The z-Transform

In the chapter, the z-transform is presented. The z-transform is developed starting from the DTFT as a generalization of the Fourier analysis. Extending the set of basis signals at other than the unit-circle results in the capability to analyze a large set of unbounded signals, which is very useful in stability analysis of systems. Examples of deriving the z-transform of useful signals are given. The properties of z-transform, which make the analysis of signals and systems much simpler, are presented. While the inverse z-transform is defined in terms of contour integration, in practice, the much simpler partial fraction and long division methods are used. A number of examples of finding the inverse z-transform are presented. The transfer function, which is the ratio of the transform of the output and that of input, is presented with examples. The pole-zero characterization of systems is presented, and the stability criterion of systems is given.