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

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

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'' points<sup>7.2</sup> 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.