causal All z n terms not include any z terms If and only if ROC is exterior of a circle and include infinity Causality f or n lt 0 causal All z n terms not include any z terms If and only if ROC is exterior of a circle and include infinity ID: 1020480
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1. CausalityROCfor n < 0causalAll z-n terms, not include any z termsIf and only if ROC is exterior of a circle and include infinity
2. Causalityfor n < 0causalAll z-n terms, not include any z termsIf and only if ROC is exterior of a circle and include infinity If H(z) is rational, then ROC is outside of the outermost pole including the infiniteH(z) is finite when z --> . A discrete-time LTI system with rational system function H(z) is causal if and only if (a) the ROC is the exterior of a circle outside the outermost pole; and (b) with the H(z) expressed as a ratio of polynomials in z, the order of the numerator can not be greater than the order of the denominator.
3. Example Not causalROC |z|>2causal(1) Exterior of circle 2(2) The order of the numerator is not larger that the denominator
4. StabilityAn LTI system is stable if and only if the ROC of the H(z) of the system function contains unit circle.At unit circle
5. StabilityA causal LTI system with rational system function H(z) is stable if and only if all the poles lie inside the unit circle, i.e. their magnitudes are all small than 1. Example Causal system Pole z=anot stable stable
6. Causal system PolesROC1Unit circlexxnot stable stable
7. LTI system characterized by linear constant difference equation ROC |z|>1/2
8. Example
9. Example
10. Example
11. Example Stable and causal systemH(z) has a pole at z = ½, and a zero on the unite circle. Other poles and zeros are unknownconverge for some h[n] has finite duration Xh[n] is realInsufficient informationis a impulse response of a stable system.