Definition: z-Transform X(z) The z-transform of x[k] for is
where is a complex variable. The sum converges to X(z) for some ROC on the z-plane as depicted in Figure C.3 for the four basic types of sequences.
The mapping from the s-plane to the z-plane when a continuous-time signal x(t) is uniformly sampled to generate the discrete-time sequence x[k] is described in Chapter 1. For the unilateral Laplace transform, the integral has a finite lower limit, usually zero:
The unilateral and bilateral z-transforms are identical when x[k] is nonzero only for , which can be emphasized by writing x[k]u[k], where u[k] is the discrete unit-step function. ...