z-Transformation

Abstract

The z-transformation emerged, at least under this curious name, in the middle of the 20th century [Jury (1977)]. The basic idea is old and is known by the name of “generating function method” among the mathematicians. Indeed, the z-transform of a sequence is the generating function of this sequence, where the independent variable z is replaced by its reciprocal 1/z. The z-transformation or z-transform today is applied to model sample-data control systems or other discrete-data systems. Its role for discrete time systems is similar to the method of Laplace transformation for continuous time systems. From a mathematical point of view, the method of z-transformation is an operational calculus for solving difference equations or systems of such equations, similar to the method of Laplace transformation in connection with differential equations. The name z-transformation or z-transform is nonsense but, unfortunately, is in common use today (it is as if the Laplace transformation would be called the s-transform or p-transform). A reasonable name for this method would perhaps be “Laurent transform or Laurent transformation” because the defining series is a Laurent series. But it is too late for that. The z-transformation is intimately connected with the discrete Laplace transformation and with other discrete transformation methods. As is the case with the Laplace transformation, there is an ordinary or one-sided z-transform and a two-sided one. In this book we study only the one-sided z-transformation. However, the one-sided z-transformation has a generalization which is called the advanced or modified z-transformation. If the latter is applied to a sequence, there is no reason for a new name. It is just the ordinary z-transform applied to another sequence depending on a supplementary parameter. But if the domain of the originals is considered to be a set of continuous time functions rather than discrete time sequences, a new name is justified. The Laplace transformation maps a continuous function of the real variable t to a function of the complex variable s. Similarly, the z-transformation maps a discrete sequence to a function of the complex variable z. Both methods benefit from the theory of functions of a complex variable.