### Region of Convergence (RoC) Properties

1.) The RoC of a Z transform consists of rings (or washers) centered about the origin in the z-plane.

2.) If X(z) is rational, aka $ X(z) = \frac {P(z)}{Q(z)} $ ,

Then the RoC does not contain any poles of X(z). A pole is a z that would make Q(z) = 0.

3.) If x[n] is of a finite duration,

Then the RoC eitherDOES NOT EXIST, or is the entire Z-plane except 0 or $ \infty $

4.) If x[n] is right sided and if |z| = $ r_0 $ is in the RoC,

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Then all finite values of z for which |z| > $ r_0 $ are also in the RoC.
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5.) If x[n] is left sided and if |z| = $ r_0 $ is in the RoC,

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Then all finite values for which 0 < |z| < $ r_0 $ are also in the RoC.
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6.) If two-sided, and if the circle |z| = $ r_0 $ is in the RoC,

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Then the RoC will consist of a ring (or washer) in the z-plane.
The ring will include the circle |z| = $ r_0 $
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7.) If X(z) is rational, aka $ X(z) = \frac {P(z)}{Q(z)} $

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Then the RoC is bounded by poles or will extend to $ \infty $
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