Z transform roc

Definition of ROC : 

The ROC stands for the region of convergence of X(Z) is set for all the values of Z for which X(Z)  attains a finite value. Every time when we find Z transform we must indicate its ROC.

The significance of ROC :
  • ROC will decide whether a system is stable or unstable
  • ROC also determine the type of sequence that means: Causal or noncausal, Finite or infinite
Properties of ROC for Z transform :
  • The ROC is a ring, whose center is at an origin
  • ROC cannot contain any pole
  • If x(n) is causal then ROC is exterior part of a circle of radius say ɖ
  • If x(n) is anticausal then ROC is interior part of a circle of radius say ɖ
  • If x(n) is  two-sided sequence then ROC is the intersection of two circles of radii ɖ and β
  • The ROC must be connected region
  • If ROC of X(Z) include unit circle then and then only the Fourier transform of DT sequence x(n) converges
  • For a finite duration sequence, x(n) the ROC is the entire Z plane except Z = 0 and Z = ∞
Summary of ROC :
  • For Causal system, ROC is entire Z plane except |Z| = 0 
  • For the anti-causal system, ROC is the entire Z plane except |Z| = ∞
  • For two-sided sequence ROC is entire Z plane except  |Z| = 0  and |Z| = ∞