Symmetries and Groups in Signal Processing: An Introduction

ISBN-13: 9789400733008 / Angielski / Miękka / 2012 / 160 str.

Virendra P. Sinha

Opis

This book deals with the subject of symmetry, and its place and role in modern signal processing. It introduces group theoretic techniques of exploiting symmetry for engineering students and explains group representation theory with the help of examples.

Recenzje

From the reviews:

"This is an modern introduction to theoretical topics in field of signal processing, which is based on representation theory of groups. It is suitable for beginning graduate students and does not require from reader solid mathematical prerequisites. ... Abstract concepts are illustrated with their concrete engineering interpretations ... . this style makes the whole book very readable and suitable as textbook." (Jaak Henno, Zentralblatt MATH, Vol. 1205, 2011)

Spis treści

Preface. 1 Signals and Signal Spaces: A Structural Viewpoint. 1.1 What is a Signal? 1.2 Spaces and Structures. 1.3 Signal Spaces and Systems. 1.4 Linearity, Shift–Invariance and Causality. 1.5 Convolutional Algebra and the Z–Transform. 1.6 Shifts, Transforms and Spectra. 2 Algebraic Preliminaries. 2.1 What’s in a Definition? 2.2 Set Theoretic Notation. 2.3 Relations and Operations. 2.4 Groups. 2.5 Vector Spaces. 2.6 Posets, Lattices, and Boolean Algebras. 2.7 Closing Remarks. 3 Measurement, Modeling, and Metaphors. 3.1 Archimedes and the Tortoise. 3.2 The Representational Approach. 3.3 Metaphors. 4 Symmetries, Automorphisms and Groups. 4.1 Introduction. 4.2 Symmetries and Automorphisms. 4.3 Groups of Automorphisms. 4.4 Symmetries of Linear Transformations. 4.5 Symmetry Based Decompositions. 5 Representations of Finite Groups. 5.1 The Notion of Representation. 5.2 Matrix Representations of Groups. 5.3 Automorphisms of a Vector Space. 5.4 Group Representations in GL(V). 5.5 Reducible and Irreducible Representations. 5.6 Reducibility of Representations. 5.7 Schur’s Lemma and the Orthogonality Theorem. 5.8 Characters and Their Properties. 5.9 Constructing Irreducible Representations. 5.10 Complete Reduction of Representations. 5.11FurtheronReduction. 6 Signal Processing and Representation Theory. 6.1 Signals as Functions on Groups. 6.2 Symmetries of Linear Equations. 6.3 Fast Discrete Signal Transforms. A Parentheses, Their Proper Pairing, and Associativity. A.1 Proper Pairing of Parentheses. A.2 Parenthese and the Associative Law. Index.

Blurb

Symmetries and Groups in Signal Processing: An Introduction deals with the subject of symmetry, and with its place and role in modern signal processing. In the sciences, symmetry considerations and related group theoretic techniques have had a place of central importance since the early twenties. In engineering, however, a matching recognition of their power is a relatively recent development. Despite that, the related literature, in the form of journal papers and research monographs, has grown enormously. A proper understanding of the concepts that have emerged in the process requires a mathematical background that goes beyond what is traditionally covered in an engineering undergraduate curriculum.

Admittedly, there is a wide selection of excellent introductory textbooks on the subject of symmetry and group theory. But they are all primarily addressed to students of the sciences and mathematics, or to students of courses in mathematics. Addressed to students with an engineering background, this book is meant to help bridge the gap.