Sigma Delta Mo dulation Of A Chaotic Signal Gary Ushaw I V N E R U S E I T H Y T O H F G E R D I N B U A thesis submitted for the degreeofDoctor of Philosophy The UniversityofEdinburgh Octob er Abstract Sigma delta mo dulation M has b ecome a widespread metho d of analogue to digital conversion however its op eration has not b een completely dened The ma jority of the analysis carried out on the circuit has b een from a linear standp oint with nonlinear analysis hinting at hidden complexities in the mo dulators op eration The sigma delta mo dulator itself is a nonlinear system consisting as it do es of a number of integrators and a one bit quantiser in a feedback lo op This conguration can b e generalised as a nonlineari ty within a feedback path which is a classic route to chaotic b ehaviour This initially raises the prosp ect that a sigma delta mo dulator may b e capable of chaotic mo des of op eration with a nonchaotic input In fact the problem do es not arise and weshowwhynot To facilitate this investigation a set of dierential equations is formulated to represent M these equations are subsequently utilised in a stability study of the sigma delta mo dulator Of more interest and more uncertainty is the eect sigma delta mo dulation mayhaveonachaotic signal If M makesachaotic signal more chaotic then this will have serious rep ercussions on the predictability of that signal In the past analysis of the circuit has tended to b e based around a steady state input or a slowly moving nonchaotic input suchasalow frequency sine wave This has greatly eased the complexity of such analyses but it do es not address the problem at hand In this thesis wepresent the results of comparing the sigma delta mo dulation of a chaotic signal to a direct quantisation of the same signal The to ol we use to investigate this is the Lyapunov sp ectrum of the time series measured using an algorithm develop ed at Edinburgh UniversityTheLyapunov exp onents of a chaotic signal are presented b efore and after b oth M and direct quantisation and it is shown that M do es not increase the chaos of the signal Indeed it is shown that M has no more eect on the predictabili ty of the signal as measured bytheLyapunov sp ectrum than direct quantisation As such we conclude that sigma delta mo dulation provides a reliable metho d for analogue to digital conversion of chaotic signals It should b e p ointed out that due to the incompleteness of rigorous analysis of M and the complex pro cesses involved in applying such analysis to a chaotic signal the results of this thesis are largely based up on exp erimentation and observation from a simulation of a sigma delta mo dulator Declaration of originality I hereby declare that this thesis and the work rep orted herein was comp osed and originated entirely bymyself in the Department of Electrical Engineering at the University of Edinburgh except for the developmentoftheLyapunov extraction algorithm describ ed in chapter whichwas carried out in collab oration with Mike Banbro ok Gary Ushaw i Acknowledgements Iwould like to thank the following p eople for their invaluable assistance during the course of this PhD Steve McLaughlin my sup ervisor for his continuous supp ort and guidance Also for reading and checking this thesis Dave Bro omhead Jerry Huke David Hughes and Robin Jones of the Defence Research Agency Malvern for useful discussions and advice Mike Banbro ok with whom I collab orated on the Lyapunov exp onent extraction algorithm describ ed in chapter and without whom etc etc The other memb ers of the Signal Pro cessing Group for their supp ort throughout my PhD DRA Malvern CASE and EPSRC for providing nancial supp ort ii Contents List of Figures vi Abbreviations ix List of Symb ols x INTRODUCTION MOTIVATION STRUCTURE OF THE THESIS CHAOS THE BACKGROUND INTRODUCTION HISTORY WHAT IS CHAOS The Henon Map The Lorenz Attractor DIMENSION Dimension As Order Of The System Emb edding A Time Series LYAPUNOV EXPONENTS CHAOS AND SIGNAL PROCESSING SIGMA DELTA MODULATION A RETROSPECTIVE INTRODUCTION ANALOGUE TO DIGITAL CONVERSION TECHNIQUES DELTA MODULATION SIGMA DELTA MODULATION EXISTING ANALYSIS OF SIGMA DELTA MODULATION Basic analysis Existing work ALTERNATIVE STRUCTURES SIGMA