EE5410 Stability, Causality and the ROC
1. Consider an LTI system with impulse response h[n]. The z-transform of h[n] is called the system function of the LTI system.
2. eg., there are 3 ROC in the pole-zero plot, namely:
|z|< 1/2, 1/2< |z| <2 and |z| >2.
3. If we state in addition that the system is stable ( or equivalently, that h[n] is absolutely summable and therefore has a Fourier transform), then the ROC must INCLUDE the unit circle.
4. Thus, stability of the system and Properties 1-8 imply that the ROC is the region 1/2 < |z| <2. Note that as a consequence, h[n] is two sided; therefore, the system is NOT causal.
5. If we state instead tat the system is CAUSAL, and therefore that h[n] is right-sided, Property 5 would require that the ROC be the region |z| >2. Under this condition, the system would NOT be stable.
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