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What are the properties of ROC for z-Transform?

Discussion in 'Information & Communication Engineering (ICE)' started by MD. Sojibul Islam, Jul 8, 2017.

  1. MD. Sojibul Islam

    MD. Sojibul Islam Administrator Staff Member

    Properties of the ROC for the z-Transform :
    1. X(z) converges uniformly if and only if the ROC of the z-transform X(z) of the sequence includes the unit circle. The ROC of X(z) consists of a ring in the z-plane centered about the origin. That is, the ROC of the z-transform of x(n) has values of z for which x(n) r-n is absolutely summable.
    2. The ROC does not contain any poles.
    3. When x(n) is of finite duration then the ROC is the entire z-plane, except possibly z=0 and/or z=infinity.
    4. If x(n) is a right sided sequence, the ROC will not include infinity.
    5. If x(n) is a left sided sequence, the ROC will not include z=0. However if x(n)=0 for all n>0, the ROC will include z=0.
    6. If x(n) is two sided and if the circle |z| = r0 is in the ROC, then the ROC will consist of a ring in the z-plane that includes the circle |z|=r0.
    7. If X(z) is rational, then the ROC extends to infinity, i.e. the ROC is bounded by poles.
    8. If x(n) is causal, then the ROC includes z=infinity.
    9. If x(n) is anti- causal, trhen the ROC includes z=0.

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