Signal Approximation Using The Bilinear Transform

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Signal Approximation using the Bilinear TransformbyArchana VenkataramanSubmitted to the Department of Electrical Engineering and Computer Sciencein partial fulfillment of the requirements for the degree ofMasters of Engineering in Electrical Engineering and Computer Scienceat theMASSACHUSETTS INSTITUTE OF TECHNOLOGYAugust 2007c Archana Venkataraman, MMVII. All rights reserved.The author hereby grants to MIT permission to reproduce and distribute publiclypaper and electronic copies of this thesis document in whole or in part.Author . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Department of Electrical Engineering and Computer ScienceAugust 10, 2007Certified by . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Alan V. OppenheimMacVicar Faculty Fellow, Ford Professor of EngineeringDepartment of Electrical Engineering and Computer ScienceThesis SupervisorAccepted by . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Arthur C. SmithProfessor of Electrical EngineeringChairman, Department Committee on Graduate Theses

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Signal Approximation using the Bilinear TransformbyArchana VenkataramanSubmitted to the Department of Electrical Engineering and Computer Scienceon August 10, 2007, in partial fulfillment of therequirements for the degree ofMasters of Engineering in Electrical Engineering and Computer ScienceAbstractThis thesis explores the approximation properties of a unique basis expansion. The expansion implements a nonlinear frequency warping between a continuous-time signal and itsdiscrete-time representation according to the bilinear transform. Since there is a one-toone mapping between the continuous-time and discrete-time frequency axes, the bilinearrepresentation avoids any frequency aliasing distortions.We devote the first portion of this thesis to some theoretical properties of the bilinearrepresentation, including the analysis and synthesis networks as well as bounds on the basisfunctions. These properties are crucial when we further analyze the bilinear approximationperformance. We also consider a modified version of the bilinear representation in whichthe continuous-time signal is segmented using a short-duration window. This segmentation procedure affords greater time resolution and, in certain cases, improves the overallapproximation quality.In the second portion of this thesis, we evaluate the approximation performance of thebilinear representation in two different applications. The first is approximating instrumental music. We compare the bilinear representation to a discrete cosine transform basedapproximation technique. The second application is computing the inner product of twocontinuous-time signals for a binary detection problem. In this case, we compare the bilinearrepresentation with Nyquist sampling.Thesis Supervisor: Alan V. OppenheimTitle: MacVicar Faculty Fellow, Ford Professor of EngineeringDepartment of Electrical Engineering and Computer Science3

AcknowledgmentsI am greatly indebted to my research advisor, Professor Alan Oppenheim. His devotionto and concern for students is unparalleled by anyone else I have come across. He hascontinually pushed me to achieve my full potential, but at the same time, he has remaineda friendly and supportive mentor. His comments and ideas have helped to expand andenrich my thesis work, and his patience and encouragement have made my first year ingraduate school an extremely positive one.I would also like to thank Professor Vivek Goyal, head of the Signal Transformation andInformation Representation group at MIT. Not only was his class (Wavelets, Approximationand Compression) instrumental in shaping my final MEng thesis, but his friendly one-on-onediscussions have been invaluable to my research work and experience.I would like to acknowledge past and present members of the Digital Signal Processing Group (DSPG): Tom Baran, Petros Boufounos, Sourav Dey, Zahi Karam, Al Kharbouch, Jon Paul Kitchens, Joonsung Lee, Charlie Rohrs, Melanie Rudoy, Joe Sikora, EricStrattman, Dennis Wei and Matthew Willsey. Over the past two years, these individualshave sustained an exciting, collaborative and friendly research environment in which I couldcontinually challenge myself and broaden my horizons. I would especially like to thank Tomand Sourav for their insightful conversations, Melanie for always being there when I neededsomeone to chat with, Zahi for keeping my temper well-honed while somehow managing tomake me smile, and lastly, Eric for running the DSPG office smoothly and efficiently.A special thanks to Austin Che, who has been my personal LaTex guru for the the pastthree months. Without his help this document would have been infinitely more dauntingand tedious to compile. Thank you, also, for being my shoulder to lean on through all thehard times here at MIT.Finally, to my family: thank you for your love and support all these years. You were theones who first nurtured my curiosity and desire to learn. Your encouragement has played amajor role in all of my successes. At the same time, you have stood besides me through allthe turmoil and trials of my life thus far. Words cannot adequately express all that I oweyou, and I hope that one day, I can repay at least a fraction of all you have done for me.4

