Modern Signal Processing

346 Pages·2004·6.899 MB·English

by  Daniel N. Rockmore, Dennis M. Healy Jr

Signalprocessingisaubiquitouspartofmoderntechnology.Itsmathematical
basis and many areas of application are the subject of this book, based on a
series of graduate-level lectures held at the Mathematical Sciences Research
Institute. Emphasis is on current challenges, new techniques adapted to new
technologies,and certainrecentadvancesin algorithms andtheory.Thebook
coverstwomainareas:computationalharmonicanalysis,envisionedasatech-
nology for efficiently analyzing real data using inherent symmetries; and the
challenges inherent in the acquisition, processing and analysis of images and
sensing data in general—ranging from sonar on a submarine to a neurosci-
entist’s fMRI study.

Mathematical Sciences Research Institute
Publications
46
Modern Signal Processing
Mathematical Sciences Research Institute Publications
1 Freed/Uhlenbeck:Instantons and Four-Manifolds,secondedition
2 Chern(ed.):Seminar on Nonlinear Partial Differential Equations
3 Lepowsky/Mandelstam/Singer(eds.):Vertex Operators in Mathematics and Physics
4 Kac(ed.):Infinite Dimensional Groups with Applications
5 Blackadar:K-Theory for Operator Algebras,secondedition
6 Moore(ed.):Group Representations, Ergodic Theory, Operator Algebras, and
Mathematical Physics
7 Chorin/Majda(eds.):Wave Motion: Theory, Modelling, and Computation
8 Gersten(ed.):Essays in Group Theory
9 Moore/Schochet:Global Analysis on Foliated Spaces
10–11 Drasin/Earle/Gehring/Kra/Marden(eds.):Holomorphic Functions and Moduli
12–13 Ni/Peletier/Serrin(eds.):Nonlinear Diffusion Equations and Their Equilibrium States
14 Goodman/delaHarpe/Jones:Coxeter Graphs and Towers of Algebras
15 Hochster/Huneke/Sally(eds.):Commutative Algebra
16 Ihara/Ribet/Serre(eds.):Galois Groups over
Q
17 Concus/Finn/Hoffman(eds.):Geometric Analysis and Computer Graphics
18 Bryant/Chern/Gardner/Goldschmidt/Griffiths:Exterior Differential Systems
19 Alperin(ed.):Arboreal Group Theory
20 Dazord/Weinstein(eds.):Symplectic Geometry, Groupoids, and Integrable Systems
21 Moschovakis(ed.):Logic from Computer Science
22 Ratiu(ed.):The Geometry of Hamiltonian Systems
23 Baumslag/Miller(eds.):Algorithms and Classification in Combinatorial Group Theory
24 Montgomery/Small(eds.):Noncommutative Rings
25 Akbulut/King:Topology of Real Algebraic Sets
26 Judah/Just/Woodin(eds.):Set Theory of the Continuum
27 Carlsson/Cohen/Hsiang/Jones(eds.):Algebraic Topology and Its Applications
28 Clemens/Koll´ar(eds.):Current Topics in Complex Algebraic Geometry
29 Nowakowski(ed.):Games of No Chance
30 Grove/Petersen(eds.):Comparison Geometry
31 Levy(ed.):Flavors of Geometry
32 Cecil/Chern(eds.):Tight and Taut Submanifolds
33 Axler/McCarthy/Sarason(eds.):Holomorphic Spaces
34 Ball/Milman(eds.):Convex Geometric Analysis
35 Levy(ed.):The Eightfold Way
36 Gavosto/Krantz/McCallum(eds.):Contemporary Issues in Mathematics Education
37 Schneider/Siu(eds.):Several Complex Variables
38 Billera/Bj¨orner/Green/Simion/Stanley(eds.):New Perspectives in Geometric
Combinatorics
39 Haskell/Pillay/Steinhorn(eds.):Model Theory, Algebra, and Geometry
40 Bleher/Its(eds.):Random Matrix Models and Their Applications
41 Schneps(ed.):Galois Groups and Fundamental Groups
42 Nowakowski(ed.):More Games of No Chance
43 Montgomery/Schneider(eds.):New Directions in Hopf Algebras
44 Buhler/Stevenhagen(eds.):Algorithmic Number Theory
45 Jensen/Ledet/Yui:Generic Polynomials: Constructive Aspects of the Inverse Galois
Problem
46 Rockmore/Healy(eds.):Modern Signal Processing
47 Uhlmann(ed.):Inside Out: Inverse Problems and Applications
48 Gross/Kotiuga:Electromagnetic Theory and Computation: A Topological Approach
49 Darmon(ed.):Rankin L-Series
Volumes 1–4 and 6–27 are published by Springer-Verlag
Modern Signal Processing
Edited by
Daniel N. Rockmore
Dartmouth College
Dennis M. Healy, Jr.
University of Maryland
Series Editor
Silvio Levy
Daniel N. Rockmore Mathematical Sciences
Department of Mathematics Research Institute
Dartmouth College 17 Gauss Way
Hanover, NH 03755 Berkeley, CA 94720
United States United States
[email protected]
MSRI Editorial Committee
Dennis M. Healy, Jr. Hugo Rossi (chair)
Department of Mathematics Alexandre Chorin
University of Maryland Silvio Levy
College Park, MD 20742-4015 Jill Mesirov
United States Robert Osserman
[email protected] Peter Sarnak
The Mathematical Sciences Research Institute wishes to acknowledge support by
the National Science Foundation. This material is based upon work supported by
NSF Cooperative Agreement DMS-9810361.
published by the press syndicate of the university of cambridge
The Pitt Building, Trumpington Street, Cambridge, United Kingdom
cambridge university press
The Edinburgh Building, Cambridge CB2 2RU, UK
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(cid:176)c Mathematical Sciences Research Institute 2004
Printed in the United States of America
