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.

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Seraj Dosenbach