I've been told that to calculate the difference equation of an LTI system, you simply take the sample values of the impulse response as the coefficients of the x[n-k] terms in the difference equation. i.e. if h[n] = {1, 0.5, 0.25, 0.125} for n>=0, then y[n] = x[n] +0.5x[n-1]+0.25x[n-2]+0.125x[n-3]. This makes sense to me since x[n] is an impulse ([itex]\delta[/itex][n]) for h[n].(adsbygoogle = window.adsbygoogle || []).push({});

However, is it true that this difference equation is only valid for x[n] = [itex]\delta[/itex][n] and no other types of input? It would certainly seem so, as if I put in x[n] = {1,1,1,1,0,...0}, take the z-transforms of x[n] and h[n], multiply them, and take then take inverse z-transform, I get:

y[n] = {1, 3/2, 7/4, 15/8, 15/8, 7/8, 3/8, 1/8, 0, 0 ... , 0} which does not correspond to the difference equation above.

Also, surely if this were true for all inputs, you could not have a recursive system since the difference equation would only contain x[n] terms and no y[n-k] terms.

Have I answered my own question?

Any answers would be greatly appreciated!

Many thanks

Paul

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# LTI Difference Equation from Impulse response

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