Signals in Natural Domain
Chapter 7 :  The Z-transform
 
Analysis of LTI system using z-transform
From the convolution property we have
$\displaystyle Y(z)=H(z)\,X(z)$

where $ X(z),\,Y(z)$are $ H(z)$ are z-transforms of input sequence $ \{x[n]\}$, output sequence $ \{y[n]\}$ and impulse response $ \{h[n]\}$ respectively. The $ H(z)$ is referred to as system function or transfer function of the system. For $ z$ on the unit circle $ (z=e^{jw})$, $ H(z)$ reduces to the frequency response of the system, provided that unit circle is in the ROC for.
A causal LTI system has impulse response $ \{h[n]\}$ such that. Thus ROC of $ H(z)$ is exterior of a circle in z-plane including. Thus a discrete time LTI system is causal if and only if ROC is exterior of a circle which includes infinity.
An LTI system is stable if and only if impulse response $ \{h[n]\}$ is absolutely summable. This is equivalent to saying that unit circle is in the ROC of.
For a causal and stable system ROC is outside a circle and ROC contains the unit circle. That means all the poles are inside the unit circle. Thus a causal LTI system is stable if and if only if all the poles inside unit circle.
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
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