Differential geometry in array processing

231 Pages·2004·5.12 MB·English

by  Manikas A.

DIFFERENTIAL GEOMETRY IN
ARRAY
PROCESSING
This page intentionally left blank
Athanassios Manikas
Imperial College London, UK
DIFFERENTIAL GEOMETRY IN
A R R A Y
P R O C E S S I N G
Imperial College Press
Published by
Imperial College Press
57 Shelton Street
Covent Garden
London WC2H 9HE
Distributed by
World Scientific Publishing Co. Pte. Ltd.
5 Toh Tuck Link, Singapore 596224
USA office: 27 Warren Street, Suite 401–402, Hackensack, NJ 07601
UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
DIFFERENTIAL GEOMETRY IN ARRAY PROCESSING
Copyright © 2004 by Imperial College Press
All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means,
electronic or mechanical, including photocopying, recording or any information storage and retrieval
system now known or to be invented, without written permission from the Publisher.
For photocopying of material in this volume, please pay a copying fee through the Copyright
Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to
photocopy is not required from the publisher.
ISBN 1-86094-422-1
ISBN 1-86094-423-X (pbk)
Typeset by Stallion Press
E-mail: [email protected]
Printed in Singapore.
34_Differential Geometry.pmd 1 10/4/2005, 6:12 PM
July6,2004 9:31 WSPC/BookTrimSizefor9inx6in fm
To my wife, Eleni
v
This page intentionally left blank
July6,2004 9:31 WSPC/BookTrimSizefor9inx6in fm
Preface
During the past few decades, there has been significant research into
sensor array signal processing, culminating in the development of super-
resolution array processing, which asymptotically exhibits infinite resolu-
tion capabilities.
Array processing has an enormous set of applications and has recently
experienced an explosive interest due to the realization that arrays have a
major role to play in the development of future communication systems,
wireless computing, biomedicine (bio-array processing) and environmental
monitoring.
However,the“heart”ofanyapplicationisthestructureoftheemployed
array of sensors and this is completely characterized [1] by the array mani-
fold.Thearraymanifold isafundamentalconceptandisdefinedasthelocus
ofalltheresponsevectorsofthearrayoverthefeasiblesetofsource/signal
parameters. In view of the nature of the array manifold and its signifi-
cance in the area of array processing and array communications, the role
of differential geometry as the most particularly appropriate analysis tool,
cannot be over-emphasized.
Differential geometry is a branch of mathematics concerned with the
application of differential calculus for the investigation of the properties of
geometric objects (curves, surfaces, etc.) referred to, collectively, as “mani-
folds”. This is a vast subject area with numerous abstract definitions,
theorems,notationsandrigorousformalproofs[2,3]andismainlyconfined
to the investigation of the geometrical properties of manifolds in three-
dimensional Euclidean space R3 and in real spaces of higher dimension.
However, the array manifolds are embedded not in real, but in
N-dimensional complex space (where N is the number of sensors). There-
fore, by extending the theoretical framework of R3 to complex spaces, the
vii
July6,2004 9:31 WSPC/BookTrimSizefor9inx6in fm
viii Differential Geometry in Array Processing
underlying and under-pinning objective of this book is to present a sum-
mary of those results of differential geometry which are exploitable and of
practicalinterestinthestudyoflinear,planarandthree-dimensionalarray
geometries.
Thanassis Manikas — London 2003
[email protected]
http://skynet.ee.imperial.ac.uk/manikas.html
Acknowledgments
This book is based on a number of publications (presented under a unified
framework) which I had over the past few years with some of my former
researchstudents.TheseareDr.J.Dacos,Dr.R.Karimi,Dr.C.Proukakis,
Dr.N.Dowlut,Dr.A.Alexiou,Dr.V.LefkaditesandDr.A.Sleiman.Iwish
toexpressmypleasureinhavinghadtheopportunityinworkingwiththem.
I am indebted to my research associates Naveendra and Jason Ng as
wellastomyMScstudentVincentChanforreading,atvariousstages,the
manuscriptandformakingconstructivesuggestionsonhowtoimprovethe
presentation material.
Finally,IshouldliketothankDr.P.Wilkinsonwhohadthepatienceto
read the final version of the manuscript. His effort is greatly appreciated.
Keywords
lineararrays,non-lineararrays,planararrays,arraymanifolds,differential
geometry, array design, array ambiguities, array bounds, resolution and
detection.
July6,2004 9:31 WSPC/BookTrimSizefor9inx6in fm
Contents
Preface vii
1. Introduction 1
1.1 Nomenclature. . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Main Abbreviations . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Array of Sensors — Environment . . . . . . . . . . . . . . . 5
1.4 Pictorial Notation . . . . . . . . . . . . . . . . . . . . . . . 8
1.4.1 Spaces/Subspaces . . . . . . . . . . . . . . . . . . . 8
1.4.2 Projection Operator . . . . . . . . . . . . . . . . . . 9
1.5 Principal Symbols . . . . . . . . . . . . . . . . . . . . . . . 11
1.6 Modelling the Array Signal-Vector and Array Manifold . . 11
1.7 Significance of Array Manifolds . . . . . . . . . . . . . . . . 17
1.8 An Outline of the Book . . . . . . . . . . . . . . . . . . . . 19
2. Differential Geometry of Array Manifold Curves 22
2.1 Manifold Curve Representation — Basic Concepts . . . . . 22
2.2 Curvatures and Coordinate Vectors in CN . . . . . . . . . . 26
2.2.1 Number of Curvatures and Symmetricity
in Linear Arrays . . . . . . . . . . . . . . . . . . . . 26
2.2.2 “Moving Frame” and Frame Matrix . . . . . . . . . 28
2.2.3 Frame Matrix and Curvatures. . . . . . . . . . . . . 29
2.2.4 Narrow and Wide Sense Orthogonality . . . . . . . . 31
2.3 “Hyperhelical” Manifold Curves . . . . . . . . . . . . . . . 33
2.3.1 Coordinate Vectors and Array Symmetricity. . . . . 37
2.3.2 Evaluating the Curvatures of Uniform Linear Array
Manifolds . . . . . . . . . . . . . . . . . . . . . . . . 38
ix