# 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.