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State and discuss the properties of the region of convergence for z-transform.

Mumbai University > EXTC > Sem 4 > Signals and Systems

Marks : 05

Year : DEC 2014

1 Answer
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(i) The properties of ROC are follows:

(ii) Property 1: The ROC of x [z] consists of a ring in the z-plane centered about the origin.

(iii) Property 2: The ROC does not contain any poles.

(iv) Property 3: If x [n] is of finite duration, then the ROC is the entire z-plane, expect possibly $z=0$ and/or $z=∞.$

(v) Property 4: If x [n] is a right-sided sequence, and if the circle $|z|=r_0$ is in the ROC, then all finite values of z for which $|z| \gt r_0$ will also be in the ROC.

(vi) Property 5: If x [n] is a left-sided sequence, and if the circle $|z|=r_0$ is in the ROC, then all values of z for which $0 \lt |z| \lt r_0$ will also be in the ROC.

(vii) Property 6: If x [n] is two-sided, and if the circle $|z|=r_0$ is in the ROC, then the ROC will consist of a ring in the z-plane that includes the circle $|z| \gt r_0.$

(viii) Property 7: If the z-transform x [z] is rational, then its ROC is bounded by poles or extends to infinity.

(ix) Property 8: If the z-transform x [z] of x[n] is rational, and if x [n] is right-sided, then the ROC is the region in the z-plane outside the outer most pole i.e., outside the circle of radius equal to the largest magnitude of the poles of x [z]. Furthermore, if x [n] is causal, then the ROC also includes $z =∞.$

(x) Property 9: If the z-transform x [z] of x[n] is rational, and if x [n] is left-sided, then the ROC is the region in the z-plane outside the inner most non-zero pole i.e., inside the circle of radius equal to the smallest magnitude of the poles of x [z] other than any at z=0 and extending inward to and possibly including z=0. In particular, if x [n] is anti-causal, then the ROC also includes z =0.

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