NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University
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Convolution Representation of LTI Filters
If is the output of an LTI filter with input and impulse response , then is the convolution of with ,
Since convolution is commutative (), we have also
Definition. The unilateral transform of the discrete-time signal is defined to be
Theorem. The convolution theorem (Papoulis ) 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. 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  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
Taking the transform of both sides, denoting the transform by , gives
using linearity and the shift theorem. Replacing by , by , and moving the terms in to the left-hand side, we obtain
Defining the polynomials
the transform of the difference equation becomes
Finally, solving for which equals the transfer function , yields