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Convolution Representation of LTI FiltersIf is the output of an LTI filter with input and impulse response , then is the

convolutionof with ,

Since convolution is commutative (), we have also

That and are transform pairs is expressed by writing or .

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

Theorem.Theconvolution theorem(Papoulis [21]) states that

In words,convolution in the time domain is multiplication in the frequency domain.Taking the transform of both sides of Eq. (B.4.3) 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[21] for transforms states that

The shift theorem can be derived immediately from the definition of the transformgiven in Eq. ():

The general difference equation for an LTI filter appears as

(B.10) (B.11)

Taking the transform of both sides, denoting the transform by , gives

(B.12) (B.13)

using linearity and the shift theorem. Replacing by , by , and moving the terms in to the left-hand side, we obtain

(B.14) (B.15)

or

(B.16) (B.17)

Defining the polynomials

(B.18) (B.19)

the transform of the difference equation becomes

Finally, solving for which equals the transfer function , yields