Signal processing and systems theory. Selected topics
In the last twenty years, the interest for functions of a complex variable analytic in a half-plane decreased gradually (for applications). At the same time, the interest grew for functions of a complex variable analytic in the unit disk. This is so because the importance of discrete signals and systems whose complex frequency domain is the unit disk increased dramatically. Many results of mathematical origin concerning the behavior of complex functions on the unit disk became of great use and importance in recent applications. The book under consideration contains some topics from Discrete Signal Processing (DSP) and Discrete System Theory (DST) which are related to the theory of functions of a complex variable analytic in the unit disk. There are only six chapters in a volume of 268 p. Here are its contents: (1) Digital signals and digital filters; (2) Linear filters; (3) Approximation in Hardy spaces; (4) Optimal Hankel-norm approximation and \(H^ \infty\) minimization; (5) General theory of Hankel-norm approximation; (6) \(H^ \infty\) optimization and system reduction for MIMO systems, References, Further reading, List of symbols and a Subject index. The first two chapters provide standard results in a rigorous exposition. The fundamental concepts of linear systems (observability, minimal realization, sensitivity) for SISO (single input-single output) as well as for MIMO (multiple input-multiple output) systems are given. The chapter 3 gives an introduction to approximation in Hardy spaces. The celebrated Nevanlinna-Pick theorem and Pick's algorithm are discussed. Chapter 4, where Hankel-norm approximation and \(H^ \infty\) minimization are related by the famous AAK theorem (Adamjan, Arov and Krein), is important. By this connection, many facts belonging to operator theory can be translated into the language of DSP or DST. The book is written with great and beautiful rigor. It is accessible to mathematicians but is less accessible to engineers unless they like and master mathematics. However, we emphasize that the work required for studying the book is worthwile because it gives a very good introduction to the operator and function of a complex variable approach for discrete systems (filters, plants to be controlled and so on).
scientific article; zbMATH DE number 555720