Properties of the region of convergence for Z-transform
A number of properties are listed in the oppenheim willsky textbook. These properties state the insights of the z-transforms region of convergence.
Property 1: The ROC of X(z) consists of a ring in the z-plane centered about the origin.
Property 2: The ROC does not contain any poles.
Property 3: If x[n] is of finite duration then the ROC is the entire z-plane except possibly z=0 and z=$ \infty $
Property 4: If x[n] is a right sided sequence and if the cirlce |z| = ro is in the ROC then all finite values of z for which |z| >ro will also be in the ROC.
Property 5: If x[n] is a left sided sequence and if the circle |z| = ro is in the ROC then all values of z for which 0< |z| <ro will also be in the ROC
Property 6: If x[n] is two sided and if the circle |z| = ro is in the ROC then the ROC will consist of a ring in the z-plane that includes the circle |z| = ro
Property 7: If the transform X(z) of x[n] is rational then its ROC is bounded by poles or extends to infinity
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 outermost pole. for example outside the circle radius equal to the largest magnitude of the poles of X(z) furthermore if x[n] is causal then the ROC also includes z=infinity
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 innermost pole. for example inside the circle radius equal to the largest magnitude of the poles of X(z) furthermore if x[n] is causal then the ROC also includes z=infinity
