Z Transform and Convergence.

Cuthbert Nyack
Consider a Z Transform with a pole at +a as shown below
When the Z transform is expressed as a series it can be that the series converges provided the condition below is satisfied.
The transform converges in the region of the Z plane shown in green. It goes from a radius a to infinity in all directions and includes the unit circle.
The same applies for any pole at r within the unit circle. ie the region of convergence includes all points with radius greater than r. For the frequency spectrum to exist, the region of convergence must include the unit circle and the pole must be inside the unit circle.
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© 2000 Cuthbert A. Nyack.