Differential Geometry in Array Processing - Manikas A. - Imperial

231 Pages·2005·6.49 MB·English

by  Athanassios Manikas

DIFFERENTIAL GEOMETRY IN
ARRAY
PROCESSING
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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
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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,
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ISBN 1-86094-422-1
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Printed in Singapore.
34_Differential Geometry.pmd 1 10/4/2005, 6:12 PM
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To my wife, Eleni
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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 thearray 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
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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.
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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
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