##
Existence of the *Z*
Transform

The *z* transform of a finite-amplitude
signal will always *exist* provided (1) the signal starts at a finite time and (2) it is
*asymptotically exponentially bounded*, *i.e.*, there exists a
finite integer , and finite real numbers and ,
such that
for all . The
bounding exponential may even be growing with (). These are
not the most general conditions for existence of the *z* transform, but they
suffice for most practical purposes.

For a signal growing as
, for , one
would naturally expect the *z* transform to be defined only in the
region
of the complex plane. This is expected
because the infinite series

*the region of convergence of the transform of a decaying exponential always includes the unit circle of the plane*.

More generally, it turns out that, in all cases of practical interest,
the domain of can be *extended* to include the
*entire complex plane*, except at isolated ``singular''
points^{7.2} at which approaches
infinity (such as at
when
).
The mathematical technique for doing this is called *analytic
continuation*, and it is described in §D.1 as applied to the
*Laplace transform* (the continuous-time counterpart of the *z* transform).
A point to note, however, is that in the extension region (all points
such that
in the above example), the signal
component corresponding to each singularity inside the extension
region is ``flipped'' in the time domain. That is, ``causal''
exponentials become ``anticausal'' exponentials, as discussed in
§8.7.

The *z* transform is discussed more fully elsewhere
[52,60], and we will derive below only
what we will need.

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Shift and Convolution Theorems

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The Z Transform