# filter coefficients

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Tom on 25 May 2012
Just a quick question - are 'filter coefficients' the coefficients of the impulse response, desired transfer function, or actual transfer function?

Walter Roberson on 25 May 2012
This is really a filter theory question, not a MATLAB question.
Matt Fig on 2 Nov 2012
Just a quick question - are 'filter coefficients' the coefficients of the impulse response, desired transfer function, or actual transfer function?

Wayne King on 25 May 2012
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]);

#### 1 Comment

Tom on 25 May 2012
Thanks Wayne - I'll check all this out over the next day or two.

Wayne King on 28 May 2012
The unit step should be treated in your DSP book.
For a sequence, the unit step sequence is the sequence which is 1 for n>=0 and 0 for all n<0.
As far as your other question, I solved the difference equation. I think if you search the web, you will find plenty of material on solving difference equations. This forum does not lend itself to me walking through that solution here.

Tom on 28 May 2012
hi wayne, i'm just having problems seeing how you got from the difference equation to the impulse response (the IIR example).
also i'm not familiar with the term 'unit step'.
if you could let me know how you did this step and what 'unit step' means that would be ace.
tom