Complex and hypercomplex analytic signals. Theory and applications
With this interesting book, the authors present the first comprehensive publication on the theory of multidimensional complex and hypercomplex signals which is illustrated by numerous examples and practical applications in digital signal and image processing. This monograph is mainly based on the research of both authors over more than 20 years. Starting point was the theory of multidimensional analytic signals with single-orthant spectra introduced by \textit{S. L. Hahn} [``Multidimensional complex signals with single-orthant spectra'', Proc. IEEE 80, No. 8, 1287--1300 (1992; \url{doi:10.1109/5.158601})]. Further in the book of \textit{S. L. Hahn} [Hilbert transforms in signal processing. Boston, MA: Artech House (1996; Zbl 0910.94003)], one can find a description of the theory of multidimensional complex analytic signals.NEWLINENEWLINENEWLINEThe book under review is divided into 10 chapters, a summary, and 8 appendices. After an introductory chapter, chapter 2 gives an overview of algebras of hypercomplex numbers, namely of Caley-Dickson algebras of quaternions and octonions as well as of Clifford algebras of biquaternions and bioctonions. Orthants of a multidimensional Cartesian space and single-orthant operators are introduced in Chapter 3. In Chapter 4, the theory of complex and hypercomplex Fourier transforms of multidimensional signals is presented. Chapter 5 explains the theory of multidimensional complex and hypercomplex analytic signals. Chapter 6 is devoted to the ranking of complex and hypercomplex analytic signals. Chapter 7 presents polar representations of \(n\)-dimensional complex and hypercomplex analytic signals for \(n=1,\,2,\,3\). Quasi-analytic signals are handled in Chapter 8. Using Wigner distribution and corresponding Woodward's ambiguity function, space-frequency representations of \(n\)-dimensional complex and hypercomplex analytic signals for \(n=1\) and \(n=2\) are given in Chapter 9. In the last chapter, the causality of multidimensional real signals is defined.NEWLINENEWLINEThis well-written book will serve as valuable source for further research of electrical engineers, physicists, and applied mathematicians.
scientific article; zbMATH DE number 6635643