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## Convolution Representation

If is the output of an LTI filter with input and impulseresponse , then is the

convolutionof with ,

Since convolution is commutative (), we have also

Definition.The-transformof the discrete-time signal is defined to be

That and are transform pairs is expressed by writing or .

Theorem.Theconvolution 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.Thetransfer functionof a linear time-invariant discrete-time filter is defined to be the -transform of the impulse response .

Theorem.Theshift theorem[Papoulis 1977] for -transforms states thatThe 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