LGENERALIZED ORTHOGONALLY MULTIPLEXED COMMUNICATION VIA WAVELET PACKET BASES/ A Dissertation Presented to The Faculty of the Russ College of Engineering and Technology Ohio University In Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy Alan R. Lindsey/ -* June 9, 1995 1995 Alan Ray Lindsey All Rights Reserved Acknowledgements I would like to sincerely thank Dr. Jeff Dill, my research advisor, for taking time away from a very busy schedule to occasionally adjust my wayward direction. His practical insight and farmboy wisdom are enviable. I would also like to thank the members of my committee individually. Dr. Dennis Irwin has been a friend and colleague in addition to his role as mentor. His support, encouragement, and assistance over my tenure as a student at Ohio University have been a tremendous blessing. And speaking of blessings, Dr. John Tague has truly filled that role with numerous discussions about non-technical things like God, the Bible, and Christianity. However, I am also grateful for his sharing of technical expertise in signal processing and communications. As the outside representative from mathematics, Dr. Jeff Connor contributed a tremendous amount of help in the theoretical developments of wavelets and wavelet packets. Without the Wavelet Research Group meetings, I would still be lost in a cesspool of notation, nested subspaces, p-norms, operators and Hilbert spaces! Many, many thanks to Dr. Connor for helping me organize and absorb that material. And finally, my sincere appreciation and respect are extended to Dr. Jerrell Mitchell, whose unique talent for symbiotically combining a practical approach with experience, and these v two with respectable analytical skills, makes him a great teacher and engineer. He also took time teaching me the art of concise and professional writing style. Denise Ragan and the ECE office staff deserve much appreciation. I have not known a more organized and well-run administration. Thank you's go also to my colleagues in the control group: Enrique, Russ, Dave and Dan. Their friendship will be forever prized. Special thanks go to the United States Air Force and the Communications branch at Rome Laboratory. Some specific people in this group who deserve mention are: Peter Leong, my supervisor and friend, who led me gently through the barrage of paperwork and required form-filling, not to mention the consistently wise counsel on matters pertaining to both studies and life; John Patti, my PK mentor, whose technical prowess is to be esteemed; and Pauline Romano, the RL office manager who works so tirelessly. My special intimate thanks go to my family, especially my wife Tammy. Her support and encouragement through the difficult days of writing, publishing, traveling, etc. have been a tremendous blessing. I continually give thanks to God for Tammy's companionship and love; and my children - Jared, Elisabeth, Benuel and Caleb will never know how I love being their daddy. And most importantly, to The One who makes everything I do and say prosper or flounder according to His will and my heart, whose names include Jehovah Jireh, God my Provider; The God of Abraham, Isaac, and Jacob; The Creator of the universe who loves me infinitely more than I could ever love Him back; Jesus the Savior of the world; My Father in Heaven. Table of Contents Acknowledgements ...................................... iv Table of Contents ....................................... vi List of Figures ......................................... viii List of Abbreviations and Symbols ............................. ix Chapter 1: Introduction .................................... 1 1.1 Contributions Leading to this Work ................... 2 1.2 Overview ................................... 3 Chapter 2: Review of Previous Work ........................... 6 2.1 Quadrature Amplitude Modulation (QAM) ............... 7 2.1.1 QAM Signal ............................. 8 2.1.2 Multi-dimensional Signaling ................... 10 2.2 Multi-Scale Modulation ........................... 11 2.2.1 Essential MRA Results ...................... 11 2.2.2 Tiling Diagram ........................... 13 2.2.3 Figures of Merit .......................... 15 2.3 M-Band Wavelet Modulation ....................... 18 2.3.1 Essential M-Band Wavelet System Results ........... 18 2.3.2 Tiling Diagram ........................... 20 2.3.3 Figures of Merit .......................... 21 Chapter 3: Wavelet Packet Modulation .......................... 23 3.1 Construction of Wavelet Packet Bases .................. 25 3.2 Example of Theorem 3.2 .......................... 31 3.3 Waveform Development .......................... 32 3.3.1 Dimensionality and Special Partitions .............. 35 3.4 An Example ................................. 