modern signal processing

modern signal processing

Signal processing is a ubiquitous part of modern technology. Its mathematical 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 certain recent advances in algorithms and theory. The book covers two main areas: computational harmonic analysis, envisioned as a tech- 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, second edition 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, second edition 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/de la Harpe/Jones: Coxeter Graphs and Towers of Algebras 15 Hochster/Huneke/Sally (eds.): Commutative Algebra 16 Ihara/Ribet/Serre (eds.): Galois Groups over 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 ro ckmore@cs.dartmouth.edu 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 dhealy@math.umd.edu 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 syndi cate of th e uni versi ty of cambridge The Pitt Building, Trumpington Street, Cambridge, United Kingdom cambridge university press The Edinburgh Building, Cambridge CB2 2RU, UK 40 West 20th Street, New York, NY 10011-4211, USA 477 Williamstown Road, Port Melbourne, VIC 3207, Australia Ruiz de Alarc´on 13, 28014 Madrid, Spain Do ck House, The Waterfront, Cape Town 8001, South Africa http://www.cambridge.org 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 Modern Signal Processing MSRI Publications Volume 46, 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. Gr ¨ unbaum An Invitation to Matrix-valued Spherical Functions 147 A. Gr ¨ unbaum, 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. ¨ Osterberg The Generalized Spike Process, Sparsity and Statistical Independence 317 N. Saito Modern Signal Processing MSRI Publications Volume 46, 2003 Hyperbolic Geometry, Nehari’s Theorem, Electric Circuits, and Analog Signal Processing JEFFERY C. ALLEN AND DENNIS M. HEALY, JR. Abstract. Underlying many of the current mathematical opportunities in 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- sign of analog signal processing circuits to minimize loss and distortion as the signal moves from its source into the propagation medium. This pa- per works the matching problem from theory to sampled data, exploiting links between H ∞ theory, hyperbolic geometry, and matching circuits. We apply J. W. Helton’s significant extensions of operator theory, convex anal- ysis, and optimization theory to demonstrate new approaches and research opportunities in this fundamental problem. 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. Healy was supported in part by ONR. 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 connected directly to this antenna, not all the power delivered to the antenna can be radiated by the antenna. If an impedance mismatch exists between the signal generator and the antenna, some of the signal power is reflected from the antenna back to the generator. To effectively use this antenna,    Courtesy of Antenna Products Figure 1 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 matching problem is to find a lossless 2-port that trans- 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. Fortunately, we only need to know how the antenna be- haves as a 1-port device. As indicated in Figure 2, the antenna’s scattering function or reflectance s L characterizes its 1-port behavior. The mathematician knows s L as an element in the unit ball of H ∞ . Figure 3 displays s L : jR → C of an HF antenna measured over the frequency 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 L (jω) is a s G s L Lossless 2-Port Matching Circuit Signal Generator Antenna Port 1 Port 2 Figure 2. An antenna connected to a lossless matching 2-port. [...]... most of this signal is reflected back from the antenna and so very little signal power is radiated Most signals are not pure tones, but may be represented in the usual way as a Fourier superposition of pure tones taken over a band of frequencies In this case, the reflectance function evaluated at each frequency in the band multiplies the corresponding frequency component of the incident signal The net... directly connecting the sinusoidal signal generator to the antenna If |sL (jω)| is near 0, almost no signal is reflected back by the antenna towards sL=lpd17fwd4_2 1 0.8 1.0 0.8 0.6 0.6 0.4 ℑ 0.2 0 −0.2 −0.4 −0.6 −0.8 −1 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 ℜ: f=9−30 MHz 0.4 0.6 0.8 1 Figure 3 The reflectance sL (jω) of an HF antenna the generator or, equivalently, almost all of the signal power passes through the... i1 vG v1 Lossless 2-port S= - i2 + s11 s12 s21 s22 v2 sL - s2 s1 Figure 6 Matching circuit and reflectances measures the response reflected from Port 2 when a unit signal is driving Port 2; s12 is the signal from Port 1 in response to a unit signal input to Port 2 If the 2-port is consumes power, it is called passive and its corresponding scattering matrix is a contraction on jR: S(jω)H S(jω) ≤ 1 0 0... after circuit at the problem and hopes for a lucky hit And there is always the nagging question: What is the best matching possible? Remarkably, “pure” mathematics has much to say about this analog signal processing problem 2 A Synopsis of the H ∞ Solution Our presentation of the impedance matching problem weaves together many diverse mathematical and technological threads This motivates beginning with... very least To take a concrete example, the circuit designer may match the HF antenna using a transformer as shown in Figure 4 If we put a signal into in Port 1 sG s1 sL Figure 4 An antenna connected to a matching transformer of the transformer and measure the reflected signal, their ratio is the scattering function s1 That is, s1 is how the antenna looks when viewed through the transformer The circuit... to the 2-port in Figure 12, define a1 r1 + a2 2-Port i1 i2 s11 s12 S= s21 s22 v1 - + r2 v2 b2 b1 Figure 12 The 2-port scattering formalism the incident signal (see [3, Eq 4.25a] and [4, page 234]): −1/2 1 a = 2 {R0 1/2 v + R0 i} (4–2) and the reflected signal (see [3, Eq 4.25b] and [4, page 234]): −1/2 b = 1 {R0 2 1/2 v − R0 i}, (4–3) with respect to the normalizing1 matrix R0 = r1 0 0 r2 The scattering... pair equals the current flowing out of the other wire We can imagine + i 1( t ) v1( t )  • • • + iN ( t ) vN ( t ) - Figure 7 The N -port characterizing such a box by supplying current and voltage input signals of given frequency at the various ports and observing the current and voltages induced at the other ports Mathematically, the N -port is defined as the collection N of voltage v(p) and current i(p)... multiplies the corresponding frequency component of the incident signal The net reflection is the superposition of the resulting component reflections To ensure that an undistorted version of the generated signal is radiated from the antenna, 4 JEFFERY C ALLEN AND DENNIS M HEALY, JR the circuit designer looks for a lossless 2-port that “pulls sL (jω) to 0 over all frequencies in the band.” As a general rule, . Analog Signal Processing JEFFERY C. ALLEN AND DENNIS M. HEALY, JR. Abstract. Underlying many of the current mathematical opportunities in digital signal processing are unsolved analog signal processing. Rockmore vii viii CONTENTS Signal Processing in Optic Fibers 301 U. ¨ Osterberg The Generalized Spike Process, Sparsity and Statistical Independence 317 N. Saito Modern Signal Processing MSRI Publications Volume. hardback Modern Signal Processing MSRI Publications Volume 46, 2003 Contents Introduction ix D. Rockmore and D. Healy Hyperbolic Geometry, Nehari’s Theorem, Electric Circuits, and Analog Signal Processing