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University of Southern California Dissertations and Theses
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An ultrawideband digital signal design with power spectral density constraints
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An ultrawideband digital signal design with power spectral density constraints
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AN ULTRAWIDEBAND DIGITAL SIGNAL DESIGN WITH POWER SPECTRAL DENSITY CONSTRAINTS by Terry Pernell Lewis A Dissertation Presented to the FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulllment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (ELECTRICAL ENGINEERING) August 2012 Copyright 2012 Terry Pernell Lewis Dedication This dissertation is dedicated to my parents: Evelyn and in memory of my dad { LeRoy Lewis. My dad was blessed to see me start this Ph.D. process but not nish it. I am very blessed to have known each of my parents. They both encouraged and prayed for me through the years. I know that my father has watched over me during my most dicult times. I love you always mom and dad. Figure 1: My wonderful parents that started me on this journey by simply giving me a subscription to the McGraw Hill Electronics Book Club when I was a small child. ii (a) My supportive friends at USC (minus a few) (b) Having fun at USC (minus a few friends) Acknowledgments I would like to thank my dream-team Ph.D. qualication and dissertation committee that consisted of Professor Robert A. Scholtz (Chair), Professor Lloyd R. Welch (Co-Chair), Professor Charles Weber, Professor Solomon Golomb and Professor Nicolai Haydn. Figure 2: My Ph.D. Committee. Back row, left to right: Professor Charles Weber, Professor Robert A Scholtz, Professor Solomon Golomb, Professor Nicolai Haydn. Front left to right: Professor Lloyd Welch and Terry Lewis. iii Professor Scholtz and my committee were very patient with me. They provided out- standing mentoring and advisement to me during this learning process. They are all very special people and they share a special place in my heart. I am eternally grateful to them for what they have done for me. While I have learned a great deal from them technically, I have also been a witness to outstanding character. While each of my committee members have had distinguished careers, they are the most humble people I have ever met. I pray that I may follow in just a couple of their footsteps. I also would like to thank Mrs. Lolly Scholtz for encouraging me though the years and helping me when and where needed. This dissertation may not have been possible without her. I especially appreciate all of her help with editing. Raytheon: I would like to thank Raytheon for the nancial support (as a Raytheon Scholar) and my colleagues at Raytheon for the emotional support and encouragement they provided, especially; Mr. Dale R. Feikema (my brother in Christ and engineering mentor), Mr. Mark Bianco, Mr. George Cassar, Mr. James (Jim) Kivett, Mr. Timo- thy Hughes, Mr. Wendell Kishaba, Mr. James Tsusaki, Mr. Kerry Frohling, Mr. Ward Bathrick, Mr. Paul Thai, Mr. Todd Taylor,Dr. Kenneth Kung, Mr. Terry Flach, Mr. David (Dave) Helsel, Mr. Tyler Ulinskas, Mr. Bob Grayson, Mrs. Marina Gurevich, Dr. John D. Olsen, Mr. Kevin Hayata, Mr. George Vardakas, Mr. Roy Acevedo, Mr. Frank Pietryka, Dr. Niraj Srivastava, Mr. Je Black, Mr. Carl Cook, Mr. Kevin Grant, Mr. John Wright, Ms. Josaphine Subido, Ms. Cheryl Reiter, Mrs. Erin Rapp and last but not least, Raytheon's CEO, Mr William H. Swanson who has being an outstanding example for me. My USC Trojan Family { The Ultra Lab: I would like to thank Professor Moe Win of MIT, Professor Won Namgoong (former Professor at USC), Dr. Meng-Hsuan iv Chung (Thanks for helping me pass the screening exam!), Dr. Majid Nemati, Dr. Robert Wilson, Dr. Sangyhun Chang, Dr. Ali Taha, Mr. Je Yang (Thanks for the help with the antennas!), Mr. Robert Weaver (Thanks for all the helpful engineering discussions. They were priceless { especially proving concepts only to nd out that someone else proved the concept already!), Dr. Yenming Chen, Dr. Yi-Ling Chao and Dr. Chee-Cheon Chui (I will never view oscillators the same.). My Trojan Family: I would like to thank Professor Irving S. Reed, Professor Robert Gagliardi (Rest in peace. He was a wonderful mentor and inspiration to me.), Professor William C. Lindsey, Professor Vijay Kumar (former Professor at USC that taught me how to love Linear Algebra!), Professor Zhen Zhang (Thanks for teaching me to love Information Theory and Randon Processes.), Professor Keith Chugg (Thanks for teaching me how to build modern receivers), Professor Urbashi (Ubli) Mitra (Thanks for the continued encouragement to learn.), Professor Todd Brun, Dr. Jun Yang, Dr. Jordan Melzer, Dr. Durai Thirupathi, Dr. On Wa Yeung, Mr. Yuan-Sheng Cheng, Mr. David Lee, Dr. Gautam Thatte, Dr. Marco Zuniga, Dr. Patricia de Rezende Barbosa, Mrs. Deidra Simon de-Godinez, Mr. Bishara Shamee, Mrs. Milly Montenegro, Ms. Mayumi Thrasher, Ms. Gerrielyn S. Ramos, Mrs. Alejandra Kalmen, Ms. Lilian Carr, Ms. Anita Fung, Mr. Sean Oconnell and Mrs. Lisa Inomata-Oconnell, Mrs. Diane Demetris (A constant source of inspiration and positive spirit. What a voice!), Associate Dean Margery Berti and the USC Distance Education Network (DEN) for all of their help, support and encouragement during the degree process. v My supportive friends (not listed above): Mr. Fred Crowder (I would not have been able to complete this degree without him. He help me learn and love mathemat- ics.), Mr. Mark Hueston, Mr. Randolph Staples, Mr. Peter Osagie, Ms. Nicole Jordan, Ms. Elaine Laird, Dr. Irene Hsia ($1.00), Ms. Thalia Ellzey and Mr. Geoery (Geo) Kelsch, Dr. Anatoli Borresynko (Thanks for the help with electromagnetics and anten- nas.), Mr Shusaku Shimada of the ANDO corporation, Dr. Robert Johnk of the National Institute of Standards and Technology (NIST), Dr Larry Stotts (Former Darpa Deputy Director of STO.) and Dr James (Jim) Freebersyser (Former Darpa Program Manager for FCS Communications Progran), Professor Robert Robinson of the Naval Post Graduate School, Professor Gilbert Strang of MIT and Professor David Pozar. Thanks for your encouragement and support through the degree process. The City of Lomita CA: I would like to thank the former Mayor Don Suminaga for extending an invitation to me to serve on the City of Lomita Railroad Museum Board of Directors and former Mayor Susan (Susie) Dever for providing unyielding support to the Lomita Railroad Museum. I would also like to thank both museum boards (i.e., Board of Directors and the Foundation Board) for their emotional support of this degree. It has been a pleasure to support and serve the citizens of Lomita at the museum. I have also beneted greatly from this civic service. Thank you for allowing me to serve as Chairman of the Board of Directors for multiple terms. The National Academies of Science (Naval Studies Board and The Com- mittee on Distributed Remote Sensing for Naval Undersea Warfare): I would especially like to thank Dr. Charles Draper{NSB Director, Dr. Miriam E. John (Chair), Dr. David A. Whelan (Vice-Chair), ADM J. Paul Reason, USN (Ret. ), Mr. James vi R. Gosler, Dr. Arul Mozhi (Study Director), Mr. Thomas (Tom) McNamara, Dr. Chip Elliot, Professor Art Bagoeror, Lieutenant General (ret. ) John Rhodes USMC, Brig Gen Leon A. Johnson, Dr. Ann Berman (Rest in peace. She was an amazing women.), Ms. Dixie Gordon, Ms. Susan Campbell and Mr. Billy M. Williams. It is an honor to have served with such wonderful and exceptional people. The National Academies of Science (Air Force Studies Board): I would like to thank Mr. Terry Jaggers{AFSB Director, Mr. Carter W. Ford (Study Director), Lieutenant General (ret. ) Brian A. Arnold USAF (Co Chair), Dr. Lawrence J. Delaney (Co-Chair), Major General (ret. ) Robert H. Lati Ph.D. USAF, Major General (ret. ) Gerald F. Perryman USAF, Brigadier general (ret. ) Michael (Mike) A. Longoria USAF, Dr. Pamela A. Drew, Mr. Collin Agee, Mrs. Melanie Austin, Ms. Zeida Patmon and Ms. Sarah Capote. I appreciate their friendship, support and the encouragement to complete this degree. My City College of New York (CCNY) and City University of NY (CUNY) family: I would like to thank Professor Donald L Schilling (Professor emeritus) for being a mentor to me and exposing me to the interesting eld of communication theory, Spread Spectrum Technology and The University of Southern California, Professor Paul Karmel (Rest in peace.), Professor Gabriel Colef (Adjunct Professor at CCNY and Professor of the NYC Maritime College), Dr Raymond L. Camisa (Rest in peace.), Dean Gerard G. Lowen, Dean Charles B. Watkins, Mr. Fred Barrios, Professor David Lieberman (Queensborough Community College) and Professor Victor Stanionious (QCC) for their encouragement, untiring support and taking the time mentor and train me technically. They all have provided support to me long after I graduated from the CUNY. vii Figure 3: My supportive family The United States Naval Reserves: Specically, I appreciate the support and mentoring from RDML Michael J Browne, CAPT Kevin Alt, CAPT John Posadas, CAPT Thomas (Tom) Kelly, CAPT Thomas Morgan, CAPT Raymond O'Toole, CAPT Phan Phan, CAPT John Klein, CAPT Robert (Bob) Decesari, CAPT John Ailes (Commander NSWC Port Hueneme), CAPT Pete (Richard) DeLong, CDR Brian Meadows, CDR Timothy Lewis, CDR Ilker Bayraktar, CDR Sheila Jenkins, CAPT(s) James (Jimmy) Cox, CDR Robert (Bob) Been, ENS Kyle Bales, The AMIIP team, Ms. Sharon Gibson, CDR Stephen and Stephanie Melvin, ET1 Micheal Folk and ET1 Peter Becton (I can x TV's if necessary). My siblings and extended family: I would like to thank Darryl and Antonia as well as my brother in-law Larry Hunt. My family: Finally and most importantly, I thank my patient, supportive and very beautiful wife Maria, my son Terry II and my daughter Maria for their continued inspiration and support. My children were not present when I started this degree. Apparently, it is easier to produce children than a thesis! viii In all, this degree may not have been possible without the support and encouragement of all the aforementioned. I am sure that I have missed someone so I would like to thank them now { Thank you! ix Table of Contents Dedication ii Acknowledgments iii List of Tables xiii List of Figures xiv Abstract xx Chapter 1: Introduction 1 1.1 Motivation and Problem Denition . . . . . . . . . . . . . . . . . . . . . . 4 1.2 Mask-Filling Eciency Measure . . . . . . . . . . . . . . . . . . . . . . . . 6 1.2.1 Pulse Waveform Selection Procedure . . . . . . . . . . . . . . . . . 8 1.3 Literature Review and Comments on Prior Work . . . . . . . . . . . . . . 9 Chapter 2: Signal Synthesis 11 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2 Signal Construction and Generation . . . . . . . . . . . . . . . . . . . . . 13 2.3 Signal Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.3.1 Waveform Processing . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.3.2 Even Digital-to-Analog Converter Search-Space Reduction . . . . . 20 2.3.3 Odd Digital-to-Analog Converter Search{Space Reduction . . . . . 23 2.4 The Signal-Path System Function . . . . . . . . . . . . . . . . . . . . . . . 25 2.4.1 DAC Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.4.2 Transmission Line Fundamentals. Initially Relaxed Conditions. . . 27 2.4.3 Terminated Transmission Line . . . . . . . . . . . . . . . . . . . . 32 2.4.4 The Terminated Transmission Line Transfer Function . . . . . . . 35 2.4.5 Distortionless Transmission Lines . . . . . . . . . . . . . . . . . . . 39 2.4.6 Requirements for a Distortionless Transmission Line System . . . . 42 2.5 The Lossless Transmission Line . . . . . . . . . . . . . . . . . . . . . . . . 43 2.5.1 The Low-Loss Transmission Line . . . . . . . . . . . . . . . . . . . 44 2.6 Power Delivered to the Load . . . . . . . . . . . . . . . . . . . . . . . . . 48 x Chapter 3: Impulse Radio Antenna Fundamentals 51 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.2 Determination of the Electric Field Intensity . . . . . . . . . . . . . . . . 57 3.2.1 Fundamental Theorems . . . . . . . . . . . . . . . . . . . . . . . . 58 3.2.2 Electric Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 3.2.3 Magnetic Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 3.2.4 Potential Functions and the Wave Equation . . . . . . . . . . . . . 68 3.2.5 Homogeneous Solutions to the Wave Equation for Point Sources . 70 3.2.6 Candidate Procedure to Determine the E-Field . . . . . . . . . . . 74 3.2.7 Homogeneous Vector Helmholtz Equations . . . . . . . . . . . . . 76 3.2.8 Plane Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 3.2.9 Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 3.2.10 Radiation Field of a Short (Hertzian) Dipole . . . . . . . . . . . . 80 3.2.11 The Eective Length of an Antenna . . . . . . . . . . . . . . . . . 84 3.2.12 Antenna Circuit Models . . . . . . . . . . . . . . . . . . . . . . . . 85 3.3 Indirectly Measuring the Electric Field Strength . . . . . . . . . . . . . . 87 3.4 Antenna Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 3.4.1 Transmitting Antenna and Radiation . . . . . . . . . . . . . . . . . 91 3.4.2 Truncated Ground Plane TEM Horn Antenna . . . . . . . . . . . . 95 3.4.3 Simulated Performance of the Large TEM Horn Antenna Post Manufacture . . . . . . . . . . . . 98 3.4.4 Diamond Dipole Antennas . . . . . . . . . . . . . . . . . . . . . . . 106 3.4.5 Eective Isotropic Radiated Power for UWB . . . . . . . . . . . . 107 3.5 Measurement Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 3.5.1 Antenna Measurements . . . . . . . . . . . . . . . . . . . . . . . . 112 3.5.2 Measurement Process . . . . . . . . . . . . . . . . . . . . . . . . . 114 3.5.3 The Dynamic Range of the Measurement Equipment . . . . . . . . 116 3.5.4 Noise Floor Determination for the Vector Network Analyzer . . . . 117 3.5.5 Computation of Antenna Gain from Measurements . . . . . . . . . 120 3.5.6 The Single Antenna Method . . . . . . . . . . . . . . . . . . . . . 122 3.5.7 The Two Antenna Method . . . . . . . . . . . . . . . . . . . . . . 123 3.5.8 The Three Antenna Method . . . . . . . . . . . . . . . . . . . . . . 124 3.6 Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 3.6.1 The Norm of a Vector . . . . . . . . . . . . . . . . . . . . . . . . . 127 3.6.2 The Norm of a Linear Operator . . . . . . . . . . . . . . . . . . . . 129 3.6.3 The Condition and Ill Conditioning of a Linear Operator . . . . . 132 3.6.4 Bounding the Relative Error . . . . . . . . . . . . . . . . . . . . . 135 3.6.5 The Spectral Decomposition of a Hermitian Matrix . . . . . . . . . 138 3.6.6 The Reliability of Computed Antenna Gains . . . . . . . . . . . . 142 3.6.7 Computed Antenna Gains and Plots . . . . . . . . . . . . . . . . . 146 xi Chapter 4: Experimental Results of the Waveform Search 151 4.1 Large TEM Horn Results using Scattering Parameter for Sequence Lengths from 1 to 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 4.2 Large TEM Horn Results using Scattering Parameters for Sequence Lengths from 1 to 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 Chapter 5: Extended S21 Results 167 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 5.2 Eective Isotropic Radiated Power Spectral Density Discussion . . . . . . 170 5.2.1 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 5.2.2 Pulse Construction Method . . . . . . . . . . . . . . . . . . . . . . 174 5.3 Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 5.4 Results and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 Chapter 6: Results Based on Computed Antenna Gains 186 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 6.2 Initial Results based on the Three Antenna Method . . . . . . . . . . . . 186 Chapter 7: Quotes heard through the Ph.D years 196 Bibliography 199 Appendices 204 Appendix A: A Review of Two-port Scattering Parameters 204 Appendix B: Root Mean Squared Values 205 B.1 RMS Voltage of a Sinusoidal Waveform . . . . . . . . . . . . . . . . . . . 206 Appendix C: FCC-Part 15 Limits 207 Appendix D: Probability, Random Processes and Statistical Inference: A Review 209 D.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 D.1.1 Statistical Averages of Real Random Variables . . . . . . . . . . . 210 D.2 Random Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 D.2.1 Stationary Random Processes . . . . . . . . . . . . . . . . . . . . . 212 D.2.2 Statistical Averages of Random Processes . . . . . . . . . . . . . . 213 D.3 Convergence, Integrals and Ergodicity of Random Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 D.3.1 Convergence Concepts . . . . . . . . . . . . . . . . . . . . . . . . . 214 D.3.2 Riemann Integral of a Random Processes . . . . . . . . . . . . . . 216 D.3.3 Stochastic Integration . . . . . . . . . . . . . . . . . . . . . . . . . 217 D.3.3.1 Existence of the Stochastic Riemann Integral . . . . . . . 218 xii D.3.4 Ergodicity in the Mean . . . . . . . . . . . . . . . . . . . . . . . . 218 D.3.5 Power and Energy Spectral Density Concepts . . . . . . . . . . . . 222 D.4 Statistical Inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 D.5 The 2D Jacobian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226 D.6 Appendix: Double integrals . . . . . . . . . . . . . . . . . . . . . . . . . . 227 D.7 Convergence of a Geometric Sequence . . . . . . . . . . . . . . . . . . . . 227 Appendix E: Generalized Binomial Theorem 229 E.1 Binomial Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 E.2 Binomial Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 Appendix F: Notes on Decibels 230 F.1 Hadamard Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 xiii List of Tables Table 2.1: Transmission Line Parameters . . . . . . . . . . . . . . . . . . . . . 28 Table 3.1: Free Space Universal Constants . . . . . . . . . . . . . . . . . . . . 54 Table 3.2: Computed Norms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 Table 3.3: Computed norms for B = A y A because A 1 is not dened. . . . . 145 Table 4.1: Parameters: Clock rate = 20GHz, N = 2 andQ =f3;1; +1; +3g. 154 Table 4.2: Parameters for the reduced example: Clock rate = 20GHz, N = 2 andQ =f3;1; +1; +3g. . . . . . . . . . . . . . . . . . . . . . . . 154 Table 4.3: The three best observed PSD eciencies for sequence lengths up to seven at the indicated clock reproduction rates. . . . . . . . . . . . 160 Table 6.1: Search statistics for a length 14 sequence at a reproduction rate of 37GHz and a cubic quantizer with 4 levels. . . . . . . . . . . . . . . 188 Table 6.2: Related sequences for the best sequences of length 14 at a repro- duction rate of 37GHz and a 4-level cubic quantizer. The two items in the table labeled itself represent a waveform that is symmetric about it's center axis and voltage levels. . . . . . . . . . . . . . . . 189 xiv List of Figures Figure 1: My wonderful parents. . . . . . . . . . . . . . . . . . . . . . . . . . ii Figure 2: My Ph.D. Committee. . . . . . . . . . . . . . . . . . . . . . . . . . iii Figure 3: My supportive family . . . . . . . . . . . . . . . . . . . . . . . . . viii Figure 1.1: FCC Outdoor and Indoor PSD Constraints . . . . . . . . . . . . . 7 Figure 2.1: System Block Diagram . . . . . . . . . . . . . . . . . . . . . . . . . 11 Figure 2.2: Digital-to-Analog Converter Lowpass Filter . . . . . . . . . . . . . 26 Figure 2.3: Dierential Transmission Line Model. . . . . . . . . . . . . . . . . 28 Figure 2.4: Terminated Transmission Line Impedance Model. . . . . . . . . . . 33 Figure 2.5: Transmission Line re ection diagram. The gure shows the internal re ections within the circuit that occur when the generator and load impedances Z g (s) and Z L (s) respectively, are not matched to the transmission line characteristic impedance Z 0 (s): . . . . . . . . . . 38 Figure 2.6: (a) Time Domain Linear Filter Model. (b) Frequency Domain Lin- ear Filter Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 Figure 3.1: Magnetic potential surface . . . . . . . . . . . . . . . . . . . . . . . 65 Figure 3.2: Antenna Equivalent Circuit Model . . . . . . . . . . . . . . . . . . 85 Figure 3.3: Measurement in the Frequency Domain. . . . . . . . . . . . . . . . 93 Figure 3.4: An equivalent decomposition of the measurement model. . . . . . . 93 Figure 3.5: The desired measurement for FCC compliance. . . . . . . . . . . . 94 Figure 3.6: Antenna S21 plots . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 Figure 3.7: Truncated Ground Plane Large TEM Horn Antenna Mechanical Design. All measurements are in inches. . . . . . . . . . . . . . . 95 xv Figure 3.8: Large TEM Horn antennas manufactured by the author to facilitate the waveform design research. . . . . . . . . . . . . . . . . . . . . 96 Figure 3.9: Measuring the Large TEM Horn Antenna in the anechoic chamber. 96 Figure 3.10: Measuring the Large TEM Horn Antenna in the anechoic chamber. 97 Figure 3.11: Truncated ground plane large TEM horn antenna rear view. All measurements are in inches. . . . . . . . . . . . . . . . . . . . . . . 99 Figure 3.12: Truncated ground plane large TEM horn antenna top view. All measurements are in inches. . . . . . . . . . . . . . . . . . . . . . . 99 Figure 3.13: Truncated ground plane model of the TEM horn antenna { per- spective view. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 Figure 3.14: Truncated ground plane TEM horn simulated input impedance. . 100 Figure 3.15: Truncated ground plane TEM horn antenna gain animation as a function of frequency. Note: This is an animated gure. Printed media will only show the antenna pattern at 1.0GHz. . . . . . . . . 101 Figure 3.16: Truncated ground plane TEM horn simulated gain at 2GHz. . . . 101 Figure 3.17: Truncated ground plane TEM horn simulated gain at 3GHz. . . . 102 Figure 3.18: Truncated ground plane TEM horn simulated gain at 4GHz. . . . 102 Figure 3.19: Truncated ground plane TEM horn simulated gain at 5GHz. . . . 103 Figure 3.20: Truncated ground plane TEM horn simulated gain at 6GHz. . . . 103 Figure 3.21: Truncated ground plane TEM horn simulated gain at 7GHz. . . . 104 Figure 3.22: Truncated ground plane TEM horn simulated gain at 8GHz. . . . 104 Figure 3.23: Truncated ground plane TEM horn simulated gain at 9GHz. . . . 105 Figure 3.24: Truncated ground plane TEM horn simulated gain at 10GHz. . . 105 Figure 3.25: Diamond dipole antenna type used in experiments. . . . . . . . . 107 Figure 3.26: Diamond dipole antenna in the anechoic chamber. The measure- ments were made in a Styrofoam test xture. . . . . . . . . . . . . 107 Figure 3.27: Scattering model of a two port network. . . . . . . . . . . . . . . . 111 Figure 3.28: Vector Network analyzer (VNA) System Model. The model shows the calibrated reference plane at the location of the VNA as well as the RF In and RF Out ports. . . . . . . . . . . . . . . . . . . . . . 114 xvi Figure 3.29: Simplied Spectrum Analyzer Model. . . . . . . . . . . . . . . . . 115 Figure 3.30: (a) Computed noise variance without VNA corrections. (b) Com- puted noise variance with VNA corrections. The measurement range was 50MHz to 20.05GHz. . . . . . . . . . . . . . . . . . . . . 120 Figure 3.31: Antenna Measurement Setup in an Anechoic Chamber. . . . . . . 123 Figure 3.32: (a) Measured S21 for Motorola antennas 1, 2 and 3. (b) Computed absolute gains. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 Figure 3.33: (a) Measured S21 for Motorola antennas 1, 2, 3 and 4. (b) Com- puted absolute gains for antennas 1, 2, 4. Note potential errors in antenna #4 (blue curve) at 11GHz and in antenna #2 (red curve) at 10GHz. . . . . . . . . . . . . . . . . . . . . . . . . . . 148 Figure 3.34: (a) Measured S21 for Skycross antennas 10, 12 and 14. (b) Com- puted absolute gains. . . . . . . . . . . . . . . . . . . . . . . . . . . 148 Figure 3.35: (a) Measured S21 for Skycross antennas 10, 12 and 15. (b) Com- puted absolute gains. . . . . . . . . . . . . . . . . . . . . . . . . . . 149 Figure 3.36: (a) Measured S21 for Large TEM Horn antennas 1, 2 and 3. (b) Computed absolute gains. . . . . . . . . . . . . . . . . . . . . . . . 149 Figure 3.37: (a) Measured S21 for Large TEM Horn antennas 1, 2 and 5. (b) Computed absolute gains for antennas 1, 2 and 5. . . . . . . . . . . 149 Figure 3.38: (a) Measured S21 for Large TEM Horn 1 and small TEM Horn antennas 1 and 2. (b) Computed absolute gains for Large TEM Horn 1 and small TEM Horn antennas 1 and 2. . . . . . . . . . . . 150 Figure 3.39: (a) Measured S21 for Small TEM Horn 2 and Diamond Dipole an- tennas 1 and 2. (b) Computed absolute gains for the Small TEM Horn 2 and Diamond Dipole antennas 1 and 2. Gains for the Dia- mond Dipoles computed for two and three antenna method. Note the signicant errors. . . . . . . . . . . . . . . . . . . . . . . . . . . 150 Figure 4.1: Search complexity of a 4 level quantizer as a function of increasing sequence length. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 Figure 4.2: The eciency measures of a length 2 sequence at a reproduction rate of 20GHz for the large TEM horn antenna. Q =f3;1; +1; +3g, best = 1:548 10 3 %, 2nd best = 4:542 10 4 %, 3rd best = 1:897 10 4 %. . . . . . . . . . . . . . . . . . . . 155 xvii Figure 4.3: The reduced set of eciency measures of a length 2 sequence at a reproduction rate of 20GHz for the large TEM horn antenna. Q =f3;1; +1; +3g, best = 1:548 10 3 %, 2nd best = 4:542 10 4 %, 3rd best = 1:897 10 4 %. . . . . . . . . . . . . . . . . . . . 155 Figure 4.4: The eciency measures of a length 2 sequence at a reproduction rate of 25.5GHz for the large TEM horn antenna. Q =f3;1; +1; +3g, best = 2:148 10 3 %, 2nd best = 4:66 10 4 %, 3rd best = 2:185 10 4 %. . . . . . . . . . . . . . . . . . . . 156 Figure 4.5: The reduced eciency measures of a length 2 sequence at a repro- duction rate of 25.5GHz for the large TEM horn antenna. Q =f3;1; +1; +3g, best = 2:148 10 3 %, 2nd best = 4:66 10 4 %, 3rd best = 2:185 10 4 %. . . . . . . . . . . . . . . . . . . 156 Figure 4.6: The eciency measures of a length 3 sequence at a reproduction rate of 13.6GHz for the large TEM horn antenna. Q =f3;1; +1; +3g, best = 2:031 10 3 %, 2nd best = 1:683 10 3 %, 3rd best = 4:597 10 4 %. . . . . . . . . . . . . . . . . . . . 157 Figure 4.7: The histogram of the eciency measures of a length 3 sequence at a reproduction rate of 13.6GHz for the large TEM horn antenna. Q =f3;1; +1; +3g, best = 2:031 10 3 %, 2nd best = 1:683 10 3 %, 3rd best = 4:597 10 4 %. . . . . . . . . . . . . . . . . . . . 157 Figure 4.8: The eciency measures of a length 3 sequence at a reproduction rate of 13.6GHz for the large TEM horn antenna. Q =f3;1; +1; +3g, best = 2:031 10 3 %, 2nd best = 1:683 10 3 %, 3rd best = 4:597 10 4 %: . . . . . . . . . . . . . . . . . . . 158 Figure 4.9: The histogram of the eciency measures of a length 5 sequence at a reproduction rate of 35GHz for the large TEM horn antenna. Q =f3;1; +1; +3g, best = 0:021%, 2nd best = 0:02%, 3rd best = 0:015%. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 Figure 4.10: The eciency measures of a length 5 sequence at a reproduction rate of 35 GHz for the large TEM horn antenna. Q =f3;1; +1; +3g; best = 0:021%; 2nd best = 0:02%; 3rd best = 0:015%: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 Figure 4.11: The histogram of the eciency measures of a length 5 sequence at a reproduction rate of 35GHz for the large TEM horn antenna. Q =f3;1; +1; +3g, best = 0:021%, 2nd best = 0:02%, 3rd best = 0:015%. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 Figure 4.12: Best result for the large TEM horn for m = 1; 2; ; 7 . . . . . . . 161 Figure 4.13: TEM Horn eciency histogram for m=6 . . . . . . . . . . . . . . . 161 xviii Figure 4.14: TEM Horn eciency search summary (n = 1; 2:::; 7) . . . . . . . . 162 Figure 5.1: (a) Generated time domain waveform at a clock reproduction rate of 25 GHz and column index n = 963. , (b) Pulse intrinsic gain, (c) D/A output waveform fed to the antenna, (d) Large TEM horn response with outdoor mask. max 4:2% . . . . . . . . . . . . . 169 Figure 5.2: Data Processing Flow Diagram. . . . . . . . . . . . . . . . . . . . . 176 Figure 5.3: (a) Small TEM horn indoor eciency summary. opt 40:14% Clock = 35 GHz, N = 11, (b) Eciency contour, (c) Observed most ecient unltered D/A time domain waveform with column index n = 619; 234, (d) Corresponding EIRPSD for the indoor mask. . . 180 Figure 5.4: (a) Small TEM Horn outdoor eciency summary. opt 31:4% Clock = 29 GHz, N = 11, (b) Eciency contour, (c) Observed most ecient unltered D/A time domain waveform with column index n = 755; 070 , (d) Corresponding EIRPSD for the outdoor mask. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 Figure 5.5: (a) Large TEM Horn outdoor eciency summary. opt 21:9% Clock = 31 GHz, N = 11, (b) Eciency contour, (c) Observed most ecient unltered D/A generated time domain waveform with column index n = 975; 211, (d) Corresponding EIRPSD for the outdoor mask. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 Figure 5.6: (a) Large TEM horn indoor eciency summary. opt 37:1% Clock = 45 GHz, N = 10, (b) Eciency contour, (c) Observed most ecient unltered D/A generated time domain waveform with column index n = 185; 118, (d) Corresponding EIRPSD for the indoor mask. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 Figure 5.7: (a) Diamond dipole outdoor eciency summary. opt 0:675% Clock = 35 GHz, N = 11,(b) Eciency contour ,(c) Observed most ecient unltered time domain waveform with column index n = 754; 663, (d) Corresponding EIRPSD for the indoor mask. . . . . . 184 Figure 5.8: (a) Diamond dipole indoor eciency summary. opt 2:1% Clock = 35 GHz, N = 8, (b) Eciency contour, (c) Observed most ecient unltered generated time domain waveform with column indexn = 16; 028, (d) Corresponding EIRPSD for the indoor mask. . . . . . 185 Figure 6.1: (a) Generated time domain waveform at a clock reproduction rate of 37GHz. A 4-level cubic quantizer was used. The waveform vec- tor is (27;1; 1; 27; 1;27;27;27; 1; 27; 27;1;27;27) t . (b) Large TEM horn response and the outdoor mask. max 9:95% (c) Histogram of the waveform eciencies in dB. . . . . . . . . . . 190 xix Figure 6.2: (a) Generated time domain waveform at a clock reproduction rate of 37GHz. A 4-level cubic quantizer was used. The waveform vector is (1;27;27; 27; 27;1;27;27; 27; 27; 1;27;27;1) t . (b) Small TEM horn response and the outdoor mask. max 24:57% (c) Histogram of the waveform eciencies in dB. . . . . . . . . . . 191 Figure 6.3: (a) Generated time domain waveform at a clock reproduction rate of 37GHz. A 4-level cubic quantizer was used. The waveform vec- tor is (1;1;27; 27; 27; 1;27;27; 1; 27; 27;27;1;1) t . (b) Diamond Dipole response and the outdoor mask. max 2:37% (c) Histogram of the waveform eciencies in dB. . . . . . . . . . . 192 Figure 6.4: (a) Generated time domain waveform at a clock reproduction rate of 37GHz. A 4-level cubic quantizer was used. The waveform vector is (27;1; 27; 27;27;27;27; 27;1;1; 27;1;1;27) t . (b) Large TEM Horn response and the indoor mask. max 42:05% (c) Histogram of the waveform eciencies in dB. . . . . . . . . . . 193 Figure 6.5: (a) Generated time domain waveform at a clock reproduction rate of 37GHz. A 4-level cubic quantizer was used. The waveform vector is (1;27; 1;1;1;1; 27; 27;1;27;27;1; 27;1) t . (b) Large TEM Horn response and the indoor mask. max 46:68% (c) Histogram of the waveform eciencies in dB. . . . . . . . . . . . . 194 Figure 6.6: (a) Generated time domain waveform at a clock reproduction rate of 37GHz. A 4-level cubic quantizer was used. The waveform vec- tor is (1;27; 27; 27;27;1;27; 27; 1;1;1;1;1;1) t . (b) Diamond Dipole antenna response and the indoor mask. max 13:11%. (c) Histogram of the waveform eciencies in dB. . . . . . 195 Figure E.1: (a) Region of integration for covariance function. (b) Transformed region of integration. . . . . . . . . . . . . . . . . . . . . . . . . . . 