Signals in Natural Domain
Chapter 7 :  The Z-transform
 
LTI systems characterized by Linear constant coefficient difference equation
For the system characterized by

$\displaystyle \sum\limits^{N}_{k=0} a_{k}y[n-k]=\sum\limits^{M}_{k=0}b_{k}x[n-k]$

We take the z-transform of both sides and use linearity and the time shift property to get
$\displaystyle \sum\limits^{N}_{k=0}a_{k}z^{-k}\,Y(z)$ = $\displaystyle \sum\limits^{M}_{k=0}b_{k}z^{-k}\,X(z)$  
$\displaystyle H(z)=\frac{Y(z)}{X(z)}$ = $\displaystyle \frac{\sum\limits^{M}_{k=0}b_{k}z^{-k} }{\sum\limits^{N}_{k=0}a_{k}z^{-k}}$  

Thus the system function is always a rational function. We can write it by inspection. Numerator polynomial coefficients are the coefficients of $ x[n-k]$ and denominator coefficients are coefficients of. The difference equation by itself does not provide information about the ROC, it can be determined by conditions like causality and stability.
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
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