# Sinha V.P. Symmetries and Groups in Signal Processing. An Introduction *PDF*

Издательство Information Science Reference, 2010, -172 pp.

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 inmy 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 on signals and 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 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 signals of interest to 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. Next I discuss 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.

Signals and Signal Spaces: A Structural Viewpoint

Algebraic Preliminaries

Measurement, Modeling, and Metaphors

Symmetries, Automorphisms and Groups

Representations of Finite Groups

Signal Processing and Representation Theory

Parentheses, Their Proper Pairing, and Associativity

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 inmy 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 on signals and 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 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 signals of interest to 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. Next I discuss 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.

Signals and Signal Spaces: A Structural Viewpoint

Algebraic Preliminaries

Measurement, Modeling, and Metaphors

Symmetries, Automorphisms and Groups

Representations of Finite Groups

Signal Processing and Representation Theory

Parentheses, Their Proper Pairing, and Associativity

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