Question

As far as I know a LTI system is caus...: "Dr.Gunther, As far as I know a LTI system is causal iff the ROC of a system function is the exterior part of the circle.Also a LTI system i..."

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Causal => ROC outside a circle in z-plane
Stable => ROC includes the unit circle in z-plane
ROC does not include poles

Putting all of this together, a causal and stable system must have all its poles inside the unit circle. Then all of the above requirements can be met.

Example: x(n) = 0.9^n u(n)

X(z) = 1/(1-0.9 z^{-1}), ROC = {z | |z| >= 0.9}

This is a causal and stable system. The pole is inside the unit circle. The ROC is everything in the z-plane outside a circle with radius 0.9. The unit circle is inside the ROC and the system is stable.