##
The *Z* Transform

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

*z*transformwhere 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

*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