Signal and System I Causality ROC for n < 0 causal All z -n terms, not include any z terms If and only if ROC is exterior of a circle and include infinity

Signal and System I Causality for n < 0 causal All z -n terms, not include any z terms If 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 infinite H(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.

Signal and System I Example Not causal ROC |z|>2 causal (1) Exterior of circle 2 (2) The order of the numerator is not larger that the denominator

Signal and System I Stability An LTI system is stable if and only if the ROC of the H(z) of the system function contains unit circle. At unit circle

Signal and System I Stability A 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=a not stable stable

Signal and System I Causal system Poles ROC 1 Unit circle x x not stable stable

Signal and System I LTI system characterized by linear constant difference equation ROC |z|>1/2

Signal and System I Example

Signal and System I Example

Signal and System I Example

Signal and System I Example Stable and causal system H(z) has a pole at z = ½, and a zero on the unite circle. Other poles and zeros are unknown converge for some  h[n] has finite duration X h[n] is real Insufficient information is a impulse response of a stable system.