221 xx Abstract Sets of ultra{wideband signals are designed and generated using a mathematical model of a zero{order hold, digital{to{analog converter. An eciency measure of lling the FCC spectral mask is developed and optimized for various choices of digital{to{analog converter parameters and waveform time duration. The proposed optimization technique takes into account ltering eects in the digital{to{analog converter as well as a measured antenna transfer function characteristic. The research presented here demonstrates that a simple ultra{wideband transmitter can be designed which consists only of a high speed digital{to{analog converter and a suitable ultra{wideband antenna. The waveform design problem is explicitly dependent on the transmitter's equivalent transfer function, the FCC spectral mask constraint, and the ultra{wideband antenna that is included as a factor in the equivalent transfer function of the system. xxi Chapter 1 Introduction Impulse{radio (IR) is a baseband digital communication technique whereby information symbols are conveyed to a receiver by an information{bearing signal without a sinusoidal carrier. Carriers are typically used in the majority of modern communication, radar and imaging systems. The ultra{wideband IR technique uses extremely short bursts of baseband pulses that are on the order of a few hundred picoseconds which imply that large bandwidths are required (on the order of a GHz). The relationship between a measurable function in time and it's frequency dependence has been well{established by the Fourier Transform. Signals that have an extremely short time duration occupy a very large bandwidth; hence the term, ultra{wideband (UWB) impulse radio. The potential simplicity of an UWB{IR makes the approach an attractive commu- nication method. Ideally, the transmitter consists of a few basic components such as a voltage source, a fast switch, and an UWB antenna that can eciently radiate the UWB pulse with high delity (i.e., minimum distortion). 1 The receiver, on the other hand, only requires an UWB antenna, a simple matched lter and a decision device. Unfortunately, it is dicult to design and manufacture UWB systems because of the extremely large bandwidth required. The United States Federal Communications Commission (FCC) has traditionally placed very tight control over the spectral emissions of radiators (in band and out of band) with the rationale of minimizing interference to other RF systems operating in well{dened frequency bands. Ideal UWB{IR would use Dirac delta functions in time (impulses). A single Dirac delta function has a at frequency spectrum. It is not pos- sible to generate a true impulse in practice. An extremely short time-duration pulse, with relatively large amplitude, serves as an approximation of the Delta function. The interested reader is referred to the excellent book by Kanwal [28] for further information about generalized functions . An approximation of a Dirac delta function imply that the frequency spectrum will not be at. The FCC regulates and mandates that the eective isotropic radiated power per MHz of an UWB signal must be compliant with an FCC dened constraint. The constraint minimizes the potential of UWB systems from causing harmful interference to other existing radio systems. The popular and prolic Global Position System (GPS) relies on a receiver estimating its position from a very weak signal plus an additive noise term. The FCC had concerns that UWB systems could signicantly degrade the performance of systems such as GPS. They dened a non-uniform spectral emission constraint on the radiated emission of any UWB system operating in the US. The non-uniformity of the constraint signicantly in uences the detailed structure and hence the design of the UWB radiated waveform in both; frequency and in space. 2 To comply with the FCC spectral density constraints, a systems approach must be used to design an FCC complaint UWB{IR system. All factors in the UWB{IR that can in uence FCC compliance must be considered. The subsystems that compose the UWB{IR cannot be considered all{pass lters because of the extremely large bandwidth involved in the pulse transmission. Each subsystem in the radio will aect the radiated emission in a frequency-dependent manner. Their impacts must be considered in the waveform design methodology. This dissertation shows that FCC{compliant UWB{IR waveforms can be synthesized from an integer-scaled, superposition of very simple basis functions. The integer{scaled basis functions are used represent a mathematical model for the output of a digital{to{ analog converter (DAC). One of the interesting attributes of the waveform design problem is that the FCC emission constraint is in the frequency domain and further pertains to a point in space. The time-domain signal synthesis occurs at the location of the transmit pulse generator. As such, the waveform design requires detailed knowledge of: the pulse at the output of the signal generator propagation through various ltering subsystems (e.g.,cables, parasitic lters, an- tennas, etc.) the waveform's radiated performance as it propagates in space Obviously, the pulse distortion eects of the subsystems must be considered since most subsystems are not spectrally at across several GHz of bandwidth. 3 This thesis demonstrates that baseband digital pulses can be optimized to a specic FCC constraint for a particular realization of an UWB system. A systems methodology was used whereby a typical system was decomposed into subsystems. The subsystems were analyzed with respect to frequency domain ltering eects and the impacts on a digitally generated pulse were analyzed. One unique approach taken in this dissertation is that real antennas were used in the analysis. The results indicate that antennas do have a signicant spectral-shaping impact on the generated and received UWB-IR pulse. The pulse design must include the spectral shaping characteristics of the antennas. The antennas cannot be ignored for an optimal pulse design. After a detailed analysis was performed for all the major subsystems, pulses of a nite time duration were mathematically constructed and ltered by each of the subsystems. A performance metric was used to assess pulse performance with respect to the FCC constraint. The FCC{mask-constrained and optimized pulse, maximizes the radiated power for a given set of system parameters. A well-designed UWB-IR waveform will have the eect of improving the signal-to-noise ratio of an UWB{IR system. 1.1 Motivation and Problem Denition The FCC [19] has imposed a constraint mask M(f) (see Figure 1.1) to bound the power spectral density S EIRP (f) of the equivalent isotropic radiated power (EIRP) of a UWB communication signal. The detailed shape of M(f) depends on the particular UWB system application (e.g., radar, imaging or communication). 4 Let x(u;t) denote a random process 1 where u denotes a point in the sample spaceU of possible events, andt denotes a time index belonging to the set of real numbersR. Let one representation of the modulated pulse train be x(u;t) = 1 X k=1 m(u;k)p(tkT s ); (1.1.1) where m(u;k)2f1g is a real random binary data sequence that is indexed by the integerk: It has been assumed that the wide-sense stationary (WSS) process m(u;k) has ensemble mean and correlation function given by Efm(u;k)g = 0; Efm(u;k)m(u;l)g =R m (kl): For the modulation format given in equation (1.1.1) and many other modulation formats, the power spectral density (PSD) of x(u;t), with pulses p(t); factors into two parts. The factorization is given by [52], [55] S x (f) =S m (f)jP (f)j 2 ; (1.1.2) whereS m (f) is a function only of the modulation process, and P (f) is the Fourier trans- form of the -limited (seconds) UWB pulse shape p(t) for 0<t, i.e., P (f) = Z 0 p(t) exp(j2ft)dt: (1.1.3) 1 A review of random processes is provided in Appendix E on page 210 5 The transformation from the modulated pulse train in the transmitter to the radiated electromagnetic far eld on which the EIRP mask constraintM(f) is imposed, is a linear transformation which we represent by the system function H eq (f). Hence the power spectral density S EIRP (f) which is bounded by the mask M(f) is of the form S EIRP (f) =S x (f)jH eq (f)j 2 =S m (f)jP (f)j 2 jH eq (f)j 2 : (1.1.4) It can be assumed that the PSD of the modulation process S m (f)a c is approximately a constant for all f in the band of interest, and without loss of generality, S EIRP (f)a c jP (f)j 2 jH eq (f)j 2 : The constanta c includes factors that convert the units ofjP (f)H eq (f)j 2 (typically volts 2 /Hertz) to the units ofM(f) (typically Watts/megahertz), in addition to scaling by the constant spectral density of the modulation. Hence the FCC requirement is that a c jP (f)j 2 jH eq (f)j 2 .M(f) (1.1.5) for all f in the measurement range [F min ;F max ] =I. 1.2 Mask-Filling Eciency Measure The mask-lling eciency of a particular signal design can be dened as the ratio of the power contained in the resulting FCC-compliant PSD S EIRP (f) to the total power 6 Figure 1.1: FCC Outdoor and Indoor PSD Constraints that could be contained under the maskM(f){both being evaluated in the measurement range. Hence, the eciency metric is dened as , a c R I jP (f)j 2 jH eq (f)j 2 df R I M(f)df : (1.2.1) This mask-lling eciency measure for a pulse waveform p(t), given the system func- tion H eq (f), also embodies the eects of the transmitter, antenna, and spatial signal propagation. LetP PG denote the countable set of possible pulse waveformsp n (t) that can be imple- mented in the pulse generator. Here, the subscriptn simply indexes the lexicographically{ ordered elements in the setP PG . A scale factor A n can be computed for each waveform inP PG , so that p(t) =A n p n (t); (1.2.2) 7 and the FCC mask bound M(f) in (1.1.5) is satised by p(t) for all f 2 [F min ;F max ], and furthermore, the bound is achieved with equality for at least one value of f in the measurement range. The constantA n can always be implemented in the transmitter as an all-band amplication/attenuation because any spectral shaping by the linear amplier can be included in the ltering process of the system. 1.2.1 Pulse Waveform Selection Procedure 1. For each possible waveform p n (t)2P P, whereP is the set over which the search is performed, nd the constant A n so that (1.1.5) is satised for all f and is achieved with equality for some f in the measurement range. This computation in dB form is equivalent to evaluating (a c A 2 n ) dB = min f2(F min ;Fmax) [(M(f)) dB ((jH eq (f)j 2 ) dB (jP n (f)j 2 ) dB ) : (1.2.3) 2. For each possible waveform p n (t)2P , calculate the mask-lling eciency n of the waveform p n (t) by evaluating n , a c A 2 n R I jP n (f)j 2 jH eq (f)j 2 df R I M(f)df : (1.2.4) 3. Determine the waveform p nopt (t) with the highest eciency max , i.e., max = max 1nN n : 8 and n opt = arg max 1nN n : This procedure describes an exhaustive search, and hence is applicable to situations in which the setP is nite and small. 1.3 Literature Review and Comments on Prior Work If there were no constraints on the pulse shapep(t), then it may be possible to determine a waveform p(t) with 100% mask-lling eciency by solving jP (f)j 2 = M(f) a c jH eq (f)j 2 (1.3.1) forP (f). This assumes that the right side of (1.3.1) can represent the squared magnitude of a nite energy waveform. Factorizations of this type that recover a pulse shape p(t) with time support [0;1) can be carried out, provided that M(f)=jH eq (f)j 2 satises the Paley{Wiener conditions [64]. Then the optimal pulse waveformp(t) can be recovered by a Fourier inverse transform of P (f). Slepian et al. studied methods to optimize the concentration of energy in a band of frequencies for functions with nite time support, using prolate spheroidal wave functions for continuous time [58] and discrete time [57] systems (see also [39], [40], [63]). Slepian's problem formulation diers from our problem statement in the following respects: 1. Here, H eq (f) is not an ideal all-pass lter. Any application of this prior work to the problem stated here must assume H eq (f) = 1; 8f2I. 9 2. Here, we are constrained by a PSD mask, while the prior work only maximizes in-band power without concern for the detailed shape of the in-band power density. 3. The allowed waveform set in the prior work is dierent fromP. The use of Prolate spheroidal functions as UWB signals has been suggested [42] and [68]. In 2003, Wu, Molisch et al. [67] published an internal Mitubishi Electric Research Laboratory (MERL) report on their waveform design eorts. The author was made aware of the report in 2009 by a personal and casual conversation with Professor Molisch. Wu, Molisch et al., investigated the UWB pulse design as an analog FIR pulse shaping problem. Their approach was to model treat the problem as a quadratic{constrained min{max problem. They used a distortion measure for to quantify pulse performance. Pulse delays and amplitudes were treated as parameters that required optimization. They treated the design problem as a two-dimensional optimization. The joint optimization entailed holding one parameter xed while optimizing the other parameter. The set of time delays k ; were treated as integer multiples of a uniform step size t: Thus, k =kt: Berger et al. [8], viewed the pulse-shaping problem as a semi-denite programming problem using nonconstant upper bounds. Their approach was based on the FIR design via semidenite programming. They showed that they can approximate any piecewise continuous bounding function via a modied Fourier expansion with non-orthogonal basis functions. They also used the eciency metric as dened in [36] to measure the mask{ lling eciency of a designed pulse. 10 Chapter 2 Signal Synthesis 2.1 Introduction Figure 2.1: System Block Diagram We begin the ultra{wideband (UWB) waveform analysis and design by assuming the model given by Figure 2.1. The model is composed of four parts: 1. The digital UWB waveform generator, 11 2. the impulse radio transmitter, 3. a measured radio frequency response, 4. the required mathematical analysis to determine FCC compliance. The Digital UWB Signal Generator models a Zero{Order{Hold (ZOH) digital-to- analog converter (DAC) and produces a waveform that is a function of time. The DAC is the waveform source for the Impulse Radio. The impulse radio is the hypothesized model of the UWB transmitter. It includes the DAC, the data modulator(s), the transmission line connecting the DAC to the transmit antenna and the transmit antenna. The data-modulated DAC waveform directly feeds the waveform from the DAC lter output signal into the UWB transmit antenna via a transmission line. The transmission line linking the DAC to the antenna must be suitably designed for minimal pulse distortion during propagation towards the antenna. The Measured Response represents a measured propagation response in the frequency domain that is inclusive of the transmit and receive antenna. The receive antenna must be included in the analysis since it modies the true pulse response. The FCC constraint is evaluated in space exclusive of any receive antenna. The last block is called Mathematical Analysis and represents the analysis that must be performed to ensure FCC compliance of the designed UWB pulse. It includes an inverse lter to normalize the expected pulse distortion caused by the receive antenna. After the receive antenna inverse lter, the waveform, that is a function of time, is converted to the frequency domain by the Fourier Transform denoted by Ffg. The magnitude square of the waveform is computed and then scaled by the parameter A so 12 that at least one point of the waveform touches the FCC mask constraint. The compare and compute eciency block a mask-lling eciency measure against a particular FCC mask constraint so that dierent waveforms may be compared against each other. The nal block in the model computes the maximum of all generated waveforms and denotes it as opt : The waveform with measure opt is the best waveform that can be constructed for the parameters provided in the model. The model in Figure 2.1 will be used throughout the dissertation. 2.2 Signal Construction and Generation LetN denote the set of natural numbers (i.e., 1; 2; 3; ) andZ denote the ring of integers (i.e., ;2;1; 0; 1; 2; ). The digital-to-analog converter (DAC) accepts an integer n and produces an integer{valued coecient vector q n 2 Z M where M denotes the dimension of the vector. The coecient vector q is modulated onto a stream M Dirac delta functions while the clock enable signal is active. The result of modulating q n 2Z M onto a train of Dirac delta functions is given by q n (t) = M1 X m=0 q m;n (tmT s ): (2.2.1) The signalq n (t) is then input into the linear-time-invariant pulse shaping lter (t) where (t) = 8 > > > > < > > > > : 1= p T s ; if 0<tT s 0; otherwise (2.2.2) 13 and in the frequency domain P (f) =Ff (t)g = Z 1 1 (t)e j2ft dt: The pulse energy is E p = Z 1 1 j (t)j 2 dt = 1: (2.2.3) The basis pulse is orthonormal for all integer i;j time shifts of T s i.e., Z 1 1 (tiT s ) (tjT s )dt = 8 > > > > < > > > > : 1; if i =j 0; otherwise; (2.2.4) which makes (t) suitable as an ideal basis function for UWB pulse construction. The pulse energy is normalized to the reciprocal of the square-root of the clock rate T s . With this normalization, (t) has unit-energy for all investigated clock rates. The pulse p n (t) has a minimum time support of MT s seconds. The output p n (t) of the pulse-shaping lter P (f) is related to the input signal q n (t) by the convolution integral given by p n (t) =q n (t) (t) = Z 1 1 q n () (t)dt = M1 X m=0 q m;n (tmT s ) for 0<tMT s (2.2.5) 14 and has Fourier transform P n (f) = M1 X m=0 q m;n P (f)e j2mTsf =P (f) M1 X m=0 q m;n e j2mTsf : (2.2.6) The term e j2mTsf in equation (2.2.6) results from the time-shifting property of the Fourier transform. A realizable DAC can only generate a nite number of waveforms. Let the setQZ denote the nite set of integer-valued amplitudes. Let q m 2Q. The number of integer elements q m 2Q is denoted asjQj and is called the order or cardinality ofQ. Consider the generation of the nite-time waveform given by equation (2.2.5). The waveformp n (t) is constructed fromM coecientsq m;n 2Q. The total number of output waveforms that can be produced by the DAC is equal toN =jQj M whereM is the vector length of q. The index n runs over the set of possible waveforms that can be produced by the DAC. The train of pulses (t) in equation (2.2.5) that are scaled by the integer coecients q m during themth time interval, produces a staircase waveform with time support MT s . Further ltering of (t) occurs within the DAC and is accounted for by H D/A (f). The lterH D/A (f) is included in the equivalent lter modelH eq (f) given by equation (1.2.1). The scaling coecients q m;n are chosen from a set of integers that are uniformly spaced. Hence, the voltage range R of this uniform quantization is given by R = (jQj 1) step ; (2.2.7) 15 where step is the magnitude of the dierence between any two adjacent elements ofQ. For a uniform symmetric (about zero) quantizer, the order of the setQ must satisfy the cardinality condition for even quantizers given by jQj mod 2 = 0; and jQj 1 mod 2 = 0 for odd quantizers. The primary focus being discussed here is the set of even quantizers. The number of distinct waveforms of time duration MT s , that the DAC can produce is N =jQj M : (2.2.8) The coecient M-tuple q m;n , m = 0;:::;M 1; used by the DAC to generate p n (t) can be generated systematically for a uniform symmetric quantizer by representingn as a basejQj number and using the elements in this representation as the coecientM-tuple. That is, q m;n = n jQj m modjQj step R 2 ; (2.2.9) where 0n<N1 and [x] represents the integer part ofx. Choosing step = 2 ensures for simplicity that the coecients q m;n are integers for even choices ofjQj and for odd choices ofjQj; step = 1. The coecients represented by q m;n can be viewed as entries of an MN matrix Q. The rows index range is 0m<M 1 and the column index range is 0n<N 1. 16 An inverse mapping from the vector q n to its matrix column index n is determined by n = 1 step M1 X m=0 (q m +R=2)jQj m : (2.2.10) The DAC performs the mapping q n !p n (t) according to (2.2.5) where q n is the n th column of Q MN . The Fourier transform of p n (t) in (2.2.5) is required to carry out the waveform selection procedure. Then P n (f) =Ffp n (t)g =P (f) M1 X m=0 q m;n exp (j2fmT s ); whereFf (t)g =P (f): 2.3 Signal Processing This section describes the signal processing of the waveforms in the frequency domain. The UWB waveform design must be compliant with the FCC spectral mask constraint in the frequency domain. The component blocks shown in Figure 2.1 are represented using frequency-domain notation because the processing methods employed here are in the frequency-domain as opposed to the time-domain. The signals owing on the lines connecting the subsystems are functions of time. After processing by the receive antenna and the inverse receive antenna lter, the time-domain waveform must be converted to the frequency-domain for further analysis. The frequency domain conversion is required to determine mask compliance. 17 2.3.1 Waveform Processing Let the column vectors x2C k and y2C k . The inner product is dened as hx;yi =x y y = X k x y: where in this notationy denotes the conjugate transpose and denotes complex conjuga- tion. In this notation, conjugation is with respect to the rst element ofh;i. The inner product property implieshx; yi =hy; xi : Proof: hx;yi = (x y y) = X k x y ! = X k x y = X k y x =hy;xi: (2.3.1) Equation (2.2.6) shows that P n (f) =P (f) M1 X m=0 q m;n e j2mTsf | {z } Pulse-shaping factor (2.3.2) and clearly, P (f) factors outside of the summation. Neglecting the pulse{shaping lter P (f) for now because it is a common lter for allP n (f), the terms in the summation can be regarded as the pulse-frequency equalization or shaping factor. The equalization factor is used in the search as compensation for the non-ideal ltering that naturally occurs in an UWB system. Dening G n (f) = M1 X m=0 q m;n e j2mTsf 18 for the nth equalization function, G n (f) can be represented as an inner product G n (f) =he(f); q n i; where e(f) = (1;e j2Tsf ; ;e j2(M1)Tsf ) t (2.3.3) for q 2 Z M and t denotes the transpose of the vector e(f). The function G n (f) is a continuous periodic function of its argument f with period F p when the domain of f 2 (1;1). Uniformly sampling P n (f) in (2.2.6) at frequencies f i = f 0 +if for i = 0; 1;:::;L 1; the complex vector p n 2C L is dened. Then, 2 6 6 6 6 6 6 6 6 6 6 4 P n (f 0 ) P n (f 1 ) . . . P n (f L1 ) 3 7 7 7 7 7 7 7 7 7 7 5 | {z } pn = 2 6 6 6 6 6 6 6 6 6 6 4 P (f 0 ) P N1 m=0 q m;n e j2mTsf 0 P (f 1 ) P N1 m=0 q m;n e j2mTsf 1 . . . P (f L1 ) P N1 m=0 q m;n e j2mTsf L1 3 7 7 7 7 7 7 7 7 7 7 5 2 6 6 6 6 6 6 6 6 6 6 4 P n (f 0 ) P n (f 1 ) . . . P n (f L1 ) 3 7 7 7 7 7 7 7 7 7 7 5 = 2 6 6 6 6 6 6 6 6 6 6 4 P (f 0 ) 0 0 0 P (f 1 ) 0 . . . . . . . . . . . . 0 0 0 P (f L1 ) 3 7 7 7 7 7 7 7 7 7 7 5 | {z } B 2 6 6 6 6 6 6 6 6 6 6 4 P M1 m=0 q m;n e j2nTsf 0 P M1 m=0 q m;n e j2nTsf 1 . . . P M1 m=0 q m;n e j2nTsf L1 3 7 7 7 7 7 7 7 7 7 7 5 : 19 2 6 6 6 6 6 6 6 6 6 6 4 P n (f 0 ) P n (f 1 ) . . . P n (f L1 ) 3 7 7 7 7 7 7 7 7 7 7 5 = B 2 6 6 6 6 6 6 6 6 6 6 4 1 e j22Tsf 0 e j2(N1)Tsf 0 1 e j22Tsf 1 e j2(N1)Tsf 1 . . . . . . . . . . . . 1 e j22Tsf L1 e j2(N1)Tsf L1 3 7 7 7 7 7 7 7 7 7 7 5 | {z } E 2 6 6 6 6 6 6 6 6 6 6 4 q 0;n q 1;n . . . q M1;n 3 7 7 7 7 7 7 7 7 7 7 5 | {z } qn : Thus, the nth pulse constructed by the DAC is succinctly written as p n = BEq n : (2.3.4) In general, because the waveform length is nite, and the waveform coecients are re- stricted to the ring of integers, all the waveforms that can be constructed by the DAC can be represented using the matrix representation P = BEQ: (2.3.5) The pulse coecients q n used to construct p n are contained in thenth column of P: The number of columns in the complex matrix P is determined by the number of possible waveforms (column vectors) in the real integer-valued matrix Q. The maximum number of vectors N that can be constructed from a nite set of values q 2 Q is given by N =jQj M . Then, P2C LN ; B2C LL ; E2C LM ; and Q2Z MN : 2.3.2 Even Digital-to-Analog Converter Search-Space Reduction To apply the waveform optimization theory, we performed an exhaustive search for DAC- generated waveforms and noted the waveforms that had the highest eciencies for a given 20 set of parameters T s , H eq (f), M and N. Uniform even-symmetric quantization in the DAC was assumed. The search process involved constructing the coecient matrix Q2 Z M Z N . For each column q n of the matrix Q, the corresponding A n p n (t),jA n P n (f)j 2 and mask-lling eciency n was computed. This chain of mappings from q n to n is many-to-one. An equivalence relation of the form q j q k () jA j P j (f)j 2 =jA k P k (f)j 2 (2.3.6) can be constructed that creates equivalence classes of coecient vectors which produce the same EIRP density and hence the same mask-lling eciency. In the search for the coecient vectors q n that produce waveforms with the highest mask-lling eciency, the eciency of only one coecient vector from each equivalence class needs to be evaluated. Under (2.3.6), two coecient vectors q j and q k are equivalent if either q j =c q k (2.3.7) for some (positive or negative) constant c, or q j;m =q k;M1m for all m2f0; 1;:::;M 1g: (2.3.8) The rst condition (2.3.7) involves scaling which is compensated in the search algorithm by adjusting the weight A n in step 1 of the selection procedure (see Section 1.2.1 on page 8). The second condition (2.3.8) is based on the fact that reversing the order of the elements in a coecient vector time-reverses the real pulse waveform, thereby conjugating 21 its Fourier transform but leaving the EIRP density unchanged. As an example, let the set Q =f3;1; 1; 3g and consider the set of ordered 2-tuples represented by the coecient matrix Q. For this example M = 2;jQj = 4; and N = 16 (see (2.2.8)), where the coecient matrix Q was formed from equation (2.2.9). Then Q = 3 1 1 3 3 1 1 3 3 3 3 3 1 1 1 1 3 1 1 3 3 1 1 3 1 1 1 1 3 3 3 3 : (2.3.9) Each column vector of Q represents a coecient vector of a waveform that can be output from the DAC. Eliminating all but one of each set of equivalent column vectors from the coecient matrix produces a matrix Q whose columns are equivalence class representatives. As an example, the matrix Q = 1 1 1 1 1 3 3 1 (2.3.10) can be formed and is a non-unique equivalence class representative of Q. The requirement for the construction of Q is that one representative from each equivalence class must appear as a column in the matrix. The equivalence class representatives in Q will produce the exact same set of eciencies as Q and without duplication. The waveform with the highest mask-lling eciency is contained in Q . The search over Q is faster than searching over the full set of vectors in Q, because signal processing computations are only performed on a single equivalent-class vector. In the example shown, the search over sixteen vectors was reduced to a search over four vectors { a seventy-ve percent reduction in the number of vectors that need to be searched. A seventy-ve percent search reduction is not always possible. Consider using the same setQ as in the example above. 22 For q2Z 3 ; there arejQj 3 = 4 3 = 64 distinct vectors. The matrix Q is composed of 17 vectors and not 64 4 = 16 which may have been expected. 2.3.3 Odd Digital-to-Analog Converter Search{Space Reduction Letk be a positive integer such thatk 1: The odd-symmetric digital-to-analog converter (DAC) requiresjQj = 2k + 1 and implies 02Q. It is of interest to generate waveforms p n (t) at the output of the DAC that have time-support MT s and q m 2Z M . Any vector q with the property q 0 = 0 or q M1 = 0 is eliminated from inclusion in the equivalent class Q . Any vector q n that has a set of leading or terminating zeros corresponds to a waveformp n (t) with a time support less thanMT s seconds, and therefore is not included in the search. Waveforms with leading or terminating zeros are generated from the lower- dimensional vector q2Z L for L<M for both q 0 ;q L1 6= 0: Only waveforms with time support MT s are investigated here. Denition 1 (Admissible Set). An admissible setA is a set of nite length vectors such that for any vector a2A, the rst and last element of a is non-zero. The nite set of vectors s2S such that s2 Z N that can be output by an odd quantizer consists of two types of vectors either s2A or s2A c , thenS =A[A c : Q = 2 6 6 4 1 0 1 1 0 1 1 0 1 1 1 1 0 0 0 1 1 1 3 7 7 5 : (2.3.11) 23 The matrix Q consist only of an admissible set of vectors and has dimension 2 2. One such representation of an admissible set is given by Q = 2 6 6 4 1 1 1 1 3 7 7 5 : (2.3.12) For M = 3 and lettingQ =f1; 0; 1g the matrix Q has dimension 3 27, whereas Q = 2 6 6 6 6 6 6 4 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 3 7 7 7 7 7 7 5 (2.3.13) and has dimension 3 5. The reduced matrix represents a signicant reduction in the number of waveforms that must be evaluated. 24 2.4 The Signal-Path System Function To complete the model for waveform optimization, we must determine the system function H eq (f) of the signal path from the signal generator output p n (t) to the strongest portion of the radiated electric far eld ~ E(t) at a distance r from the radiating antenna. The power spectral density of the amplitudeE(t) of this eld is multiplied by 4r 2 to produce the measured isotropic radiated power spectral density S EIRP (f). The system function H eq (f) has several factors that are identied in Figure 2.1 on page 11. The signal- path system function H eq (f) encompasses the totality of all distortion eects from the generated pulse p n (t) to the radiated electric eld E(t). 2.4.1 DAC Filtering The waveformp n (t) is ltered by the digital-to-analog converterH D/A (f): The frequency- domain lter output is Ffp n (t)h D/A (t)g =P n (f)H D/A (f): (2.4.1) For the sample computations here, the system function H D/A (f) was determined from a measured response of a real UWB digital synthesizer. The step response of the signal generator was measured and it was determined that the signal generator had a 90% risetime of 41 ps. The shape of the leading edge of the response was determined to t well with the shape of a Gaussian cumulative distribution function. The Guassian distribution is fully characterized by a mean and variance 2 : The properties of random variables were used to compute the parameters of a zero-mean Gaussian density function. 25 The lter's impulse response variance was calculated to be 2 D/A = 1:25E 11 sec 2 , and thus the DAC lter is completely determined by h D/A (t) = 1 p 2 e (t t ) 2 2 2 ; (2.4.2) where for simplicity, the mean t = 0. The specied Gaussian density lter (see Figure 2.2) has a gain of -3 dB at approximately 10 GHz. The normalized (at DC) transfer function of H D/A (f) is Ffh D/A (t)g = H D/A (f) = 1:0 exp[7:8 10 23 (2f) 2 ]: (2.4.3) The lter gain is shown in Figure 2.2 0 5 10 9 × 1 10 10 × 1.5 10 10 × 2 10 10 × 15 - 10 - 5 - 0 Frequency (Hz) Gain (dB) Figure 2.2: Digital-to-Analog Converter Lowpass Filter 26 2.4.2 Transmission Line Fundamentals. Initially Relaxed Conditions. Pulses synthesized at the UWB generator will travel a distanced via a transmission line to the antenna which represents the circuit load. A thorough understanding of transmission line theory is warranted. Only under specic conditions outlined in this chapter will pulses synthesized at the generator travel to, and be radiated by an antenna in an undistorted form. Otherwise, all pulses synthesized at the generator will undergo distortions of the source pulse. The distortions will impact the radiated waveform seen at the point in space where the FCC constraint is applicable. In the assumed dierential transmission line circuit, lumped elements appear in the dierential crcuit model of a transmission line. Further, the circuit analysis must take into account that Maxwell's equations dene resistance, capacitance and inductance in a circuit. When the wavelengths are large compared to the device, the simplistic Ohm's law model (lumped element model) suces; otherwise, Maxwell's equations must be used to accurately describe the parameters of the system (see Plosney and Collin in [43]). A transmission line is a waveguide that has the purpose of eciently delivering power from a source to a load. Steady state wave propagation in a transmission line has been well studied (see Cheng [16], Pozar [44], Gonzalez [24], Collin [18], Mansour and Brenner [27], and Karmel, Camisa and Colef [29] ). The author studied under the latter three and learned immensely from all cited. Transmission line theory is reviewed and presented here because it is critical to the pulse generation problem and many system engineers are not familiar with the topic. 