z-Transformation: Further Topics

Abstract

The z-transform of a sequence f(kT) was introduced in Chapter 2. It is a function of the complex variable z and is defined by The domain of F(z) is the exterior of some circle centered at the origin. In the applications to control problems, the values f(kT) are the sampled-data values of a time function f(t), sampled with the sampling period T > 0. Instead of sampling a time function at the discrete times 0, T, 2T, etc., we may sample at the instants τ, τ + T, τ + 2T, …, where we consider τ as a fixed parameter restricted to 0 ≤ τ < T. The z-transform of the sequence f(kT + τ) then is simply and nothing is really new. Thus, the z-transformation is a mapping which maps a sequence to a function. Things may be different when we do not consider τ as a fixed parameter anymore, but as a new independent variable, ranging in the interval [0, T). From this new point of view the original object is no longer the sequence f(kT + τ) with k = 0, 1, 2, … but again the continuous time function f(t), where t varies over the entire real axis. As for the Laplace transformation, we assume that f(t) = 0 on the negative part of the real axis. A new name now is justified.