is critically sampled so you have only two points in one period of cosine. This does not impact your result but your signal is actually a zigzag line rather than a sinusoid.

2. I'd like to also point out that sinc function is actually the continuous Fourier transform of a boxcar and in the discrete system, it should be a Dirichlet function. Again, this doesn't really affect your result much but I figure it is better if you align everything correctly. Below is an example if you want to use this approach

 ns = numel(t);
 ns = ns/4;
 f = -fm/2:fm/ns:fm/2-fm/ns;
 h1 = ifftshift(diric(f/fm,2*ns));

 x_p1 = filter(h1,1,x);
 plot(x_p1);

HTH

[Juan]

Thanks Honglei Chen, makes a lot of sense. I knew that boxcar is the dft of sinc but I was trying attack the problem from the time domain.

Just another question, I noticed that when appling the filter I get a line close to zero but I also have a little sinc function present in one corner. Is there any way to improve the filter's performance?

Thanks a lot!