11. Convolution Representation

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

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NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Introduction to Digital Filters with Audio Applications", by Julius O. Smith III, Copyright © 2017-11-26 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

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Convolution Representation

If $y(n)$ is the output of an LTI filter with input $x(n)$ and impulseresponse $h(n)$, then $y$ is the convolution of $x$ with $h$,


Since convolution is commutative ($x\), we have also

Definition. The $z$-transform of the discrete-time signal $x(n)$ is defined to be


That $x(n)$ and $X(z)$ are transform pairs is expressed by writing $X(z)={\ or $X(z)\.

Theorem. The convolution theorem (Papoulis [Papoulis 1977]) states that


In words, convolution in the time domain is multiplication in the frequency domain.

Taking the $z$-transform of both sides of Eq. (2.2.1) and applying the convolution theorem gives


where $H(z)$ is the $z$-transform of the filter impulse response. Thus the$z$-transform of the filter output is the $z$-transform of the input times the $z$-transform of the impulse response.

Definition. The transfer function $ H(z) $ of a linear time-invariant discrete-time filter is defined to be the $z$-transform of the impulse response $ h(n)$.

Theorem. The shift theorem [Papoulis 1977] for $z$-transforms states that


The general difference equation for an LTI filter appears as

$\ $\ $\ (44)
  $\ $\ (45)

Taking the $z$-transform of both sides, denoting the transform by ${\ gives
$\ $\ $\ (46)
    $\ (47)

using linearity and the shift theorem. Replacing ${\ by $Y(z)$, ${\ by $X(z)$, and solving for $Y(z)/X(z)$, which equals the transfer function $H(z)$, yields

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