27 Rz Lz Gz Cz i(z;t) i(z + z;t) z + z z v(z;t) v(z + z;t) Figure 2.3: Dierential Transmission Line Model. Consider a dierential length of transmission line of total length z with zero initial conditions as shown in Figure 2.3. The length of transmission line is assumed to have a uniformly distributed series resistanceRz, series inductanceLz, a shunt conductance Gz and a shunt capacitance Cz, all per unit length of transmission line. Symbol Unit Name Units Unit Breakdown R Ohms/meter /m V/Am G Siemens/meter 1 /m A/Vm L Henries/meter H/m Vs/Am Z 0 Ohms V/A Table 2.1: Transmission Line Parameters The voltage drop across the conductanceGz per unit length that is in parallel with the capacitance Cz per unit length is v(z + z;t) = (Gz +Cz)i(z;t): 28 Applying Kirchho's voltage law around the circuit results in the partial dierential equation v(z;t) Rzi(z;t) +Lz @i(z;t) @t +v(z + z;t) = 0 Dividing the result by z and combining terms and then invoking a limit results in lim z!0 v(z + z;t)v(z;t) z + Ri(z;t) +L @i(z;t) @t = 0 @v(z;t) @z + Ri(z;t) +L @i(z;t) @t = 0: (2.4.4) A similar equation results when Kirchho's current law is used. The resulting equation is i(z;t) + Gzv(z + z;t) +Cz @v(z + z;t) @t +i(z + z;t) = 0: Following the same procedure used to develop equation (2.4.4) results in the partial dierential equation for the current given by @i(z;t) @z + Gv(z;t) +C @v(z;t) @t = 0: (2.4.5) To simplify the solutions of the partial dierential equations, we assume that the input voltage and the in-phase current functions are factorable (i.e., using phasors), and thus it is assumed that v(z;t) = Refv(z)e j!t g and i(z;t) = Refi(z)e j!t g: (2.4.6) 29 Using the assumed factorization for the voltages and currents, the partial dierential equa- tions become ordinary linear dierential equations that can be solved using the Fourier Transform operator Ffg. The frequency domain approach alleviates dealing directly with the time derivatives, because derivatives with respect to the time domain are equal to multiplication byj! in the frequency domain. Thus, the space-time domain dierential equations can be written as space-frequency equations dv(z) dz =(R +j!L)i(z) (2.4.7) di(z) dz =(G +j!C)v(z): (2.4.8) If the rst order equations are dierentiated with respect to the coordinate z; a set of second order dierential equations is derived d 2 v(z) dz 2 =(R +j!L) di(z) dz and di 2 (z) dz 2 =(G +j!C) dv(z) dz : (2.4.9) These two second order equations are related to the rst order equations. Substitution of (2.4.7) and (2.4.8) into (2.4.9) results in d 2 v(z) dz 2 = (R +j!L)(G +j!C)v(z) (2.4.10) and di 2 (z) dz 2 = (R +j!L)(G +j!C)i(z): (2.4.11) 30 Two independent second order equations have been determined: one equation is a function of the voltage only and the other equation involves only the current. Dening 2 = (R +j!L)(G +j!C); where = +j = p (R +j!L)(G +j!C): (2.4.12) The variable (Np=m) represents the attenuation coecient of the line and (rad=m) represents the phase coecient. The variables and are typically not constants. They tend to depend on! in a complicated manner which is clearly demonstrated by expansion of (2.4.12). Using 2 in (2.4.10) and (2.4.11) simplies the equations to d 2 v(z) dz 2 = 2 v(z) (2.4.13) and di 2 (z) dz 2 = 2 i(z): (2.4.14) A general solution to for (2.4.13) is V (z) =V + e z +V e z ; (2.4.15) where the rst term represents a wave propagating in the forward direction (towards the load from the generator) and the second term is a wave traveling in the reverse direction 31 (towards the generator from the load). Under an innite transmission line constraint, only the forward propagating wave exists because the re ected wave would increase without bound due to the second term in (2.4.15). A general solution for the current is similar to and is determined by dierentiating (2.4.15) with respect to thez-coordinate and then substituting the result into (2.4.8) and then solving for the current I(z). The results are I(z) =[I + e z +I e z ] (2.4.16) = 1 Z o [V + e z V e z ]; (2.4.17) where I + = V + Z o ; I = V Z o ; and Z o = s (R +j!L) (G +j!C) : (2.4.18) The termsI + ; I andZ o represent the current ow towards the load from the generator, the current ow from the load back to the generator, and the frequency-dependent char- acteristic impedance of the transmission line respectively. The termsV + ; V ; I + ; I are determined from appropriate boundary conditions. 2.4.3 Terminated Transmission Line Consider a voltage sourceV g (f) with an internal impedanceZ g (f) be connected in series to an load impedanceZ L (f): The generator's internal and load impedances are separated 32 Z g (f) Z L (f) V g (f) z = 0 z =d Figure 2.4: Terminated Transmission Line Impedance Model. a distance d with the voltage refernces at the terminals of the impedances. The voltage at the generator's output is V (z;f)j z=0 =V g (f)Z g (f)I(z;f)j z=0 V + +V =V g (f) Z g (f) Z 0 (f) V + e z V e z j z=0 =V g (f) Z g (f) Z 0 (f) [V + V ]; (2.4.19) where (2.4.15) was evaluated at the boundaryz = 0 and used on the left-hand side (LHS) and (2.4.16) was used on the right-hand side (RHS) for the current. Solving for the forward propagating wave V + yields V + = Z 0 (f)V g (f) Z g (f) +Z 0 (f) + g (f)V ; (2.4.20) where g (f) = Z g (f)Z 0 (f) Z g (f) +Z 0 (f) (2.4.21) 33 is dened as the generator re ection coecient. Application of the boundary condition at z =d yields the solution for V . The voltage V (z;f) evaluated at z =d is V (d;f) =Z L (f)I(d;f) V + e d +V e d = Z L (f) Z 0 (f) h V + e d V e d i ; (2.4.22) and when solved for V yields V = L (f)V + e 2 d ; (2.4.23) where L (f) = Z L (f)Z 0 (f) Z L (f) +Z 0 (f) : (2.4.24) The term L (f) is dened as the load re ection coecient. Substitution of (2.4.23) into (2.4.20) and then solving for V + in terms of the input voltage V g (f) yields V + =V g (f) Z 0 (f) Z g (f) +Z 0 (f) 1 (1 g (f) L (f)e 2 d ) : (2.4.25) The voltage as a function of line positionz and frequencyf is determined by substituting the solutions for (2.4.23) and (2.4.25) into the general propagating voltage equation given by V (z;f) =V + e z +V e z : 34 Thus, the general solution for the voltage anywhere on the line is V (z;f) =V g (f) Z 0 (f) Z g (f) +Z 0 (f) " e z + L (f)e (2dz) 1 g (f) L (f)e 2 d # : (2.4.26) The current I(z;f) anywhere on the transmission line is similarly determined as V (z;f) by using (2.4.16). The current is determined to be I(z;f) =V g (f) 1 Z g (f) +Z 0 (f) " e z L (f)e (2dz) 1 g (f) L (f)e 2 d # : (2.4.27) 2.4.4 The Terminated Transmission Line Transfer Function The voltage V (z;f) anywhere on the transmission line as a function of frequency f and position z is determined by equation (2.4.26). Using Figure 2.4 as a model, the transfer functionH(z;f) at any position on the line can be determined by dividingV (z;f) by the input voltage V g (f). Then, H(z;f) = V (z;f) V g (f) =G(f) " e z + L (f)e (2dz) 1 g (f) L (f)e 2 d # : (2.4.28) The transfer function H(z;f) is dened for any point z on the line. Let G(f) = Z 0 (f) Zg(f)+Z 0 (f) : Putting x = g (f) L (f)e 2 d in (E.7.6) which is given by 1 1x = 1 +x +x 2 + +x n +x n+1 + (2.4.29) = 1 X k=0 x k ; (2.4.30) 35 results in the geometric series expansion of the denominator in (2.4.28). Thus H(z;f) =G(f) h e z + L (f)e (2dz) i [1 + g (f) L (f)e 2 d + ( g (f) L (f)e 2 d ) 2 + ]: (2.4.31) Equation (2.4.31) clearly demonstrates that the terminated transmission line is an Innite Impulse Response (IIR) lter when both the generator internal impedanceZ g (f) and the load impedanceZ L (f) are not matched to the transmission line characteristic impedance Z 0 (f). Let e (2dz) =e d e (zd) ; then, H(z;f) =G(f) h e z + L (f)e d e (zd) i 1 X k=0 h g (f) L (f)e 2 d i k ; which can be written as H(z;f) =G(f) 1 X k=0 8 > > < > > : e z h g (f) L (f)e 2 d i k | {z } Re ects wave towards load +e d e (zd) k g (f) k+1 L (f)e 2 dk | {z } Re ects wave toward generator 9 > > = > > ; : Expanding the rst three terms of the summation of H(z;f) results in H(z;f) =G(f)e z h 1 + g L e 2 d + ( g L e 2 d ) 2 + ( g L e 2 d ) 3 i +G(f)e d h L e 2 d + g ( L e 2 d ) 2 + 2 g ( L e 2 d ) 3 + i e (zd) ; (2.4.32) 36 and clearly shows the innite impulse response eect of mismatch when both impedances (Z g (f) and Z L (f) ) are not matched to the transmission line. The eects of mismatch are shown in gure 2.5 when the circuit is excited by a pulse V g (s). The gure does not show the actual waves which are \usable" by the load because only a fraction of the load's incident wave is actually delivered to the load. It is then prudent to dene a transmission coecient T (f) =T which represents the portion of the wave that is not re ected back towards the generator. The load voltage at position d on the line is V (d;s) =T L V + e d ; (2.4.33) whereT L is dened as the transmission coecient. Equating the solution for the voltage anywhere on the line given by (2.4.15) with (2.4.33) for z =d results in V (d;s) =T L V + e d V + e d +V e d =T L V + e d V + e d + L V + e 2 d e d =T L V + e d (Using equation (2.4.23)) V + e d + L V + e d =T L V + e d ) T L (f) = 1 + L (f): (2.4.34) 37 Z g (s) Z L (s) V g (s) Time g - Plane L - Plane z = 0 z =d Vg(s)G(s)e z Vg(s)G(s)e d L e (zd) Vg(s)G(s)e z g L e 2 d Vg(s)G(s)g 2 L e (zd) e 3 d Vg(s)G(s)e z (g L e 2 d ) 2 Vg(s)G(s) 2 g 3 L e (zd) e 5 d Vg(s)G(s)e z (g L e 2 d ) 3 0 2T 4T 6T T 3T 5T 7T Legend = (s) L = L (s) g = g (s) Figure 2.5: Transmission Line re ection diagram. The gure shows the internal re ections within the circuit that occur when the generator and load impedances Z g (s) and Z L (s) respectively, are not matched to the transmission line characteristic impedance Z 0 (s): 38 x(t) h(t) y(t) (a) X(f) H(f) Y (f) (b) Figure 2.6: (a) Time Domain Linear Filter Model. (b) Frequency Domain Linear Filter Model. 2.4.5 Distortionless Transmission Lines The convolution integral denes the output y(t) of a linear, time-invariant lter h(t) caused by a bounded excitation x(t) as shown in Figure 2.6(a). The convolution is dened as y(t) = Z 1 1 x()h(t)d = Z 1 1 x(t)h()d (2.4.35) and has Fourier transformFfy(t)g =Y (f) =X(f)H(f). A distortionless lter has an output of the form, y(t) =cx(t d ); (2.4.36) wherec is a constant and d is a time delay. The distortionless lter outputy(t) is merely a time-delayed and amplitude-scaled exact replica of the inputx(t). The frequency domain representation of (2.4.36) is Ffy(t)g =Ffcx(t d )g =cX(f)e 2f d : (2.4.37) 39 Recall that H(f) = Y (f)=X(f). Equation (2.4.36) shows that the distortionless lter function H(f) is H(f) =ce j(f) (2.4.38) =ce j2f d : (2.4.39) The lterH(f) exhibits constant gainc for all frequencies and the phase(f) = 2f d is a linear function of frequency. Ideally, c = 1 which means that the lter is a pure delay of the input. Recall that = +j (2.4.40) and Z o = s R +j!L G +j!C : (2.4.41) Let R L = G C ; (2.4.42) then the transmission line is distortionless. 40 Proof: = p (R +j!L)(G +j!C) = s LC R L +j! G C +j! = s LC R L +j! R L +j! = s LC R L +j! 2 = p LC R L +j! =R r C L +j! p LC =R r C L +j2f p LC; which shows thatRef g =R q C L , is a constant for all frequencies andImf g = 2f p LC, is a linear function of frequency. Substituting (2.4.42) into the transmission line charac- teristic impedance given by equation (2.4.41) results in Z o = s R +j!L G +j!C = s L C R L +j! G C +j! = s L C R L +j! R L +j! = r L C ; which shows that the line exhibits a pure resistive impedance which is independent of frequency. 41 2.4.6 Requirements for a Distortionless Transmission Line System This section summarizes the requirements for a generated pulse to be delivered to the loadZ L (s) without distortion. The requirements are derived from equation (2.4.32) which shows that H(z;s) represents an innite impulse response lter when the generator and load impedances are not matched to the transmission characteristic impedance. Source and load mismatch is undesirable when distortionless pulse propagation down a trans- mission line is required. When an impedance mismatch condition exists, the transmission line not only scales the pulse amplitude and induces a propagation delay in the pulse at the location of the load, the mismatch condition also lengthens the time support of the propagating pulse as measured across the load Z L (s). The following conditions are necessary for distortionless pulse transmission and can be found in [29] Karmel et. al. The conditions are: 1. The load Z L (s) must be matched to the characteristic impedance Z 0 of the trans- mission line so that L (s) = 0: Then, H(z;s) = Z 0 (s) Z g (s) +Z 0 (s) e (s)z : (2.4.43) 2. Comparing (2.4.43) with (2.4.37) c = Z 0 (s) Z g (s) +Z 0 (s) (2.4.44) 42 and implies that Z g (s) =Z 0 (s). Then, H(z;s) = 1 1 + e (s)z : (2.4.45) 3. Finally, lettings = +j! for! = 2f and comparing (2.4.43) with (2.4.37) shows that j! = (s)z (2.4.46) = p (R +j!L)(G +j!C): (2.4.47) When R =G = 0 = p LCz (seconds): (2.4.48) Dening v p = 1= p LC, which has the units of meters-per-seconds (refer to Table 2.1 on page 28) simplies (2.4.49) to = z v p (seconds): (2.4.49) 2.5 The Lossless Transmission Line When R =G = 0, the transmission line is lossless. Then, = p (R +j!L)(G +j!C)j R=G=0 =j! p LC 43 and shows that Ref g = = 0 and Imf g = =! p LC. The phase velocity is v p = ! = 1 p LC and the characteristic impedance given by Z 0 = s R +j!L G +j!C R=G=0 = r L C : This shows that the lossless transmission line is distortionless. 2.5.1 The Low-Loss Transmission Line It is benecial to analyze a low-loss transmission line because all known passive materials to date are known to exhibit an impedance with loss. This analysis is focused on the situation in which the lossy elements are small in comparison to the frequencies used. The analysis is useful because transmission lines used for UWB will typically be on the order of a few millimeters or centimeters long, and it is expected that the resistance and conductances will be small. The propagation constant is = p (R +j!L)(G +j!C); 44 and can be written as =j! s LC R j!L + 1 G j!C + 1 =j! p LC R j!L + 1 1=2 G j!C + 1 1=2 : The two factors can be expanded into an innite series by using the generalized binomial expansion (see (F.2.1)). Recall that (1 +x) r = 1 +rx + r(r 1) 2! x r2 + : Then R j!L + 1 1=2 = 1 + 1 2 R j!L 1 8 R j!L 2 + H.O.T 1 ; (2.5.1) and the condition R << !L applies for convergence to hold. The term G j!C + 1 1=2 is similarly expressed with the convergence condition G<<!C that must hold. Then, j! p LC 1 + R 2j!L 1 + G 2j!C (2.5.2) 1 H.O.T=Higher order terms 45 where only the rst two terms in the binomial expansion was used becauseR<<!L and G<<!C and implies that the H.O.T rapidly approach zero. Then, =j! p LC R j!L + 1 1=2 G j!C + 1 1=2 j! p LC 1 + 1 2j! R L + G C p LC 2 R L + G C +j! p LC = R 2 r C L + G 2 r L C +j! p LC; where the quadratic terms of ! were neglected. The result shows that R 2 r C L + G 2 r L C and ! p LC: (2.5.3) The transmission line is not lossless; however, the line is distortionless. 46 The frequency-dependent characteristic impedance is Z 0 (f) =r(f) +jx(f) = s R +j!L G +j!C = r L C R j!L + 1 1=2 G j!C + 1 1=2 r L C 1 + 1 2 R j!L 1 8 R j!L 2 + H.O.T ! 1 1 2 G j!G + 1 8 G j!C 2 + H.O.T ! r L C 1 + 1 2 R j!L 1 1 2 G j!C (neglecting H.O.T of !) r L C 1 + 1 2j! R L G C (neglecting H.O.T of !) r L C j 2! r L C R L G C ; which shows that the RefZ(f)g =r(f) r L C (2.5.4) and ImfZ(f)g =x(f) 1 2! r L C R L G C : (2.5.5) Clearly, the real part RefZ(f)g is independent of frequency, however; the reactive part ImfZ(f)g = x(f) is frequency-dependent. It is typically the case that r(f) << x(f), and thus the term x(f) will have a negligible impact on pulse propagation because the transmission line will be dominated by a purely resistive term. If the line is engineered so that (R=L) (G=C), then the transmission line will be independent of frequency. This situation is identical to the lossless transmission line case. 47 It is assumed for the sample computations performed in this dissertation, that trans- mission line behaves like an ideal all-pass lter across the measurement band. Then jH(d;s)j = 1 for alls; and thus the eects ofH(d;s) can ignored. It is also assumed that the generator impedance Z g (s) = Z 0 , so that c = 1=(1 +) which is not a function of frequency. Further, it is assumed that the transmission line to the antenna exhibits a constant very low loss constant impedance (typically 50 ) across a wide frequency band of operation. Thus, the transmission line and generator combination behave like an ideal all-pass lter across the measurement band. Distortion of ultra-fast pulses in transmis- sion lines has been studied by Barth and Richner [6]. Their results have shown that distortions do occur if the transmission line is not well engineered. 2.6 Power Delivered to the Load The power delivered to the load is determined from the assumed voltage and current solutions of the general transmission line equations given by V (z;s) =V + e z +V e z and I(z;s) = V + Z 0 e z V Z 0 e z : (2.6.1) The average power P ave at any point on the line is P ave = 1 2 RefV (z;s)I (z;s)g (2.6.2) = 1 2 Re V + e z +V e z V + Z 0 e z V Z 0 e z : (2.6.3) 48 Recall from (2.4.23) thatV = L (f)V + e 2 d , which can be substituted into P ave to yield P ave = 1 2 Re V + e z + L (f)V + e 2 d e z V + Z 0 e z V + Z 0 L (f)e 2 d e z = 1 2 Re jV + j 2 Z 0 e z + L (f)e (z2d) e z L (f)e (z2d) : Let W =e z , X = (f)e (z2d) and P =XW , then P ave = jV + j 2 2 Re 1 Z 0 (W +X)(WX) = jV + j 2 2 Re 1 Z 0 (WW +W XWX XX ) = jV + j 2 2 Re 1 Z 0 jWj 2 +PP jXj 2 = jV + j 2 2 Re 1 Z 0 jWj 2 + 2jImfPgjXj 2 : (2.6.4) The complex terms W and X have squared magnitudesjWj 2 = exp (2z) andjXj 2 = j L j 2 exp (2(zd)) and thus P ave = jV + j 2 2 Re 1 Z 0 e 2z + 2jImfPgj L j 2 e 2(zd) : (2.6.5) Letting Z 0 =R 0 +jX 0 ; then P ave = jV + j 2 2R 0 e 2z j L j 2 e 2(zd) (2.6.6) 49 and reduces to P ave = jV + j 2 2R 0 1j L j 2 (2.6.7) when the real part of the propagation constant = +j is zero (lossless transmission line). The term P inc = jV + j 2 2R 0 (2.6.8) represents the incident power on the line traveling towards the load. When L = 0; all of the incident power is delivered to the load. 50 Chapter 3 Impulse Radio Antenna Fundamentals 3.1 Introduction The mask constraints considered in this research are on the eective isotropic radiated power per megahertz. Therefore an understanding of the electric eld intensity at a distancer from a source antenna is necessary. We begin with the postulates of Maxwell's equations that describe eld phenomena for source excitations. Transmission antennas in their most fundamental form are electromagnetic transduc- ers that convert guided waves, induced by a time-varying voltage within a transmission line, to waves that radiate in a propagation medium. Receiving antennas are also trans- ducers. They convert an impinging electric eld into a voltage within a waveguide. The 51 fundamental time-domain electromagnetic equations are known as Maxwell's equations and are given by [60] rE = @B @t ; (3.1.1) rH = @ @t D +J T ; (3.1.2) rD = T (t); (3.1.3) rB = 0: (3.1.4) The script font (i.e.,E , E(x;y;z;t)) is used to indicate a time-varying vector eld. The subscript \T" is used to denote the \total" as in the total current densityJ T or total charge T (t). Maxwell's equations can be simplied by suppressing the time-dependent factors. It is typical to dene E = Re Ee j!(t) ; H = Re He j!(t) ; etc. The variable is dened as a time delay and the phasor quantities E; H; D; B; T ; J T are complex functions of spatial coordinates only (i.e., e j!t time dependence is sup- pressed). Typically, the sources T (t) and I T (t) are taken to vary sinusoidally with time and the radian frequency ! = 2f is constant. Maxwell equations are linear dierential equations and implies that the radiated elds are also varying sinusoidally with time and at the same radian frequency [17]. Further, sinusoids are eigenfunctions of Maxwell's equations because they are linear dierential 52 equations. For IR-UWB, the excitation source is inherently impulsive in nature and thus one may question the validity of the narrowband approach to Maxwell equations when applied to UWB excitations. Using the phasor quantities and taking the Fourier transforms of (3.1.1) { (3.1.4), results in the frequency domain representation given by r E = j!B; (3.1.5) r H = j!D + J T ; (3.1.6) r D = T (!); (3.1.7) r B = 0: (3.1.8) The total current density is J T = E + J where E is called the conduction current density term and J is the current vector. The electric displacement D and the magnetic eld B are dened by D =E (3.1.9) and B =H: (3.1.10) The material's permittivity has units of Farads per meter (F/m). The permeability has units of Henries per meter (H/m). In free-space the permittivity and permeabil- ity are denoted by o and o respectively. Table 3.1 on page 54 summarizes relevant electromagnetic constants. 53 Universal Constants Symbol Value Units Unit Breakdown Permittivity o 1 36 10 9 8:854 10 12 F/m (As)/V Permeability o 4 10 7 H/m (Vs)/A Speed of light c ( o o ) 1=2 = 3 10 8 m/s |{ Intrinsic impedance o q o o = 120 377 V/A Table 3.1: Free Space Universal Constants Maxwell equations are valid for all radian frequencies ! provided the transformed equations meet the criteria of being Fourier transformable. A set of sucient condi- tions for the existence of Fourier transforms are known as the Dirichlet conditions [38]. When more than one frequency is present, as it is in UWB applications, the principal of superposition follows for each frequency present in the source excitation function. Ad- ditionally, the time-varying form of the electromagnetic quantities can be determined by inverse transform of the solutions of (3.1.5){(3.1.8) as a function of radian frequency! or since UWB signals have short duration in time, the time domain equations may be used directly. This principal is used when modeling UWB antennas by numerical analysis. The instantaneous time-varying Poynting vector is dened as S = 1 2 EH: (3.1.11) The average complex power owing (radiating power) through a closed surface s (typically a sphere) is determined by evaluation of P = 1 2 Z Z s E H ds; where 54 ds = ^ nda is a normal unit vector directed outward relative to the closed dierential surface area da, and S = 1 2 E H (3.1.12) is dened as the Poynting vector . The Poynting vector is a spatial power density measured in W=m 2 . The spatially distributed power ows outward in the direction that is normal to a convenient ctitious volume enclosing the radiating antenna. The factor 1/2 appears in equation (3.1.12) because the electric and magnetic elds represent peak amplitudes of a sinusoidal function. The 1/2 factor should not be used if rms quantities are used for the elds. The real power owing through the surface is given by P = Re 8 < : ZZ s Sds 9 = ; = 1 2 ZZ s E H ds: (3.1.13) The radiation intensity U = Sr 2 is the time-averaged power per unit solid angle and is measured in units of Watts per steradian. In spherical coordinates,ds = ^ a R r 2 sin()dd. Dening d = sin()dd, the power can be written in terms of the radiation intensity P = ZZ s Sds (3.1.14) = ZZ s U(;)d : (3.1.15) 55 The radiation intensity of a ctitious isotropic radiator is U iso = P 4 : (3.1.16) Directivity The directive gain G D (;) of an antenna at a xed frequency, is dened as the dimen- sionless ratio of the radiation intensity U(;) of the particular antenna to that of an isotropic radiator U iso and thus D = U(;) U iso : (3.1.17) The maximum directive gain D of an antenna for a xed frequency is called the directivity of the antenna. It is the ratio of the maximum radiation intensity to the average radiation intensity of an isotropic radiator. Thus, D = U max U iso (3.1.18) = 4U max U iso ; (3.1.19) where U max =U( max ; max ) and max ; max are the azimuth and elevation respectively of maximum radiation intensity for a single xed frequency. It is important to recognize that the directive gain concept is for a given xed frequency. It is highly probable that the directive gain will be dierent for each frequency under test for a particular UWB antenna. This is a very important concept because UWB signals considered here occupy several decades of bandwidth. The implications are that the directivity of many antennas vary as a function of frequency. This implies that as the UWB signal is transmitted in time, 56 the signal will vary in its spatial coordinates at a distant point away from the antenna [7]. Spatial ltering will impact the design and performance of the matched ltering operation in an UWB receiver. The signal shape at the receiver critically depends on the receiver's spatial coordinates relative to the transmitter's antenna. Optimal matched ltering is dicult in UWB communication systems. 3.2 Determination of the Electric Field Intensity The electric eld can be determined via analytic expressions of Maxwell's equations. A particular clever method calculates the electric eld from the current density vector J. To use this method, the current density must be known on the surface of the radiator. The cleverness of the method relies on the introduction and use of a ctitious auxiliary eld vector A. The divergence-free postulate of the magnetic eld B states thatr B = 0 and implies that B is solenoidal. A solenoidal vector can be written as the curl of another vector eld quantity. Let A denote the magnetic potential with units of Webers per meter (Wb/m) [17], then B =r A: (3.2.1) Equation (3.2.1) is used in section 3.2.3 which begins on page 64. The next section introduces Maxwell's equations that are fundamental to the study of electromagnetic phenomena. The theorems are stated without proof because they are well published. The interested reader is referred to the the excellent book by David Cheng [17] which was a primary source of theory contained in this chapter. 57 3.2.1 Fundamental Theorems Theorem 1 (Divergence Theorem (Gauss Theorem)). The volume integral of the diver- gence of a vector eld equals the total outward ux of the vector through the surface that bounds the volume then Z V r Adv = I S r Ads: (3.2.2) Theorem 2 (Stokes Theorem). The surface integral of the curl of a vector eld over an open surface S is equal to the closed line integral of the vector along the contour C bounding the surface. Then Z S (r A) ds = I C A dl: (3.2.3) The starting point for evaluating the electric eld intensity is electrostatics for a single point charge q. The permittivity of a material is a constant in a homogeneous media and thus (3.1.7) can be written r E = q : (Using (3.1.9) to relate E to D) (3.2.4) The electric eld intensity due to a charge q can be determined by integrating (3.2.4) over the volume V that encloses the point charge. The volume integral is related to the surface integral by the Divergence theorem and thus Z V r Edv = I S E ds: (3.2.5) 58 The dierential surface ds is given by ds = ^ a R R 2 sin()dd for 2f0;g;2f;g in spherical coordinates. The term ^ a R represents a unit vector pointing outward and normal to the bounding surface. Spherical coordinates were chosen because of spherical symmetry about the charge q. This choice of coordinates simplies that analysis of the problem. The total charge bounded by the volume V , which is scaled by the permittivity of free space, is given by q = Z Z 0 ^ a R E R ^ a R R 2 sin()dd = 4R 2 E R ; where (3.2.5) was evaluated. A single stationary charge q in free space will result in a static E-eld given by E = ^ a R q 4 o R 2 : (3.2.6) This result is critical because it is used to determine the magnetic potential A by using a method that is similar to the determination of the vector eld E from the scalar potential V given by (3.2.8) on page 61. In general, we could have solved the E-eld starting from dE = dq 4 o R 2 59 caused by a dierential chargedq. The resulting E-eld from such a dierential charge is determined by the superposition of all such dierential charges and thus the total charge Q is Q = Z dq R d l; (for a line of charge) = Z dq R d s; (for a surface of charge) = Z dq R d v: (for a volume of charge): The prime (') appearing in the equations denotes coordinates of the physical line, surface or volume of charge. 3.2.2 Electric Potential The E-eld can be expressed in terms of the gradient of a scalar potential function V . To see this, we start with a useful identity called the Null Identity. Identity 1. Null Identity The curl of the gradient of any scalar eld is identically zero. It has been assumed that the rst derivative of the potential function exists every- where. Then, rrV = 0; (3.2.7) where (in Cartesian coordinates) r, ^ a x @ @x + ^ a y @ @y + ^ a z @ @z 60 and thus rV = ^ a x @V @x + ^ a y @V @y + ^ a z @V @z : The Null Identity can be proved by performing the indicated curl operation in Cartesian coordinates. The Null Identity enables us to write rrV ) Z S (rrV ) ds = Z C rVdl = 0: (by Stokes Theorem) Dening dV =rVdl, the contour integral of both sides results in Z C rVdl = Z C dV = 0: This result states that the surface integral ofrrV over any surface is zero which conversely states that if a vector eld is curl free (as electric elds are in free space) then it can be expressed as the gradient of a scalar eld. It is customary to write E =rV: (3.2.8) The negative sign is chosen by convention. 61 The scalar potential dierence between the pointsP 1 andP 2 is determined by the line integral of (3.2.8) given by Z p 2 p 1 Edl = Z p 2 p 1 rVdl (3.2.9) = Z p 2 p 1 dV =V 2 V 1 : Evaluation of the left-hand side of (3.2.9) in spherical coordinates for a single point charge q located at the origin is determined by V R V 1 = Z R 1 Edl = Z R 1 ^ a R q 4 o R 2 ^ a R dR = q 4 o R ; where V R and V 1 = 0. The potential at innity is dened to be 0 and hence we get a positive potential by dening E =rV . Thus, V (R) = q 4 o R : 62 The fundamental concept called the Vector Magnetic potential is similarly developed. The vector magnetic potential will also be used to derive the E-eld in section 3.2.3 on page 64. In general, the charge q represents the total charge, and thus we can write V = 1 4 o Z V 0 v dv 0 R ; (volume of charge) V = 1 4 o Z S 0 s ds 0 R ; (surface of charge) V = 1 4 o Z l 0 l dl 0 R : (line of charge) (3.2.10) In most cases, it is easier to determine the scalar potential V rst and then determine the E-eld from E =rV . Poisson's equationr 2 V == relates the scalar potential V to the static charge q and is derived by r D = rE = r (rV ) = ( is constant for homogeneous media) and thus r 2 V = : (3.2.11) 63 Equations (3.2.10) are solutions to Poisson's equation and they will be used in the next section to determine the magnetic potential. 3.2.3 Magnetic Potential This section develops the results for the magnetic potential which is similar to the devel- opment used in section 3.2.2. Starting from a second null identity given by r (r A) = 0: (3.2.12) Equation (3.2.12) states that the divergence of the curl of any vector is identically zero. The proof of (3.2.12) is determined by using the divergence theorem given on page 58. Thus Z V r (r A)dv = I r A ds = Z S 1 r A ^ a n 1 ds + Z S 2 r A ^ a n 2 ds; where ^ a n 1 and ^ a n 2 are outward normal unit vectors to the surfaces S 1 and S 2 (e.g., see Figure 3.1). By Stokes Theorem, we can write Z S 1 r A ^ a n 1 ds + Z S 2 r A ^ a n 2 ds = I C 1 Adl + I C 2 Adl = 0 64 Figure 3.1: Magnetic potential surface whereC 1 andC 2 are equal and opposite closed contours around the surface and thus the integrals are equal in magnitude. This result proves the null identity and thus the volume integral Z V r (r A)dv = 0: The Null identity shows that if a vector eld is divergenceless (e.g.r B = 0) then it can be expressed as the curl of another vector eld. Then r B = 0 ) B =r A: (3.2.13) Equation (3.2.13) enables us to write the curl of B as the curl of another vector A and thus r B =rr A: (3.2.14) The next identity is called the Curl-Curl Identity. It allows equation (3.2.14) to be expanded further. 