DELTA MODULATORS AND CHAOS CONTINUOUS MODEL OF A SIGMA DELTAMODULATOR INTRODUCTION CONTINUOUS MODEL First order M Second order M nth order M iii RUNGE KUTTA SOLUTION OF DIFFERENTIAL EQUATIONS REPRES ENTING M FIRST ORDER SIGMA DELTA MODULATOR WITH CONSTANT INPUT Numerical Approach Analytical Approach Fixed Point Analysis PRELIMINARYANALYSIS OF THE LYAPUNOV EXPONENTS OF A SIGMA DELTA MODULATED SIGNAL CONCLUSIONS STABILITY AND CONFIGURATION INTRODUCTION SIGMA DELTA MODULATION AS DIFFERENTIAL EQUATIONS OPERATIONAL BOUNDARIES FROM SIMULATION STABILITY ANALYSIS OF DIFFERENTIAL EQUATIONS General Dynamics Fixed Point Metho d Laplace Transform Metho d DISCUSSION OF RESULTS FOR SECOND ORDER MODULATION ASYMPTOTIC BEHAVIOUR OR INSTABILITY THIRD ORDER MODULATION FIRST ORDER MODULATION A REPRISE HIGHER ORDER MODULATORS GENERALISED SIGMA DELTA MODULATOR DISCUSSION CONCLUSIONS LYAPUNOV EXPONENTS INTRODUCTION LYAPUNOV SPECTRA LYAPUNOV EXPONENTS FROM TIME SERIES Overview Time Series Emb edding Cho osing The Neighb ourho o d Calculating The Tangent Maps Applying Lo cal Noise Reduction ToTangent Mapping Averaging The Exp onents USING THE ALGORITHM The Lorenz time series an example of applying the algorithm DISCUSSION EFFECT OF MONLYAPUNOV EXPONENTS INTRODUCTION A SINE WAVE TIME SERIES GENERATED FROM LORENZ SYSTEM FOUR DIMENSIONAL ANALYSIS iv TIME SERIES GENERATED FROM DUFFING SYSTEM TIME SERIES GENERATED FROM ROSSLER SYSTEM LOCAL SINGULAR VALUE DECOMPOSITION GAUSSIAN NOISE SECOND ORDER MODULATION SIGMA DELTA MODULATION WITH TONE SUPPRESSION DISCUSSION CONCLUSIONS AND DISCUSSION PRELIMINARY DISCUSSION ACHIEVEMENTS LIMITATIONS FURTHER WORK References A Original publications v List of Figures Sigma delta modulation of a chaotic signal what happens Examples of attractors generatedbyapendulum with angular velocity plotted against angular displacement a a simple undampedpendulum ie periodic b a dampedpendulum and c a driven pendulum involving a morecomplicated period The Henon attractor a chaotic discrete mapping Detail of sections of the Henon attractor The Lorenz Attractor a chaotic ow Lorenz attractor viewedasitrotates about the Yaxis A torus showing a a section of the time series and b the phase spaceplot Generic Analogue to Digital Converter Delta modulation and demodulation Input and output of sigma delta modulator First order sigma delta converter Time series for sinusoidal input to rst order sigma delta converter showing a the analogue input to the modulator b the output of the integrator c the output from the quantiser and from the modulator as a whole and d the output from the modulator after decimation Higher order sigma delta modulation and demodulation Laplacetransform of rst order sigma delta modulator Ztransform of rst order sigma delta modulator Noise added by one bit quantiser for sinusoidal input Athree stage MASH converter Reconstructed attractor from the integrator output in a rst order sigma delta modulator with a sinusoidal input Short section of reconstructedattractor from integrator output in rst order mod ulator with sinusoidal input First Order Continuous Sigma Delta Modulator Schematic of a discrete integrator and a continuous integrator First Order Continuous Sigma Delta Modulator Time series of a utbxtcyt and d decimated yt for sinusoidal input to continuous rst order sigma delta converter calculatedbyRunge Kutta method Time series of a utbxtcyt and d decimated yt for sinusoidal input to continuous second order sigma delta converter calculatedbyRunge Kutta method Time series of xt for constant input of to continuous rst order sigma delta converter calculatedbyRunge Kutta method Plot of xt Periodofxt Limit cycle of rst order equation Output vs scaled dc input Graphical solution for Graphical solution for Lyapunov exponents calculatedfrom the dierential equation representing rst order sigma delta modulation with a constant input vi Second Order Continuous Sigma Delta Modulator Plot of input to quantiser x t during acorrect operation and basymptotic operation Operational plot of Runge Kutta solution to dierential equations representing second order CM with varying a abscissa and a ordinate Operational plot of second order M with varying a abscissa and a ordinate Schematic