SIGNALS AND COMMUNICATION TECHNOLOGY

Transcription

1 SIGNALS AND COMMUNICATION TECHNOLOGY Forothertitlespublishedinthisseries,goto

2 Virendra P. Sinha Symmetries and Groups in Signal Processing An Introduction 123

3 Prof.Virendra P. Sinha Dhirubhai Ambani Institute of Information and Comm. Tech. Near Indroda Circle Gandhinagar, Gujarat, India vp ISBN e-isbn DOI / SpringerDordrechtHeidelbergLondonNewYork Library of Congress Control Number: c Springer Science+Business Media B.V No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Cover design: SPi Publisher Services Printed on acid-free paper Springer is part of Springer Science+Business Media (

4 To my grandsons Rohan, Ishaan, and Shloak, for a whiff of algebra

5 Preface The field of signal processing, as it stands today, abounds in varied generalizations of system theoretic concepts that can be said to rest on the notion of symmetry, and on group theoretic methods of exploiting symmetries. A wide range of such generalizations and developments rely centrally on a transition from the classical Fourier theory to the modern theory of non-commutative harmonic analysis, with its roots in the representation theory of groups. In the framework that emerges through this transition, all the basic notions transforms, convolutions, spectra, and so on, carry over in a form that allows a wide variety of interpretations, subsuming the old ones and admitting new ones. This book is an introductory treatment of a selection of topics that together serve to provide in my view a background for a proper understanding of the theoretical developments within this framework. Addressed primarily to beginning graduate students in electrical and communication engineering, it is meant to serve as a bridge between what they know from their undergraduate years, and what lies ahead for them in their graduate studies, be it in the area of signal processing, or in related areas such as image processing and image understanding, coding theory, fault diagnostics, and the theory of algorithms and computation. I assume that the reader is familiar with the theory of linear time invariant continuous time and discrete time systems as it is generally taught in a basic undergraduate course onsignalsand systems. There are no mathematical prerequisites beyond what they would have learnt in their undergraduate years. Familiarity with rudiments of linear algebra would be helpful, but even that is not necessary; whatever of it is needed in the book, they can pick up on their own as they go along. A point about pedagogy. In teaching mathematical concepts to engineering students, a plan of action that is commonly followed is to separate what is regarded as mathematics per se from its applications, and to introduce the two separately in alternation. Thus one first introduces them to differential equations, linear or modern algebra, or discrete mathematics, on abstract lines as they would appear in a mathematics text, and then one turns to their applications in solving engineering problems. This plan works well, perhaps just about, when the students are fresh to their engineering studies. But at a stage when they have already had their first exposure to basic engineering principles, it has an inhibiting influence, both on their pace of learning and on their motivation for it. Faced with a new abstract concept at vii

6 viii Preface this stage, they instinctively begin to look for a pattern in which the new will fit in smoothly, and through analogies and metaphors, with what they already intuitively know of their main subjects. They look for the sort of experience that, for instance, they had at the time they learnt their elements of Euclidean geometry, when they saw how the theorem on triangle inequality, logically derived from the axioms, agreed with what they knew all along about triangles as they drew them on paper. It is the same experience which they had while learning elements of graph theory concurrently with network analysis. More generally, they look for a backdrop of intuition against which they would like the abstractions to be set and to unfold. Study of new mathematical structures becomes, as result, an easier and more pleasant task for such students if the abstractions are presented seamlessly with their concrete engineering interpretations. I have tried to keep this point in mind in my presentation in this book. The contents of the book are organized as follows. Chapter 1 is devoted to an overview of basic signal processing concepts in an algebraic setting. Very broadly, it is an invitation to the reader to revisit these concepts in a manner that places in view their algebraic and structural foundations. The specific question that I examine is the following: How should system theoretic concepts be formulated or characterized so that they are, in the first instance, independent of details such as whether the signalsof interestto us are discrete, discrete finite, one dimensional,or multi dimensional. Implicit in this question is a finer question about representation of signals that I discuss first, focussing attention on the distinction between what signals are physically, and the models by which they are represented. NextIdiscuss those aspects of linearity, translation invariance, causality, convolutions, and transforms, that are germane to their generalizations, in the context of discrete signals. Chapter 2 presents in a nutshell those basic algebraic concepts that are relied upon in a group theoretic interpretation of the concept of symmetry. In Chapter 3, the points made in Chapter 1 about the choice of mathematical models is taken up again. Chapter 4 is about symmetry and its algebraic formalization. Representation theory of finite groups is introduced in Chapter 5. Chapter 6 gives a final look at the role of group representation theory in signal processing. Acknowledgements This book has grown out of notes written for a transition course offered to beginning graduate students in the Department of Electrical Engineering of IIT Kanpur, and also for a similar course at DA-IICT Gandhinagar. Creating these courses has been a very fruitful learning experience for me, and I have benefitted enormously from discussions and interactions with the participating students at both the places. I can not thank them enough for their active involvement. My special thanks are due to Ratnik Gandhi and Pratik Shah, both currently working for their doctoral degrees at DA-IICT, for their regular interactions and feedback. Ratnik has been my know-all

