# 11. Convolution Representation

## GUIDE: Elementary Digital Filter Theory - Julius O. Smith III. Convolution Representation

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### Convolution Representation

If is the output of an LTI filter with input and impulseresponse , then is the convolution of with , Since convolution is commutative ( ), we have also Definition. The -transform of the discrete-time signal is defined to be That and are transform pairs is expressed by writing or .

Theorem. The convolution theorem (Papoulis [Papoulis 1977]) states that In words, convolution in the time domain is multiplication in the frequency domain.

Taking the -transform of both sides of Eq. (2.2.1) and applying the convolution theorem gives where is the -transform of the filter impulse response. Thus the -transform of the filter output is the -transform of the input times the -transform of the impulse response.

Definition. The transfer function of a linear time-invariant discrete-time filter is defined to be the -transform of the impulse response .

Theorem. The shift theorem [Papoulis 1977] for -transforms states that The general difference equation for an LTI filter appears as   (44)  (45)

Taking the -transform of both sides, denoting the transform by gives   (46) (47)

using linearity and the shift theorem. Replacing by , by , and solving for , which equals the transfer function , yields << Previous page  TOC  INDEX  Next page >>