To my family5

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Contents1 Introduction1.115Signal Representation Using Basis Expansions . . . . . . . . . . . . . . . . .151.1.1Properties of an Orthonormal Signal Representation . . . . . . . . .161.1.2Properties of a Biorthogonal Signal Representation . . . . . . . . . .171.2The Necessity for Signal Approximation . . . . . . . . . . . . . . . . . . . .181.3Signal Representation using the Bilinear Transform . . . . . . . . . . . . . .181.4Thesis Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .192 The Bilinear Signal Representations2.12.2The Orthonormal Bilinear Signal Representation . . . . . . . . . . . . . . .212.1.1The Orthonormal Basis Functions . . . . . . . . . . . . . . . . . . .222.1.2Analysis and Synthesis Networks for the Orthonormal Expansion . .22The Biorthogonal Bilinear Signal Representation . . . . . . . . . . . . . . .242.2.1The Primal and Dual Basis Functions . . . . . . . . . . . . . . . . .252.2.2Analysis and Synthesis Networks for the Biorthogonal Expansion . .263 Properties of the Bilinear Signal Representations3.13.22131The Orthonormal Representation . . . . . . . . . . . . . . . . . . . . . . . .313.1.1Orthonormal Signal Space . . . . . . . . . . . . . . . . . . . . . . . .323.1.2Bounding the Basis Functions . . . . . . . . . . . . . . . . . . . . . .32The Biorthogonal Representation . . . . . . . . . . . . . . . . . . . . . . . .333.2.1Weighted Energy Relationship . . . . . . . . . . . . . . . . . . . . .333.2.2Biorthogonal Signal Space . . . . . . . . . . . . . . . . . . . . . . . .343.2.3Computing Inner Products using the Biorthogonal Expansion . . . .363.2.4Bounds on the Primal and Dual Basis Functions . . . . . . . . . . .377

4 MATLAB Implementation of Bilinear Analysis and Synthesis394.1The Analysis Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .394.2The Synthesis Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .404.3Limitations of the Continuous Time Approximations . . . . . . . . . . . . .404.3.1The Analysis Network . . . . . . . . . . . . . . . . . . . . . . . . . .414.3.2The Synthesis Network41. . . . . . . . . . . . . . . . . . . . . . . . .5 Approximation Properties of the Bilinear Representations5.143Approximation of Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . .445.1.1Linear vs. Nonlinear Approximation . . . . . . . . . . . . . . . . . .445.1.2Error Metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .455.1.3Qualitative Measure of Approximation Performance . . . . . . . . .465.2Effect of the Parameter, a . . . . . . . . . . . . . . . . . . . . . . . . . . . .465.3Exact Representation of a Signal using a Finite Number of DT ExpansionCoefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .495.4Deviating from an Exact Representation . . . . . . . . . . . . . . . . . . . .505.5Additional Signal Characteristics which Affect the Bilinear Approximation5.6Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .545.5.1Distribution of Energy over Time . . . . . . . . . . . . . . . . . . . .545.5.2Isolated Discontinuities . . . . . . . . . . . . . . . . . . . . . . . . .575.5.3Signal Decay Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . .60Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .646 The Windowed Bilinear Representation656.1The Window Function w(t) . . . . . . . . . . . . . . . . . . . . . . . . . . .666.2Relationship to Original Representation . . . . . . . . . . . . . . . . . . . .686.3Approximation Using the Windowed Representation . . . . . . . . . . . . .696.3.1Segmenting with a Rectangular Window . . . . . . . . . . . . . . . .706.3.2Comparison of a Rectangular, Bartlett and Raised Cosine Window .727 Approximation of Audio Signals777.1Details of the Simulation Setup . . . . . . . . . . . . . . . . . . . . . . . . .777.2Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .788

7.3Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .8 The Binary Detection Problem8.18.28087The Nyquist Signal Representation . . . . . . . . . . . . . . . . . . . . . . .898.1.1Ideal (Unconstrained) Nyquist Detection . . . . . . . . . . . . . . . .898.1.2Constrained Nyquist Detection . . . . . . . . . . . . . . . . . . . . .90Bilinear Matched Filtering - A Theoretical Analysis . . . . . . . . . . . . .918.2.1The Orthonormal Representation . . . . . . . . . . . . . . . . . . . .928.2.2The Biorthogonal Representation . . . . . . . . . . . . . . . . . . . .928.2.3Variations of the Bilinear Representations . . . . . . . . . . . . . . .949 Matched Filtering Simulations959.1Details of the Simulation Setup . . . . . . . . . . . . . . . . . . . . . . . . .959.2Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .979.3Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .102A The Family of Generalized Laguerre Polynomials105A.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .105A.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .106A.2.1 Orthogonality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .106A.2.2 Recurrence Relation . . . . . . . . . . . . . . . . . . . . . . . . . . .107A.2.3 Signal Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .107A.2.4 Bounds on the Generalized Laguerre Polynomials . . . . . . . . . . .108B Additional Properties of the Bilinear Representations109B.1 Representing Anti-causal Signals . . . . . . . . . . . . . . . . . . . . . . . .109B.1.1 The Orthonormal Representation . . . . . . . . . . . . . . . . . . . .109B.1.2 The Biorthogonal Representation . . . . . . . . . . . . . . . . . . . .110B.2 Conditions to Preserve Correlation . . . . . . . . . . . . . . . . . . . . . . .111B.3 Noise Analysis for the Analysis and Synthesis Networks . . . . . . . . . . .112B.3.1 The Low-pass Filter Stage . . . . . . . . . . . . . . . . . . . . . . . .112B.3.2 The All-pass Filter Stage . . . . . . . . . . . . . . . . . . . . . . . .114B.3.3 Combined Effect of Input and Component Noise . . . . . . . . . . .1159