65 Identity 2. Curl Curl Identity rr A =r(r A)r 2 A (3.2.15) The Curl Curl identity is proved by performing the indicated operations and using rr =r 2 and thus rr A =r(r A)rrA =r(r A)r 2 A: For magnetostatics, Maxwell's magnetic intensity curl equation is r H = J T r B =J T (B =H) r (r A) =J T (B =r A) r(r A)r 2 A =J T (Curl Curl Identity): Since the divergence of A was not dened, a choice ofr A = 0 for convenience results in a result called the Laplacian of A given by r 2 A =J T : (3.2.16) 66 In Cartesian coordinates, the Laplacian of A is r 2 A = ^ a x r 2 A x + ^ a y r 2 A y + ^ a z r 2 A z : (3.2.17) The Laplacian of A has produced a decoupled result. The three coordinates of the Laplacian can be treated as three independent scalar equations. They are each similar to the scalar potential Poisson equation given at the end of section 3.2.2. The three independent equations are r 2 A x =J x ; r 2 A y =J y ; r 2 A z =J z :: (3.2.18) Using equations (3.2.10) and (3.2.11) on pages 63 and 63, the solutions of (3.2.18) are similar to (3.2.10) and are given by A x = 4 Z X 0 J x R dx 0 A y = 4 Z Y 0 J y R dy 0 A z = 4 Z Z 0 J z R dz 0 : 67 The observation range variable R cannot be readily taken outside the integral because it depends on the source's coordinates. When there is a single source q, the range variable R may be taken outside of the integral because the observation range R from the source is a constant. The range will vary with the dierential current on the conductor. In general, the vector potential can be written in compact vector form as A = 4 Z V J R dv 0 (Wb/m): (3.2.19) 3.2.4 Potential Functions and the Wave Equation The wave equation is derived from Maxwell's equations using the time-dependent form, because time derivatives are involved in the development; however, we stay with the phasor notation to be consistent with Cheng's development of the material for this section. Here, we use the concept of the magnetic potential for the development where B =r A and r E = @B @t : Then, r E =r @A @t r E +r @A @t = 0 r E + @A @t = 0: (Curl free condition) (3.2.20) 68 The curl free condition means that we can writeE + @A @t as the gradient of a scalar and thus E + @A @t =rV: Then E =rV @A @t (V/m). (3.2.21) The wave equation denes UWB waveform propagation. The wave equation can be developed for propagation in a linear, isotropic media from the curl of the magnetic intensity which is given by r H = J + @D @t r B =J + @E @t (Sub B =H and D =E) rr A =J + @E @t (Sub B =r A) r(r A)r 2 A =J + @E @t (Use Curl Curl identity 3.2.15) r(r A)r 2 A =J + @(rV @A @t ) @t (Using E =rV @A @t ) r(r A)r 2 A =JrV @V @t @ 2 A @t 2 : Rearranging terms and dening the Lorentz condition for potentials as in Cheng [16] results in r A + @V @t = 0: (3.2.22) 69 The equation is valid because constraints have not been placed on the divergence of A. We are at liberty to dene it for convenience. The Lorentz condition for potentials is dened as r 2 A @ 2 A @t 2 =J (3.2.23) which is a non-homogeneous wave equation for the vector potential A. Similarly, a non- homogeneous wave equation can be derived for the scalar potential V in an isotropic medium. The wave equation is developed by starting with r D = r E = (D =E) r rV + @A @t = (isotropic condition for and eq (3.2.20)) r 2 V + @ @t (r A) = r 2 V + @ @t @V @t = (Using Lorentz Condition forr A) and the nal result for the scalar potential is given by r 2 V @ 2 V @t 2 = : (3.2.24) 3.2.5 Homogeneous Solutions to the Wave Equation for Point Sources The homogeneous solution to the wave equation for point sources can be determined for the scalar potential V or the magnetic potential A. 70 Assume that a dierential charge distribution (t)v 0 is located at the origin of a spherical coordinate system. The homogeneous solution of the wave equation can be solved by the method of superposition. This approach has diculties because of the spatial distribution of the point charges about the origin. The far-eld solution is the vector sum of the charges. The solution depends on the relative location of each charge. An easier method that can be used to solve the wave equation is to determine the scalar potentialV for a charge at the location of the source origin. For a charge distribution, the scalar potential for all charges are similarly determined and then superposition is applied for each scalar potential. This method is less complicated because the electric eld is a scalar. It avoids vector addition. The scalar potential V for a single point charge q only varies radially about the point charge because of symmetry about the charge. The solution will not depend on nor . The homogeneous wave equation in spherical coordinates for the radial component of the potential is 1 R 2 @ @R R 2 @V @R @ 2 V @t 2 = 0: (3.2.25) Any twice dierentiable function subject to appropriate boundary conditions is a valid solution to the wave equation. The solution is valid everywhere except at the origin because of the point source located at the origin. 71 Assume a general function V (R;t) =f(R;t)=R where the factor 1=R is suggested by the form of the radial wave equation. The potential at innity is dened to equal zero as a boundary condition. Plugging V (R;t) into (3.2.25) results in 0 = 1 R 2 @ @R R 2 @ @R f(R;t) R R @ 2 f(R;t) @t 2 = 1 R @ @R R @f(R;t) @R f(R;t) @ 2 f(R;t) @t 2 : Then @ 2 f(R;t) @R 2 @ 2 f(R;t) @t 2 = 0: (3.2.26) Eigenfuctions are solutions to homogeneous dierential equations. The eigenfunction given by f(R;t) =e j!(t R ) (3.2.27) is a solution to (3.2.26) where ! = 2f 0 . The wave equation for Fourier transformable UWB signals, in a linear isotropic media, can be solved by the principle of superposition. Substitution of equation (3.2.27) into (3.2.26) and solving for (the time factor cancels) results in 1 =) = 1 p (m=s); (3.2.28) which is dened as the velocity constant. In free space, = o and = o (3.2.29) 72 and thus = 1 p o o (3.2.30) = 4 10 7 1 36 10 9 1=2 =c (m=s); (3.2.31) wherec is the speed of light in free space. The wave equation solutionf(R;t) =e j!(t+ R ) does not correspond to any meaningful realizable solution and thus f(R;t) =e j!(t R ) is the realizable solution. To determine the specic function f, we know from the static charge solution that the electric potentialV is given by (3.2.10) on page 63. The dierential potential voltage for a dierential volume of charge distribution (t)v 0 is given by V (R;t) = (t)v 0 4R : Then f t R = t R v 0 4 : Integrating V (R;t) = f t R R results in V (R;t) = 1 4 Z V 0 t R R dv 0 (V/m): (3.2.32) 73 A similar result follows for the vector potential which is given by A(R;t) = 4 Z V 0 J t R R dv 0 (Wb/m): (3.2.33) Recall the use of phasors where for example Ee j!t = Re(E(x;y;z)e j!t ): Using phasors to express (3.2.32) and (3.2.33) results in V (R) = 1 4 Z V 0 e jkR R dv 0 (3.2.34) and A(R) = 4 Z V 0 Je jkR R dv 0 ; (3.2.35) where k = ! =! p (3.2.36) has been used. 3.2.6 Candidate Procedure to Determine the E-Field This section outlines the detailed process used to determine theE-eld caused by a source of excitation. A summary of the procedure is as follows: 74 1. Determine A(R) from (3.2.35), 2. Determine H(R) = 1 r A, 3. Finally, E(R) = 1 j! (r H J): The last item results from use of Maxwell's equation (3.1.6) on page 53. Alternatively, the elds can be determined by 1. Determine V (R) and A(R) from (3.2.34) and (3.2.35). 2. Determine E(R) =rVj!A; and B(R) =r A: 3. The last step is to determine E(R;t) = RefE(R)e j!t g and similarly for B(R;t). The two procedures summarized above are standard and only depends on knowledge of either the current density J or the charge distribution of the point sources. The ability 75 of computing the integrals given by (3.2.34) and (3.2.35) may not be tractable. For these reasons a more tractable method for measuring and predicting the E-eld is sought. A suitable method of indirectly measuring the E-eld is given in section 3.3 on page 87. 3.2.7 Homogeneous Vector Helmholtz Equations In a source-free isotropic media, Maxwell's equations are given by r E = @H @t r E = 0 r H = @E @t r H = 0: The wave equation is determined as in section 3.2.4. The results are r 2 E 1 2 @ 2 E @t 2 = 0 and r 2 H 1 2 @ 2 H @t 2 = 0: where 2 = 1=(). These results are called the Helmholtz equations and will be used to introduce the plane wave concept. 3.2.8 Plane Waves A uniform plane wave is the particular solution to Maxwell's equations where the E-eld has the same direction, magnitude and phase in innite planes that are perpendicular to 76 the direction of wave propagation. Consider the frequency domain source-free Helmholtz equation for the E-eld given by r 2 E + ! 2 2 E = 0 r 2 E +k 2 E = 0; where k =!=. When E = ^ a x E x , the source-free Helmholtz equation reduces to d 2 E x dz 2 +k 2 E x = 0: (3.2.37) The wave propagates in the positivez-direction for this example. The partial derivatives with respect to x and y are identically zero. A generalized wave equation solution is E x (z) =E + x (z) +E x (z) =E + o e jkz +E o e jkz : The two terms E + o and E o are complex constants and the superscripts () are used to denote forward and backward propagation. The time-varying form for the forward propagating wave is E + (z;t) = RefE x (z)e j!t g = RefE + o e j(!tkz) g =E + o cos(!tkz): 77 The phase must be constant for a plane wave and therefore !tkz = constant)z = !t constant k : Then dz dt = ! k = 2f 2f = 1 p : which is the wave's phase velocity. For free space conditions = o and = o , the phase velocity =c, the speed of light. Once the E-eld is known, the H-eld can be determined from H = 1 j! (r E): Assuming propagation in the z-direction, H = 1 j! ^ a x ^ a y ^ a z @ @x @ @y @ @z E + x (z) 0 0 ; 78 wherejj denotes the determinant and E = ^ a x E x has been used. Then H =^ a y k ! E + o e jkz ; which can be simplied to give (in free space) H =^ a y Re r o o n E + o e jkoz o (3.2.38) =^ a y 1 o n E + o e jkoz o (see table 3.1) (3.2.39) =^ a y E + o o cos(!tk o z): (3.2.40) This result shows that for a uniform plane wave propagating in free space that the E and H elds are transverse to each other. Further, B and H are perpendicular to the direction of propagation. The elds are also in time phase. Propagating waves that exhibit these characteristics are called transverse electromagnetic mode waves (TEM). The E-eld and H-eld of a TEM wave are related by the intrinsic impedance of free space o . This result is important. The intrinsic impedance of free space appears in (3.3.12) of section 3.3 on page 87. Equation (3.3.12) is a important because it allows us to determine the E-eld experimentally. In general, it can be shown [17] that a TEM wave can be described by the general result E(R) = E o e jkR = E o e jk^ anR ; 79 where R, ^ a x x + ^ a y y + ^ a z z (i.e., a radius vector at the origin) and k, ^ a x k x + ^ a y k y + ^ a z k z : (wave number) 3.2.9 Polarization The polarization of a wave is dened in terms of the time-varying behavior of the E-eld. When the E-eld is uniform and oriented in a single direction, say ^ a x E x , then we say that the E-eld is linearly polarized in the ^ a x direction. Since the H-eld is transverse, a description is not required. Further information about types of polarization can be found in Cheng [17]. We are primarily concerned here with linear polarization. 3.2.10 Radiation Field of a Short (Hertzian) Dipole The short dipole is fundamental for antenna analysis. An understanding of the H and E elds are critical in order to derive E i (!) = s j! o e j!R=c 2RcZ o V t (!)V r (!) (V/m); which relates the E-eld to measured voltages. The analysis begins with the short dipole because it is fundamental to the analysis of complicated antenna structures. It is assumed that the dipole is oriented in the z-direction, center fed, innitesimally thin (to avoid complicated elds from the radius) and has a length z << . It is also assumed that 80 the current varies sinusoidally. The retarded vector magnetic potential will be used and was derived in section 3.2.4 equation (3.2.35). The retarded potential is A = 4 Z V 0 e jkR J R dv 0 = 4 Z V 0 e jkR I(x 0 )(y 0 ) R dx 0 dy 0 dz 0 (() is the Dirac delta function) = ^ a z 4 Z z=2 z=2 e jkR I R dz 0 : A Hertizain dipole's length is very short when compared to the excitation wavelength. Because the length is very short, the distance from the dipole to an observation point can be considered approximately constant, and thus the range variable R can be taken outside of the integral. The range variable for the E-eldR cannot be considered constant because the far-eld phase critically depends on the range from the source of excitation to the observation point in space. The range variable is a factor in the complex exponential phase term. This however does not present a problem. The dipole is very short and the current is approximately constant, and thus the result of the integral is given by A = ^ a z Ize jkR 4R : (3.2.41) This equation exhibits spherical symmetry, therefore it is best to use spherical coordinates where the unit vector is given by ^ a z = ^ a R cos() ^ a sin(): 81 In spherical coordinates A = ^ a R A R + ^ a A + ^ a A where A R =A z cos(); A =A z sin(); A = 0: It was assumed that the variation in the direction is zero. The corresponding H-eld is determined from H = 1 r A = 1 R 2 sin() ^ a R ^ a R ^ a R sin() @ @R @ @ @ @ A R RA (R sin()A =^ a Iz sin()e jkR k 2 4 1 jkR + 1 (jkR) 2 : The nal result was determined by simplications and detailed algebra. 82 The E-eld is similarly determined from E = 1 j! r H = 1 j! 1 R 2 sin() ^ a R ^ a R ^ a R sin() @ @R @ @ @ @ H R = 0 RH = 0 (R sin()H =^ a R E R ^ a E where E R = Iz sin()e jkR k 2 2 cos() 4 r 1 (jkR) 2 + 1 (jkR) 3 and E = Iz sin()e jkR k 2 2 cos() 4 r 1 (jkR) + 1 (jkR) 2 : Observations In free space and in the far eld (k o R = (2=) for R >> 1), the magnetic and electric eld intensities are H = jIz sin()k o e jkoR 4R (3.2.42) and E = jIz sin()k o o e jkoR 4R : (3.2.43) 83 The high order terms of R have been neglected because the terms with high orders of R approach zero rapidly. 3.2.11 The Eective Length of an Antenna The eective length of an antenna is a useful concept and is applicable in the far eld. The electric eld radiated or received by an antenna is proportional to it's length. The eective length relates an open circuit voltage V oc to the E-eld impinging upon the antenna. The actual form of the eective length depends on the particular antenna and the current distribution on the structure. The eective length L e is dened in terms of the open circuit voltage V oc =EL e : (3.2.44) The minus sign is chosen by convention because the electric potential opposes change that results from the E-eld. It should be noted that the particular denition of the eective length depends on the problem. The eective length is described in Stuzman and Thiele [60], Cheng [17] and the eective height in discussed in Kraus [31]. The eective length is helpful when circuit models are used to model antennas. The next section introduces antenna circuit models. 84 The concept of the receiving antenna sensitivity function is closely related to the eective length idea. The receiving antenna sensitivity function is dened by Clark in references [49,50]. The sensitivity function is H r (!) = V r (!) E i (!) =L r (!) Z r (!) Z r (!) +Z L (!) ; (meters) (3.2.45) whereV r is the voltage at the terminals of the receive antenna,Z r is the receive antenna's impedance, Z L is the load impedance in the equivalent series receiving circuit and E i (!) is the incident E-eld that impinges onto the receiving antenna. The sensitivity function has units of length. 3.2.12 Antenna Circuit Models Figure 3.2: Antenna Equivalent Circuit Model Figure 3.2 shows a circuit model of a transmitting and receiving antenna system. The model is used to derive an equation that relates the E-eld the source voltage V g . The 85 derived equation can then be used to relate a measurement to the E-eld. The circuit model assumes that the transmitter and receiver can be represented as Thevinin equiv- alent series circuits. The transmitter circuit has a voltage source V g ; an internal source impedanceZ g and a transmit antenna. The tranmit antenna is represented as a complex impedance Z t . The transmit antenna radiates a portion of the incident voltage. The voltage not accepted by the transmit antenna is re ected back towards the source. The radiated eld propagates in space and a fraction of the radiated eld impinges upon the receive antenna. In the series receiving circuit model, the impinging E-eld produces an open circuit voltage V oc across the receiving antenna terminals. The receiving antenna is modeled in the circuit as an equivalent complex load impedance Z r that is in series with a load impedance Z L . The circuit model just described is called the weak approxi- mation model because it neglects the coupling impedance Z 12 . The coupling impedance represents the connection between the transmitter and receiver. It is noteworthy that the circuit approximation has limited value. Care should be ex- ercised when interpreting any results concerning re-radiation (scattering) of power. This issue was discussed in several papers by Love and Collin in correspondence between the two in open literature. It was shown by Collin that the Thevenin equivalent circuit does not provide the correct results concerning scattered power by an antenna; unfortunately, many textbooks are incorrect [5, 31, 62]. Geyi [22] has presented a corrected Thevenin receiving circuit which addresses the issue of antenna scattering. Scattering is not a concern here, hence the simplied antenna circuit is valid. 86 3.3 Indirectly Measuring the Electric Field Strength Robertson and Morgan [48] derived a useful result that can be used to indirectly measure the electric eld. Their result is veried and derived here from fundamental principles. The determination of the E-eld of a plane wave, based on measurements, begins by using an identical pair of polarization matched antennas. The antennas are assumed to be \well" matched to the characteristic impedance Z o of the transmission line. The derivation begins by using equation (3.2.43) given by E = jIz sin()k o o e jkoR 4R : (3.3.1) An expression for the current I is determined by using the circuit model in Figure 3.2 on page 85. Writing the current in terms of the complex impedances and the voltage generator V g results in I(!) = V g (!) Z g (!) +Z t (!) : (3.3.2) Dening the eective length as in Stuzman and Thiele [60] L 1 = z sin() (3.3.3) and then substituting (3.3.2) and (3.3.3) into (3.2.43) results in E = jk o o e jkoR 4R L 1 V g (!) Z g (!) +Z t (!) : (3.3.4) 87 LetL 2 be dened as the eective length of the receive antenna. The open circuit voltage at the receiving antenna is V oc (!) =EL 2 (3.3.5) = jk o o e jkoR 4R L 1 V g (!) Z g (!) +Z t (!) L 2 : (3.3.6) The usual negative sign on the E-eld has been dropped because the receive antenna can be oriented for a positive voltage if required. The voltage V r caused by the impinging E-eld is the voltage that would be measured across the receiving antenna's complex impedance Z r : The voltage is given by the voltage divider V r (!) = V oc (!)Z r (!) Z r (!) +Z L (!) : (3.3.7) Substitution of (3.3.5) into (3.3.7) results in V r (!) = jk o o e jkoR 4R L 1 V g (!) Z g (!) +Z t (!) L 2 Z r (!) Z r (!) +Z L (!) : (3.3.8) Assuming an impedance matched condition to the transmission line's characteristic impedance Z o = 50 implies Z g =Z t =Z r =Z L =Z o ; 88 and V t = 1 2 V g ) V g = 2V t : Substituting these results into (3.3.8) yields V r (!) = jk o o e jkoR 8RZ o V t (!)L 1 L 2 : (3.3.9) The voltage transfer function relates the received voltageV r (!) to the transmitter voltage V t (!). The voltages are measured across the antenna terminals. The transfer function is V r (!) V t (!) = jk o o e jkoR 8RZ o jL 2 j 2 ; (3.3.10) where an assumption of similar antennas was used to dene L 1 = L 2 . Solving for the eective length results in L 2 = s 8RZ o jk o o e jkoR V r (!) V t (!) : (3.3.11) Obviously, this result depends on frequency and thus it is convenient to use the sensitivity function denition dened by (3.2.45). Thus H r (!) = s 2RZ o e j!R=c j! o =c V r (!) V t (!) = s 2RcZ o e j!R j! o =c V r (!) V t (!) (meters) 89 and E i (!) = s jk o o e jkoR 2RZ o V t (!)V r (!) = s j! o e j!R=c 2RcZ o V t (!)V r (!) (V/m) (3.3.12) where the factor of 2 appears under the radical because the assumed impedance-matched condition and c is the speed of light in free space. The result shown in (3.3.12) depends only on measured voltages and can be used to indirectly measure the E-eld intensity at a distance R. To determine the electric eld intensity at a distance R, caused by the radiation of an UWB pulse p n (t) through a linearly polarized antenna, we measure the E-eld under the assumption that the transmit and receive impedances are equal and matched perfectly to Z o . The EIRP spectral density can be determined from (3.3.12) by taking the magnitude squared response of E i (!) and dividing it by the intrinsic impedance of free space and thus S EIRP (!) = jE i (!)j 2 o (3.3.13) = 1 o E i (!)(E i (!)) (3.3.14) = !jV t (!)V r (!)j 4RcZ o : (3.3.15) When the conditions that were used to derive (3.3.12) are met, the resulting EIRP spec- tral density is accurate. One problem that can be encountered using equation (3.3.12) is that the transmit and receive antenna impedances are not perfectly matched to the 90 transmission line. Further, the transmission and receive antennas may appear physically identical; however, electrically they are unique. 3.4 Antenna Types 3.4.1 Transmitting Antenna and Radiation The transfer function of an antenna must be accurately known either by analysis, mea- surement or simulation [11], [9] to support an UWB waveform design. David Pozar used analytical models for dipole antennas over a wide bandwidth [45]. His results indicate that the antenna transfer function can exhibit a non-uniform transfer function when a very wide bandwidth is considered. His work implies that some antennas can have a spectral shaping eect on an UWB waveform. In general it is dicult to design and construct simple UWB antennas that exhibit a relatively at frequency response. Three types of antennas were used for the sample computations performed in this dissertation. The antennas used were: 1. Transverse Electromagnetic Mode (TEM) Horn, 2. Small Transverse Electromagnetic Mode (TEM) Horn, 3. Diamond dipole. Transverse Electromagnetic Mode Horn Antennas: TEM horn antennas ex- hibit relatively constant gain, a good impedance match and linear phase over several 91 decades of frequency. Their performance is very predictable using mathematical mod- els [49], [50], [48]. The features of TEM horns are typically dicult to achieve in simple antennas. Small TEM Horn Antennas: The small TEM horn antennas were designed to have a slightly higher cut-on frequency relative to the larger TEM horn antennas. Both types of TEM horns were designed with truncated ground planes for compactness of size, causing somewhat degraded behavior. Diamond Dipole Antennas: The diamond dipole antenna is a relatively small antenna when compared to the TEM horns. It was previously proposed for UWB, albeit for a dierent frequency band than what was specied by the FCC's 2002 Report and Order [19]. The antenna was used here because it was readily available and it could serve as an engineering reference. It was expected that the antenna would alway perform worst than the TEM horns. The TEM horns were designed but not optimized for the FCC R&O spectral mask. The measured S 21 responses for the three antennas used are shown in Figure 3.6 on page 94. The measurements were taken in an anechoic chamber with a calibrated network analyzer. The calibration reference plane was at the antenna input terminals. The number of data points used in all measurements was 1601 uniformly-spaced frequency samples. The observation bandwidth was from 50 MHz to 20.05 GHz. In all cases, pairs of similar looking antennas were used in the measurement process. A few relatively large re ections were observed around 18 GHz. The re ections were believed to be caused by the limitation of the SMA connectors. Later, measurements were taken to determine the VNA's noise oor. The data suggests that the corrected VNA noise could have 92 X(f) H meas (f) Y (f) Figure 3.3: Measurement in the Frequency Domain. X(f) H tx (f) H ch (f) H rx (f) Y (f) Figure 3.4: An equivalent decomposition of the measurement model. contributed to the relatively large spectral lines observed around 18 GHz (See Figure 3.30 on page 120). The re ections were not believed to be artifacts of the antennas and thus the re ections were dampened for the analysis. Dening H meas (f),S 21 , it is possible to decompose this measured function as H meas (f) =H tx (f)H ch (f)H rx (f); (3.4.1) where H tx (f);H ch (f) andH rx (f) are the system functions of the transmit antenna, free space environment and the receiving antenna respectively. The decomposed measurement model is shown in Figure 3.4.1. The last factorH rx (f) is the transfer function from the electric eld in the vicinity of the receiving antenna to the receiving antenna terminals, and its eect must be removed (if known) by calibration or analytically [48], [4]. The removal of H rx (f) is performed by multiplying (3.4.1) by H 1 rx (f) which explains the use of the inverse receive antenna transfer function in Figure 2.1. 93 X(f) H tx (f) H ch (f) ~ Y (f) Figure 3.5: The desired measurement for FCC compliance. Figure 3.6: Antenna S21 plots In summary, ~ Y (f) = Y (f)H 1 rx (f) removes any distortion that is introduced by the receiving antenna which has H rx (f) as a transfer function. The equivalent ltering that must be considered for the UWB waveform design problem is given by H eq (f) =H D/A (f)H TxLine (f)H meas (f)H 1 rx (f): (3.4.2) All of these factors must be considered in the process of designing FCC-compliant wave- forms. 94 TEM Horn Geometry a) Side view b) Top view z 5-1/4 y 1-7/16 β/2 7-13/16 x z s 7-1/16 α/2 Styrofoam support ground plane conductining plane center conductor of coax feed dielectric of coax feed c) Front view d) Transition 7-13/16 y w h x 7-3/4 a SMA connector Figure 3.7: Truncated Ground Plane Large TEM Horn Antenna Mechanical Design. All measurements are in inches. 3.4.2 Truncated Ground Plane TEM Horn Antenna A truncated ground plane transverse electromagnetic (TEM) horn antenna consists of an upper triangular conducting plate at an optimized angle 0 < 180 relative to the voltage input point. The current conducting plate is elevated at an optimized angle 0 < 180 relative to a ground image plane. The optimizations of and are relative to a desired antenna size and to a desired impedance match to a transmission line characteristic impedance Z 0 (!). The impedance Z 0 (!) is explicitly to be a function of the radian frequency ! because a constant impedance for all frequencies is dicult to achieve for UWB systems. Typically, TEM horn antennas are constructed from two matched triangular plates without a ground plane between the two plates. However, similar performance can be achieved by using a single upper conducting plate over a conducting ground plane of innite extent. In practice, an innite ground is approximated by a ground plane that is at least 20 max in dimension and max corresponds to the largest wavelength to be transmitted by the antenna. The ground plane in the TEM horn antennas discussed here 95 are approximately square (See Figure 3.8). Figures 3.9 and 3.10 show the antennas in the anechoic chamber during the measurement process. Figure 3.8: Large TEM Horn antennas manufactured by the author to facilitate the waveform design research. Figure 3.9: Measuring the Large TEM Horn Antenna in the anechoic chamber. They have an upper conducting plate where the front of the antenna is ush with the front of the ground plane as shown in the side view of Figure 3.7 1 . Styrofoam was 1 The TEM horn antennas used in this research were jointly designed by Mr Je Yang and the author. 96 Figure 3.10: Measuring the Large TEM Horn Antenna in the anechoic chamber. used as a mechanical supporting structure because it has a relative permittivity e r 1:0 and hence, it is transparent to the radiating E-eld. The characteristic impedance of the transmission line connected to the antenna was 50 . The feed point of the antenna was its apex. The connector used to connect the antenna and the transmission line was an SMA connector. Figure 3.7 shows the antenna-feed structure. Key features of the TEM horn are: High directivity, Performs single time dierentiation on transmit but not on receive, The opening of the horn is proportional to 1 4 of the lowest operating frequency. The low frequency limit is also proportional to the size of the horn aperture; how- ever, a large aperture limits high frequency operation because higher order modes are stimulated. The high frequency limit is dictated by the input coaxial-to-horn transition. 97 TEM horn antennas have been extensively studied in the literature (e.g., see Lee and Smith [35]). 3.4.3 Simulated Performance of the Large TEM Horn Antenna Post Manufacture The large TEM horn antennas were simulated after they were designed and manufactured to determine the expected gain and voltage standing wave ratio (VSWR). The VSWR was determined by detailed computation of the antenna's impedance as a function of frequency. Antenna simulations were performed after they were manufactured because of an expressed interest in the patterns at various frequencies. Three dimensional pattern measurements at multiple frequencies are time consuming. The simulations were sucient for the research performed here. Prior to the simulations being performed, it was assumed that the direction of maximum radiation would be in thez-direction (see 3.7 on page 95). The simulation results that are shown in Figures 2 3.15{3.24 indicate that the maximum gain is in thezdirection. The simulated results should be compared to actual measured data. Unfortunately, measurements could not be performed (prior to the time of thesis publication) so that a comparison could be made between simulated and measured results. The Method of Moments (MoMs) was used to determine the 3-dimensional radiation patterns for the large TEM horm antennas. There were 2,834 unknowns in the computa- tions. The simulations were run on a dual core CPU personal computer using 4 MBs of RAM. The MoM software was custom-written. The expected accuracy of the simulation was 5% below 4 GHz, 10% up to 7 GHz, and up to 15% beyond 7 GHz. The accuracy 2 The gures shown here were simulated by Dr Anatoliy Boryssenko. The simulations were based on the physical characteristics of the large TEM horn antenna after it was made. 98 Figure 3.11: Truncated ground plane large TEM horn antenna rear view. All measure- ments are in inches. Figure 3.12: Truncated ground plane large TEM horn antenna top view. All measure- ments are in inches. varied with frequency because of the phase accuracy. Accurate phase computations at very high frequencies require high resolution which implies a high density of the sample points. 99 Figure 3.13: Truncated ground plane model of the TEM horn antenna { perspective view. Figure 3.14: Truncated ground plane TEM horn simulated input impedance. 100 Figure 3.15: Truncated ground plane TEM horn antenna gain animation as a function of frequency. Note: This is an animated gure. Printed media will only show the antenna pattern at 1.0GHz. Figure 3.16: Truncated ground plane TEM horn simulated gain at 2GHz. 101 Figure 3.17: Truncated ground plane TEM horn simulated gain at 3GHz. Figure 3.18: Truncated ground plane TEM horn simulated gain at 4GHz. 102 Figure 3.19: Truncated ground plane TEM horn simulated gain at 5GHz. Figure 3.20: Truncated ground plane TEM horn simulated gain at 6GHz. 103 Figure 3.21: Truncated ground plane TEM horn simulated gain at 7GHz. Figure 3.22: Truncated ground plane TEM horn simulated gain at 8GHz. 104 Figure 3.23: Truncated ground plane TEM horn simulated gain at 9GHz. Figure 3.24: Truncated ground plane TEM horn simulated gain at 10GHz. 105 3.4.4 Diamond Dipole Antennas Diamond dipole antennas [66] are antennas that are constructed from two triangular conducting metals (e.g., copper) separated a very small distance apart as shown in Figure 3.25. The upper conducting plate was connected to the center conductor of a coaxial transmission line. The lower plate was attached to ground. These antennas were proposed for use in UWB communications by a company. The S 21 of the antennas were measured in an anechoic chamber at a distance of 215 inches (see Figure 3.26). An UWB low-noise amplier was used in the meaurement. The S 21 results are shown in Figure 3.6. The far-eld distance requirement for reliable antenna measurements is given by r> 2D 2 (3.4.3) where r is the distance between antennas, D is the maximum linear distance of the antenna aperture, is the wavelength. The linear length of the diamond dipoles was measured to be approximately 10cm. The antennas were swept with a sinusoid to perform the S 21 measurement. The frequency of the sinusoid was linearly varied from 50MHz { 20.05GHz with a step size of 12.5MHz. Considering equation (3.4.3), the antennas were measured in the far eld. 106 Figure 3.25: Diamond dipole antenna type used in experiments. Figure 3.26: Diamond dipole antenna in the anechoic chamber. The measurements were made in a Styrofoam test xture. 