The z transform of a 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 be growing with (). These are
not the most general conditions for existence of the z transform, but they
suffice for most practical purposes.

One would naturally expect the z transform to be defined only in the
region
of the complex plane, where
is the asymptotically bounding exponential envelope
for discussed in the previous paragraph. This expectation is
reasonable because the infinite series

requires
to ensure convergence. Since
for a decaying exponential, we see that the region of
convergence of the transform always includes the unit circle of
the plane.

More generally, it turns out that, in all practical cases, the domain
of can be extended to include the entire complex plane
except for isolated ``singular'' points called poles at which
approaches infinity. The mathematical technique for doing this
is known as analytic continuation, and it is discussed in
§B.1 as applied to Laplace transforms (the
continuous-time counterpart of z transforms).

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