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Signal Processing and Communication with Solitons

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Signal Processing and Communication with Solitons Signal Processing and Communication with Solitons Andrew C. Singer RLE Technical Report No. 599 June 1996 The Research Laboratory of Electronics MASSACHUSETTS INSTITUTE OF TECHNOLOGY CAMBRIDGE, MASSACHUSETTS 02139-4307 This research was generously supported in part by the Defense Advanced Research Projects Agency under Grant N00014-89-J-1489, in part by Lockheed Martin Sanders, in part by the U.S. Navy Office of the Chief of Naval Research under Grant N00014-93-1-0686, and in part by U.S. Air Force Office of Scientific Research under Grant AFOSR-91-0034. Signal Processing and Communication with Solitons by Andrew Carl Singer Submitted to the Department of Electrical Engineering and Computer Science on May 14, 1996, in partial fulfillment of the requirements for the degree of Doctor of Philosophy Abstract Traditional signal processing algorithms rely heavily on models that are inherently linear. Such models are attractive both for their mathematical tractability and their applicability to the rich class of signals that can be represented with Fourier methods. Nonlinear systems that support soliton solutions share many of the properties that make linear systems attractive from an engineering standpoint. Although nonlinear, these systems are solvable through inverse scattering, a technique analogous to the Fourier transform for linear systems. Solitons are eigenfunctions of these systems which satisfy a nonlinear form of superposition and display rich signal dynamics as they interact. By using solitons for signal synthesis, the corresponding nonlinear systems become specialized signal processors which are naturally suited to a number of complex signal processing tasks. Specific analog circuits can generate soliton signals and can be used as natural multiplexers and demultiplexers in a number of potential soliton-based wireless communication applications. These circuits play an important role in investigating the effects of noise on soliton behavior. Finally, the soliton signal dynamics also provide a mechanism for decreasing transmitted signal energy while enhancing signal detection and parameter estimation performance. Thesis Supervisor: Alan V. Oppenheim Title: Distinguished Professor of Electrical Engineering and Computer Science Acknowledgments I would like to express my sincere gratitude to Professor Alan Oppenheim, for both putting me out on a limb, and for making sure that I did not fall. Through his many roles as an advisor, a mentor, and above all, as a friend, he has had a tremendous impact on both my professional and personal development. I look forward to exploring many more trees together. For his guidance, encouragement, and friendship, I also wish to thank Professor Gregory Wornell. Our many discussions have greatly influenced the way I think about research problems in general and this thesis in particular. I am sincerely grateful to Professor Ruben Rosales for his many insightful comments, his interest and his encouragement. Thanks also to Dr. Jim Kaiser of Bellcore and Dr. Doug Mook of Lockheed Sanders for numerous discussions relating to this work. Lloyd D'Souza deserves thanks for his many hours of circuit hacking and development. The friendly atmosphere and productive research environment of the Digital Signal Processing Group have made my experiences as a graduate student both challenging and rewarding. Thanks to John Buck, Babis Papadopoulos, and Kathleen Wage for their help with various parts of this thesis and to Paul Beckmann, Steve Isabelle, Stephen Scherock, and Kambiz Zangi for many years of friendship and thought- provoking discussion. Thanks also to Jeff Ludwig, Lon Sunshine and Yakov Royter for their camaraderie and our many athletic pursuits. I am especially grateful to Giovanni Aliberti both for his computer expertise and friendship. Special thanks go to Maggie Beucler, Janice Zaganjori and Sally Bemus for administrative support and for help in preparing this document. This research has been generously supported by the Defense Advanced Research Projects Agency, Lockheed Martin Sanders, the Department of the Navy Office of the Chief of Naval Research, and the Air Force Office of Scientific Research. Finally, I would like to thank my parents for their love and support and for being my very first teachers. Also, thanks to my sister Amy for reminding me what it's like to be a big brother. Most importantly, thanks to my loving wife, Cathy, for her unending love and encouragement. Contents 1 Introduction 11 1.1 Outline of the Thesis ............. .. 12 2 Soliton Systems 15 2.1 The Toda Lattice ............... 16 .............. 2.2 The Inverse Scattering Transform ...... .............. 