7 Preface ix man for the subtleties of latex, and has acted as a sounding board for me at various stages of writing. Writing has its lows, when one is held back by bouts of perfectionism. Constant nudging to get on with the job is in such times a pragmatic antidote. My deep appreciation for that to Dr. A.P. Kudchadker, former Director of DA-IICT, and to Dr. S.C. Sahasrabudhe, the present Director. Amongst colleagues, and former students, there are many who have directly and critically influenced my thought processes that have prompted this text. I gratefully acknowledge receiving constructive inputs from Drs. S.D. Agashe, S. Chatterji, S.K. Mullick, P. Ramakrishna Rao, K.R. Sarma, M.U. Siddiqi, V.R. Sule, and K.S. Venkatesh. From the time I first put forth my book proposal to Springer in November 2008, it has been a pleasure interacting with Editor Mark de Jongh. My thanks go to him, and to Mrs. Cindy Zitter, his Senior Assistant, for benignly putting up with delays in my self-imposed deadlines, and for all the meticulous support. Finally, to my wife, Meera, and daughters, Shubhra and Shalini, I am immeasurably grateful for being at one with me in negotiating the rhythms of academic life. Gandhinagar April, 2010 Virendra P. Sinha

8 Contents Preface vii 1 Signals and Signal Spaces: A Structural Viewpoint What Is a Signal? Spaces and Structures Signal Spaces and Systems Linearity, Shift Invariance and Causality Linearity Shift Invariance Causality Characterization Convolutional Algebra and the Z Transform Shifts, Transforms and Spectra Shift Invariance on Finite Index Sets Transforms and Spectra Algebraic Preliminaries What s in a Definition? Set Theoretic Notation Relations and Operations Equivalence Relations and Partitions Operations Groups Groups Within Groups Group Morphisms Groups and Geometry Vector Spaces Matrices of Vectors and Linear Transformations Direct Sums of Subspaces Posets, Lattices, and Boolean Algebras From Posets to Lattices Complemented and Distributive Lattices Lattice of Subspaces of a Vector Space Closing Remarks xi

9 xii Contents 3 Measurement, Modeling, and Metaphors Archimedes and the Tortoise The Representational Approach Measuring Lengths From Measurement to Modeling Time and Space Models in General Metaphors Symmetries, Automorphisms and Groups Introduction Symmetries and Automorphisms Groups of Automorphisms Symmetries of Linear Transformations Symmetries and Symmetry Operations Translation Operators Symmetry Based Decompositions Block Diagonalizability and Invariant Subspaces Transformation Groups and Their Invariant Subspaces Transformations with Symmetries Representations of Finite Groups The Notion of Representation Matrix Representations of Groups Automorphisms of a Vector Space Group Representations in GL(V) Reducible and Irreducible Representations Reducibility of Representations Schur s Lemma and the Orthogonality Theorem Characters and Their Properties Constructing Irreducible Representations Complete Reduction of Representations Further on Reduction Signal Processing and Representation Theory Signals as Functions on Groups Symmetries of Linear Equations Fast Discrete Signal Transforms A Parentheses, Their Proper Pairing, and Associativity 151 A.1 Proper Pairing of Parentheses A.2 Parentheses and the Associative Law Index 157