Existence of the Z Transform

The z transform of a finite-amplitude signal will always exist provided (1) the signal starts at a finite time and (2) it is asymptotically exponentially bounded, i.e., there exists a finite integer , and finite real numbers $A\geq 0$ and $\sigma$ , such that $\left\vert x(n)\right\vert<A\exp(\sigma n)$ for all $n\geq n_f$ . The bounding exponential may even be growing with ( $\sigma>0$ ). These are not the most general conditions for existence of the z transform, but they suffice for most practical purposes.

For a signal growing as $\exp(\sigma n)$ , for $\sigma>0$ , one would naturally expect the z transform to be defined only in the region $\left\vert z\right\vert>\exp(\sigma)$ of the complex plane. This is expected because the infinite series

$\displaystyle \sum_{n=0}^\infty e^{\sigma n} z^{-n} = \sum_{n=0}^\infty \left(\frac{e^{\sigma}}{z}\right)^n$

requires $\left\vert z\right\vert>\exp(\sigma)$ to ensure convergence. Since $\sigma<0\,\Leftrightarrow\,\exp(\sigma)<1$ for a decaying exponential, we see that the region of convergence of the transform of a decaying exponential always includes the unit circle of the plane.

More generally, it turns out that, in all cases of practical interest, the domain of can be extended to include the entire complex plane, except at isolated ``singular'' points^7.2 at which $\vert X(z)\vert$ approaches infinity (such as at $z=\exp(\sigma)$ when $x(n)=\exp(\sigma n)$ ). The mathematical technique for doing this is called analytic continuation, and it is described in §D.1 as applied to the Laplace transform (the continuous-time counterpart of the z transform). A point to note, however, is that in the extension region (all points such that $\left\vert z\right\vert<\exp(\sigma)$ in the above example), the signal component corresponding to each singularity inside the extension region is ``flipped'' in the time domain. That is, ``causal'' exponentials become ``anticausal'' exponentials, as discussed in §8.7.

The z transform is discussed more fully elsewhere [52,60], and we will derive below only what we will need.

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Next: Shift and Convolution Theorems

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About the Author: 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.

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Introduction to Digital Filters
Transfer Function Analysis
Existence of the Z Transform