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:

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

or, using operator notation,

which can be abbreviated as

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 signal

has the z transform .

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written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.