Symmetries and Groups in Signal Processing: An Introduction

By Virendra P. Sinha

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.

• Format: Hardcover
• Language: English
• ISBN: 9048194334
• ISBN13: 9789048194339
• Release Date: August 2010
• Publisher: Springer
• Length: 160 Pages
• Weight: 0.93 lbs.
• Dimensions: 0.4" x 6.1" x 9.2"