Using sinc as a filter
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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)
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Honglei Chen
2014-5-7
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
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