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List of Figures2-1 λn (t) for different index values using a 1 . . . . . . . . . . . . . . . . . . .232-2 λn (t) for different values of a using n 5 . . . . . . . . . . . . . . . . . . .242-3 Orthonormal analysis network to compute the expansion coefficients. . . . .242-4 Orthonormal synthesis network to reconstruct a continuous-time signal fromits bilinear expansion coefficients. . . . . . . . . . . . . . . . . . . . . . . . .252-5 φn (t) for different index values using a 1 . . . . . . . . . . . . . . . . . . .262-6 φn (t) for different values of a using n 5 . . . . . . . . . . . . . . . . . . .272-7 hn (t) for different index values using a 1 . . . . . . . . . . . . . . . . . . .282-8 hn (t) for different values of a using n 5 . . . . . . . . . . . . . . . . . . .282-9 Biorthogonal analysis network to compute the bilinear expansion coefficients. 292-10 Biorthogonal synthesis network to reconstruct a continuous-time signal fromits bilinear expansion coefficients. . . . . . . . . . . . . . . . . . . . . . . . .293-1 First-order cascade used to compute the coefficients bg [n] for the biorthogonalinner product. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .364-1 Orthonormal basis functions using a 10 . . . . . . . . . . . . . . . . . . .425-1 Orthonormal Bilinear Expansion Coefficients for f (t) sin(10t) . . . . . . .475-2 Metrics of Linear and Nonlinear Approximation Performance when using theOrthonormal Expansion for f (t) sin(10t) . . . . . . . . . . . . . . . . . .485-3 Sorted Bilinear Coefficients for fk (t) tk e at , a 20 . . . . . . . . . . . . .515-4 Sorted Bilinear Coefficients for fk (t) t3 e kt , a 20 . . . . . . . . . . . . .525-5 Partial Sum Sequences for fk (t) t3 e kt , a 20 . . . . . . . . . . . . . . .535-6 Sorted Bilinear Coefficients for fk (t) e at sin(kt), a 20 . . . . . . . . . .555-7 Partial Sum Sequences for fk (t) e at sin(kt), a 20 . . . . . . . . . . . .5611

5-8 Sorted Bilinear Coefficients for Shifted sinc Functions, a 10 . . . . . . . . .585-9 Partial Sum Sequences for Shifted sinc Functions, a 10 . . . . . . . . . . .595-10 Original Signal (black) and its Bilinear Reconstructions (color) . . . . . . .605-11 Sorted Coefficients and Partial Sum Sequences for Rectangular Pulse, a 62.8 615-12 Sorted Bilinear Coefficients for f1 (t) and f2 (t) . . . . . . . . . . . . . . . . .625-13 Partial Sum Sequences for f1 (t) and f2 (t) . . . . . . . . . . . . . . . . . . .636-1 Segmenting the Original CT Signal using a Non-Overlapping Window. . . .656-2 Rectangular, Bartlett and Hanning Windows to Segment a CT Signal. . . .676-3 Original Continuous-Time Signal and its Shifted Version. . . . . . . . . . .687-1 Normalized Error f (t) fˆM (t) for the Piano Sound Clip . . . . . . . . . . .827-2 Normalized Error f (t) fˆM (t) for the Violin Sound Clip . . . . . . . . . . .837-3 Reconstructed Audio Segment of a Piano Sound Clip . . . . . . . . . . . . .847-4 Reconstructed Audio Segment of a Violin Sound Clip . . . . . . . . . . . . .858-1 Binary detection scenario. The transmitted signal r(t) is corrupted by AWGN.The received signal x(t) consists either of noise or the signal s(t) plus noise.879-1 Magnitude of S(jω) and Noise Power Spectrum with Sample Spacing T . . .959-2 ROC Curves for s1 (t); 5 DT Multiplies, a 100 . . . . . . . . . . . . . . . .989-3 ROC Curves for s2 (t); 25 DT Multiplies, a 100 . . . . . . . . . . . . . . .999-4 ROC Curves for s3 (t); 25 DT Multiplies, a 100 . . . . . . . . . . . . . . .1009-5 ROC Curves for s4 (t); 5 DT Multiplies, a 62.8 . . . . . . . . . . . . . . .101(α)106A-2 Ln (x) for different values of a using n 5 . . . . . . . . . . . . . . . . . .(α)107B-1 Orthonormal analysis network for anti-causal CT signals. . . . . . . . . . .110B-2 Orthonormal synthesis network for anti-causal CT signals. . . . . . . . . . .110B-3 Biorthogonal analysis network for anti-causal CT signals. . . . . . . . . . .111B-4 Biorthogonal synthesis network for anti-causal CT signals. . . . . . . . . . .111B-5 Block and Implementation Diagram for the Low-pass Filter Stage. . . . . .113B-6 Block and Implementation Diagram for the All-pass Filter Stage. . . . . . .114A-1 Ln (x) for different index values using α 0 . . . . . . . . . . . . . . . . .12

