z-TRANSFORMS
• For a discrete-time signal
, the two-sided z-transform is defined by
. The one-sided z-transform is defined by
. In both cases, the z-transform is a polynomial in the complex variable
.
• The inverse z-transform is obtained by contour integration in the complex plane
. This is usually avoided by partial fraction inversion techniques, similar to the Laplace transform.
• Along with a z-transform we associate its region of convergence (or ROC). These are the values of
for which
is bounded (i.e., of finite magnitude).
• Some common z-transforms are shown below.
Table 1: Common z-transforms
Signal
|
z-Transform
|
ROC
|
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1
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All
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• The z-transform also has various properties that are useful. The table below lists properties for the two-sided z-transform. The one-sided z-transform properties can be derived from the ones below by considering the signal
instead of simply
.
Table 2: Two-sided z-Transform Properties
Property
|
Time Domain
|
z-Domain
|
ROC
|
Notation
|
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Linearity
|
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At least the intersection of
and
|
Time Shifting
|
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That of
, except
if
and
if
|
z-Domain Scaling
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Time Reversal
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z-Domain
Differentiation
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Convolution
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At least the intersection of
and
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Multiplication
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At least
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Initial value theorem
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causal
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