Advanced topics in system and signal theory. A mathematical approach
The monograph provides an in-depth analysis of selected methods in signal and system theory with applications to problems in communications, stochastic processes and optimal filter theory. It is intended primarily for engineers working in the area of robustness problems under a casualty constraint. At the beginning it presents the mathematical methods, necessary to approach these problems. Then, some related classical results concerning the boundedness and continuity of the Hilbert transform and Riesz projection are presented. Finally, these methods and results are applied to selected topics from signal processing. The first part of the monograph gives a very brief introduction to the main mathematical methods used later in the book. This part serves primarily as a review of results needed in later chapters, so that the work becomes essentially self-contained. The different topics are only covered as far as they will be needed and proofs are sometimes omitted. Appropriate references are given for those who want a more detailed introduction to the different topics. This work presupposes a working knowledge of real and complex analysis as well as some basic elements of functional analysis. The second part collects the basic abstract results concerning the continuity and the boundedness of the Hilbert transform and the Riesz projection on different Banach spaces. These results are the basis for the applications discussed in the third part of this monograph. Here four applications from signal processing are investigated in some detail, namely the expansions of transfer functions in orthonormal bases, the linear approximation from measured data, the calculation of the Hilbert transform, and the spectral factorization. All these topics are essentially problems of recovery or approximation of causal function from measured data, which are generally corrupted by small errors. It is investigated how these errors influence the possibility of recovering the desired signal from the measurements under the restriction that this recovery is based only on past and present measurements, i.e. under the requirement of causality.
scientific article; zbMATH DE number 5639721