Introduction to orthogonal transforms. With applications in data processing and analysis.

*(English)*Zbl 1272.94001
Cambridge: Cambridge University Press (ISBN 978-0-521-51688-4/hbk). xxii, 568 p. (2012).

Orthogonal transforms play an essential role in signal processing, data analysis and communication. The aim of this book is to present a systematic and unified treatment of orthogonal transform methods, which guides the reader from mathematical theory to problem solving in practice. The transform methods covered in this book are a collection of both old and new ideas ranging from the classical Fourier series expansion, that goes back almost 200 years, to some relatively recent thoughts such as the various origins of the wavelet transform. The book examines each transform method in depth, emphasizing the common mathematical principles and essential properties of each method in terms of signal decorrelation and energy compaction. It contains plenty of practical examples with online Matlab and C codes and exercises at the end of each chapter, which makes it more suitable for use in a graduate course.

The chapter titles are: “Signals and systems”; “Vector spaces and signal representation”; “Continuous-time Fourier transform”; “Discrete-time Fourier transform”; “Application of the Fourier transforms”; “The Laplace and \(z\)-transforms”; “Fourier-related orthogonal transforms”; “The Walsh-Hadamard, slant, and Haar transforms”; “Karhunen-Loève transform and principal component analysis”; “Continuous- and discrete-time wavelet transforms”; “Multiresolution analysis and discrete wavelet transform”.

There are also appendices with brief reviews of linear algebra and random variables. The book is an excellent text/reference resource for graduates and professionals in applied mathematics, electrical engineering and signal processing.

The chapter titles are: “Signals and systems”; “Vector spaces and signal representation”; “Continuous-time Fourier transform”; “Discrete-time Fourier transform”; “Application of the Fourier transforms”; “The Laplace and \(z\)-transforms”; “Fourier-related orthogonal transforms”; “The Walsh-Hadamard, slant, and Haar transforms”; “Karhunen-Loève transform and principal component analysis”; “Continuous- and discrete-time wavelet transforms”; “Multiresolution analysis and discrete wavelet transform”.

There are also appendices with brief reviews of linear algebra and random variables. The book is an excellent text/reference resource for graduates and professionals in applied mathematics, electrical engineering and signal processing.

Reviewer: Wenchang Sun (Tianjin)

##### MSC:

94-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to information and communication theory |

42-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to harmonic analysis on Euclidean spaces |

94A11 | Application of orthogonal and other special functions |

94A12 | Signal theory (characterization, reconstruction, filtering, etc.) |

42C15 | General harmonic expansions, frames |

42C40 | Nontrigonometric harmonic analysis involving wavelets and other special systems |