The z-Transform
The z transform is a discrete time counterpart of Laplace transform. The Laplace transform is used on continuous signal while z transform is used for the discrete signal.
The z- transform of a general discrete signal x[n] is defined as
$ X(Z) = \sum_{n=-\infty}^{\infty}x[n]z^{-n} $
Region of Convergence(ROC)
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 ROC is the entire z-plane, except possibly z=0 and/or $ z=\infty $
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|>$ r_0 $ will also be in the ROC.
Property 5:- If x[n] is a left sided sequence and if the circle |z|=$ r_0 $ is in the ROC, then all vaues of z for which 0<|z|<$ r_0 $ will also be in the ROC.
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|=$ r_0 $.
Property 7:- If the z-transform X(z) of x[n] is rational, then its ROC is bounded by pole 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.
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.
