The z transform of a 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 be growing with (). These are not the most general conditions for existence of the z transform, but they suffice for most practical purposes.
One would naturally expect the z transform to be defined only in the region of the complex plane, where is the asymptotically bounding exponential envelope for discussed in the previous paragraph. This expectation is reasonable because the infinite series
More generally, it turns out that, in all practical cases, the domain of can be extended to include the entire complex plane except for isolated ``singular'' points called poles at which approaches infinity. The mathematical technique for doing this is known as analytic continuation, and it is discussed in §B.1 as applied to Laplace transforms (the continuous-time counterpart of z transforms).
The z transform is discussed more fully elsewhere [52,60], and we will derive below only what we will need.