3.4.5 Eective Isotropic Radiated Power for UWB The Eective Isotropic Radiated Power (EIRP) is dened as the power P t at the input terminals of an antenna, multiplied by the transmitting antenna's gain G t and thus EIRP =P t G t : (3.4.4) 107 In 1946, Friis [20] provided system engineers with a gift that greatly simplied the esti- mation of a radio's link performance via a simplied link equation given by P r = P t A r A t (4d) 2 2 A isotr (3.4.5) where P t = the power in Watts that are fed into the transmitting antenna terminals. P r = the power in Watts that are available at the receiving antenna's output terminals. A r = the receiving antenna's eective area (m 2 ). A t = the transmitting antenna's eective area (m 2 ). d = the distance (in meters) separating the transmitting and receiving antennas. = the wavelength (in meters) of the sinusoidal source used to excite the antennas under consideration. Friis' equation is only applicable in the far eld where the distance d > 2D 2 must hold. Recall that D is the largest linear dimension of the transmit or receive antenna. Friis' formula is more readily recognized as P r = P t G r G t 2 (4d) 2 ; (3.4.6) however, Friis original form provides more insight into the the received power P r because it shows that the received power critically depends on the eective apertures of both 108 the receiving and transmitting antennas. The received power also depends on the eec- tive aperture of the ctitious isotropic radiator given by A isotr = 4 and thus the more recognizable form given by equation (3.4.6) results. In UWB systems, the EIRP is dicult to evaluate because of the potential of extreme variations of radiated powerP rad . Further, the antenna gain is dicult to measure because of the large bandwidth used for UWB. One approach that can be used to determine the EIRP of an UWB system is to use discrete wavelengths (i.e., i ). The gain can be determined at each discrete wavelength by measuring the transmitted and received power and then solving for the individual gains at each frequency. When the transmitting and receiving antennas are electrically identical, the gains are also identical and thus P r = P t G 2 t 2 (4d) 2 : (3.4.7) In the original paper, Friis dened the eective area of a receiving apparatus as A e , P r P o ) P r ,P o A e (3.4.8) where P r is the power received at a measurement reference point and P o is the power ow per unit area, which is outside of the receiving antenna. This denition accounts for all anomalies such as impedance mismatch, power dissipated by the transmit or receive antenna. As such, equation (3.4.7) greatly simplies link performance evaluation. The equation is "simple" as the author Friis suggests. All the parameters of equation (3.4.7) are known except the gain G t . The gain can be determined by solving (3.4.7) for G t at 109 each wavelength. OnceG t is determined, the EIRP at each wavelength can be computed. The resulting equation for the gain of an antenna is given by G t = 4d r P r P t : (3.4.9) The result given by (3.4.9) was derived only from manipulations of Friis' free space equa- tion and the assumption of electrically identical transmit and receive antennas. It has been recently discovered by the author that equation (3.4.9) appeared in 1946 in [13] followed by Kraus [32] in chapter 18-6. Substituting (3.4.9) into the equation for EIRP (3.4.4) results in EIRP = 4d s P 2 t P r P t = 4d p P t P r = 4df c p P t P r ; (3.4.10) wheref is the frequency andc is the speed of light. This version of EIRP is useful because only the powers that are transmitted and received are required. The received power is measured across the receiver's load impedance which is approximately matched to the transmission line. Unfortunately, this result is actually dicult to use for UWB because one cannot assume that a perfect match exists between a transmitter's waveform generator and the transmit antenna. The transmit antenna represents a complex series impedance load. In general, UWB antennas are not perfectly matched to the transmission line. Thus, the power presented to the antenna's input terminals will not be completely converted 110 into desirable RF radiation. Further, antennas are not perfect, ecient radiators. Some of the incident power is converted into Ohmic heat. In summary, the power is lost in three forms: P rad is the power radiated, P re is the power re ected back to the source, P loss is the power loss in the form of heat. The power radiated is the only desirable loss. The other two losses represent wasted energy. 3.5 Measurement Procedures The measurement procedures for the EIRP Spectral Density are dened in the FCC-RO. Unfortunately, the actual measurement apparatus is not well dened. The FCC RO does mention that the measuring antenna that is used to measure the UWB system under test must be calibrated; meaning that the measured response can be approximated by backing out any in uence that the testing apparatus has placed on the measurement. For the measurements performed in this work, the measurements of the AUT were performed in the \far eld." Unless otherwise noted, pairs of \like" antennas were used.. Figure 3.27: Scattering model of a two port network. 111 The power-wave scattering matrix was derived by Kurokawa [34] for any N-port network. The two-port scattering network can be represented by the matrix given by 2 6 6 4 b 1 b 2 3 7 7 5 = 2 6 6 4 S 11 S 12 S 21 S 22 3 7 7 5 2 6 6 4 a 1 a 2 3 7 7 5 ; (3.5.1) wherefa 1 ; a 2 g andfb 1 ; b 2 g are the inputs and outputs of the two-port network respec- tively (e.g., see Figure 3.27). Kurokawa dened a i = V i +Z i I i 2 p jRefZ i gj and b i = V i +Z i I i 2 p jRefZ i gj ; (3.5.2) as incident and re ected power-waves respectively. According to Kurokawa, the terms V i and I i are the voltage and the current owing into the ith port of a junction and Z i is the impedance looking out from the ith port. The positive real value is chosen for the square root in the denominators. Here, we are only interested in a 2-port network which is fully characterized by the matrix given in equation (3.5.1). Our measurements are based on two-port networks. 3.5.1 Antenna Measurements Friis' freespace equation [20] is used to relate the power received P Rx and delivered to a load impedanceZ L ; due to a powerP Tx that is transmitted by an antenna. Friis' equation is P Rx = Tx Rx (1j Rx j 2 )(1j Tx j 2 )P Tx G Tx G Rx c 2 (4fr) 2 ; (3.5.3) where 112 Tx Rx denote the dot product of the polarization vector between the transmit and receive antennas respectively, Tx and Rx are the complex re ection coecients of the transmitter and receiver respectively, P Rx is the received power by the receiving antenna, P Tx is the transmitted power by the transmit antenna, G Tx is the power gain of the transmit antenna, G Rx is the power gain of the receiving antenna, c is the speed of light, f is the specic frequency under test, r is the range between the transmit and receive antennas. Friis' equation given in [20] assumes that the transmit and receive antenna impedances are ideally matched to a transmission line impedance (typically 50 ) so that maximum power can be delivered from the transmitting source to the receiving load. When the impedance of both the transmit and receive antennas are matched to their respective transmission lines, and the loads are also impedance matched thenj Tx j =j Rx j = 0. Maximum power transfer occurs from a source to a load when the respective impedances are matched. Antennas used in this research are linearly polarized. During the measurements, the antennas were oriented so that the polarization dot product was approximately unity (i.e., 113 VNA DUT Bsin(2πf i t+θ i ) RF In Port RF Out Port Asin(2πf i t) Reference Plane Figure 3.28: Vector Network analyzer (VNA) System Model. The model shows the calibrated reference plane at the location of the VNA as well as the RF In and RF Out ports. Tx Rx = 1). Further, it was assumed that measurements were taken in the bore-sight direction where the gain was maximum over all frequencies. For the antennas used in this work, the gain terms in equation (3.5.3) were not known. Section 3.5.5 starting on page 120 provides the method used to determine the frequency- dependent unique gains of each antenna. 3.5.2 Measurement Process An Agilent 8720 series Vector Network Analyzer (VNA) was used for the scattering parameter measurements. The VNA takes measurements of a device under test (DUT) in the frequency domain. The VNA is similar to a spectrum analyzer except that the VNA can output a frequency set ofi accurately known internal sinusoidal stimuli signals of the form s i (t). It is possible to measure the complex input/output relationship of a device under test (DUT) because the VNA has an accurately known internal stimulus. A properly calibrated VNA can accurately measure the relative (to the internally generated stimulus) amplitude and phase of a DUT at the location of the calibration reference plane 114 s(t) H RBW (f) ( ·) 2 H vid (f) r(t) 1 Figure 3.29: Simplied Spectrum Analyzer Model. (see Figure 3.28 ). The VNA measurements are based on the well-known 2{port complex S ij parameters [65] where i;j2f1; 2g. It is assumed for the results that follow that the VNA's transmitted signal is of the form s i (t) =A sin(2f i t + i ) i2f1; 2; ;Ng (3.5.4) where the phase i is accurately known at the detector and thus the transmitter reference phase can be taken to be zero. The detected phase represents the phase shift because of the DUT. The frequenciesff i g denote the set of discrete test frequencies (pure tones) used in the measurement process. The received signal at the receiver front end (VNA input port) is of the form r i (t) = i s i (t + i ) +n(u;t) u2U; t2T; (3.5.5) where i denotes the received amplitude at the ith frequency, i represents the signal propagation delay induced by the DUT and test environment,U denotes the set of random events andT denotes the time index set. The noisen(u;t) was assumed to be wide-sense stationary (WSS) Gaussian noise and statistically independent of the signals i (t). The statistics of the noise at the receiver front end are assumed to be zero mean ( n = 0) with variance 2 n ==2 where is a constant. 115 Prior to detection, the signal is ltered by h res (t) which has the purpose of combating excess noise and setting the minimum resolution for an observation after ltering. If for an example, two Dirac delta functions are relatively close together in time (closer than the time support of the lter) were input into h res (t), the individual impulses would not be observed at the lter output; instead, a single \blurred" image would be observed. In general, the particular shape ofh res (t) is unimportant. The lter can be represented by an equivalent rectangular lter with a bandwidth called the noise-equivalent bandwidth given by B n = 1 max f H res (f) Z 1 1 jH res (f)j 2 df: (3.5.6) When white Gaussian noise is the input of the lter with bandwidth B n , the output noise power is simply B n =kTB n , where k 1:381 10 23 J K 1 denotes Boltzmans's constant and T denotes the thermal temperature of the equipment which is measured in Kelvin. For all of the measurements discussed here B n = 30Hz and T = 300K. For these parameters, the 2-sided noise power is159:06 dBm. 3.5.3 The Dynamic Range of the Measurement Equipment The dynamic range of the measuring system is dened to be the range of power that the measuring device can accurately measure. For passive devices (non amplifying devices), the dynamic range can be taken to be P ref P min such that P ref > P min : The term P ref is the nominal output power of the device whereas P min is the minimum input power the device can measure. This model assumes that the harmonics of the source signal 116 (internally generated) are not present. The assumption is important because source harmonics could mask signals of interest. The dynamic range critically depends on the so{called noise oor of the measuring apparatus. There are several accepted methods that can be used to characterize the noise oor. One method uses the root mean squared (RMS) voltage value (e.g., see Witte in [65] or Agilent's application note in [1]). Given a receiver's RMS voltage noise oor, the receiver's sensitivity can be dened as [37] [21] K = 10 log(kTB n ) +F dB + S N dB ; (3.5.7) where S=N is the desired signal-to-noise ratio at the detector and F dB = 10 log S in =N in S out =N out (3.5.8) is the noise gure of the receiver. In the equation above, the subscripts \in" and \out" refer to the SNR at the input and output of a device respectively. When (S=N) = 0 dB in (3.5.7), the signal power is equal to the noise power. 3.5.4 Noise Floor Determination for the Vector Network Analyzer The dynamic range for the Agilent 8720ET VNA is reported [2] to have a minimum dynamic range of 102dB in the frequency range of 50MHz to 840MHz. For frequencies greater than 840MHz and up to the VNA max frequency of 20.05GHz, the dynamic range is 104dB. The performance specication assumes that the VNA was calibrated and the reference plane is at the instrument terminals. The quoted dynamic range require an IF 117 bandwidth of 10Hz. Measurements taken with a larger IF BW and cabling will increase the noise oor and reduce the dynamic range of the VNA. To determine the so-called noise oor of the calibrated VNA, a vector n2 C p of S 21 (f i ) noise measurements were observed at each test frequency f i where p = 1001 and i2f1; 2;:::; 1601g. It was assumed that the noise samples were statistically indepen- dent, identically distributed (iid) and WSS with nite means and variances. The noise measurements can be arranged into a complex matrix N2C pm wherem = 1601 which are the number of test tones. Each column vector of N has 1001 data points that are observed for the mth test frequency. The variable m indexes the columns of N from the lowest to the highest frequencies used. As a comparison, two measurements were performed; calibrated and uncalibrated. The uncalibrated measurements were measured to determine the state of the noise oor when the VNA calibration was o and similarly for when the VNA calibration was on. Antenna measurements were always performed with correction turned on. The noise observations were made with the VNA in the same conguration as when the antenna measurements were made. The only exception was that the transmit and receive ports of the VNA was terminated with a calibrated 50 load at the location of the reference plane (i.e., at the ends of the cables). During the antenna tests, the stimulus signals (pure tones) were set to 10 dBm to account for free-space propagation loss (see equation (3.5.4)). During the noise measurements, the tones were also generated; however, the transmitter was terminated into a 50 load after propagation via the same length of cable that was used for the antenna measurements. The calibrated 50 termination fully absorbed the incident signal. Because of the transmit termination, the receiver could only 118 receive and detect internally generated noise contained in the receiver. The receive port of the VNA was also terminated with a calibrated impedance. The rationale for receive- side termination was to minimize any potential re ections or spurious emissions. It was unknown at the time of measurement if the VNA generated harmonics that would be detected. Noise oor bias was also a concern since it would have the eect of decreasing the dynamic range of the VNA. Termination of the transmit and receive ports of the VNA ensured that the VNA would only measure pure internally generated noise. Prior to processing the measured data, it was assumed that the noise process was ergodic in the mean as described in section E.2.2. The complex sample mean of the noise, for the lth frequency, was estimated by X l = 1 p p X k=1 n kl fl = 1; 2;:::;mg (3.5.9) = 1 p p X k=1 (a kl +jb kl ); (3.5.10) where a kl and b kl are the real and imaginary parts respectively of the noise. It is known that the sequence of partial sums converges to the true mean by the strong law of large numbers (SLLNs) and certainly converges via the weak law of large numbers (WLLNs) . The sample variance was estimated by the unbiased estimator given by ^ 2 l = 1 p 1 p X k=1 (n kl ^ l )(n kl ^ l ) : (3.5.11) Equations (3.5.9) and (3.5.11) are consistent estimators for the true mean and variance 2 of the noise. The computed noise mean and variance are shown in Figure 3.30 for each 119 0 5 10 9 × 1 10 10 × 1.5 10 10 × 2 10 10 × 2.5 10 10 × 140 − 130 − 120 − 110 − 100 − 3 kHz 1 kHz 100 Hz 30 Hz 10 Hz Noise Power Relative 0dB Signal. Correction Off Frequency Variance dB (a) 0 5 10 9 × 1 10 10 × 1.5 10 10 × 2 10 10 × 2.5 10 10 × 140 − 120 − 100 − 80 − 60 − 3 kHz 1 kHz 100 Hz 30 Hz 10 Hz Noise Power Relative 0dB Signal. Correction On Frequency Variance dB (b) Figure 3.30: (a) Computed noise variance without VNA corrections. (b) Computed noise variance with VNA corrections. The measurement range was 50MHz to 20.05GHz. frequency. The sample mean and variance was computed for the following resolution bandwidths: 10Hz, 30Hz, 100Hz, 1kHz and 3kHz. The gure clearly shows that the noise oor increases with bandwidth. The gures also show that the VNA corrections enhances the noise around 18GHz. The gures also demonstrate the kTB n relationship. 3.5.5 Computation of Antenna Gain from Measurements The Friis free space equation is given by P Rx (f) = P Tx (f)G Tx (f)G Rx (f) 2 (4r) 2 : (3.5.12) It is the primary equation used to determine antenna gain from measurements. The equation assumes that a polarization and an impedance matched condition exist in the communication system. The frequency dependence is implicitly shown to remind the reader that the terms are frequency dependent. The subscripts Tx and Rx are used 120 to distinguish the variables that relate the transmit and receive antennas under test. Expressing Friis' equation in dB form results in 10 log P Rx (f) P Tx (f) = 10 logG Tx (f) + 10 logG Rx (f) + 20 log 4r : (3.5.13) Equation (3.5.13) can be rewritten by taking the last term to the left hand side. Thus, 10 log P Rx (f) P Tx (f) 20 log 4r | {z } b(f) = 10 logG Tx (f) + 10 logG Rx (f) (3.5.14) b(f) =G Tx (f) +G Rx (f) (dB): (3.5.15) The frequency variablef2 [f min ;f max ]. The set of frequencies used in the measurements were [50MHz to 20.05GHz] with f = 12:5MHz selected as the frequency spacing between adjacent samples. The maximum frequency resolution of the Vector Network Analyzer was f. The sampled form of (3.5.14) is given by b = g 1 + g 2 ; where b; g 1 ; g 2 are column vectors and b is not known and must be determined. When a \calibrated" antenna reference is not available to compute the amplitude and phase of an antenna, other approaches are required. There are typically three approaches that can be used which are: The single antenna method. The two antenna method where the antennas are exactly the same. 121 The three antenna method. These three methods are discussed in the next three sections. 3.5.6 The Single Antenna Method The single antenna method was proposed by two papers; Glimm et. al. [23] and Krieger et. al., [33]. Glimm et. al., focused on the measurement of antenna gain only and Krieger et. al. explored gain and phase measurements. In both papers, a single antenna under test (AUT) is excited in the presence of a large re ecting ground plane. The ground plane was located a distance R away from the AUT. It's image within the ground plane is at distance r = 2R from the physical antenna. Since there was an excited eld in the presence of the ground plane, a near perfect re ection results from the excitation and was returned to the AUT. The re ected wave would appear to have been emanated from an identical antenna located the exact same distance R from ground plane as the AUT. Thus, Friis equation can be used with G Tx =G Rx and thus the vector b = 2g ) g = b 2 (dB) (3.5.16) can be formed by uniformly sampling (in frequency) equation (3.5.14). Since all the parameters of Friis equation are known, the gain can be directly computed. This method requires switching or directional coupler equipment so the transmit and receive modes can be readily observed. 122 3.5.7 The Two Antenna Method For the measurements reported in this thesis, the measurements were performed in an anechoic chamber (see Figure 3.31 on page 123 ) using two antennas. Each antenna was connected to the VNA by two lengths of high quality phase-stabilized cables measuring approximately 13ft and 18ft each. The cables were calibrated out during the VNA cal- ibration process. The reference plane was at the location of the SMA connectors of the antennas under test. R A B Figure 3.31: Antenna Measurement Setup in an Anechoic Chamber. The two-antenna method is similar to the single antenna method except two antennas with similar characteristics are used. It has been observed that while antennas may 123 physically appear to be the same, dierences can be observed when the pair of AUT are place under rigorous scrutiny. The errors in the method results from a variety of sources such as: Variations in the manufacturing process which prevents antennas from being elec- trically identical, Range measurement error between the pair of antennas under test, Finally, antenna orientation error which is manifested as polarization errors. The two-antenna measurement method assumes that the pair of antennas are electrically identical and implies G Tx = G Rx = G. Assuming that P Tx (f i ) and P Rx (f i ) are mea- sured at a particular frequency f i and the range r is accurately known, the gain can be determined because G is the only unknown variable. The computed gain represents the average gain (in dB) of the antennas because G = 4fr c r P Rx P Tx (3.5.17) and in dB G = 10 log 4rf c + 10 2 log P Rx P Tx for f > 0: (3.5.18) 3.5.8 The Three Antenna Method The three antenna method is considered when a pair of \identical" antennas are not available. The method assumes that the antennas are measured in pairs. The antenna gain of a single antenna can be determined if at least one of the gains terms are provided 124 using the two antenna method; otherwise, if both gains are unknown, the gains cannot be determined because there are two unknowns and only one equation. When at least three or more antennas (denoted antenna A, antenna B and antenna C) are available, a three antenna test method can be used. Using (3.5.3) in dB form results in a sum of terms given by 10 log P B P A + 20 log 4fr c | {z } P AB =G A +G B ; (3.5.19) where the subscripts Tx and Rx have replaced with subscripts A and B. The pair of gains G A and G B are in dB. The LHS of (3.5.19) is known from measurements and the RHS must be determined. Performing the experiment with antenna pairs A and C results in 10 log P C P A + 20 log 4fr c | {z } P AC =G A +G C (3.5.20) which results in two equations and three unknowns. An additional experiment between antennas B and C is required to solve the system of equations which results in 10 log P C P B + 20 log 4fr c | {z } P BC =G B +G C : (3.5.21) It has been assumed that the antennas are reciprocal meaning that a measurement from A to B is approximately equal to the measurement from B to A. We say approximately because in order to show (by measurement) reciprocity, the antennas must be physically moved and measured in the new conguration. It is dicult to reproduce experimental results exactly however, the Reciprocity Theorem as stated by Kraus [32] is 125 If an emf is applied to the terminals of an antenna A and the current mea- sured at the terminals of another antenna B, then an equal current (in both amplitude and phase) will be obtained at the terminals of antenna A if the same emf is applied to the terminals of antenna B. The reciprocity theorem enables equations (3.5.19) { (3.5.21) to be written in matrix form as 2 6 6 6 6 6 6 4 P AB P AC P BC 3 7 7 7 7 7 7 5 | {z } p = 2 6 6 6 6 6 6 4 1 1 0 1 0 1 0 1 1 3 7 7 7 7 7 7 5 | {z } A 2 6 6 6 6 6 6 4 G A G B G C 3 7 7 7 7 7 7 5 | {z } g ; (3.5.22) where p and g are column vectors. The full rank binary matrix A is Hermitian because A y = A: (3.5.23) The symbol \y" denotes the complex conjugate transpose operation. Obviously A in equation (3.5.22) is real and thus conjugation is not required. It is well-known [59] that Hermitian matrices have real eigenvalues. In fact, the eigenvalues of A are 1 =1; 2 = 1; 3 = 2. The set of three eigenvalues show that A is \well conditioned" because they are close in magnitude and none of them are extremely large nor extremely small. When the full rank matrix A has an eigenvalue that is extremely small, A 1 has an eigenvalue that is quite large because A 1 / 2 6 6 6 6 6 6 4 1 1 0 0 0 1 2 0 0 0 1 3 3 7 7 7 7 7 7 5 : 126 A large eigenvalue of A 1 implies that the system is sensitive to perturbations which will be made clear in section 3.6.4 on page 135 where bounds on the relative error is derived. It will be demonstrated that the condition number (A) of a linear system is an important concept in numerical analysis. It determines how sensitive the solution x = A 1 b is to changes in A or b. To make the term \well conditioned" more precise, the concept of the norm of a matrix is required. The matrix norm will then be used to dene (A) for the system given in (3.5.22). 3.6 Norms 3.6.1 The Norm of a Vector In this section, some basic concepts from Linear Algebra are reviewed. For more infor- mation on the subject, the reader is referred to the excellent texts of Gilbert Strang [59] and Thomas Shores [54]. Results are generalized to the complex domain because a real vector space is a subset. The column vector v is denoted v = 2 6 6 6 6 6 6 6 6 6 6 4 v 1 v 2 . . . v n ; 3 7 7 7 7 7 7 7 7 7 7 5 (3.6.1) and has conjugate transpose v y = (v 1 ;v 2 ; ;v n ); (3.6.2) 127 where denotes complex conjugation. Denition 2. The Vector Norm: The norm on a vector spaceV is a functionkk that assigns to each vector v2V a real numberkvk such that for any scalarc2C and vectors u; v2V the following apply: 1. kuk 0;kuk = 0 if and only if u = 0. 2. kcuk =jcjkuk where c2C is any complex number. 3. ku + vkkuk +kvk (Triangle inequality). A vector spaceV together with a normkk is called a metric space or a normed space. Given any two vectors u; v2V, a distance measure d(u; v) is dened asku vk: Denition 3. The pnorm : LetV denote a metric space onC n : Let v2V and p 1; be a real number then the pnorm is dened as kvk p = (jv 1 j p ;jv 2 j p ; ;jv n j p ) 1=p (3.6.3) wherejv i j =v i v i ; for i = 1; 2; ;n: From the pnorm, the following three norms can be dened for p = 1; 2;1: Denition 4. The 1-Norm: The 1-norm (p = 1) is dened as kvk 1 =jv 1 j +jv 2 j + +jv n j: (3.6.4) 128 Denition 5. The Euclidean Norm: The Euclidean norm (p = 2) is dened as kvk 2 = p v y v (3.6.5) = p jv 1 j 2 +jv 2 j 2 + +jv n j 2 : (3.6.6) Denition 6. The1Norm: The1-norm (p =1) is dened as the limit as p!1 and is given by lim p!1 kvk p =kvk 1 = maxfjv 1 j;jv 2 j; ;jv n jg: (3.6.7) Thepnorm dened in denition 3 is used to dene induced norms on matrices which are provided in the next section. 3.6.2 The Norm of a Linear Operator Let x2V (a normed vector space) and consider the equation Ax = b: (3.6.8) The matrix A is a linear operator on the vector x and thus takes x! b. The vector Ax has an induced norm on the linear operator A. The operator norm is an important concept to consider because it can be used to explain when computed results are accurate and thus can be trusted. The interested reader is referred to the 1966 textbook by Keller [26] for proofs and further exposition. Denition 7. The least upper bound of a set of real numbers is called the supremum and is denoted \sup." 129 The supremum may or may not be in the set under consideration. Denition 8. Let x2V denote any ndimensional complex vector with real normkk. The vector x induces (generates) an operator norm on A2C mn and is dened by kAk = sup 8x:x6=0 kAxk kxk = sup 8x:x6=0 A x kxk = sup 8^ x:k^ xk=1 kA^ xk; (3.6.9) where ^ x is a unit vector which means that it has one unit of measure in some norm. Denition 9. A pair of matrices A; B are conformal for addition if and only if A; B are mn matrices, then the matrix sum is dened as C = A + B where C is an mn matrix. Denition 10. A pair of matrices A; B are conformal for multiplication if and only if A is amp matrix and B is apn matrix, then the matrix product is dened as C = AB where C is an mn matrix. Denition 11. The Matrix Norm: A vector normkk that is dened on the vector space C mn of mn matrices, for any pair m;n, is a matrix norm if for all pairs of matrices A; B that are conformal for multiplication, the following properties hold: 1. kAk 0;kAk = 0 if and only if A = 0. 2. kcAk =jcjkAk where c is a real number. 3. kA + BkkAk +kBk (Triangle inequality). 4. kABkkAkkBk (Schwartz inequality). 130 The rst three properties are the same as in denition 2. The fourth denition for the matrix norm is required because the product of two matrices is an allowed operation whereas the product of two vectors is not dened. All vector norms do not induce the same operator norm. This fact is follows because of the Schwartz inequality. As an example, the vectorkvk 1 does not extend tokAk 1 . Similar to vector pnorms, operator pnorms can be dened for matrices. Prior to dening matrix norms, the spectral radius of a matrix is introduced. Denition 12. The spectral radius (A) of any square matrix A with eigenvalues 1 ; 2 ; ; n is dened as (A) = maxfj 1 j;j 2 jj n jg: The largest eigenvalue (in magnitude) is called the dominant eigenvalue of A. The spectral radius will be used in the denition of a matrix norm below. The following matrix norms satisfy properties 1{4 above. Denition 13. For any mn matrix A the following norms are dened: 1. Maximum Absolute Column Sum (L 1 {norm) kAk 1 = max 0jn P m i=1 ja i;j j 2. EuclideanL 2 norm (2{norm)kAk 2 = max i [ i (A y A)] 1=2 = p (A y A): The notation i (A y A) denotes the ith eigenvalue of A y A. 131 3. Maximum Absolute Row Sum (L 1 {norm)kAk 1 = max 1in P n j=1 ja i;j j Denition 14. The Frobenius Norm: The Frobenius norm of A2C mn is dened by kAk F = 0 @ m X i=1 n X j=1 ja i;j j 2 1 A 1=2 : (3.6.10) Considering the Euclidean norm, when A = A y (i.e., Hermitian), the spectral radius is equal to the Euclidean norm because kAk 2 = q (A y A) = p ((A)) 2 =(A): (3.6.11) The above result can be demonstrated more clearly when spectral decomposition is per- formed on the invertible matrix A y A: The spectral radius of A is only equal to the Euclidean norm when A = A y (i.e., Hermitian). In general, the spectral radius in not equal to the Euclidean norm. 3.6.3 The Condition and Ill Conditioning of a Linear Operator The condition of a linear operator is an important concept. It is directly related to the reliability of matrix inversions even though the matrix may be invertible. Denition 15. The condition number of a full rank linear operator A is dened by the real number (A),kAkkA 1 k: (3.6.12) 132 Theorem 1. The condition number (A) 1: Proof. Let A and B be two square matrices under norm denitions given in denition 13, then kABkkAkkBk: (3.6.13) Suppose that B = A 1 then kAA 1 kkAkkA 1 k kIkkAkkA 1 k 1kAkkA 1 k (A) ) (A) 1: The theorem is also true for the Frobenius norm except that the norm of the identity matrix iskIk F = p n. The condition number (A) indicates how perturbations in the equation Ax = b (3.6.14) aects the solution x = A 1 b: (3.6.15) 133 It may be expected that any small perturbation in Ax = b would manifest as a small perturbation in the solution x. Mathematically, we are interested in A^ x = ^ b A(x +x) = b +b (3.6.16) where ^ x = x+x, ^ b = b+b andx;b represent perturbations. The perturbations could represent errors from round o, number truncation or random noise in the measurement process. The term b +b represent errors in the data. It is of interest to determine a measure of the error magnitude in a given computed solution due to the perturbation. The following two error denitions are provided using the concepts of norms. Denition 16. Let x be dened as the true vector which is to be determined. Let ^ x be an approximation of x which could have resulted from errors in the computation of x. Let b and x denote the perturbations. The absolute and relative errors are dened by " a =kx ^ xk (3.6.17) and " r = kx ^ xk kxk (3.6.18) respectively. It should be noted that while the absolute error is often used, it can be misleading. Kincaid [15] and Keller [26] both showed that a small absolute error does not imply a small error and thus the relative error is typically used in numerical analysis. 