The bilateral z transform of the discrete-time signal is
defined to be

(7.1)

where is a complex variable. Since signals are typically defined
to begin (become nonzero) at time , and since filters are often
assumed to be causal, the lower summation limit given above may be
written as 0 rather than to yield the unilateral z transform:

Note that the mathematical representation of a discrete-time signal
maps each integer
to a complex number
(
) or real number (
). The z transform of
, on the other hand, , maps every complex number
to a new complex number
. On a higher
level, the z transform, viewed as a linear operator, maps an entire
signal to its z transform . We think of this as a ``function to
function'' mapping. We express the fact that is the
z transform of by writing

or, using operator notation,

which can be abbreviated

One also sees the convenient but possibly misleading notation
, in which and must be understood as
standing for the entire domains
and
, as
opposed to denoting particular fixed values.

The z transform of a signal can be regarded as a polynomial in
, with coefficients given by the signal samples. For example,
the finite-duration signal