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Existence of the Z Transform

The z transform of a finite-amplitude signal $ x$ 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 $ n_f$ , 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 $ n$ ($ \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 $ x(n)$ growing as $ \exp(\sigma n)$ , for $ \sigma>0$ , one would naturally expect the z transform $ X(z)$ 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 $ z$ transform of a decaying exponential always includes the unit circle of the $ z$ plane.

More generally, it turns out that, in all cases of practical interest, the domain of $ X(z)$ can be extended to include the entire complex plane, except at isolated ``singular'' points7.2 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 $ z$ 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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``Introduction to Digital Filters with Audio Applications'', by Julius O. Smith III, (September 2007 Edition)
Copyright © 2024-05-20 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University