##
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 and , such that for all . The bounding exponential may even be growing with (). 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 , for , one would naturally expect the

*z*transform to be defined only in the region of the complex plane. This is expected because the infinite series

*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 approaches infinity (such as at when ). 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 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.

**Next Section:**

Shift and Convolution Theorems

**Previous Section:**

The Z Transform