## Freitag, 24. Juni 2011

### 9 properties of the ROC for the Z-transform

9 properties of the ROC for the Z-transform

9 properties of the Region of Convergence ( ROC) for the Z-transform

1. The ROC of X(z) consists of a ring in the z-plane centered about the origin.

2. The ROC does NOT contain any poles.

3. If x[n] is of finite duration, then the ROC is the ENTIRE z-plane, except possibly z=0 and/or z = undefined.

4. If x[n] is a Right-sided sequence, and if the circle | z | = r0 is in the ROC, then all finite values of z for which | z |> r0 will also be in the ROC.

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 |< r0 will also be in the ROC.

6. If x[n] is two sided, and if the circle |z|=r0 is in the ROC, hthen the ROC will consist of a RING in the z-plane that includes the circle | z| = r0.

7. If the z-transform X(z) of x[n] is rational, then its ROC is bounded by poles or extends to infinity.

8. If the z-transform X(z) of x[n] is ration, and if x[n] is right sided, then the ROC is the region in the z-plane OUTSIDE the outermost pole -- i.e., outside the circle of radius equal to the largest magnitude of the poles of X(z).

Furthermore, if x[n] is causal ( i.e., if it is right sided and eual to 0 for n <0), then the ROC also inclides z = undefined.

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 INSIDE the innermost nonzero pole --

i.e., inside the circle of radius equal to the smallest magnitude of the poles of X(z) other than any at z=0. and extending inward to and possibly including z=0.

In particular, if x[n] is anti-causal (i.e., if it is left sided and equal to 0 for n >0 ), then the ROC also includes z= 0.

Every Little Thing - Pray

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