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m18 - Region of Convergence for the Z-Transform

Module by: C. Sidney Burrus. E-mail the author

Summary: Because the z transform is defined as an infinite summation, questions of convergence of the sum are important. The region of convergence in the region in the complex z plane where the sum converges.

Region of Convergence for the Z-Transform

Since the inversion integral must be taken in the ROC of the transform, it is necessary to understand how this region is determined and what it means even if the inversion is done by partial fraction expansion or long division. Since all signals created by linear constant coefficient difference equations are sums of geometric sequences (or samples of exponentials), an analysis of these cases will cover most practical situations. Consider a geometric sequence starting at zero.

f ( n ) = u ( n ) a n f ( n ) = u ( n ) a n
(1)
with a z-transform
F ( z ) = 1 + a z 1 + a 2 z 2 + a 3 z 3 + + a M z M . F ( z ) = 1 + a z 1 + a 2 z 2 + a 3 z 3 + + a M z M .
(2)
Multiplying by a z 1 a z 1 gives
a z 1 F ( z ) = a z 1 + a 2 x 2 + a 3 z 3 + a 4 z 4 + + a M + 1 z M 1 a z 1 F ( z ) = a z 1 + a 2 x 2 + a 3 z 3 + a 4 z 4 + + a M + 1 z M 1
(3)
and subtracting from (2.32) gives
( 1 a , z 1 ) F ( z ) = 1 a M + 1 z M 1 ( 1 a , z 1 ) F ( z ) = 1 a M + 1 z M 1
(4)
Solving for F ( z ) F ( z ) results in
F ( z ) = 1 a M + 1 z M 1 1 a z 1 = z a ( a z ) M z a F ( z ) = 1 a M + 1 z M 1 1 a z 1 = z a ( a z ) M z a
(5)
The limit of this sum as M M is
F ( z ) = z z a F ( z ) = z z a
(6)
for | z | > | a | | z | > | a | . This not only establishes the z-transform of f ( n ) f ( n ) but gives the region in the z z plane where the sum converges.

If a similar set of operation is performed on the sequence that exists for negative n n

f ( n ) = u ( n 1 ) a n = { a n n < 0 0 n 0 f ( n ) = u ( n 1 ) a n = { a n n < 0 0 n 0
(7)
the result is
F ( z ) = z z a F ( z ) = z z a
(8)
for | z | < | a | | z | < | a | . Here we have exactly the same z-transform for a different sequence f ( n ) f ( n ) but with a different ROC. The pole in F ( z ) F ( z ) divides the z-plane into two regions that give two different f ( n ) f ( n ) . This is a general result that can be applied to a general rational F ( z ) F ( z ) with several poles and zeros. The z-plane will be divided into concentric annular regions separated by the poles. The contour integral is evaluated in one of these regions and the poles inside the contour give the part of the solution existing for negative n n with the poles outside the contour giving the part of the solution existing for positive n n .

Notice that any finite length signal has a z-transform that converges for all z z . The ROC is the entire z-plane except perhaps zero and/or infinity.

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