Sinha Symmetries and Groups in Signal Processing

An Introduction
1. Auflage 2010
ISBN: 978-90-481-9434-6
Verlag: Springer Netherlands
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 160 Seiten

Reihe: Signals and Communication Technology

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.

Autoren/Hrsg.

Sinha, Virendra P.

Inhaltsverzeichnis

1;Preface;6
2;Contents
;9
3;Chapter 1:
Signals and Signal Spaces: A Structural Viewpoint;11
3.1;1.1 What Is a Signal?;11
3.2;1.2 Spaces and Structures;13
3.3;1.3 Signal Spaces and Systems;17
3.4;1.4 Linearity, Shift–Invariance and Causality
;22
3.4.1;1.4.1 Linearity;22
3.4.2;1.4.2 Shift–Invariance
;25
3.4.3;1.4.3 Causality;26
3.4.4;1.4.4 Characterization;27
3.5;1.5 Convolutional Algebra and the Z–Transform
;30
3.6;1.6 Shifts, Transforms and Spectra;34
3.6.1;1.6.1 Shift–Invariance on Finite Index Sets
;36
3.6.2;1.6.2 Transforms and Spectra;41
3.7;References;48
4;Chapter 2:
Algebraic Preliminaries;51
4.1;2.1 What's in a Definition?;51
4.2;2.2 Set Theoretic Notation;52
4.3;2.3 Relations and Operations;55
4.3.1;2.3.1 Equivalence Relations and Partitions;55
4.3.2;2.3.2 Operations;56
4.4;2.4 Groups;58
4.4.1;2.4.1 Groups Within Groups;59
4.4.2;2.4.2 Group Morphisms;60
4.4.3;2.4.3 Groups and Geometry;61
4.5;2.5 Vector Spaces;62
4.5.1;2.5.1 Matrices of Vectors and Linear Transformations;62
4.5.2;2.5.2 Direct Sums of Subspaces;65
4.6;2.6 Posets, Lattices, and Boolean Algebras;69
4.6.1;2.6.1 From Posets to Lattices;69
4.6.2;2.6.2 Complemented and Distributive Lattices;71
4.6.3;2.6.3 Lattice of Subspaces of a Vector Space;72
4.7;2.7 Closing Remarks;74
4.8;References;75
5;Chapter 3:
Measurement, Modeling, and Metaphors;77
5.1;3.1 Archimedes and the Tortoise;77
5.2;3.2 The Representational Approach;78
5.2.1;3.2.1 Measuring Lengths;78
5.2.2;3.2.2 From Measurement to Modeling;80
5.2.3;3.2.3 Time and Space;82
5.2.4;3.2.4 Models in General;84
5.3;3.3 Metaphors;84
5.4;References;86
6;Chapter 4:
Symmetries, Automorphisms and Groups;88
6.1;4.1 Introduction;88
6.2;4.2 Symmetries and Automorphisms;90
6.3;4.3 Groups of Automorphisms;97
6.4;4.4 Symmetries of Linear Transformations;100
6.4.1;4.4.1 Symmetries and Symmetry Operations;100
6.4.2;4.4.2 Translation Operators;101
6.5;4.5 Symmetry Based Decompositions;103
6.5.1;4.5.1 Block–Diagonalizability and Invariant Subspaces
;104
6.5.2;4.5.2 Transformation Groups and Their Invariant Subspaces;105
6.5.3;4.5.3 Transformations with Symmetries;107
6.6;References;109
7;Chapter 5:
Representations of Finite Groups;111
7.1;5.1 The Notion of Representation;111
7.2;5.2 Matrix Representations of Groups;111
7.3;5.3 Automorphisms of a Vector Space;114
7.4;5.4 Group Representations in GL(V);114
7.5;5.5 Reducible and Irreducible Representations;118
7.6;5.6 Reducibility of Representations;121
7.7;5.7 Schur's Lemma and the Orthogonality Theorem;125
7.8;5.8 Characters and Their Properties;128
7.9;5.9 Constructing Irreducible Representations;132
7.10;5.10 Complete Reduction of Representations;136
7.11;5.11 Further on Reduction;145
7.12;References;147
8;Chapter 6:
Signal Processing and Representation Theory;148
8.1;6.1 Signals as Functions on Groups;149
8.2;6.2 Symmetries of Linear Equations;150
8.3;6.3 Fast Discrete Signal Transforms;152
8.4;References;154
9;Appendix A:
Parentheses, Their Proper Pairing, and Associativity ;156
9.1;A.1 Proper Pairing of Parentheses;156
9.2;A.2 Parentheses and the Associative Law;158
9.3;References;160
10;Index;161