filter coefficients

Question

Just a quick question - are 'filter coefficients' the coefficients of the impulse response, desired transfer function, or actual transfer function?

John Michell · Accepted Answer

That depends. The filter coefficients are the coefficients of the difference equation. If your filter is an FIR filter, then the filter coefficients are the values of the impulse response.
 
If you have an IIR filter, then the filter coefficients are not the same as the impulse response. Remember in that case the impulse response is infinite.
 
For example:
 
b = fir1(10,0.2);
stem(b)
h = impz(b);
stem(h)
isequal(b',h)
but
   [b,a] = butter(10,0.2);
   h = impz(b,a);
But the ratio of Z-transforms of the numerator coefficients to denominator coefficiens is equal to the Z-transform of the impulse response.
 
For example - consider the IIR system with the following difference equation
 
y(n)-0.8*y(n-1) = x(n)
 
So the filter coefficients are:
 
 
   A = [1 -0.8]; B =1;
The impulse response is:
 
h(n) = 0.8^n*u(n) where u(n) is the unit step. But compare:
 
   h1 = impz(B,A);
   subplot(211)
   stem(h1,'color',[1 0 0]);
   subplot(212)
   n = 0:length(h1)-1;
   h = 0.8.^n;
   hold on;
   stem(h,'color',[0 0 1]);

filter coefficients

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