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
requires
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''
points7.2 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.