Properties of the z-transform
Time shifting
$ x_1[n-n_0] \longrightarrow Z^{-n_0}X(Z) $
ROC:R except for possible addition of origin or infinity
Convolution
$ x_1[n] * x_2[n] \longrightarrow X_1(Z)X_2(Z) $
$ ROC continuous R_1 \bigcap R_2 $
ROC of z-transform
1.The ROC of a z-transform consists z-plane centered about the origin.
2.ROC does not include any poles of X(z) if X(z) is rational.
3.ROC can be infinite or entire z-plane except possibly z=0 and/of z=infinite , if x[n] is finite duration.
4.if x[n] is right - sided sepuence and if $ \mid z \mid = r_0 $ is in the ROC, then all finite values of z which $ \mid z \mid > r_0 $ will also in the ROC.
5.if x[n] is left - sided sepuence and if $ \mid z \mid = r_0 $ is in the ROC, then all finite values of z which $ 0<\mid z \mid < r_0 $ will also in the ROC.
6.if x[n] is two sided, and if the circle $ \mid z \mid = r_0 $ is in the ROC, then it will consistof a ring in the z-plane than include the circle $ \mid z \mid =r_0 $
