Z Transform of Convolution
From Eq.(5.5), we have that the output
from a linear
time-invariant filter with input
and impulse response
is given
by the convolution of
and
, i.e.,
where ``
![$ \ast $](http://www.dsprelated.com/josimages_new/filters/img568.png)
![$ \,$](http://www.dsprelated.com/josimages_new/filters/img94.png)
where H(z) is the z transform of the filter impulse response. We may divide Eq.
![$ \,$](http://www.dsprelated.com/josimages_new/filters/img94.png)
![$ X(z)$](http://www.dsprelated.com/josimages_new/filters/img306.png)
![$\displaystyle H(z) \eqsp \frac{Y(z)}{X(z)} \;\isdef \; \hbox{transfer function}.
$](http://www.dsprelated.com/josimages_new/filters/img664.png)
![$ h(n)$](http://www.dsprelated.com/josimages_new/filters/img536.png)
![$ H(z)=Y(z)/X(z)$](http://www.dsprelated.com/josimages_new/filters/img305.png)
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Z Transform of Difference Equations
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Shift and Convolution Theorems