(New page: == Properties of the z-transform == Time shifting <math>x_1[n-n_0] \longrightarrow Z^{-n_0}X(Z)</math> ROC:R except for possible addition of origin or infinity Convolution <math>x_1[n]...) |
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3.ROC can be infinite or entire z-plane except possibly z=0 and/of z=infinite , if x[n] is finite duration. | 3.ROC can be infinite or entire z-plane except possibly z=0 and/of z=infinite , if x[n] is finite duration. | ||
| − | 4. | + | 4.if x[n] is right - sided sepuence and if <math>\mid z \mid = r_0</math> is in the ROC, then all finite values of z which <math>\mid z \mid > r_0 </math> |
| + | will also in the ROC. | ||
| + | |||
| + | 5.if x[n] is left - sided sepuence and if <math>\mid z \mid = r_0</math> is in the ROC, then all finite values of z which <math>0<\mid z \mid < r_0 </math> | ||
| + | will also in the ROC. | ||
| + | |||
| + | 6.if x[n] is two sided, and if the circle <math>\mid z \mid = r_0</math> is in the ROC, then it will consistof a ring in the z-plane than include the circle <math>\mid z \mid =r_0</math> | ||
Latest revision as of 11:42, 3 December 2008
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 $