134 The perturbed system in (3.6.16) can be written as a sum of the true system plus the additive error because of linearity. Each term can be analyzed individually. Using (3.6.16), it is known that Ax = b (3.6.19) and Ax =b: (3.6.20) Assuming that A is full rank implies that x = A 1 b (3.6.21) and x = A 1 b (3.6.22) which has respective norms kxk =kA 1 bkkA 1 kkbk (3.6.23) and kxk =kA 1 bkkA 1 kkbk: (3.6.24) 3.6.4 Bounding the Relative Error It will now be demonstrated that the ratio kxk=kxk can be bounded and that the condition number (A) is a critical factor of the bound. 135 Multiplying the right hand side of equation (3.6.24) by 1 = kbk kbk ; results in kxkkA 1 kkbk kbk kbk kA 1 kkbk kAxk kbk kA 1 kkbk kAkkxk kbk (3.6.25) which implies that kxk kxk kAkkA 1 k kbk kbk =(A) kbk kbk (3.6.26) and thus kxk kxk (A) kbk kbk (3.6.27) which is the desired result. The result clearly shows the amplifying eect of the condition number (A). Ideally (A) 1. When (A) is very large, the computed solution for x is unreliable even though A is invertible and the computed solution for x is unique. It is generally accepted that a small condition number (i.e., approximately less than 10) represents a well-conditioned system while a large condition number is an indication of an ill-conditioned system. Unfortunately, the threshold between a well{conditioned vs ill{conditioned system is not dened. It is interesting to note that the bound in (3.6.27) not only depends on the condition number (A) but also critically depends on the ratio ofkbk=kbk which is a function of the data. Whenkbkkbk, the ratio is very large and thus the large factor multiplies the condition number. As a result, the right hand side of the bound can be large even though (A) is close to one. Hence, the computed solution is once again, unreliable. 136 However, it should be noted thatkbk is typically an increasing function of the data. The norm ofkbk will be exactly zero if and only if b = 0 which follows from the denition of the norm. The likelihood of this event happening is extremely small (even in the absence of a signal) because noise is always present and the probability of observing a noise vector with zero norm is zero with high probability. The condition number gives an indication on the reliability or accuracy of a computed x. The reliability is based on both A and b even though A is full rank. As an example of an ill{conditioned system, consider the 3 3 Hilbert matrix given by Cheney and Kincaid [15] H 3 = 2 6 6 6 6 6 6 4 1 1 2 1 3 1 2 1 3 1 4 1 3 1 4 1 5 3 7 7 7 7 7 7 5 ; (3.6.28) which has a large condition number(H 3 ) 524:06. Another example of ill{conditioned systems are the comparable matrices given by Ralston [46] Ax = b 2 6 6 4 2 6 2 6:0001 3 7 7 5 2 6 6 4 x 1 x 2 3 7 7 5 = 2 6 6 4 8 8:00001 3 7 7 5 (3.6.29) and 2 6 6 4 2 6 2 5:99999 3 7 7 5 2 6 6 4 x 1 x 2 3 7 7 5 = 2 6 6 4 8 8:00002 3 7 7 5 : (3.6.30) 137 The two equations are strikingly similar diering by only 0.00002 in a 22 and by 0.00001 in b 2 . One would (incorrectly) think that the solution (x 1 ;x 2 ) in each case would dier by a similar small amount. In fact, the solution of (3.6.29) is (x 1 = 1;x 2 = 1) while the solution to (3.6.30) is (x 1 = 10;x 2 =2). The solutions are very dierent. Equations (3.6.29) and (3.6.30) have the same condition number; namely (A) = 4 10 5 which is very large. The large condition number demonstrates that very small perturbations can have large eects in the solution x. We will see how the condition number aects the solution in section 3.6.5. Systems with condition numbers near one are ideal. 3.6.5 The Spectral Decomposition of a Hermitian Matrix The spectral decomposition of a Hermitian matrix provides detailed insight into the linear transformation Ax = b. It shows in detail how the vector x is modied by the linear operator A. Spectral decomposition can only be performed when the matrix A can be diagonalized which implies A has a full set of eigenvectors e that span the space. Any point in the space can be represented by a linear combination of the normalized eigenvectors. The spectral decomposition of the matrix A given in (3.5.22) is A = EE y (3.6.31) 138 = 2 6 6 6 6 6 6 4 1 p 6 1 p 2 1 p 3 2 p 6 0 1 p 3 1 p 6 1 p 2 1 p 3 3 7 7 7 7 7 7 5 2 6 6 6 6 6 6 4 1 0 0 0 1 0 0 0 2 3 7 7 7 7 7 7 5 2 6 6 6 6 6 6 4 1 p 6 2 p 6 1 p 6 1 p 2 0 1 p 2 1 p 3 1 p 3 1 p 3 3 7 7 7 7 7 7 5 where the unitary matrix E 2 R 33 has column vectors composed of the Euclidean- normalized eigenvectors of A. The matrix is a diagonal matrix with i as elements on the main diagonal. The three orthonormal eigenvectors e 1 = 1= p 6(1;2; 1) t ; e 2 = 1= p 2(1; 0; 1) t ; e 3 = 1= p 3(1; 1; 1) t in E are ordered corresponding to the eigenvalues 1 ; 2 ; 3 appearing in . The inverse of a spectrally decomposed matrix is simple to determine because A 1 = (EE y ) 1 = (E y ) 1 1 E 1 where it is recognized that E 1 = E y because E is a unitary real matrix and 1 is determined by replacing the elements i of with their reciprocals 1 . Representing the set of antennas with labels A, B, C, equation (3.5.22) can be uniquely solved for the gain vector g = (g A ; g B ; g C ) t 2R 3 by multiplying both sides of (3.5.22) by A 1 = 1 2 2 6 6 6 6 6 6 4 1 1 1 1 1 1 1 1 1 3 7 7 7 7 7 7 5 (3.6.32) = E 1 E y 139 = 2 6 6 6 6 6 6 4 1 p 6 1 p 2 1 p 3 2 p 6 0 1 p 3 1 p 6 1 p 2 1 p 3 3 7 7 7 7 7 7 5 2 6 6 6 6 6 6 4 1 0 0 0 1 0 0 0 1=2 3 7 7 7 7 7 7 5 2 6 6 6 6 6 6 4 1 p 6 2 p 6 1 p 6 1 p 2 0 1 p 2 1 p 3 1 p 3 1 p 3 3 7 7 7 7 7 7 5 which yields the \unique" solution g =A 1 p =E 1 E y p; (3.6.33) where p = (p AB ; p AC ; p BC ) t 2R 3 . The notationp AB indicates the measured result from antennas A and B. To form each element of p (e.g., p AB ), let r(f k ) = r R (f k ) +jr I (f k ) denote the complex S 21 value at frequency f k then, p AB =10 log(r(f k )r(f k ) ) =20 log(jr(f k )j): The S 21 measurement r(f k ) contains the desired signal plus additive, complex circular white Gaussian noise. The equation holds for other pairs of antennas under test. The 140 solution given by equation (3.6.33) is used to determine the gain vector at each xed frequency f k . As an example, the gain of antennasfA; B; Cg is determined from g =A 1 p (3.6.34) 2 6 6 6 6 6 6 4 g A g B g C 3 7 7 7 7 7 7 5 = 1 2 2 6 6 6 6 6 6 4 1 1 1 1 1 1 1 1 1 3 7 7 7 7 7 7 5 2 6 6 6 6 6 6 4 p AB p AC p BC 3 7 7 7 7 7 7 5 (3.6.35) For the ultrawideband work performed here, approximately 20GHz of bandwidth is considered and thus a vector observation is required; one observation at each test fre- quency f k for k =f1; 2; ;Ng. Assuming a vector observation of k uniformly spaced frequencies for each antenna pair, then p! P2 R 3k ; g! G2 R 3k and (3.6.33) becomes G = A 1 P = E 1 E y P; (3.6.36) with transpose given by G y = (PA 1 ) y = (PE 1 E y ) y = E 1 E y P y : (3.6.37) 141 The computed gain samples for antennas A, B and C are in the rst, second and third columns of G y respectively. The solution G y applies to any set of three antennas that are permuted. Once again, performing spectral decomposition on A provides additional insight into the inversion process because it shows how the gains are projected onto the eigenvectors. 3.6.6 The Reliability of Computed Antenna Gains Norm kAk kA 1 k (A) kk 1 2 3 6 kk 2 p (A y A) = 2 1 2 kk 1 2 3 6 kk F p 6 2:45 3 p 6 7:35 Table 3.2: Computed Norms. To determine the reliability of the results given in (3.6.36) or equivalently eq. (3.6.37), the condition number(A) will provides insight. Table 3.2 contains the computed condi- tion numbers for the four norms previously dened. The computed norms demonstrates that the matrix A given in (3.5.22) and on page 126, is well conditioned. The maximum of all computed condition numbers is (A) < 7:4. It is then expected that the system Ax = b is not sensitive to perturbations and thus the computed results should be accu- rate; however, the characteristics of the data and the respective norms of the data vector b should also be considered because b represents a noisy measurement. 142 Given the spectral decomposition of A, it is very easy to see that the spectral radius equalsj max j for the Euclidean norm because kAk 2 2 =(A y A) =((EE y ) y EE y ) =((E y ) y y E y EE y ) =(EE y EE y ) =(EIE y ) =(E 2 E y ) = 2 max : Finally, kAk 2 =j max j; (3.6.38) which shows that the Euclidean norm is equal to the spectral radius when A = A y . The spectral radius for a given p{norm determines the maximum stretching of x after the linear transformation by A: This eect is readily observed from the spectral decomposition of A and then observing sup 8x:kxk=1 kAxk 2 : 143 Assume now that there is a set of four antennas A, B, C, D. One may believe that it is possible to take four measurements and thus construct a 4 4 matrix given by A = 2 6 6 6 6 6 6 6 6 6 6 4 1 1 0 0 0 1 1 0 0 0 1 1 1 0 0 1 3 7 7 7 7 7 7 7 7 7 7 5 2 6 6 6 6 6 6 6 6 6 6 4 G A G B G C G D 3 7 7 7 7 7 7 7 7 7 7 5 = 2 6 6 6 6 6 6 6 6 6 6 4 P AB P BC P CD P AD 3 7 7 7 7 7 7 7 7 7 7 5 (3.6.39) which seems reasonable; however, the matrix A given by (3.6.39) has rank r(A) = 3 which shows that it is not invertible. For this case, it is not possible to construct a full rank matrix A2R 44 , determine its inverse and solve for the \unique" gains of the four antennas. Instead, the number of equations required is given by the binomial 4 2 = 6 and thus A2R 64 . The system of equations has the form 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 1 1 0 0 1 0 1 0 1 0 0 1 0 1 1 0 0 1 0 1 0 0 1 1 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 2 6 6 6 6 6 6 6 6 6 6 4 G A G B G C G D 3 7 7 7 7 7 7 7 7 7 7 5 = 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 P AB P AC P AD P BC P BD P CD 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 ; (3.6.40) 144 which is immediately identied as a projection problem which can be solved (meaning estimated) by the method of least squares. The optimal solution ^ x ls in the least squares sense, is determined from A^ x ls =b (A y A) 1 A y A^ x ls =(A y A) 1 A y b ^ x ls =(A y A) 1 A y b: (3.6.41) While norms are dened for the rectangular matrix A, the condition number is not dened because A 1 is not dened for a rectangular matrix. Instead, the condition number of B = A y A in investigated and provided in table 3.3. Norm kBk kB 1 k (B) kk 1 4 4 16 kk 2 p (B) = 6 3 18 kk 1 4 4 16 kk F 6.11 6.11 6:11 2 37:33 Table 3.3: Computed norms for B = A y A because A 1 is not dened. The matrix given in (3.6.40) has rank r(A) = 4 so the solution x ls 2C(A), i.e., belongs to the column space of A: Since x ls 2C(A), a unique solution (possibly not 145 correct) can be determined by using nm the left inverse L of the mn rectangular matrix A. In this case, LAx = Lb Ix = Lb x = Lb: (3.6.42) 3.6.7 Computed Antenna Gains and Plots In this section, the S 21 (dB) measured data for several measured antennas are presented along with the computed gains. The gains were computed using the three antenna method via (3.5.22) on page 126. In some cases, four similar antennas (labeled A; B; C; D) were available and they were measured using the three antenna method. First, the set fA; B; C; Dg was partitioned into two sets: the setA =fA; B; Cg andB =fA; B; Dg: Two matrix solutions G A and G B were determined using (3.6.36) on page 141. Thus, G A = A 1 P B G B = A 1 P B ; which are of the same form as equation (3.6.36) on page 141. Obviously, the dierence matrix D =jG y A G y B j should contain the zero vector in columns 1 and 2 of D because two of the three antennas are the same (i.e., antennas A and B). The third column of D is ignored because it represents the gain dierence between two disimilar antennas. The antenna gain plots demonstrate errors either in the computation, experimental setup or model used to determine the gain. For example, errors are readily observed in plots 146 that contained two of the same antennas (e.g., see Figure 3.32(b) and Figure 3.33(b)). A comparison of the two gures shows slight variations in the computed results. In most cases, the errors occur at the location of nulls in the spectrum of 20 log(jS 21 j). The nulls represent zeros in the gain spectrum. The matrix inversion process attempts to invert the measured values that are close to zero (negative innity in dB) which of course is a large number in comparison to the other numbers in the inversion process. Also, noise is always present in the measurement process and thus adds to the estimation error. The signal to noise ratio is a well-known method of evaluating the quality of an estimate. The noise oor estimates, as a function of the discrete frequencies, are provided in section 3.5.4 on page 117. Dierences are observed in the computed gains of the small TEM Horn antenna in Figures 3.38(b) and 3.39(b). Similar discrepancies exist for other plots shown. Letp denote the normalized received power. WhenpG>>N o B N the estimates are of high quality; however, when pGN o B N , the estimates exhibit what appear to be errors. It is known that the condition number (A)< 10 and thus the mathematical inversion process can be considered accurate. When the measurements have a low signal- to-noise ratio the estimation error is large. Further, errors are also large in the regions of spectral nulls. 147 0 5 10 15 20 25 120 100 80 60 40 S21 Motorola Pair 1 to 2 S21 Motorola Pair 1 to 3 S21 Motorola Pair 2 to 3 S21 Motorola Pair 1 to 2 S21 Motorola Pair 1 to 3 S21 Motorola Pair 2 to 3 Frequency (GHz) S21 (dB) (a) 0 5 10 15 20 25 60 40 20 0 20 Motorola # 1 Motorola # 2 Motorola # 3 Motorola # 1 Motorola # 2 Motorola # 3 Frequency (GHz) Absolute Gain (dB) (b) Figure 3.32: (a) Measured S21 for Motorola antennas 1, 2 and 3. (b) Computed absolute gains. 0 5 10 15 20 25 120 100 80 60 40 S21 Motorola Pair 1 to 2 S21 Motorola Pair 1 to 4 S21 Motorola Pair 2 to 4 S21 Motorola Pair 1 to 2 S21 Motorola Pair 1 to 4 S21 Motorola Pair 2 to 4 Frequency (GHz) S21 (dB) __________________________________________________________________________________ __________________________________________________________________________________ (a) 0 5 10 15 20 25 60 40 20 0 20 Motorola # 1 Motorola # 2 Motorola # 4 Motorola # 1 Motorola # 2 Motorola # 4 Frequency (GHz) Absolute Gain (dB) __________________________________________________________________________________ __________________________________________________________________________________ (b) Figure 3.33: (a) Measured S21 for Motorola antennas 1, 2, 3 and 4. (b) Computed absolute gains for antennas 1, 2, 4. Note potential errors in antenna #4 (blue curve) at 11GHz and in antenna #2 (red curve) at 10GHz. 0 5 10 15 20 25 120 100 80 60 40 S21 Skycross Pair 10 to 12 S21 Skycross Pair 10 to 14 S21 Skycross Pair 12 to 14 S21 Skycross Pair 10 to 12 S21 Skycross Pair 10 to 14 S21 Skycross Pair 12 to 14 Frequency (GHz) S21 (dB) (a) 0 5 10 15 20 25 60 40 20 0 20 Skycross # 10 Skycross # 12 Skycross # 14 Skycross # 10 Skycross # 12 Skycross # 14 Frequency (GHz) Absolute Gain (dB) (b) Figure 3.34: (a) Measured S21 for Skycross antennas 10, 12 and 14. (b) Computed absolute gains. 148 0 5 10 15 20 25 120 100 80 60 40 S21 Skycross Pair 10 to 12 S21 Skycross Pair 10 to 15 S21 Skycross Pair 12 to 15 S21 Skycross Pair 10 to 12 S21 Skycross Pair 10 to 15 S21 Skycross Pair 12 to 15 Frequency (GHz) S21 (dB) (a) 0 5 10 15 20 25 60 40 20 0 20 Skycross # 10 Skycross # 12 Skycross # 15 Skycross # 10 Skycross # 12 Skycross # 15 Frequency (GHz) Absolute Gain (dB) __________________________________________________________________________________ __________________________________________________________________________________ (b) Figure 3.35: (a) Measured S21 for Skycross antennas 10, 12 and 15. (b) Computed absolute gains. 0 5 10 15 20 25 90 80 70 60 50 40 S21 Large TEM 1 to 2 S21 Large TEM 1 to 3 S21 Large TEM 2 to 3 S21 Large TEM 1 to 2 S21 Large TEM 1 to 3 S21 Large TEM 2 to 3 Frequency (GHz) S21 (dB) (a) 0 5 10 15 20 25 30 20 10 0 10 20 Large TEM 1 Large TEM 2 Large TEM 3 Large TEM 1 Large TEM 2 Large TEM 3 Frequency (GHz) Absolute Gain (dB) (b) Figure 3.36: (a) Measured S21 for Large TEM Horn antennas 1, 2 and 3. (b) Computed absolute gains. 0 5 10 15 20 25 90 80 70 60 50 40 S21 Large TEM 1 to 2 S21 Large TEM 1 to 3 S21 Large TEM 2 to 5 S21 Large TEM 1 to 2 S21 Large TEM 1 to 3 S21 Large TEM 2 to 5 Frequency (GHz) S21 (dB) (a) 0 5 10 15 20 25 30 20 10 0 10 20 Large TEM 1 Large TEM 2 Large TEM 5 Large TEM 1 Large TEM 2 Large TEM 5 Frequency (GHz) Absolute Gain (dB) __________________________________________________________________________________ __________________________________________________________________________________ (b) Figure 3.37: (a) Measured S21 for Large TEM Horn antennas 1, 2 and 5. (b) Computed absolute gains for antennas 1, 2 and 5. 149 0 5 10 15 20 25 90 80 70 60 50 40 S21 Large TEM 1 to Small TEM 1 S21 Large TEM 1 to Small TEM 2 S21 Small TEM 1 to Small TEM 2 S21 Large TEM 1 to Small TEM 1 S21 Large TEM 1 to Small TEM 2 S21 Small TEM 1 to Small TEM 2 Frequency (GHz) S21 (dB) __________________________________________________________________________________ __________________________________________________________________________________ (a) Frequency (GHz) 0 5 10 15 20 25 30 20 10 0 10 20 Large TEM 1 Small TEM 1 Small TEM 2 Large TEM 1 Small TEM 1 Small TEM 2 Frequency (GHz) Absolute Gain (dB) __________________________________________________________________________________ __________________________________________________________________________________ (b) Figure 3.38: (a) Measured S21 for Large TEM Horn 1 and small TEM Horn antennas 1 and 2. (b) Computed absolute gains for Large TEM Horn 1 and small TEM Horn antennas 1 and 2. 0 5 10 15 20 25 120 100 80 60 40 S21 SmallTEM # 2 to Diamond # 1 S21 SmallTEM # 2 to Diamond # 2 S21 Diamond Pair 1 to 2 S21 SmallTEM # 2 to Diamond # 1 S21 SmallTEM # 2 to Diamond # 2 S21 Diamond Pair 1 to 2 Frequency (GHz) S21 (dB) (a) 0 5 10 15 20 25 60 40 20 0 20 Small TEM # 2 Diamond # 1 Diamond # 2 Pair of Diamonds with 2-Ant Method Small TEM # 2 Diamond # 1 Diamond # 2 Pair of Diamonds with 2-Ant Method Frequency (GHz) Absolute Gain (dB) (b) Figure 3.39: (a) Measured S21 for Small TEM Horn 2 and Diamond Dipole antennas 1 and 2. (b) Computed absolute gains for the Small TEM Horn 2 and Diamond Dipole antennas 1 and 2. Gains for the Diamond Dipoles computed for two and three antenna method. Note the signicant errors. 150 Chapter 4 Experimental Results of the Waveform Search 4.1 Large TEM Horn Results using Scattering Parameter for Sequence Lengths from 1 to 7 The measured S 21 of the antennas was used in the waveform search to determine the most ecient waveform based on the waveform's eciency measure : The experimental results in this section were initial attempts to determine the complexity and potential performance which could be achieved via a search method. These results were obtained in the early part of 2003. Approximately 24 Intel computers were used to perform the computations using MathCad version 11. The computers were not required for sequences less than length four; however, they were required for sequences with a longer length. The search was performed for various sets of parameters. The results of the search are provided in Table 4.3 on page 160 and Figures 4.12{4.14 on page 162. Figure 4.12 on page 161 shows the best waveforms observed out of all the waveforms tested. Figure 4.13 indicates that the best waveforms p opt (t)2P are rare events and was typical for all three antennas investigated. Figure 4.14 on 162 is a visualization of the data contained in Table 4.3. 151 The table summarizes the best search results for the large TEM Horn antenna over the search-space considered. The surface plots were unique for each antenna type used in the search. This section only shows the results for large TEM horn. The data in the plots was generated using the same waveform parameters. The only dierence in the experiments was the transmit and receive antennas used. The S 21 parameter was always measured using pairs similar antennas. Three dierent antenna types were used which was; a large TEM horn, a small TEM horn and a diamond dipole antenna. As previously stated, the same search parameters were used for all three antennas. The three antennas produced dierent optimal waveforms for the same input parameters e.g., vector lengths and clock reproduction rates and mask constraints. In all cases, low levels of eciencies were achieved for the parameters chosen. The eort illustrated the complexity of the design space, and the seemingly limitless variety of hardware representations of H eq (f) that can aect the optimal pulse waveform. The eciencies of the best waveforms observed in the experiment were low and indicate that signicant improvements could be achieved by: Decreasing the reproduction time T s . Increasing the number of quantization levelsjQj. Increasing the time-support of the pulse waveform p(t). Inserting high-pass lters in the signal path to constrain the radiated emissions for f < 3:1 GHz. Jointly optimize the spectral response of the radiating antenna to the mask con- straint and the waveform. 152 When the number of signal coecients are increased, the search-space increases asjQj N (see 4.1. Thus, the search-space is dominated by the number of levels in the quantizer jQj and the sequence length N. They both make the problem intractable for small to moderatejQj and N: Other methods should be investigated to optimize the eciency measure when both largejQj and N are required. Figure 4.1: Search complexity of a 4 level quantizer as a function of increasing sequence length. 153 This section provides results of the search for a limited data set. The results provides details that will assist the reader in understanding how Table 4.3 was determined. A few representative results of the search are shown in Figures 4.2{4.11. The reproduction rate, the vector length N and the voltage levels are provided in the caption. When N = 2, jQj = 4 andjQj 2 = 16 which is the maximum number of non-unique waveforms that can be produced. The waveform coecients are shown in table 4.1 on page 154 with the associated eciency measure. Only the best three eciencies from Table 4.1 are contained in Table 4.3 which is used then used to produce the 3D plot in Figure 4.14 on page 162. Exhaustive Vector Set Vector Index 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 -3 -1 +1 +3 -3 -1 +1 +3 -3 -1 +1 +3 -3 -1 +1 +3 -3 -3 -3 -3 -1 -1 -1 -1 +1 +1 +1 +1 +3 +3 +3 +3 (%) 0.017 0.019 0.045 0.155 0.019 0.017 0.155 0.045 0.045 0.155 0.017 0.019 0.155 0.045 0.019 0.017 Table 4.1: Parameters: Clock rate = 20GHz, N = 2 andQ =f3;1; +1; +3g. As previously discussed, Table 4.1 can be reduced to a non-unique equivalent vector set given by Equivalent Vector Class Vector Index 0 1 2 3 -3 -1 +1 +3 -3 -3 -3 +3 (%) 0.017 0.019 0.045 0.155 Table 4.2: Parameters for the reduced example: Clock rate = 20GHz, N = 2 andQ = f3;1; +1; +3g. [!htb] Figure 4.8 clearly demonstrates that the most ecient vector does not neces- 154 0 2 4 6 8 10 12 14 16 0 0.05 0.1 0.15 0.2 Vector Column Index Mask Filling Efficiency ( %) Figure 4.2: The eciency measures of a length 2 sequence at a reproduction rate of 20GHz for the large TEM horn antenna. Q =f3;1; +1; +3g, best = 1:548 10 3 %, 2nd best = 4:542 10 4 %, 3rd best = 1:897 10 4 %. 1 0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 0 0.05 0.1 0.15 0.2 Vector Column Index Mask Filling Efficiency ( % ) Figure 4.3: The reduced set of eciency measures of a length 2 sequence at a reproduction rate of 20GHz for the large TEM horn antenna. Q =f3;1; +1; +3g, best = 1:548 10 3 %, 2nd best = 4:542 10 4 %, 3rd best = 1:897 10 4 %. 155 0 2 4 6 8 10 12 14 16 0 0.05 0.1 0.15 0.2 0.25 Vector Column Index Mask Filling Efficiency ( %) Figure 4.4: The eciency measures of a length 2 sequence at a reproduction rate of 25.5GHz for the large TEM horn antenna. Q = f3;1; +1; +3g, best = 2:148 10 3 %, 2nd best = 4:66 10 4 %, 3rd best = 2:185 10 4 %. 1 0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 0 0.05 0.1 0.15 0.2 0.25 UWB Pulse (Q=4, range(-3,-1,+1 +3)) Vector Column Index Mask Filling Efficiency ( % ) Figure 4.5: The reduced eciency measures of a length 2 sequence at a reproduction rate of 25.5GHz for the large TEM horn antenna. Q =f3;1; +1; +3g, best = 2:148 10 3 %, 2nd best = 4:66 10 4 %, 3rd best = 2:185 10 4 %. 156 0 10 20 30 40 50 60 0 0.05 0.1 0.15 0.2 0.25 Vector Column Index Mask Filling Efficiency ( %) Figure 4.6: The eciency measures of a length 3 sequence at a reproduction rate of 13.6GHz for the large TEM horn antenna. Q =f3;1; +1; +3g, best = 2:031 10 3 %, 2nd best = 1:683 10 3 %, 3rd best = 4:597 10 4 %. 0 5 . 10 4 0.001 0.0015 0.002 1 10 100 Efficiency bins (10 bins) Count per bin Figure 4.7: The histogram of the eciency measures of a length 3 sequence at a repro- duction rate of 13.6GHz for the large TEM horn antenna. Q =f3;1; +1; +3g, best = 2:031 10 3 %, 2nd best = 1:683 10 3 %, 3rd best = 4:597 10 4 %. 157 0 2 4 6 8 10 12 14 16 0 0.05 0.1 0.15 0.2 0.25 Vector Column Index Mask Filling Efficiency ( % ) Figure 4.8: The eciency measures of a length 3 sequence at a reproduction rate of 13.6GHz for the large TEM horn antenna. Q =f3;1; +1; +3g, best = 2:031 10 3 %, 2nd best = 1:683 10 3 %, 3rd best = 4:597 10 4 %: sarily occur at the center column vector as it did in Figures 4.3 and 4.5. Therefore, the search space includes the condition 0 < n jQj L 2 wherejQj represents the number of quantizer levels and L denotes the vector length. 0 100 200 300 400 500 600 700 800 900 1000 0 0.5 1 1.5 2 2.5 UWB Pulse (Q=4, range(-3,-1,+1 +3)) Vector Column Index Mask Filling Efficiency ( %) Figure 4.9: The histogram of the eciency measures of a length 5 sequence at a repro- duction rate of 35GHz for the large TEM horn antenna. Q =f3;1; +1; +3g, best = 0:021%, 2nd best = 0:02%, 3rd best = 0:015%. 158 0 50 100 150 200 250 0 0.5 1 1.5 2 2.5 UWB Pulse (Q=4, range(-3,-1,+1 +3)) Vector Column Index Mask Filling Efficiency ( % ) Figure 4.10: The eciency measures of a length 5 sequence at a reproduction rate of 35 GHz for the large TEM horn antenna. Q =f3;1; +1; +3g; best = 0:021%; 2nd best = 0:02%; 3rd best = 0:015%: 0 0.005 0.01 0.015 0.02 0.025 1 10 100 1 . 10 3 Efficiency bins (10 bins) Count per bin Figure 4.11: The histogram of the eciency measures of a length 5 sequence at a repro- duction rate of 35GHz for the large TEM horn antenna. Q =f3;1; +1; +3g, best = 0:021%, 2nd best = 0:02%, 3rd best = 0:015%. 4.2 Large TEM Horn Results using Scattering Parameters for Sequence Lengths from 1 to 11 The results that are contained in this section were generated by software written by the author in C++. In order to provide the best understanding of the code, a description 159 Vector Length Clock Rate (GHz) 1 2 3 4 5 6 7 1.0 Best 7.36E-03 2.50E-02 3.64E-02 3.21E-02 3.38E-02 3.53E-02 4.10E-02 Second best 0.0 1.22E-02 2.43E-02 3.03E-02 3.20E-02 3.29E-02 3.30E-02 Third best 0.0 6.03E-03 1.70E-02 2.85E-02 3.17E-02 3.19E-02 3.30E-02 5.0 Best 8.54E-03 8.27E-03 5.10E-02 4.30E-02 7.70E-02 8.20E-02 1.27E-01 Second best 0.0 8.15E-03 1.80E-02 3.80E-02 7.30E-02 8.10E-02 0.1076 Third best 0.0 6.03E-03 1.70E-02 2.85E-02 3.17E-02 3.19E-02 3.30E-02 6.85 Best 1.20E-02 1.80E-02 1.40E-01 1.63E-01 1.90E-01 1.81E-01 1.90E-01 Second best 0.0 1.50E-02 8.10E-02 1.50E-01 1.50E-01 1.79E-01 0.18 Third best 0.0 8.63E-03 3.70E-02 1.35E-01 1.49E-01 1.69E-01 0.17 10.0 Best 1.70E-02 4.40E-02 4.61E-01 4.70E-01 7.88E-01 1.31 3.2 Second best 0.0 2.80E-02 1.10E-01 3.53E-01 4.45E-01 5.05E-01 2.2 Third best 0.0 1.10E-02 6.00E-02 3.28E-01 3.32E-01 4.54E-01 2.2 13.6 Best 2.10E-02 8.20E-02 2.03E-01 1.45 8.54E-01 1.27 2.46 Second best 0.0 3.80E-02 1.68E-01 5.98E-01 8.31E-01 1.09 1.69 Third best 0.0 1.40E-02 9.60E-02 3.59E-01 4.72E-01 9.01E-01 1.33 15.0 Best 2.20E-02 9.70E-02 2.24E-01 2.13E+00 8.21E-01 1.14 1.92 Second best 0.0 4.00E-02 1.21E-01 8.51E-01 7.03E-01 1.10 1.32 Third best 0.0 1.50E-02 1.05E-01 4.86E-01 6.50E-01 1.04 1.27 20.0 Best 2.50E-02 1.55E-01 2.46E-01 1.11 1.92 2.03 2.15 Second best 0.0 4.50E-02 1.23E-01 8.74E-01 9.03E-01 1.31 2.09 Third best 0.0 1.90E-02 8.50E-02 4.49E-01 6.96E-01 1.18 2.04 25.0 Best 2.70E-02 2.10E-01 2.21E-01 9.08E-01 2.14 4.69 3.18 Second best 0.0 4.70E-02 1.22E-01 5.89E-01 1.43 2.31 2.37 Third best 0.0 2.20E-02 1.10E-01 4.30E-01 1.19 1.76 2.20 25.5 Best 2.70E-02 2.15E-01 2.17E-01 9.67E-01 1.89 4.42 3.18 Second best 0.0 4.70E-02 1.22E-01 5.76E-01 1.54 2.49 2.24 Third best 0.0 2.20E-02 1.12E-01 4.46E-01 1.25 1.47 2.23 50.0 Best 3.10E-02 3.72E-01 1.71E-01 7.40E-01 6.88E-01 1.45 3.11 Second best 0.0 4.10E-02 1.45E-01 6.57E-01 6.88E-01 1.31 2.71 Third best 0.0 2.80E-02 1.22E-01 5.90E-01 6.43E-01 1.24 2.64 Table 4.3: The three best observed PSD eciencies for sequence lengths up to seven at the indicated clock reproduction rates. of the code is provided in pseudocode. The algorithm used to generate and test the sequences in this section is provided below. The vector sequence can be envisaged as elements an MP matrix where P =jqj N andjqj denotes the cardinality of the set of voltage levels. The matrix can be constructed by a systematic process in that each signal vector can be identied by its column number p: Signal Generator tracks all generated sequences and their relatives via a boolean searched key array Initialize the searched bool 160 Figure 4.12: Best result for the large TEM horn for m = 1; 2; ; 7 Figure 4.13: TEM Horn eciency histogram for m=6 array to all false and counter = 0 When waveform sequences are generated, only one sequence is output. All related sequences (time reversed, scaled etc.) are marked true in the master search key index. The master search key index keeps track of all sequences that have been searched. If a sequence has an indicating mark of true, the sequence is skipped until a sequence that has not been searched is observed. The untested sequence 161 Figure 4.14: TEM Horn eciency search summary (n = 1; 2:::; 7) is then tested and the related sequences are all marked as true. The process repeats until all sequences have been tested. 162 Algorithm 1 Sequence Generator Function 1: Inputs: Column Index col, Sequence Length N, Number of quantization levels q. 2: Output: A vector of integers of length N. 3: function Sequence Generator(col;N;q) 4: step = ( 2 for an even symmetric quantizer (q is even); 1 for an odd quantizer (q is odd). (4.2.1) 5: bias = (q1) step 2 . Used to center the quantizer about zero voltage 6: s 0 . Assign all zeros to s. 7: for row 0 to N 1 do . Index over the rows of the vector 8: temp h col q row i . Compute the integer part of the fraction 9: s row (temp mod q) step bias 10: end for 11: return s 12: end function 163 Algorithm 2 Determine related sequences 1: Inputs: Column Vector w2Z N . 2: Output: The subsetR of integers with elements r 0 ;r 1 : The elements ofR corre- sponds to the column indices of related vectors in an ordered matrix. . R is an NN anti-diagonal matrix with ones going from the bottom left to top right on the diagonal and zeros elsewhere; that is, r i;j = 08 r i;j such that i +j6= N + 1 otherwise, r i;j = 1. 3: function Related(w) 4: R = . AssignR the empty set. 5: g jG(w)j . Let G(w) denote the GCD of the elements of w. 6: if g6= 1 then 7: x 0 g 1 w . If thejG(w)j = 1, then x 0 = w: 8: R R S Related(x 0 ) 9: else 10: x 0 w 11: end if 12: i 0 13: while x 0 x i = 0 and iN 1 do 14: i i + 1 . Counts the number of identical elements of x 0 . 15: end while 16: if i6=N then . Elements are not identical 17: x 1 Rw . Index reverse w 18: x 2 w . Negate w 19: x 3 Rw . Negate and index reverse w 20: R R S 3 i=0 Index(x i ). Convert the vectors to column indices and add them to the setR. 21: else . i =N and the elements are identical 22: R Index(w) S Index(w) S R 23: end if 24: returnR . Returns the set of unique related vectors of the ordered matrix 25: end function 164 Algorithm 3 Convert a vector x2Z N of integers to an integer column index 1: Inputs: Column Vector x2Z N of an ordered matrix X: 2: Known: The number of levels of the quantizer q2Z: The distance between adjacent levels. The quantizer centering bias bias: 3: Output: The column index n = argfx n g. 4: function Index(x) 5: a;b 0 6: for k 0 to N 1 do 7: a (x k +bias)q k 8: b a +b 9: end for 10: n = b 11: return n2N. 12: end function Algorithm 4 Next Sequence Generator 1: Inputs: Sequence length N and q; the total number of quantizer voltage levels. 2: Output: A unique vector s of integers of length N each time the function is called. 3: procedure Vector Sequence 4: Initialize the static vector key dim(key) =q N and initialize elements to False if this is the rst time the algorithm is called. For N > 1; q N is the total number of vectors in the signal set. 5: Initialize static (persistent) variable col 0;done =False at rst run. 6: while key col =True and 0<col<q N do 7: col col + 1 . Increase the key until a sequence has not been generated. 