The Z Transform Next  |  Prev  |  Up  |  Top  |  Index  |  JOS Index  |  JOS Pubs  |  JOS Home  |  Search

## The Z Transform

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: (7.2)

For inverting z transforms, see §6.8.

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 has the z transform .

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