36 3.5 Waveform Figures of Merit ........................ 38 3.5.1 Power Spectral Density ...................... 38 3.5.2 Bandwidth Efficiency ....................... 42 Chapter 4: Implementation .................................. 44 4.1 The Discrete Wavelet Packet Transform and its Inverse ....... 45 4.1.1 The WPM Transceiver ....................... 47 4.2 Translation Between Tiling Diagram and Filter Bank ......... 51 4.2.1 Gray Coding of Frequency Bins ................. 51 4.2.2 Translation Computations ..................... 53 4.2.3 Why Gray Coding? ......................... 55 Chapter 5: Supersymbol Tuning ............................... 57 5.1 T-F Diagram Principles .......................... 58 5.2 Supersymbol Tuning Rules ......................... 59 5.3 Wavelet Packet Library Combinatorics ................. 61 5.4 The Simplest T-F Jammer . 111 ...................... 62 5.5 N, / N, Channels .A General Supersymbol Tuning Algorithm . 65 5.6 Enhancements to the General Tuning Algorithm ............ 75 5.6.1 111 Noise Scenarios Provide Predictable Best-Level ..... 76 5.6.2 Efficiency Improvement via Another Reverse-Tuning Stage .................................78 Chapter 6: Conclusions and Future Work ......................... 80 6.1 Conclusions ................................ 80 6.2 Future Work ................................. 83 6.2.1 Timing and Synchronization ................... 83 6.2.2 Parameterization of Scaling Filter Coefficients ........ 84 6.2.3 Coding Via Variable Constellation Geometries With Fixed Symbol Counts ........................... 85 6.2.4 Generalized P-adic Filter Banks and Associated Packets . 85 6.2.5 Improvement of Frequency Localization ............ 86 6.2.6 Continued Development of Supersymbol Tuning ....... 86 6.2.6.1 Other Noise Classes ............... 87 6.2.6.2 Better Cost Functions .............. 87 6.2.7 Spread-Spectrum Application of WPM ............. 88 6.3 Epilogue .................................... 88 References ...........................................89 Appendix A ...........................................103 List of Figures Figure 2.1. Low-Pass Equivalent Modulation Diagram ................ 9 Figure 2.2. Tiling Diagram for MultiScale Modulation ................ 15 Figure 2.3. Tiling Diagram for M-Band Wavelet Modulation ............ 21 Figure 3.1: Uniform Analysis Filter Bank corresponding to basic wavelet packet bases with m=3 . Nonuniform bank corresponding to standard wavelet bases is shown in grey.shades ............................ 24 Figure 3.2. Analysis filters for signal decomposition .................. 26 Figure 3.3. Synthesis filters used in signal expansion ................. 26 Figure 3.4. Wavelet packet decomposition by two-channel filter bank ....... 28 Figure 3.5. Generalized subspace decomposition .................... 29 Figure 3.6: Example application of theorem 3.2, moving from partition to filter bank . Translation to the tiling diagram is shown for completeness ..... 32 Figure 3.7: Comparison of modulation methods for a given interference environment consisting of time impulse and tone jammers . Grey-shaded areas indicate corrupted symbols .......................... 37 Figure 4.1: Single stage of the (a) Wavelet Packet Transform (b) Inverse Wavelet Packet Transform .............................. 47 Figure 4.2. WPM Transmitter 1 Receiver Model ................... 48 Figure 4.3: Connection between natural ordering of filter bank nodes and sequency or Gray coded ordering of frequency bins .............. 54 Figure 5.1. Tuning procedure illustrated step by step ................. 63 Figure 5.2. Flow Diagram for General Tuning Algorithm .............. 67 Figure 5.3. Best Level Supersymbol for N=6 ..................... 71 Figure 5.4. Forward-Tuned Supersymbol ........................ 72 Figure 5.5. Optimally Tuned Supersymbol: Cost =30 ................. 73 Figure 5.6. Optimally Tuned Partition for N=8: Cost =58 ............. 74 Figure 5.7. Optimally Tuned Partition, N =9: Cost =245 .............. 75 List of Abbreviations and Symbols 1 2(2) The space of square-summable sequences L 2(R) The space of all square-integrable functions on 1W z The set of integers 0' Partition defining a supersymbol RMS bandwidth Af At RMS duration @ Mutually exclusive vector space sum, i.e., "Direct Sum" AWGN Additive white Gaussian noise bps Bits per second BPSK Binary Phase Shift Keying DSPN Direct Sequence Pseudo Noise DWPT Discrete Wavelet Packet Transform IDWPT Inverse Discrete Wavelet Packet Transform IS1 Inter-symbol interference LPIID Low Probability of InterferenceIDetection MAXLEVELS Maximum level of the wavelet packet filter bank MBWS M-Band Wavelet System MCM Multi-Carrier Modulation MRA Multiresolution Analysis MSM Multi-Scale Modulation MWM M-Band