##
The *Z* Transform

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

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,

^{7.1}the lower summation limit given above may be written as 0 rather than to yield the

*unilateral*:

*z*transform(7.2) |

The unilateral

*z*transform is most commonly used. For inverting

*z*transforms, see §6.8.

Recall (§4.1) 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 may say is the *z* transform of by writing

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

*z*transform .

**Next Section:**

Existence of the Z Transform

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Time Domain Representation Problems