22 2.3 Other Systems Exhibiting Solitons ..... .............. 28 2.4 Soliton Hierarchies ............. .............. 30 2.A A Soliton Solution .............. .............. 34 2.B Multi-soliton Solutions ........... .............. 35 2.C Single Pole Reflection Coefficient ...... .............. 36 3 New Electrical Analogs for Soliton Systems 39 3.1 Toda Circuit Model of Hirota and Suzuki . · . .. .. 41 3.2 Diode Ladder Circuit Model for Toda Lattice . .. .. 43 3.3 Circuit Simulation ................. .. .. 45 3.4 Circuit Implementation .... .. .. 48 3.5 Circuit Model for Discrete-KdV . .. .. 52 3.6 Further Considerations .... .. .. 57 4 Communication with Soliton Signals 61 4.1 Soliton Modulation .............. .............. 62 4.2 Soliton Multiplexing ............. .............. 63 4.3 Spectrum of Toda Lattice Solitons ...... .............. 66 4.4 Low Energy Signaling. ............ .............. 69 4.5 Gain Normalization ............. .............. 74 4.6 Other Potential Applications ......... .............. 77 5 Noise Dynamics in Soliton Systems 81 5.1 Toda Lattice Small Signal Model ...... .. .. 83 5.2 Linearized Model ............... .. .. 85 5.3 Simulation of the Lattice in Noise ...... .. .. 87 5.4 Noise Correlation ............... .. .. 89 5.5 Noise Dynamics for the Diode Ladder Circuit . .. .. .. 95 5.6 Inverse Scattering-Based Noise Modeling .... 97 5 6 Estimation of Soliton Signals 103 6.1 Soliton Parameter Estimation: Bounds ..... ..... 104 6.2 Multi-soliton Parameter Estimation: Bounds ..... 107 6.3 Estimation Algorithms ............. ..... 110 6.3.1 General Approach ............ ..... 110 6.3.2 Position Estimation ........... ..... 114 6.3.3 Velocity Estimation ........... ..... 115 6.3.4 Estimation Based on Inverse Scattering . ..... 116 6.4 Summary ..................... ..... 120 7 Detection of Soliton Signals 123 7.1 General Approach ................ 123 7.2 Simulations .................... 127 7.3 Summary and Further Considerations ...... 128 8 Conclusions and Directions for Future Work 131 8.1 Future Directions ................ 133 6 List of Figures 2-1 The Toda Lattice .......................................... 16 2-2 A propagating wave solution to the Toda lattice equations. Each trace corresponds to the force f (t) stored in the spring between mass n and n-1 . ....................... 17 2-3 Two solitary wave solutions to the Toda lattice ................. 20 2-4 Spectrum of eigenvalues of the matrix L(t) for the Toda lattice. ... 24 2-5 Schematic solution to linear evolution equations ............. 25 2-6 Schematic solution to soliton equations .................. 25 2-7 Soliton-like solutions to the automata of Ablowitz et al ......... 31 3-1 Two-soliton signal processing by a soliton system ............ 40 3-2 Nonlinear LC ladder circuit of Hirota and Suzuki ............ 41 3-3 Diode ladder network ..................................... 43 3-4 Double capacitor circuit diagram ..................... 44 3-5 HSPICE simulation of the evolution of a two-soliton signal through the diode lattice. Each horizontal trace shows the current through one of the diodes 1, 3,4 and 5 . 47 3-6 Precision bipolar current source ...................... 49 3-7 Diagram for the double capacitors used in the diode ladder circuit. Analog switches, placed in parallel with the capacitors, are used to reset the circuit after each processed signal ............... 50 3-8 Hardware implementation of the diode ladder circuit. The first col- umn from the left contains the pulse-generation circuitry; the second contains the voltage-controlled current source; each of the last three contains four stages of diodes, series resistors and double capacitors.. 51 3-9 Oscilloscope traces for two solitons in the diode ladder circuit. The traces correspond to the currents in the first four diodes ....... 52 3-10 Diode currents measured from the diode ladder circuit in operation. The input signal consists of three square pulses of different areas. The spike that appears in the figure near t = -1 ms is a result of the signal that resets the lattice ..................................... 53 3-11 Illustration of the relationship between two adjacent Toda lattices, fi,, i = 1,2 and the discrete-KdV equation. This process suggests a possible implementation of the dKdV equation using two adjacent Toda lattice circuits ............................ 55 3-12 Collection of nodes for the discrete-KdV circuit ............. 56 7 3-13 To the left, the normalized node capacitor voltages, vn(t)/vt for each node is shown as a function of time. To the right, the state of the circuit is shown as a function of node index for five different sample times. The bottom trace in the figure corresponds to the initial condition .. 58 4-1 A soliton carrier signal for the Toda lattice ............... 62 4-2 Modulating the relative amplitude or position of soliton carrier
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