8: if col =q N then 9: done =True . Stopping condition. All sequences has been generated. 10: end if 11: end while 12: if done6=True then 13: Sequence Generator(col;N;q) . Generate the sequence in location col. 14: R Related(col) . Determine the set of all related sequences. 15: for k = 1 tojRj do 16: key r2R =True . For all related sequences, set their key location to True to avoid generation again. 17: end for 18: return s . The sequence that will be tested by the system. 19: else 20: Stop. . Condition to stop the search. 21: end if 22: end procedure 165 Algorithm 5 Expanded sequence generator to accommodate non-linear quantizers 1: Inputs: Quantizer type (linear, Quadratic, Cubic), Quantization Levels, Sequence length. 2: Output: All unique vector sequences s k with integer coecients. 3: procedure Sequence Generator(f, Number of levels, Sequence Length) . Outputs all \unique" integer vector sequences of length N 4: q Number of Quantization Levels 5: N Length of generated vectors 6: SetSize jqj N . The number of vectors in the set for N 1 7: function Generate Vector(column index) 8: for c 0;SetSize do . Indexes over the columns of the signal matrix 9: Generate s c according to equation (2.2.9) . Generate a vector sequence with row index i and column index c. 10: 11: end for 12: end function 13: r a modb 14: while r6= 0 do . We have the number if r = 0 15: b c 16: b r 17: r a modb 18: end while 19: return s k . The kth sequence of length L 20: end procedure Algorithm 6 Euclid's algorithm 1: procedure Euclid(a;b) . The GCD of a and b 2: r a modb 3: while r6= 0 do . We have the number if r = 0 4: b c 5: b r 6: r a modb 7: end while 8: return b . The GCD is b 9: end procedure 166 Chapter 5 Extended S21 Results 5.1 Introduction In [36], Lewis and Scholtz dened an FCC [19] Ultrawideband (UWB) mask lling e- ciency metric , a c R Fmax F min jP (f)j 2 jH eq (f)j 2 df R Fmax F min M(f)df (5.1.1) to determine how well a radiated UWB waveform matched the FCC mask constraint M(f): The metric was required, to assess, out of all the possible signal designs which are possible, which signal had the most power while still being FCC compliant (see unpublished results from [36] in Fig.5.1). The eciency measure is a true eciency since a c jP (f)j 2 jH eq (f)j 2 .M(f)8f2 [F min ;F max ] (5.1.2) anda c is a constant scale factor to ensure that (5.1.2) is satised for everyf2 [F min ;F max ] and thus 0< 1. The factors P (f) =Ffp(t)g andH eq (f) =Ffh eq (t)g are the Fourier transforms of the nite-time, information bearing, pulse shape at the generator and the 167 system's equivalent lter function respectively. The equivalent lter takes into account all the ltering operations that connect the signal generator to the waveform propagating in space. The eciency measure can be used to compare waveforms generated by methods which are dierent than the search we proposed in [36], assuming that the evaluation interval andH eq (f) is the same. Of course, diering equivalent ltersH eq (f) will produce dierent waveform eciency results for the same waveform setfp(t) : p(t)2P PG g as demonstrated here. Typically, the time support (suppe(t)) of the radiated pulse e(t) =F 1 f a c P (f)H eq (f)g (5.1.3) is greater than the pulse synthesized at the signal generator due to impedance mismatch of both the generator and antenna to the connecting transmission line. The measure of mismatch is determined by the re ection coecients gen (f) = Z gen (f)Z o Z gen (f) +Z o and ant (f) = Z ant (f)Z o Z ant (f) +Z o : (5.1.4) The termsZ gen (f) andZ ant (f) are the Th evenin equivalent impedances of the generator and antenna, Z o is the constant characteristic line impedance. When a perfect match exists, (f) = 0 and no re ected wave exists on the line. When the source and load are not matched to the transmission line, a frequency dependent re ected wave will exist. The multiply re ected wave will have the eect of lengthening the generated UWB pulse depending on the pulse bandwidth and transmission line length when gen (f)6= ant (f)6= Z o . The multiply re ected wave is attenuated because we assume that the system has 168 1 0 1 2 3 4 5 6 7 4 3 2 1 0 1 2 3 4 p(t,963) p(t,963) Generated Pulse from D/A w/o Filtering time (Clock Normalized) Voltage (V) (a) 0 5 . 10 9 1 . 10 10 1.5 . 10 10 2 . 10 10 2.5 . 10 10 3 . 10 10 3.5 . 10 10 4 . 10 10 4.5 . 10 10 5 . 10 10 30 18 6 6 18 30 PulseGain(f,963) PulseGain(f,963) Complex Expontial Gain (Pulse Gain) Frequency (Hz) Intrinsic Gain (dB) (b) 0 5 . 10 9 1 . 10 10 1.5 . 10 10 2 . 10 10 2.5 . 10 10 3 . 10 10 3.5 . 10 10 4 . 10 10 4.5 . 10 10 5 . 10 10 100 75 50 25 0 25 D/A Gaussian Filter Basis Pulse Filtered Basis Pulse Antenna Input Waveform D/A Gaussian Filter Basis Pulse Filtered Basis Pulse Antenna Input Waveform Composite Waveform Presented to the Transmit Antenna Terminal Frequency (Hz) Normalized Power (dB) (c) 0 4 . 10 9 8 . 10 9 1.2 . 10 10 1.6 . 10 10 2 . 10 10 100 80 60 40 FCC Outdoor Mask LG TEM Horn S_21 Antenna Input Best Observed Waveform FCC Outdoor Mask LG TEM Horn S_21 Antenna Input Best Observed Waveform CLK=25GHz (Dim = 6, Q={-3, -1, 1, 3} Frequency (Hz) dBm (d) Figure 5.1: (a) Generated time domain waveform at a clock reproduction rate of 25 GHz and column index n = 963. , (b) Pulse intrinsic gain, (c) D/A output waveform fed to the antenna, (d) Large TEM horn response with outdoor mask. max 4:2% 169 real ohmic resistance, which has the eect of damping the multiple re ections between the source and the load. Mismatch can be quite severe in an UWB system since the transmission line match must be maintained over the frequency band of interest, which is several GHz. At the frequencies of mismatch, a ltered re ected pulse will exist on the line and the residual transmitted [16] [44] with a fraction also lost to heat. Due to multiple re ections, antenna discontinuities and antenna bandwidth, the time support relationship suppp(t) suppe(t) (5.1.5) is justied. 5.2 Eective Isotropic Radiated Power Spectral Density Discussion The FCC constraint on the eective isotropic radiated power (EIRP) as measured in a resolution bandwidth of 1 MHz includes the UWB transmit antenna (denition of EIRP). Antennas are known to have frequency dependent gain G Tx (f) and thus can increase the emitted EIRP spectral density (EIRPSD) in a frequency-selective manner. It is curious that many UWB waveform design methods do not include one of the most important elements - the antenna, which is part of the EIRPSD denition. Apparently, the importance of the antenna is not yet \well known" in this area of research even though outstanding papers have discussed the issue [10,14]. 170 Historically, narrowband EIRP, evaluated for a xed frequency f 0 , is dened as S EIRP (;),P Tx G Tx (;) (W); (5.2.1) whereP Tx is the power accepted and radiated by a transmit antenna andG Tx (;) is the gain of the antenna. The antenna gain is a function of azimuth and elevation angle in space. The narrowband EIRP denition assumes a matched condition between both the source and the antenna to the transmission line. In the context of UWB systems, due to the large bandwidths involved, the frequency dependent re ection coecients (f) for the generator and antenna must be used in the EIRPSD denition [53] S EIRP (f), (1j gen (f)j 2 )(1j ant (f)j 2 )P Tx (f)G Tx (f; max ; max ) =(1j gen (f)j 2 )(1j ant (f)j 2 )P Tx (f)G max (f) (W/Hz): (5.2.2) The left hand side of (5.2.2) is not function of the spatial coordinates and because mask compliance is measured in the spatial direction where the gain is maximum over all frequencies and spatial coordinates, hence the subscript max. The direction of maximum gain in space is non-trivial to determine because all antennas are spatial lters [7] which implies that the EIRPSD will change as an observer takes measurements in dierent xed locations about the transmitter. For the results presented here, we assumed that the maximum antenna gain is in the \perceived" bore-sight direction. It is dicult to 171 design UWB antennas that exhibit constant gain across all the frequencies of interest and thus we explicitly note the frequency dependence in the gain term. The output EIRPSD S EIRP (f) of a linear time invariant lter H eq (f), is related to the input random process m(u;t) by S EIRP (f) =S m (f)jH eq (f)j 2 ; (5.2.3) where S m (f) is the power spectral density (PSD) of the input random process m(u;t), where u2U denotes the sample space, t2R denotes time. 5.2.1 Assumptions Let a synchronous information bearing signal be given by m(u;t) = 1 X n=1 a n p(tnT p ) (5.2.4) where each pulse in the vicinity of the generator has time support T p = NT s . The rst and second order statistics of the information sequence are given by Efa n g = 0 and R a (mn) = Efa m a n g which imply that the real modulating sequencefa n g 1 1 is widesense stationary (WSS) 1 . It has been shown [56] that the PSD of the modulation process can be written as S m (f) =S a (f)S p (f); (5.2.5) 1 A WSS constraint is not required to satisfyM(f). It is considered here because it covers a wide range of communication applications and the solutions are well known. 172 where S a (f) and S p (f) are the PSDs of the modulation sequencefa n g 1 n=1 and of the pulse shapep(t) (see [36]) respectively. The PSD ofp(t) is the time average of the Fourier transform given by S p (f) = 1 T p jFfp(t)gj 2 = 1 T p jP (f)j 2 : (5.2.6) When factorizations of S m (f) exist as in (5.2.5), attempts can be made to whiten the information sequence a n (see [41]) so that S m (f) can be written as S m (f) =k a S p (f) (5.2.7) withS a (f) =k a (some constant) for all frequencies in the band under consideration. The assumption that the modulating sequencefa n g is spectrally white signicantly simplies the waveform optimization since onlyjP (f)H eq (f)j 2 needs to be investigated and certainly jH eq (f)j 2 6= 1 8f2 [f min ;f max ]. The EIRPSD given in (5.2.2) is determined solely from the pulse shapep(t) andH eq (f). In certain modulation processes, the PSDS m (f) consists of a continuous spectrum plus an additive discrete spectrum which may be undesirable; however, lines in the spectrum are useful for synchronization purposes at the receiver. Modulations for which spectral factorizations of the type given in (5.2.5) apply are provided in [56], [61]. It has been known since at least 1959, from the work of Titsworth and Welch [61], that modulation processes called Negative Equally Probable (NEP) pro- cesses have a spectral density which is merely the weighted sum of the energy spectra of the signals over one basic time unit; there are no spectral spikes present. The results presented here apply to modulations where spectral factorization is possible. 173 5.2.2 Pulse Construction Method We use the same real, orthonormal, basis pulse construction method used in [36], given by p n (t) = N1 X k=0 s k ' k (t) = N1 X k=0 s k '(tkT s ); (5.2.8) wheref' k (t)g N1 k=0 t2 R is the real orthonormal basis, the coecients s k 2Q are a nite set of integers that represent the output of a nite level, uniform even-symmetric quantizer. The total number of voltage levels of the quantizer is equal to the cardinality of the set of voltage levels given byjQj =M. Using a small number of amplitudes decreases the generator complexity, design and cost. For the work presented here, as in [36], M = 4 and the setQ =f3;1; 1; 3g. Other sets of integers are considered in future work. The central problem previously introduced was: Given a nite set of voltage levels jQj =M that can be output from a digital to analog converter (DAC), what is the best nite time sequence of time duration 0 < (NT s ) = T p , where N are the number of integer output values,T s is the inverse of the output reproduction rate (clock) of the DAC, such that the UWB antenna produces a eld strength e opt (t) that has maximal power while still FCC-compliant. Note that the sequences are constructed at the generator but mask compliance is measured post transmit antenna. The eciency of a waveform for a particular antenna is a function of the reproduction clock rateT s and the number of componentsN used in the signal coecient vector s2Z. We are interested in achieving the highest eciency possible for a particular vector length and reproduction frequency. The waveform optimization is a maximization of . It is possible that there are two or more waveforms that share the same eciency. However, for 174 the searches performed here, it was never the case that there were two or more maximally ecient waveforms contained inP PG . A contributing factor for this was that eorts were made to exclude searching related waveforms in the search space [36]. Due to the search space reduction, all sequences performed worse than the maximally ecient waveform for the given set (T s ;N). The coecient vector s that produces the highest eciency for a given (T s ;N) is the upper bound sequence for all s2P PG given H eq (f). 5.3 Processing The search was performed entirely in the frequency domain. The DAC lter output represents the input to the transmit antenna when the transmission line is well engineered and distortionless and thusjH TxLine (f)j 2 = 18f2 [F min ;F max ]: A distortionless but lossy transmission line can be achieved if the line parameters satisfy R L = G C ; (5.3.1) a relationship given by Pozar [44] and attributed to Oliver Heaviside. The parameters R;G;L;C are the line resistance, conductance, inductance and capacitance, all per unit length of the transmission line. We assume a well engineered transmission line which is not unrealistic. The DAC output is P D/A (f) =P (f)H D/A (f) N1 X k=0 s k e j2fTs | {z } A(f),pulse intrinsic gain factor : (5.3.2) 175 We considerjA(f)j 2 the pulse intrinsic gain since the term is performing an equalization function to produce the best radiated waveform matched to the constraint M(f). The pulse intrinsic gain compensates for non-ideal antennas, the DAC lter, and transmission line ltering eect { none ideally matched to M(f). For the simulations performed here, the frequency dependent functions in (5.3.2) were uniformly sampled where the specic sampling interval was determined from actual antenna measurements in frequency domain. Considering the entire reduced coecient matrix ~ S where each coecient vector s is a column in ~ S, the sampled form is represented as P D/A = PH D/A E ~ S: (5.3.3) The matrices P and H D/A are diagonal matrices with 1601 rows and columns, E is a 1601N complex rectangular matrix and ~ S =P PG is the NM reduced coecient matrix with MjQj N =4. s2P PG Delta Function Generator '(f) H D/A (f) Ant 1 M id (f) M od (f) Ant 2 M id (f) M od (f) Ant 3 M id (f) M od (f) max Ts;N f id (T s ;N)g max Ts;N f od (T s ;N)g max Ts;N f id (T s ;N)g max Ts;N f od (T s ;N)g max Ts;N f id (T s ;N)g max Ts;N f od (T s ;N)g Figure 5.2: Data Processing Flow Diagram. 176 In order to save computational complexity (i.e., E ~ S), it was prudent to test multiple masks (indoor and outdoor) as well as multiple (3) dierent antennas (see Fig.5.2). Once a candidate waveform was generated, at considerable computational cost, the ltered waveform was input into three dierent transmit antennas. The antenna outputs were then input and each signal scaled to the indoor and outdoor mask creating six outputs. From each output, opt = max f max g was saved and plotted in section 5.4 for the given (T s ;N) pair. For each waveform generated, the mask and antenna pair deter- mined the waveform with the highest eciency and not merely the comparison of the candidate waveform at the generator to the mask constraint. In [36], we used the measured S 21 response as the antenna transfer function because the true antenna transfer functions (related to S 21 ) were not known at the time and S 21 provided an estimate of the spectral shaping performed by a pair of antennas. The EIRPSD clearly is a function of a single transmit antenna and not any pair. However, any measured EIRPSD will be a result of a pair of antennas and thus the transfer function for the measuring probe antenna must be known. We have investigated this problem and the results will be published in a dissertation. Here, we continued to use H meas (f) =S 21 . The results indicated that the maximum achievable eciency are antenna dependent; as shown in section 5.4. Results that use the true EIRPSD given in (5.2.2) will be published in the future. 177 5.4 Results and Conclusions Once the parameters are set (i.e., H eq ;Q are known), the eciency of a particular waveform depends only on the clock reproduction rate T s and N = dim s. Forfp n (t) : p n (t)2P t g which could be produced for the xed pair (T s ;N), there exist an optimal waveform p opt (t;T s ;N)2P t in the 2-dimensional space with opt = maxf max g. When (T s ;N) was changed, a new max was extracted. When N was xed, the matrix of coecient vectors was xed and thus the setfp n (t)g2P t of possible waveforms was also xed for all possible values of T s except for time scaling. The parameter T s sets the null to null bandwidth of the basic pulse shape and scales (contracts or expands in frequency) the waveform's pulse intrinsic gain. It is well known from Fourier transform theory that p(at) F ! 1 jaj P f a : Unfortunately, whileP was known and all of the intrinsic gains were also known after the rst computation, the waveforms could not be saved due to the large sizes of each set when N was large. The implication was that for each new value of T s ; we did not know how eciently all of the new waveforms would t under the mask since the original PSDs had been frequency scaled by the parameter a. As such, each waveform was regenerated and tested again for each new reproduction rate (i.e.,the classic block test). We tested three antennas and two masksM in (f);M out (f) corresponding to the indoor and outdoor masks respectively. For each antenna and mask pair, we constructed two plots; a 3-D eciency plot and an accompanying contour plot. Each point in the plot represents the maximum eciency observed for the search for a particular vector length 178 and clock rate. Obviously, vector length is an integer and no data points exists between the integers. The points between the samples was interpolated to produce the continuous plots for visualization of the ctitious eciency surface. Along the clock rate axis, we chose approximately uniformly spaced clock rates, such as 1GHz, 2GHz, etc. If one were attempting to make an educated guess of where to search next based on past search observations, one would chose coordinates (T s ;N) near the fastest increasing- eciency slope. This method was used for length 11 and 12 sequences since we did not have the computational resources to exhaustively search various clock rates. The data for those lengths are sparse in the provided plots. From the 3-D and contour plots, we observed the following: Dierent antenna and mask pairs change the best observed waveform. Antennas that are not well designed to be matched to the mask will require longer vector lengths to provide increased degrees of freedom of the pulse intrinsic gain for a \good" mask t. Waveforms tted to the outdoor mask always performed worse than the indoor mask because the outdoor mask is an tighter constraint. The optimal generated pulse is clearly antenna dependent. Antenna and sequence data may be procured by request from the author or via the UltRa Lab's website at http://ultra.usc.edu/. 179 (a) Efficiency for Small TEM Horn for Indoor Mask Large_TEM_ID_Data.dat (b) 1 0 1 2 3 4 5 6 7 8 9 10 11 12 4 3 2 1 0 1 2 3 4 p(t, 619,234) p(t, 619,234) Time (Clock Normalized) Voltage (V) t sup 0.314 ns = (c) 0 5 . 10 9 1 . 10 10 1.5 . 10 10 2 . 10 10 2.5 . 10 10 100 80 60 40 FCC Indoor Mask Small TEM Horn S21 Antenna Input Best Observed Waveform FCC Indoor Mask Small TEM Horn S21 Antenna Input Best Observed Waveform CLK=35GHz, Dim = 11, |Q|=4, RNG(-3,3) Frequency (Hz) EIRP per MHz (dBm) (d) Figure 5.3: (a) Small TEM horn indoor eciency summary. opt 40:14% Clock = 35 GHz, N = 11, (b) Eciency contour, (c) Observed most ecient unltered D/A time domain waveform with column index n = 619; 234, (d) Corresponding EIRPSD for the indoor mask. 180 (a) Efficiency for Small TEM Horn for Outdoor Mask (b) 1 0 1 2 3 4 5 6 7 8 9 10 11 12 4 3 2 1 0 1 2 3 4 p(t, 755,070) p(t, 755,070) Time (Clock Normalized) Voltage (V) t sup 0.379 ns = (c) 0 5 . 10 9 1 . 10 10 1.5 . 10 10 2 . 10 10 2.5 . 10 10 100 80 60 40 FCC Indoor Mask Small TEM Horn S21 Antenna Input Best Observed Waveform FCC Indoor Mask Small TEM Horn S21 Antenna Input Best Observed Waveform CLK=29GHz, Dim = 11, |Q|=4, RNG(-3,3) Frequency (Hz) EIRP per MHz (dBm) (d) Figure 5.4: (a) Small TEM Horn outdoor eciency summary. opt 31:4% Clock = 29 GHz, N = 11, (b) Eciency contour, (c) Observed most ecient unltered D/A time domain waveform with column index n = 755; 070 , (d) Corresponding EIRPSD for the outdoor mask. 181 (a) Efficiency for Large TEM Horn for Outdoor Mask (b) 1 0 1 2 3 4 5 6 7 8 9 10 11 12 4 3 2 1 0 1 2 3 4 p(t, 975,211) p(t, 975,211) Time (Clock Normalized) Voltage (V) t sup 0.355 ns = (c) 0 5 . 10 9 1 . 10 10 1.5 . 10 10 2 . 10 10 2.5 . 10 10 100 80 60 40 FCC Outdoor Mask Large TEM Horn S21 Antenna Input Best Observed Waveform FCC Outdoor Mask Large TEM Horn S21 Antenna Input Best Observed Waveform CLK=31GHz, Dim = 11, |Q|=4, RNG(-3,3) Frequency (Hz) EIRP per MHz (dBm) (d) Figure 5.5: (a) Large TEM Horn outdoor eciency summary. opt 21:9% Clock = 31 GHz, N = 11, (b) Eciency contour, (c) Observed most ecient unltered D/A generated time domain waveform with column index n = 975; 211, (d) Corresponding EIRPSD for the outdoor mask. 182 (a) Efficiency for Large TEM Horn for Indoor Mask (b) 1 0 1 2 3 4 5 6 7 8 9 10 11 4 3 2 1 0 1 2 3 4 p(t, 185,118) p(t, 185,118) Time (Clock Normalized) Voltage (V) t sup 0.222 ns = (c) 0 5 . 10 9 1 . 10 10 1.5 . 10 10 2 . 10 10 2.5 . 10 10 100 80 60 40 FCC Indoor Mask Small TEM Horn S21 Antenna Input Best Observed Waveform FCC Indoor Mask Small TEM Horn S21 Antenna Input Best Observed Waveform CLK=45GHz, Dim = 10, |Q|=4, RNG(-3,3) Frequency (Hz) EIRP per MHz (dBm) (d) Figure 5.6: (a) Large TEM horn indoor eciency summary. opt 37:1% Clock = 45 GHz, N = 10, (b) Eciency contour, (c) Observed most ecient unltered D/A generated time domain waveform with column index n = 185; 118, (d) Corresponding EIRPSD for the indoor mask. 183 (a) Efficiency for Diamond Dipole and Outdoor Mask (b) 1 0 1 2 3 4 5 6 7 8 9 10 11 12 4 3 2 1 0 1 2 3 4 p(t, 754,663) p(t, 754,663) Time (Clock Normalized) Voltage (V) t sup 0.314 ns = (c) 0 5 . 10 9 1 . 10 10 1.5 . 10 10 2 . 10 10 2.5 . 10 10 100 50 FCC Outdoor Mask Diamond Dipole S21 Antenna Input Best Observed Waveform FCC Outdoor Mask Diamond Dipole S21 Antenna Input Best Observed Waveform CLK=35GHz, Dim = 11, |Q|=4, RNG(-3,3) Frequency (Hz) EIRP per MHz (dBm) (d) Figure 5.7: (a) Diamond dipole outdoor eciency summary. opt 0:675% Clock = 35 GHz, N = 11,(b) Eciency contour ,(c) Observed most ecient unltered time domain waveform with column index n = 754; 663, (d) Corresponding EIRPSD for the indoor mask. 184 (a) Efficiency for Diamond Dipole and Indoor Mask (b) 1 0 1 2 3 4 5 6 7 8 9 4 3 2 1 0 1 2 3 4 p(t, 16,028) p(t, 16,028) Time (Clock Normalized) Voltage (V) t sup 0.229 ns = (c) 0 5 . 10 9 1 . 10 10 1.5 . 10 10 2 . 10 10 2.5 . 10 10 120 100 80 60 40 20 FCC Indoor Mask Diamond Dipole S21 Antenna Input Best Observed Waveform FCC Indoor Mask Diamond Dipole S21 Antenna Input Best Observed Waveform CLK=35GHz, Dim = 8, |Q|=4, RNG(-3,3) Frequency (Hz) EIRP per MHz (dBm) (d) Figure 5.8: (a) Diamond dipole indoor eciency summary. opt 2:1% Clock = 35 GHz, N = 8, (b) Eciency contour, (c) Observed most ecient unltered generated time domain waveform with column index n = 16; 028, (d) Corresponding EIRPSD for the indoor mask. 185 Chapter 6 Results Based on Computed Antenna Gains 6.1 Introduction The results in this section are based on the three antenna method. A computer program was written and and data was logged for three antenna types and two FCC masks. The program collected results for various vector lengths and reproduction rates. The data discussed and shown in this section was randomly selected from a rather large data set. The search performed included over one hundred reproduction rates (i.e., 1GHz through 50GHz in 0.5GHz increments). Further, the search included vector lengths from 1 through 14. The search was performed for a uniform quantizer, a square law and a cubic law quantizer. The length 14 search was incomplete because of the amount of processing time required. The data from the complete search will be available if requested. 6.2 Results Based on the Three Antenna Method The results in this section are based on the three antenna method. Unlike the previous sections, this section is based on the Eective Isotropic Radiated Power (EIRP) which 186 only depends on the input power spectral density and the transmit antenna character- istic over the frequency range of interest. The receiving antenna has been removed by computational method described in section 3.5.8 on page 124. Re ections were observed around 18GHz and they were dampened prior to performing the simulations. While the re ections were dampened, they were not completely removed. Residual artifacts are readily observed hitting the FCC mask in Figure 6.2 on page 191. The re ections were believed to be caused by the limitations of the SMA connectors around 18GHz. The search results in this section was selected at random from a vector length of fourteen. The total search-space results form a rather large data set. Unfortunately, the data set was too large to be included in this dissertation. For the results provided here, the search was performed for the six trials listed in the run-time summary report given by: The program start time was Wed Dec 29, 09:22:59 2010. The program terminated on Sat Jan 1, at 17:15:58 2011. The run time was = 79.8831 hours on a 2.4 GHz AMD dual core processor with one search job on each processor (i.e., 37GHz on core number 1 and 37.5GHz on core number 2.) The clock reproduction rate was 37GHz. The number of quantization levels was 4. The cubic quantizer was used with valuesQ =f27;1; 1; 27g: The time sequence length was 14. 187 The number of sequences tested was 67,112,896 out of a total of 268,435,456 se- quences. The fraction of the space searched was 25.0015%. The number of antennas tested was 4. The number of FCC masks tested was 2. The sample statistics of the search are contained in Table 6.1. The statistics were collected in dB to accommodate the large variations observed during several trial runs. The maximum statistic represents the eciency of best observed waveform in the set. Ideally, the optimal waveform would have a value of 0dB which would indicate a perfect match to the mask. A value of -3dB would mean that the best waveform had an eciency of 50%. Table 6.2 provides all of the related sequences for the best observed sequences. A histogram function was included in the code to capture the distribution of the search because it was not practical to save the eciencies of each tested waveform vector. The histogram had a bin resolution of 1dB between bin centers. The histograms all indicate that good waveforms are rare events and the best waveform is an extremum. Sample Statistics in dB Mask Type Antenna Type Max Min Mean Variance Range Indoor TEM 1 -3.76 -38.24 -19.01 -34.99 -3.76 Indoor Small TEM 2 -3.31 -35.17 -13.89 -27.68 -3.31 Indoor Diamond 2 -8.83 -44.46 -31.92 -57.17 -8.83 Outdoor TEM 1 -10.02 -37.77 -22.88 -49.44 -10.03 Outdoor Small TEM 2 -6.10 -34.70 -19.47 -41.47 -6.10 Outdoor Diamond 2 -16.25 -43.99 -33.43 -67.92 -16.26 Table 6.1: Search statistics for a length 14 sequence at a reproduction rate of 37GHz and a cubic quantizer with 4 levels. 188 Related Sequences Mask Type Antenna Type 1 2 3 4 Indoor TEM 1 24,494,324 32,519,636 235,915,819 243,941,131 Indoor Small TEM 2 121,763,170 124,390,370 144,045,085 146,672,285 Indoor Diamond 2 89,572,594 118,408,874 150,026,581 178,862,861 Outdoor TEM 1 8,258,276 28,838,864 239,596,591 260,177,179 Outdoor Small TEM 2 70,191,041 71,110,529 197,324,926 198,244,414 Outdoor Diamond 2 87,952,325 180,483,130 Itself Itself Table 6.2: Related sequences for the best sequences of length 14 at a reproduction rate of 37GHz and a 4-level cubic quantizer. The two items in the table labeled itself represent a waveform that is symmetric about it's center axis and voltage levels. 189 1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 40 30 20 10 0 10 20 30 40 Time (Clock Normalized) Voltage (V) t sup 0.378ns = (a) 0 5 10 15 20 25 100 80 60 40 FCC Outdoor Mask Large TEM Horn Antenna Input Best Observed Waveform FCC Outdoor Mask Large TEM Horn Antenna Input Best Observed Waveform Frequency (GHz) EIRP per MHz (dBm) (b) 40 − 30 − 20 − 10 − 0 10 100 1 10 3 × 1 10 4 × 1 10 5 × 1 10 6 × 1 10 7 × 1 10 8 × Bin Value (dB) Efficiency Bin Count (c) Figure 6.1: (a) Generated time domain waveform at a clock reproduction rate of 37GHz. A 4-level cubic quantizer was used. The waveform vector is (27;1; 1; 27; 1;27;27;27; 1; 27; 27;1;27;27) t . (b) Large TEM horn response and the outdoor mask. max 9:95% (c) Histogram of the waveform eciencies in dB. 190 1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 40 30 20 10 0 10 20 30 40 Time (Clock Normalized) Voltage (V) t sup 0.378ns = (a) 0 5 10 15 20 25 100 80 60 40 FCC Outdoor Mask Small TEM Horn Antenna Input Best Observed Waveform FCC Outdoor Mask Small TEM Horn Antenna Input Best Observed Waveform Frequency (GHz) EIRP per MHz (dBm) (b) 40 − 30 − 20 − 10 − 0 1 10 100 1 10 3 × 1 10 4 × 1 10 5 × 1 10 6 × 1 10 7 × 1 10 8 × Bin Value (dB) Efficiency Bin Count (c) Figure 6.2: (a) Generated time domain waveform at a clock reproduction rate of 37GHz. A 4-level cubic quantizer was used. The waveform vector is (1;27;27; 27; 27;1;27;27; 27; 27; 1;27;27;1) t . (b) Small TEM horn re- sponse and the outdoor mask. max 24:57% (c) Histogram of the waveform eciencies in dB. 191 1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 40 30 20 10 0 10 20 30 40 Time (Clock Normalized) Voltage (V) t sup 0.378ns = (a) 0 5 10 15 20 25 100 80 60 40 FCC Outdoor Mask Diamond Dipole Antenna Input Best Observed Waveform FCC Outdoor Mask Diamond Dipole Antenna Input Best Observed Waveform EIRP per MHz (dBm) (b) 50 − 40 − 30 − 20 − 10 − 0 1 10 100 1 10 3 × 1 10 4 × 1 10 5 × 1 10 6 × 1 10 7 × Bin Value (dB) Efficiency Bin Count (c) Figure 6.3: (a) Generated time domain waveform at a clock reproduction rate of 37GHz. A 4-level cubic quantizer was used. The waveform vector is (1;1;27; 27; 27; 1;27;27; 1; 27; 27;27;1;1) t . (b) Diamond Dipole response and the outdoor mask. max 2:37% (c) Histogram of the waveform eciencies in dB. 192 1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 40 30 20 10 0 10 20 30 40 Time (Clock Normalized) Voltage (V) t sup 0.378ns = (a) 0 5 10 15 20 25 100 80 60 40 FCC Outdoor Mask Large TEM Horn Antenna Input Best Observed Waveform FCC Outdoor Mask Large TEM Horn Antenna Input Best Observed Waveform Frequency (GHz) EIRP per MHz (dBm) (b) 40 − 30 − 20 − 10 − 0 100 1 10 3 × 1 10 4 × 1 10 5 × 1 10 6 × 1 10 7 × Bin Value (dB) Efficiency Bin Count (c) Figure 6.4: (a) Generated time domain waveform at a clock reproduction rate of 37GHz. A 4-level cubic quantizer was used. The waveform vector is (27;1; 27; 27;27;27;27; 27;1;1; 27;1;1;27) t . (b) Large TEM Horn re- sponse and the indoor mask. max 42:05% (c) Histogram of the waveform eciencies in dB. 193 1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 40 30 20 10 0 10 20 30 40 Time (Clock Normalized) Voltage (V) t sup 0.378ns = (a) 0 5 10 15 20 25 100 80 60 40 FCC Outdoor Mask Small TEM Horn Antenna Input Best Observed Waveform FCC Outdoor Mask Small TEM Horn Antenna Input Best Observed Waveform Frequency (GHz) EIRP per MHz (dBm) (b) 40 − 30 − 20 − 10 − 0 1 10 100 1 10 3 × 1 10 4 × 1 10 5 × 1 10 6 × 1 10 7 × Bin Value (dB) Efficiency Bin Count (c) Figure 6.5: (a) Generated time domain waveform at a clock reproduction rate of 37GHz. A 4-level cubic quantizer was used. The waveform vector is (1;27; 1;1;1;1; 27; 27;1;27;27;1; 27;1) t . (b) Large TEM Horn response and the indoor mask. max 46:68% (c) Histogram of the waveform eciencies in dB. 194 1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 40 30 20 10 0 10 20 30 40 Time (Clock Normalized) Voltage (V) t sup 0.378ns = (a) 0 5 10 15 20 25 100 80 60 40 FCC Outdoor Mask Diamond Dipole Antenna Antenna Input Best Observed Waveform FCC Outdoor Mask Diamond Dipole Antenna Antenna Input Best Observed Waveform Frequency (GHz) EIRP per MHz (dBm) (b) 50 − 40 − 30 − 20 − 10 − 0 10 100 1 10 3 × 1 10 4 × 1 10 5 × 1 10 6 × 1 10 7 × Bin Value (dB) Efficiency Bin Count (c) Figure 6.6: (a) Generated time domain waveform at a clock reproduction rate of 37GHz. A 4-level cubic quantizer was used. The waveform vector is (1;27; 27; 27;27;1;27; 27; 1;1;1;1;1;1) t . (b) Diamond Dipole antenna re- sponse and the indoor mask. max 13:11%. (c) Histogram of the waveform eciencies in dB. 195 Chapter 7 Quotes heard through the Ph.D years James T. Kirk {Star Trek (The Kobayashi Maru): I don't believe in the no-win scenario. Doctor McKoy { Star Trek: Dammit Jim, I'm a doctor...not an engineer! A professor: It's well known by those that know it well. A friend of Professor Weber: I just pasted my PhD Defense in mathematics. I feel like I am smart but I know that I'm not! A Ph.D Student: Water is slow to boil. Meaning that learning takes a long time but once it gets going, you start to learn fast. A Ph.D Student: I had a low GPA...3.8. A Ph.D Student: I must do something really signicant. Most students that have this as an idea will not achieve their goal. Worst, they tend to not graduate at all. 196 CDR Brian Meadows PhD: An admiral once said...Brian, you are a smart guy but think of us as pig farmers. The comment was directed to how you should approach complicated presentations to senior executives. Sergio Verdu': [2010 Information Theory workshop at USC] It is not what you dene, it is what you prove (that matters). Dr Jordan Melzer: All code errors are three lines of code......the problem is deter- mining which three lines! Mr. Dale Feikema: Don't wrestle with pigs. Why not? Because you will get dirty....it serves no purpose.....and the pigs like it! An Admiral: I'm not happy unless your not happy. Note: He was in the process of cutting budgets across the board in a fair manner. Simon Ramo: A very optimistic engineer { When one missile rose about 6 inches be- fore toppling over and exploding, Ramo reportedly beamed and said: Well, Benny, now that we know the thing can y, all we need to do is improve its range a bit. Ms Melani Austin: Don't make my life better without you. A PhD Student: This communications PhD boils down to a simple equation - y = x +n. Who knew I would be studying addition this long! (Men of Honor) Jo: Why do you want this so bad?.....Carl Brasher:Because they said I couldnt have it. 197 (Men of Honor) [Sunday regards a picture of Carl's late father] Billy Sunday: What the hell did he ever say to make you try so hard? Carl Brashear: Be the best. Billy Sunday: Well, you are. A program manager: Don't let physics get in the way of sales! A Navy Captain: That which does not kill you, makes you stronger. A program manager: I did not appreciate the idiots I had until I got new idiots. Lieutenant General Brian Arnold USAF (ret): A closed mouth gathers no feet! Professor Bob Gagliarardi (rest in peace): Terry, you cannot be a good Dr. studying part time! Professor Irving S. Reed: Terry, there are no smart people. There are only people that work hard. You see the results of their hard work and believe them to be smart. Work hard! 198 Bibliography [1] Agilent, \Understanding network analyzer dynamic range," Agilent Application Note 1363-1, p. 7, September 2000. 