John wiley sons fourier analysis on finite groups with applications in signal processing (2005) ddu ocr 7 0 lotb

Fourier Analysis on Finite Groups with Applications in Signal Processing and System Design Radomir S Stankovid Claudio Moraga Jaakko T Astola IEEE PRESS A JOHN WILEY & SONS, MC., PUBLICATION This Page Intentionally Left Blank Fourier Analysis on Finite Groups with Applications in Signal Processing and System Design IEEE Press 445 Hoes Lane Piscataway, NJ 08854 IEEE Press Editorial Board Stamatios V Kartalopoulos, Editor in Chief M Akay J B Anderson R J Baker J E Brewer M E El-Hawary R Leonardi M Montrose M S Newman F M B Periera C Singh S Tewksbury G Zobrist Kenneth Moore, Director of Book and Information Services (BIS) Catherine Faduska, Senior Acquisitions Editor Anthony VenGraitis, Project Editor Fourier Analysis on Finite Groups with Applications in Signal Processing and System Design Radomir S Stankovid Claudio Moraga Jaakko T Astola IEEE PRESS A JOHN WILEY & SONS, MC., PUBLICATION Copyright 2005 by the Institute of Electrical and Electronics Engineers, Inc All rights reserved Published by John Wiley & Sons, Inc., Hoboken, New Jersey Published simultaneously in Canada No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvcrs, MA 01923, (978) 750-8400, fax (978) 646-8600, or on the web at www.copyright.com Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 11 River Street, Hoboken, NJ 07030, (201) 748-601 I , fax (201) 748-6008 or online at http://www.wiley.comlgo/permission Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in 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in electronic format For information about Wiley products, visit our web site at www.wiley.com Library of’Congress Cataloging-in-Publication is available ISBN-13 978-0-471-69463-2 ISBN-10 0-471-69463-0 Printed in the United States of America Preface We believe that the group-theoretic approach to spectral techniques and, in particular, Fourier analysis, has many advantages, for instance, the possibility for a unified treatment of various seemingly unrelated classes of signals This approach allows to extend the powerful methods of classical Fourier analysis to signals that are defined on very different algebraic structures that reflect the properties of the modelled phenomenon Spectral methods that are based on finite Abelian groups play a very important role in many applications in signal processing and logic design In recent years the interest in developing methods that are based on Finite non-Abelian groups has been steadily growing, and already, there are many examples of cases where the spectral methods based only on Abelian groups not provide the best performance This monograph reviews research by the authors in the area of abstract harmonic analysis on finite non-Abelian groups Many of the results discussed have already appeared in somewhat different forms in journals and conference proceedings We have aimed for presenting the results here in a consistent and self-contained way, with a uniform notation and avoiding repetition of well-known results from abstract harmonic analysis, except when needed for derivation, discussion and appreciation of the results However, the results are accompanied, where necessary or appropriate, with a short discussion including comments concerning their relationship to the existing results in the area The purpose of this monograph is to provide a basis for further study in abstract harmonic analysis on finite Abelian and non-Abelian groups and its applications V vi PREFACE V Chapter > Fitrictroriol E\pres\ror7s or1 Qiiotrr 171onGroirps Fig 0.7 Relationships among the chapters The monograph will hopefully stimulate new research that results in new methods and techniques to process signals modelled by functions on finite non-Abelian groups Fig 0.1 shows relationships among the chapters RADOMIRS STANKOVIC, CLAUDIOMORAGA,JAAKKOT ASTOLA NiS Dorrrniind Tirrrpere Acknowledgments Prof Mark G Karpovsky and Prof Lazar A Trachtenberg have traced in a series of publications chief directions in research in Fourier analysis on finite non-Abelian groups We are following these directions in our research in the area, in particular in extending the theory of Gibbs differentiation to non-Abelian structures For that, we are very indebted to them both The first author is very grateful to Prof Paul L Butzer, Dr J Edmund Gibbs, and Prof Tsutomu Sasao for continuous support in studying and research work The authors thank Dragan JankoviC of Faculty of Electronics, University of NiS, Serbia, for programming and performing the experiments partially reported in this monograph A part of the work towards this monograph was done during the stay of R S StankoviC at the Tampere International Center for Signal Processing (TICSP) The support and facilities provided by TICSP are gratefully acknowledged R.S.S.,C.M, J.T.A Vii 222 HILBERT TRANSFORM ON FINlTE GROUPS Note that there are functions from l1 whose Hilbert transforms defined by (8.2) not belong to l l An example is f(x) = as is noted in [2] Recall that & t1.p ( / m exp (*dx)) Jz;; 03 = Lrn -2i lim - t-0 Jz;r sinrx) dx From there it can be written at least formally: (8.4) where F is the Fourier transform operator, * denotes the convolution product on R, and the convolution integral is understood in the sense of Cauchy principal value In the case of periodic functions with the period equal to 27r the convolution kernel used to define the corresponding Hilbert transform is {cot ;} The approach of defining the Hilbert transform in transform domain as the multiplication by a sign function, that is by employing the relation (8.3) was used as the starting point for the introduction of a discrete Hilbert transform, i.e., the Hilbert transform for functions on finite Abelian groups It is important to note that definitions appearing in [3] and [ I ] [4], [ ] , [6] are based upon the differently defined sign functions and they coincide only in the case of cyclic groups Recalling that the real line R exhibits the structure of a locally compact Abelian group, it can be concluded that the Hilbert transform for real-variable functions and the discrete Hilbert transform can be considered uniformly as the Hilbert transform on Abelian groups However, the above discussed group-theoretic approach of introducing the Hilbert transform on Abelian groups through the product in the transform domain by a suitably defined sign function, can hardly be used further to extend the concept to finite non-Abelian groups That fact becomes obvious if we recall that unlike Abelian groups, the domain r of the Fourier transform S f of a function f on a finite not necessarily Abelian group G may not have any algebraic structure suitable to define a multiplication in it which in turn can be mapped into a convolution in the group Therefore, we have suggested in [ ] just the opposite way, we have defined a Hilbert transform on a finite non-Abelian group as the pointwise multiplication of a given function by a suitably defined sign function in the group As was shown in [8], an analysis of the properties of the thus defined transform justifies to consider it as a proper counterpart of the Hilbert transform on R or on finite Abelian groups Therefore, we are encouraged to suggest this "opposite way to be actually a "proper" way to define a Hilbert-like transform for functions on both Abelian and non-Abelian groups permitting the considerations of these two cases in a uniform way Recall that the same approach was already used for Hilbcrt transform on R in some particular engineering applications as for example signal filtering In that way two aims are reached First, the main properties of the "classically" defined Hilbert transform on Abclian groups are preserved by the "new" transform, SOME RESULTS OF FOURIER ANALYSIS ON FINITE NON-ABELIAN GROUPS 223 and the concept is extended to finite non-Abelian groups Further, as it will be shown below, the same approach can be used to introduce a Hilbert-like transform for functions mapping a given finite non-Abelian group into a finite field admitting the existence of a Fourier transform 8.1 SOME RESULTS OF FOURIER ANALYSIS ON FINITE NON-ABELIAN GROUPS For the sake of completeness of presentation in this section we disclose several further results of Fourier analysis on finite non-Abelian groups which are somewhat restricted counterparts of the corresponding results on finite Abelian groups Recall again that the Fourier transform SF is defined on r and, thus, cannot be regarded as a function on a group and, therefore, some of the properties valid on Abelian groups are non-existent when the group is no longer commutative Recalling the bijection V from non-Abelian group G of order g onto the subset M = (0, ,g - l} of integers adopted in this monograph, note that the natural ordering "