List of Tables6.1δON for f (t) sinc(100(t 0.5)) . . . . . . . . . . . . . . . . . . . . . . . .706.2δBiO for f (t) sinc(100(t 0.5)) . . . . . . . . . . . . . . . . . . . . . . . .716.3ǫ when using a Rectangular window for f (t) sinc(50(t 0.5)) . . . . . . .746.4ǫ when using a Bartlett triangular window for f (t) sinc(50(t 0.5)) . . .756.5ǫ when using a Raised Cosine window for f (t) sinc(50(t 0.5)) . . . . . .767.1Normalized Reconstruction Error for the Piano Sound Clip, a 8820 . . . .797.2Normalized Reconstruction Error for the Violin Sound Clip, a 14700 . . .8013

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Chapter 1IntroductionRecent technological advancements have allowed for a rapid growth in the speed and capabilities of digital processors. These improvements have made it easier, and often more efficient,to perform many signal processing operations in the digital domain. However, much of thevaluable real-world information encountered remains analog or continuous-time in nature.1.1Signal Representation Using Basis ExpansionsWe can circumvent the above dilemma by representing a continuous-time (CT) signal f (t)using a basis expansion as follows:f (t) Xf [n]ψn (t)(1.1)n The set of functions {ψn (t)} n is a countable set of basis functions and the coefficientsf [n] are the expansion coefficients with respect to a chosen basis.Since the basis functions are known a priori, the CT signal f (t) is completely specifiedby the discrete-time (DT) sequence of expansion coefficients f [n]. Therefore, it can now bemanipulated using DT operations.We assume that the reader is familiar with the following concepts related to basis expansions,as they will be referenced throughout the remainder of this thesis:We refer to analysis as the process of computing the expansion coefficients from theoriginal continuous-time signal and the given set of basis functions. Similarly, we refer15

to synthesis as the process of reconstructing a continuous-time signal from its expansioncoefficients according to Equation (1.1).Additionally, we define the standard inner product, in continuous and discrete time, ashf, gic hf, gid Z Xf (t)g(t)dt(1.2)f [n]g[n](1.3)n We consider two signals to be orthogonal if their standard inner product is zero, and weshall denote the signal energy as the standard inner product of a signal with itself.In the following subsections we introduce two commonly-used types of basis expansions;specifically the orthonormal and the biorthogonal signal representations. We assume thatall signals and all sequences have finite energy and are real-valued.1.1.1Properties of an Orthonormal Signal RepresentationIn an orthonormal expansion, the set of basis functions {λn (t)} n are chosen to satisfythe condition 1,hλn , λm ic 0,n m(1.4)otherwiseIn particular, Equation (1.4) indicates that the basis functions are mutually orthogonal toeach other and have unit energy.One advantage of an orthonormal representation is that the expansion coefficients canbe obtained via an inner product with the respective basis function. Mathematically, ifPf (t) m f [m]λm (t), then it follows from Equation (1.4) thathf, λn ic Xmf [m]hλm , λn ic f [n](1.5)An orthonormal representation will also preserve the standard inner product betweenPPcontinuous time and discrete time. Namely, if f (t) n f [n]λn (t) and g(t) n g[n]λn (t),then we havehf, gic hf, gid(1.6)The above property is extremely useful in our consideration of matched filtering applications16

since we can compute the inner product of two CT signals as the inner product of theirexpansion coefficient sequences.1.1.2Properties of a Biorthogonal Signal RepresentationIn a biorthogonal expansion, the continuous-time signal is represented asf (t) Xf [n]φn (t)(1.7)n For convenience, we refer to the set of function

continuous-time signals for a binary detection problem. In this case, we compare the bilinear representation with Nyquist sampling. Thesis Supervisor: Alan V. Oppenheim Title: MacVicar Faculty Fellow, Ford Professor of Engineering De

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