11. Convolution Representation

GUIDE: Elementary Digital Filter Theory. Convolution Representation

It appears that you are using AdBlocking software. The cost of running this website is covered by advertisements. If you like it please feel free to a small amount of money to secure the future of this website.

<< Previous page  TOC  INDEX  Next page >>

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
\

<< Previous page  TOC  INDEX  Next page >>

 

© 1998-2017 – Nicola Asuni - Tecnick.com - All rights reserved.
about - disclaimer - privacy