A catalogue record for this book is available from the British Library.
Library of Congress Cataloging in Publication data available
ISBN 0 521 82706X hardback
ModernSignalProcessing
MSRIPublications
Volume46,2003
Contents
Introduction ix
D. Rockmore and D. Healy
Hyperbolic Geometry, Nehari’s Theorem, Electric Circuits, and Analog
Signal Processing 1
J. Allen and D. Healy
Engineering Applications of the Motion-Group Fourier Transform 63
G. Chirikjian and Y. Wang
Fast X-Ray and Beamlet Transforms for Three-Dimensional Data 79
D. Donoho and O. Levi
Fourier Analysis and Phylogenetic Trees 117
S. Evans
Diffuse Tomography as a Source of Challenging Nonlinear Inverse
Problems for a General Class of Networks 137
A. Gru¨nbaum
An Invitation to Matrix-valued Spherical Functions 147
A. Gru¨nbaum, I. Pacharoni and J. Tirao
Image Registration for MRI 161
P. Kostelec and S. Periaswamy
Image Compression: The Mathematics of JPEG 2000 185
Jin Li
Integrated Sensing and Processing for Statistical Pattern Recognition 223
C. Priebe, D. Marchette, and D. Healy
Sampling of Functions and Sections for Compact Groups 247
D. Maslen
The Cooley–Tukey FFT and Group Theory 281
D. Maslen and D. Rockmore
vii
viii CONTENTS
Signal Processing in Optic Fibers 301
U. O¨sterberg
The Generalized Spike Process, Sparsity and Statistical Independence 317
N. Saito
ModernSignalProcessing
MSRIPublications
Volume46,2003
Hyperbolic Geometry, Nehari’s Theorem,
Electric Circuits, and Analog Signal Processing
JEFFERY C. ALLEN AND DENNIS M. HEALY, JR.
Abstract. Underlyingmanyofthecurrentmathematicalopportunitiesin
digital signal processing are unsolved analog signal processing problems.
For instance, digital signals for communication or sensing must map into
an analog format for transmission through a physical layer. In this layer
we meet a canonical example of analog signal processing: the electrical
engineer’s impedance matching problem. Impedance matching is the de-
signofanalogsignalprocessingcircuitstominimizelossanddistortionas
the signal moves from its source into the propagation medium. This pa-
per works the matching problem from theory to sampled data, exploiting
linksbetweenH∞ theory,hyperbolicgeometry,andmatchingcircuits. We
applyJ.W.Helton’ssignificantextensionsofoperatortheory,convexanal-
ysis,andoptimizationtheorytodemonstratenewapproachesandresearch
opportunitiesinthisfundamentalproblem.
Contents
1. The Impedance Matching Problem 2
2. A Synopsis of the H∞ Solution 4
3. Technical Preliminaries 8
4. Electric Circuits 12
5. H∞ Matching Techniques 27
6. Classes of Lossless 2-Ports 35
7. Orbits and Tight Bounds for Matching 39
8. Matching an HF Antenna 42
9. Research Topics 47
10. Epilogue 52
A. Matrix-Valued Factorizations 52
B. Proof of Lemma 4.4 55
C. Proof of Theorem 6.1 56
D. Proof of Theorem 5.5 56
References 59
Allen gratefully acknowledges support from ONR and the IAR Program at SCC San Diego.
HealywassupportedinpartbyONR.
1
2 JEFFERY C. ALLEN AND DENNIS M. HEALY, JR.
1. The Impedance Matching Problem
Figure 1 shows a twin-whip HF (high-frequency) antenna mounted on a su-
perstructure representative of a shipboard environment. If a signal generator is
connecteddirectlytothisantenna,notallthepowerdeliveredtotheantennacan
nn
beradiatedbytheantenna. Ifanimpedance mismatch existsbetweenthesignal nn
nn
generatorandtheantenna,someofthesignalpowerisreflectedfromtheantenna
back to the generator. To effectively use this antenna,
a matching circuit must be inserted between the signal
generator and antenna to minimize this wasted power.
Figure 2 shows the matching circuit connecting the
generator to the antenna. Port 1 is the input from the
generator. Port 2 is the output that feeds the antenna.
The matching circuit is called a 2-port. Because the
2-port must not waste power, the circuit designer only
considers lossless 2-ports. The mathematician knows
the lossless 2-ports as the 2×2 inner functions. The
matchingproblemistofindalossless2-portthattrans-
fers as much power as possible from the generator to
the antenna.
The mathematical reader can see antennas every-
where: on cars, on rooftops, sticking out of cell phones.
A realistic model of an antenna is extremely complex
because the antenna is embedded in its environment.
CourtesyofAntennaProducts
Fortunately,weonlyneedtoknowhowtheantennabe-
Figure 1
haves as a 1-port device. As indicated in Figure 2, the
antenna’s scattering function or reflectance s characterizes its 1-port behavior.
L
The mathematician knows s as an element in the unit ball of H∞.
L
Figure3displayss :jR →C ofanHFantennameasuredoverthefrequency
L √
range of 9 to 30 MHz. (Here j = + −1 because i is used for current.) At
each radian frequency ω = 2πf, where f is the frequency in Hertz, s (jω) is a
L
Antenna
s
G
Port 1 Port 2
Signal Lossless
2-Port
Generator
Matching
Circuit
s
L
Figure 2. An antenna connected to a lossless matching 2-port.