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The coecientsa i ;b i wherei2f1; 2g, are input and output voltages respectively. When a 2 = 0, the input a 1 at port 1 is related to the observed output b 2 at port 2 by S 21 = b 2 a 1 : (A.0.2) When the impedances at the input and output ports of a two port network are equal, the ratio represented by S 21 is a transfer function of the independent frequency variable f and is denoted as S 21 (f) =H S21 (f). The magnitude squared is the power gain given by jS 21 (f)j 2 =jH S21 (f)j 2 : (A.0.3) 204 Appendix B Root Mean Squared Values The root{mean{squared voltage of a periodic voltage v(t) =v(t +T ) is dened as V rms , s 1 T Z T 0 v(t)v (t)dt = s 1 T Z T 0 jv(t)j 2 dt: (B.0.1) The instantaneous power delivered to a load resistance R is p(t) = v(t)v (t) R = jv(t)j 2 R : (B.0.2) The total energy in one period is E = Z T 0 p(t)dt: (B.0.3) The time{averaged power is P ave = E T = 1 T Z T 0 p(t)dt: (B.0.4) Since the time averaged power is positive, it can be written in terms of V rms given in (B.0.1) which results in P ave = V 2 rms R : (B.0.5) 205 B.1 RMS Voltage of a Sinusoidal Waveform The RMS voltage of a periodic waveform is often of interest. Let v(t) =A cos(!t +): (B.1.1) The RMS value of v(t) can be shown to be V rms = A p 2 : (B.1.2) 206 Appendix C Notes on Decibels Given a power measurement X that is measured in Watts, the power in decibels is given by X dBW = 10 log(X): (C.0.1) When the power is small, the power is typically measured with respect to 1 mW and thus X can be dened with respect to a 1mW reference as X dBm =10 log X 10 3 (C.0.2) =10 log(X) 10 log(10 3 ) =10 log(X) + 30 log(10) =10 log(X) + 30 =X dB + 30: (C.0.3) The above result shows that in order to convert from dBW to dBm, we only have to add a constant of 30 to the value in dBW. 207 Appendix D FCC-Part 15 Limits This section is based on a set of unpublished notes by Professor Robert A. Scholtz. The FCC part-15 radiated limits states that any radiating device under test (DUT) at a range of three meters from the measuring apparatus, the radiated electric eld intensity, in any 1MHz of bandwidth, are constrained by E max 500V m MHz 1=2 : (D.0.1) Assuming that the radiated eld is a plane wave, it was shown in section 3.2.8 that the E-eld is related to the H-eld by the intrinsic impedance of free-space o 377 - a constant. Assuming that the E-eld is in the ^ a x and H-eld is in the ^ a y then the resulting EM wave propagates in the ^ a z direction. The power density (over space) is given by the Poynting vector P =E H = ^ a x E x ^ a y E x = ^ a z jE x j 2 ; which can be written as P rms = jE max j 2 o (500V=m) 2 o W m 2 MHz : (D.0.2) Assuming isotropic radiation, the surface area of the sphere is given by 4r 2 wherer = 3m is the range of the DUT to the detector. Dividing the power density by the surface area of an isotropic radiator, the result is P DUT 4r 2 j r=3m = jE max j 2 o (500V=m) 2 o W m 2 MHz : (D.0.3) 208 and thus P DUT 4(3m) 2 (500V ) 2 o W m 2 MHz (D.0.4) 43 2 (500V ) 2 377 W MHz (D.0.5) 7:496 10 8 W MHz ; (D.0.6) or in dB form P DUT 71:25 dBW MHz (D.0.7) =41:25 dBm MHz : (D.0.8) Under Part 15 rules and regulation, the maximum power allowable in any B MHz (in dB-MHz) bandwidth is P DUT 41:25 dBm MHz +B MHz 41:25 dBm: 209 Appendix E Probability, Random Processes and Statistical Inference: A Review E.1 Preliminaries In this review section, the reader is introduced to the notation used to describe random variables and random processes. This section is not meant to be an extensive exposition since space is limited and the concepts are not new. Readers that are not familiar with this subject is referred to one of the many excellent textbooks on random variables and random processes. The author has beneted greatly from several authors such as my adviser Robert A Scholtz, Papoulis, Garcia, Bob Gray, Stark and Woods, David Middleton, Davenport and Root and many others. Denition 17. (Gerald Folland, Real Analysis) An algebra of sets on X is a nonempty collectionA of subsets of X that is closed under nite unions and complements; that is, E 1 ;:::;E n 2A; and n [ j=1 E j 2A; and if E2A then E c 2A: Denition 18. A sigma algebraF is an algebra that is closed under countable unions, intersections and combinations thereof. Denition 19. A random variable is a mapping of an event u2U to the real line R by a function X(u): The random variable X(u) can be denoted as fu2UjX(u)!Rg: (E.1.1) It is well known (to people that study this subject) that random variables are dened by the three setsU;F;P; where u2U denotes an event taken from all possible events that can occur,F denotes the algebra dened onU; andP denotes the probability measure of events. The probability measure of a particular eventu2U is determined by the distribution function F X (x) =P (X(u)x) =Pfu2UjX(u)2Rg where P () denotes a probability measure. 210 Since X(u) is a random variable, let x 2 R be dened as any xed real number (assuming that X(u)2 R), then the event X(u) = x is dened asfu2UjX(u) = xg. Similarly, the events 1. (X(u)x) =fu2UjX(u)xg, 2. (X(u)>x) =fu2UjX(u)>xg, 3. 8x 1 ;x 2 2R; (x 1 X(u)x 2 ) =fu2Ujx 1 X(u)x 2 g are dened. The probability of the above events are dened by the probability measure of the event u2U and thus 1. F X (x) =P (X(u)x) =Pfu2UjX(u)xg, 2. 1F X (x) =P (X(u)>x) =Pfu2UjX(u)>xg, 3. 8x 1 ;x 2 2R; F X (x 2 )F X (x 1 ) =P (x 1 X(u)x 2 ) =Pfu2Ujx 1 X(u)x 2 g. The results are also extensible to random variables dened on the complex plane where Z(u)2C such that Z(u) =X(u) +jY (u) where j = p 1 and (X(u);Y (u))2R: E.1.1 Statistical Averages of Real Random Variables Statistical (or ensemble) averages of random variables are also called moments. The rth moment of X(u) is EfX r (u)g = Z 1 1 x r f X (x)dx 8r2N: (E.1.2) Whenr = 0; the integral integrates to unity. Forr = 1; the average is called the statistical mean X . The rth central moment of X(u) is Ef(X(u) X ) r g = Z 1 1 (x X ) r f X (x)dx 8r2N: (E.1.3) When r = 2, the rth central moment (about the mean) is called the variance m 2 X = 2 X and the standard deviation is given by X = q 2 X (i.e., the positive square root). Let X(u) and Y (u) denote random variables that are dened on the same probability space. The rth joint moments EfX m (u)Y n (u)g = Z 1 1 x m y n f XY (x;y)dxdy; (E.1.4) with mixed central moments Ef(X(u) X ) m (Y (u) Y ) n g = Z 1 1 (x X ) m (y Y ) n f XY (x;y)dxdy for m +n =r2N; (E.1.5) 211 where f XY (x;y) is the joint probability density function. When m = n = 1, the mixed central moment is called the covariance. Probability theory has been extended to include events that depend on time and the study of time dependent random events are called Random or Stochastic Processes. E.2 Random Processes Denition 20. A random (or stochastic) process is set of indexed random variables X(u;t) dened on the 4{tuple spacefU;F;P;Tg where the time index t is a member of the setT: There are many possibilities for the setT which can include the entire real line, the set of natural numbers or the ring of integers. Many more sets forT can be considered; however, when any particular t2T is provided, X(u;t) is an ordinary random variable. There are particular names given to the random process that depends on the time index setT: When the time index is: 1. the set of real numbers, the stochastic process is called a random process. 2. the ring of integers, the random process is called a random sequence. 3. a nite set of natural numbers, the random process is called a random vector. In general, most processes here are called random processes where it is expected that the domainT should be understood and clear. The probability measure for a random process is determined by the distribution func- tion F X (x 1 ;t 1 ) =P (X(u;t 1 )x 1 ) =Pfu2U;t 1 2TjX(u;t 1 )x 1 g (E.2.1) for a particular x 1 2 R. The partial derivative of the distribution function denes the probability density function. Thus, p(x;t) = @F (x;t) @x : (E.2.2) The meaning ofF (x =x 0 ;t =t 0 ) implies when two real numbers x 0 ;t 0 are provided, the distribution function F (x 0 ;t 0 ) denes the probability that X(u;t 0 )x 0 . The second order distribution is F X (x 1 ;x 2 ;t 1 ;t 2 ) =P (X(u;t 1 )x 1 ;X(u;t 2 )x 2 ); (E.2.3) with joint density p(x 1 ;x 2 ;t 1 ;t 2 ) = @ 2 F (x 1 ;x 2 ;t 1 ;t 2 ) @x 1 @x 2 : (E.2.4) In general, an ndimensional random process, with the set of observations X(u;t 1 );X(u;t 2 ); ;X(u;t n ); (E.2.5) 212 has an nth order distribution function F (x 1 ;x 2 ; ;x n ;t 1 ;t 2 ; ;t n ) = (E.2.6) P (X(u;t 1 )x 1 ;X(u;t 2 )x 2 ; ;X(u;t n )x n ); with the nth-order density function f(x 1 ;x 2 ; ;x n ;t 1 ;t 2 ; ;t n ) = @ n F (x 1 ;x 2 ; ;x n ;t 1 ;t 2 ; ;t n ) @x 1 @x 2 @x n : (E.2.7) Using the description provided by Rowe [51], when the probability density function p X (x;t) for the random process X(u;t) is multiplied by a dierential element dx, the resultant p X (x;t)dx has the interpretation that p x (x 1 ;t 1 )dx 1 = Probability that x 1 <X(u)<x 1 +dx 1 at time t 1 ; p x (x 1 ;x 2 ;t 1 ;t 2 )dx 1 dx 2 = The joint probability that x 1 <X(u)<x 1 +dx 1 at time t 1 ; x 2 <X(u)<x 2 +dx 2 at time t 2 ; simultaneously; p x (x 1 ;x 2 ; ;x n ;t 1 ;t 2 ; ;t n )dx 1 dx 2 dx n = The joint probability that x 1 <X(u)<x 1 +dx 1 at time t 1 ; x 2 <X(u)<x 2 +dx 2 at time t 2 ; . . . x n <X(u)<x n +dx n at time t n simultaneously. E.2.1 Stationary Random Processes In general, the statistics of a random process are time varying. When the statistics of an nth order random process does not vary with time, the process is called nth order stationary. Denition 21. Annth order random process is called Strict Sense Stationary if and only if, for any 2R F (x 1 ;x 2 ; ;x n ;t 1 ;t 2 ; ;t n ),F (x 1 ;x 2 ; ;x n ;t 1 +;t 2 +; ;t n +): (E.2.8) The nth order joint stationary PDF is given by @ n F (x 1 ;x 2 ; ;x n ;t 1 ;t 2 ; ;t n ) @x 1 @x 2 @x n = p(x 1 ;x 2 ; ;x n ;t 1 ;t 2 ; ;t n ),p(x 1 ;x 2 ; ;x n ;t 1 +;t 2 +; ;t n +): 213 E.2.2 Statistical Averages of Random Processes Similar to the statistical averages of random variables, statistical averages of random processes are similarly dened. Typically, the rst and second order statistics (the mean, correlation and covariance) are of interest for most engineering problems. In general, it is assumed that the random processes are complex. Denition 22. Let X(u;t)2 C be dened on the spacefU;F;P;Tg with both t;s2 T , then the mean Z (t); correlation R X (t;s) and covariance K X (t;s) functions of the complex random process are X (t) =EfX(u;t)g; (E.2.9) R X (t;s) =EfX(u;t)X(u;s) g; (E.2.10) K X (t;s) =Ef(X(u;t) X (t))(X(u;s) X (s)) g; (E.2.11) where denotes complex conjugation. Denition 23. Let both X(u;t);Y (u;t)2C be dened on the same space fU;F;P;Tg with botht;s2T , then the cross-correlationR XY (t;s) and cross-covariance K XY (t;s) functions of the complex random processes are R XY (t;s) =EfX(u;t)Y (u;s) g; (E.2.12) K XY (t;s) =Ef(X(u;t) X (t))(Y (u;s) Y (s)) g: (E.2.13) The moment descriptions provided extends to random sequences and random vectors. For n-dimensional complex random vectors X2C n ; Y2C n X = 2 6 6 6 4 X(u; 1) X(u; 2) . . . X(u;n) 3 7 7 7 5 Y = 2 6 6 6 4 Y (u; 1) Y (u; 2) . . . Y (u;n); 3 7 7 7 5 the mean vector X , correlation and covariance matrices R X and K X are X =EfXg R X =E n XX y o K X =E n (X X )(X X ) y o ; (E.2.14) where y denotes the conjugate transpose operation. The cross-correlation and cross- covariance matrices are R XY =E n XY y o K X =E n (X X )(Y Y ) y o : (E.2.15) Denition 24. A random process is said to be wide sense stationary (WSS) if the rst moment EfX(u;t)g = X (t) = X is a constant (i.e., time invariant) and the second order moments R X (t;s) = EfX(u;t)X (u;s)g; K X (t;s) = EfX(u;t) X )(X(u;s) X ) g; depends only on the time dierence =ts. 214 E.3 Convergence, Integrals and Ergodicity of Random Processes Often it is desired to estimate the moments of a random process X(u;t) from a nite time observation of a single realization of the process. In general, the observation interval is over a time duration ofTtT and thus the total observation time is 2T . When the probability density function p X (x) ofX(u) is known, all the moments of the random variable may be computed; the same is true for a random process X(u;t) when a specic time instant t2T is known. The most commonly computed moments of a random process are the rst and second order moments. The mean and correlation and covariance functions are given by x (t) =EfX(u;t)g, Z 1 1 xp X (x;t)dx; (E.3.1) and R X (t;s) =EfX(u;t)X (u;s)g, Z 1 1 Z 1 1 x 1 x 2 p X (x 1 ;x 2 ;t;s)dx 1 dx 2 ; (E.3.2) and K X (t;s) =Ef[X(u;t)(t)][X(u;s)(s)] g , Z 1 1 Z 1 1 [x 1 (t)][x 2 (s)] p X (x 1 ;x 2 ;t;s)dx 1 dx 2 (E.3.3) for time instants t and s. The averages discussed thus far are the so-called ensemble averages. That is, the computed averages are computed with respect to the governing probability distribution function (PDF)p X (x) or the joint PDF when joint moments are to be determined. When the PDF is not known (typically the case), attempts are made to estimate the rst and second moments of a random process by using short term time averages. It is of interest to determine if these time averages are useful; meaning that the time average is getting close to (converging to) the true parameter of interest as the observation interval becomes innite. To determine a measure of \goodness" of the estimate, the following theories and denitions are required since they provide theoretical models for experiments. E.3.1 Convergence Concepts Theorem 2. Cauchy Criterion A sequence of complex numbersx n converges to a limit if and only if lim m!1 n!1 jx n x m j! 0: (E.3.4) The interested reader is referred to the Theory of Linear Operators in Hilbert Space by Akhiezer and Glazman [3] for the proof and additional details. 215 Denition 25. (Convergence in Probability) Let X(u; 1);X(u; 2);:::; be a random sequence. The sequence of random variables converges to X(u) in probability if for every > 0; lim n!1 P (jX(u;n)X(u)j)! 0; (E.3.5) or equivalently lim n!1 P (jX(u;n)X(u)j<)! 1: (E.3.6) The the weak law of large numbers is a consequence of convergence in probability (weak convergence). Theorem 3. (The Weak Law of Large Numbers) LetX(u; 1);X(u; 2);:::; denote a sequence of independent and identically distributed (iid) random variables each with nite mean X and nite variance 2 X , then for every > 0, lim n!1 P (jX n j<) P ! 1; (E.3.7) where X n = 1 n n X k=1 X(u;k) (E.3.8) and is not a function of n: The proof is omitted and can be found in many texts on probability theory. Obviously, since the ensemble expectation was not taken, X n is a random variable. The dependence on the sample space u2U was suppressed. The implication of the WLLNs is that the sample mean converges in probability to the true mean. The proof of the WLLN is a direct application of Chebychev's Inequality. The requirement for an iid sequence in the theorem simplies the computation of the sample mean and variance. The proof is omitted. Theorem 4. (Chebychev's Inequality) Let X(u) be a random variable and g(x) be any nonnegative function, then for any a> 0, P (g(X(u))a) Efg(X(u))g a : (E.3.9) See Cassella and Berger [12] for a straight forward proof. The WLLNs can be used to show that the sample variance S 2 n = 1 n 1 n X k=1 (X(u;n)X n ) 2 P ! 2 : (E.3.10) That is P (jS 2 n 2 X j) VarS 2 n 2 ; (E.3.11) where (E.3.11) is a form of Chebychev's Inequality and VarS 2 n denotes the variance of the the random variable S 2 n . 216 Denition 26. (Almost Sure Convergence ) The random sequenceX(u;n) converges to the random variable X(u) almost surely (a.s) if P (j lim n!1 X(u;n) X j>) a:s ! 0 (E.3.12) or equivalently P (j lim n!1 X(u;n) X j) a:s ! 1 (E.3.13) Almost sure convergence is also referred to as the Strong Law of Large Numbers (SLLNs). Denition 27. Convergence in the Mean Square Sense: The random sequence X(u;n) converges to the random variable X(u) in the mean square sense (m.s) if lim n!1 EfjX(u;n)X(u)j 2 ! 0g: (E.3.14) Mean square convergence is also denoted l:i:m n!1 X(u;n) =X(u) or lim n!1 X(u;n) =X(u) (m.s): Mean square convergence is based on the fact that Cauchy sequences are convergent in a Hilbert Space which is true because a Hilbert space is complete in the sense that it contains all its limit points. E.3.2 Riemann Integral of a Random Processes Recall that the Riemann integral of a function f on the domain x2 R over the nite interval [a;b]; is used to determine the area L under the curve f(x) and is given by L = Z b a f(x)dx: An approximation s n of the area L is determined by partitioning the the interval a = x 0 ;x 1 ;:::;x n1 ;x n =b into n parts. Let I 1 =x 0 x 1 ;I 2 =x 2 x 1 ;:::;I n =x n x n1 and let x k = x k x k1 be the length of the kth subinterval. Form an interior point x k 2I k for eachkth interval. The area of each rectangle is determined by the evaluation of f(x k )x k . Since the intervals are non overlapping, the area under the curve can be approximated by the Riemann sum L n = n X k=1 (x k )x k : For a given x k , a norm of the length iskxk = maxfx k : 1kng. It follows that ba kxk n 217 is satised for any general partition (including a uniform partition) of the interval [a;b]. As the number of subintervalsfI k : 1 k ng becomes small,kxk! 0 implies that the setfI k g is becoming dense in [a;b] and the total number of intervals are increasing without bound. In other words, n!1. Let L be the limit of the Riemann sum (if it exists), then lim kxk!0 n X k=1 f(x k )x k =L; which is to say that the limit converges absolutely. Restating the limit in the form of an absolute metric, the limit of the Riemann sum converges to the limit L if for any "> 0, there exists a > 0 such that jL n X k=1 f(x k )x k j<" for any choice x k 2I k : Denition 28. (Riemann Integral) If a functionf is dened on the interval [a;b] and the limit of the Riemann sum of f exists, then it is said that f is Riemann integrable on [a;b] and is denoted as lim kxk!0 n X k=1 f(x k )x k , Z b a f(x)dx: (E.3.15) E.3.3 Stochastic Integration Often it is desired to determine L(u) = Z b a X(u;t)dt; (E.3.16) that is, the output of a Riemann integral over a time interval [a;b]. The output of the integral is a random variable and the integral is called a stochastic integral. The stochastic integral is developed similarly to the Riemann integral with a few notable distinctions. Similar to the development of the deterministic Riemann integral, the stochastic Riemann partial sum L(u;n) is determined as L n (u) =L(u;n) = n X k=1 X(u;t k )t k ; (E.3.17) with t k 2 I k and[ n k=1 I k = [a;b]. In the development of the Riemann integral, the convergence criterion used was thel 1 {norm (absolute norm). Instead of thel 1 {norm, the l 2 {norm is preferred here since the l 2 {norm is used to determine the energy or power of a process. The stochastic integral is developed using mean square convergence. 218 Denition 29. Stochastic Integral The Stochastic Riemann integral of a random pro- cess X(u;t) over the interval [a;b] exists if the stochastic Riemann sum L n (u) converges to L(u) in the mean square sense as lim ktk!0 . Thus, l.i.mL n (u) =L(u) as ktk! 0 (E.3.18) lim ktk!0 E 8 < : n X k=1 X(u;t k )t k L(u) 2 ! 0 9 = ; ; then L(u) m:s = Z b a X(u;t)dt: (E.3.19) The stochastic integral is a linear operator which has the implication that the expec- tation operatorE commutes with the integral sign under most situations. E.3.3.1 Existence of the Stochastic Riemann Integral The stochastic integral I(u) = Z T 2 T 1 X(u;t)dt (m.s) (E.3.20) exists if the second moment of I(u) is nite. The necessary and sucient condition for the existence of the stochastic Riemann integral is the existence of Z T 2 T 1 Z T 2 T 1 R X (t;s)dtds<1 (E.3.21) which is determined by evaluating the correlation function of (E.3.20). E.3.4 Ergodicity in the Mean Random process theory is concerned with mathematical models of nature. Typically, data is gathered from a well designed experiment and the rst and second moments are to be determined from the observed data if possible. Ergodic theory provides the necessary and sucient conditions when short observations of a random process can be used as a reliable estimates. Most measurements of random phenomena rely on the fact that the processes under observation are WSS and ergodic. Denition 30. A WSS random process X(u;t) is said to be ergodic in the mean if the limit of the time average converges to the ensemble average in the mean square sense. Mathematically, this means lim T!1 1 2T Z T T X(u;t)dt m:s ! EfX(u;t)g = X : (E.3.22) A more thorough investigation of ergodicity can be found in Gray [25]. Obviously, in equation E.3.22, the expected value of the left hand side equals the right hand side 219 which implies that the time average is an unbiased estimator of the mean. The mean square convergence criteria places constrains of the second moment of the time average. It makes since to investigate the m.s convergence of the short time average estimate of X (u) m:s = 1 2T Z T T X(u;t)dt (E.3.23) which is 2 X =Ef( X (u) x )( X (u) x ) g =E ( 1 2T Z T T X(u;t)dt x 2 ) =E ( 1 2T Z T T (X(u;t) x )dt 2 ) = 1 4T 2 E Z T T (X(u;t) x )dt Z T T (X(u;s) x ) ds = 1 4T 2 E Z T T Z T T (X(u;t) x )(X(u;s) x ) dtds = 1 4T 2 Z T T Z T T Ef(X(u;t) x )(X(u;s) x ) dtdsg = 1 4T 2 Z T T Z T T K X (t;s)dtds = 1 4T 2 Z T T Z T T K X (ts)dtds; (E.3.24) which is a function of only the time dierence ts. The double integral degenerated to a single variable =ts: The transformation of the double integral to a single integral as a function of can be shown by by setting =t +s and =ts in equation (E.3.24) (see Appendix E.5). The transformation takes all points (t;s)2 (the rectangular region) bounded by the four points (T;T ); (T;T ); (T;T ); (T;T ) in the (t;s) plane to all point (;)2 (the diamond shaped region) with corners (0; 2T ); (2T; 0); (0;2T ); (2T; 0) in the (;) plane. The region is determined by the bounds2T 2T and2T 2T which are determined from the extreme points of t ands. The equations that bound is determined by straight lines which are determined by solving the equation of a familiar form = i () =m i +b i i = 1; 2; 3; 4; (E.3.25) or = i () =m i +b i i = 1; 2; 3; 4; (E.3.26) where m i and b i are the slope of the line and the axis intercept respectively in the ith quadrant in the the (;) plane. The equations of the bounding lines are shown in Figure E.1(b) where equation (E.3.26) was used. The region dened by is complicated and 220 s t T T T T (T;T) (T;T) (T;T) (T;T) (a) s t 3 ()=2T 2T 2 ()=2T 2T 1 ()=+2T 2T 4 ()=+2T 2T d d (b) Figure E.1: (a) Region of integration for covariance function. (b) Transformed region of integration. therefore it is decomposed into two parts = + [ where + =f : 0g and is the complement of + (i.e., = c + ). Then, 1 4T 2 Z T T Z T T K X (ts)dtds = 1 4T 2 2 6 6 6 6 4 ZZ + K X (;)jJjd d | {z } I + + ZZ K X (;)jJjd d | {z } I 3 7 7 7 7 5 221 Integrating over the region + results in I + = 1 4T 2 ZZ + K X () 1 2 d d = 1 8T 2 Z 2T 0 K X () Z 1 () 2 () d d (Interchange order of integration) = 1 8T 2 Z 2T 0 K X () Z +2T 2T d d = 1 8T 2 Z 2T 0 K X ()(2 + 4T )d = 1 4T 2 Z 2T 0 K X ()( + 2T )d = 1 2T Z 2T 0 K X () 1 2T d; where the Jacobian J was determined by jJ(;)j = abs @t @ @s @ @t @ @s @ = @ @t @ @s @ @t @ @s 1 = 1 1 1 1 1 = 1=2: (E.3.27) Similarly, I = 1 4T 2 ZZ K X () 1 2 d d = 1 8T 2 Z 0 2T K X () Z 4 () 3 () d d = 1 8T 2 Z 0 2T K X () Z +2T 2T d d = 1 8T 2 Z 0 2T K X ()(2 + 4T )d = 1 4T 2 Z 0 2T K X ()( + 2T )d = 1 2T Z 0 2T K X () 1 + 2T d: Adding the results of I + and I + results in the compact form X = 1 2T Z 2T 2T 1 jj 2T K X ()d: (E.3.28) 222 The necessary and sucient condition for a random process to be ergodic in the mean is X = lim T!1 1 2T Z 2T 2T 1 jj 2T K X ()d! 0: (E.3.29) A similar necessary and sucient condition can be derived for a mean ergodic random sequence. In this work, it is assumed that the measured processes are mean ergodic. E.3.5 Power and Energy Spectral Density Concepts The energy and power is considered here in terms of a simple series circuit with a voltage generator v(t) connected to a pure resistance R. The voltage across the load R induces a current i(t) owing in the series circuit. The instantaneous power is p(t) = v(t)i(t) where denotes complex conjugation. The time averaged power P t , over the nite time interval t 1 tt 2 , due to the deterministic source v(t)2C, is P t = 1 t 2 t 1 Z t 2 t 1 p(t)dt = 1 t 2 t 1 Z t 2 t 1 jv(t)j 2 R dt (W ): (E.3.30) The energy E t over the same interval is E t = Z t 2 t 1 p(t)dt = Z t 2 t 1 jv(t)j 2 R dt (J=s): (E.3.31) Similarly, when the voltage source has innite time support, the total energy and time averaged power, for a normalized resistance R = 1, is dened as E = Z T T p(t)dt = Z T T jv(t)j 2 dt (J=s) (E.3.32) and P = lim T!1 1 2T Z T T p(t)dt (W ): (E.3.33) For a positive constant M > 0, for E < M <1 equations (E.3.32) and (E.3.33) state that power P = 0. When v(t) =v(t +T ) the function is periodic and the average energy is E T = Z T 0 p(t)dt; (E.3.34) and average power P ave = 1 T Z T 0 p(t)dt = q R T 0 1 T v 2 (t) R dt; (E.3.35) 223 where the rms voltage is given by V rms = s Z T 0 1 T v 2 (t): (E.3.36) Parseval's Theorem states that E = Z 1 1 jv(t)j 2 dt = Z 1 1 jV (f)j 2 df; (E.3.37) where V (f) =Ffv(t)g is the Fourier transform of v(t). Parseval's theorem is an equiva- lence relationship of energy or power. The energy or power can be computed in time or the frequency domain. Often, it is of interest to determine the detailed spectrum (shape) of either the energy/power spectral density and thus Parseval's theorem is very useful. Parseval's theorem is directly applicable to random processes and is commonly employed to analyze systems. The power spectral density (PSD) of a random process X(u;t) is dened as S X (f), lim T!1 E FfX T (u;t)g 2T 2 (W/ Hz); (E.3.38) where the subscript on S X (f) denotes a label for the random process X(u;t), and FfX T (u;t)g denotes the nite time Fourier transform over the interval (T;T ). The expectation operatorE is over u2U. Theorem 5. Wiener-Khintchine The power spectral densityS X (f) of a WSS random process X(u;t) is the Fourier transform of the correlation function R X (), assuming S X (f) exists. Thus S X (f) =FfR X ()g = Z 1 1 R X ()e j2f d: (E.3.39) The Wiener-Khintchine theorem can be proved by expanding equation (E.3.38) and forming a result similar to the method used to derive the right hand side of (E.3.29). Once the right hand side of (E.3.29) is obtained, it must be demonstrated that the term jjR X () 2T ! 0 as T!1. E.4 Statistical Inference In this section, basic denitions and processing procedures are provided so that statistical inferences can be made of data collected from a non-deterministic experiment. The term non-deterministic means that the outcome of an experiment is not known a priori even though the experiment has been performed many times before. A typical example is the measurement of a small voltage across a resistance R with a very precise voltmeter. Due to internally generated random noise caused by thermal electron excitation in the resistor, the voltage measurement will vary from measurement to measurement. In order 224 to characterize the voltage, a statistical model f(xj ) is conjectured where 2 is a member of the parameter space of the probability distribution function (pdf) f(xj): In some PDF models, the parameter space is vector valued meaning 2R n . It is typical that parameters of the PDF are not known and they must be inferred from observations of the data. Denition 31. (Cassella and Berger) The random variables X(u; 1);X(u; 2);:::;X(u;n) are called a random sample of size n from the population f(x) if X(u; 1);X(u; 2);:::;X(u;n) are mutually independent random variables and the marginal pdf or probability mass func- tion (pmf) of each X i is the same function f(x). Alternatively, X(u; 1);X(u; 2);:::;X(u;n) are called independent and identically distributed (iid) random variables with pdf or pmf f(x). Denition 32. (Cassella and Berger) Let X(u; 1);X(u; 2);:::;X(u;n) be a random sample from a population and let T(X(u; 1);X(u; 2);:::;X(u;n)) be a real or vector valued function whose domain includes the sample space u2U. Then, the random variable or random vector Y(u) = T(X(u; 1);X(u; 2);:::;X(u;n)) is called a statistic. The probability distribution of the statistic Y(u) is called the sam- pling distribution of Y(u). It should be emphasized that the statistic T(X(u; 1);X(u; 2);:::;X(u;n)) is not a function of the parameter space 2 . Denition 33. A Sucient Statistic. (Cassella and Berger) A statistic is a suf- cient statistic for if the conditional distribution of the sample X(u) given the value T(X(u)) does not depend on . Denition 33 does not provide a method of determining T(X(u)). It can be shown that both the sample mean and sample variance are sucient statistics from the next denition. 225 Theorem 6. Determination of a sucient statistic via a constant as a func- tion of . (Cassella and Berger) If p(x(u)j) is the joint pdf or pmf of X(u) and q(tj) is the pdf or pmf of T(X(u)), then T(X(u)) is a sucient statistic for if for every u2U, the ratio p(xj)=q(T(x)j) is a constant as a function of . Theorem 6 can be used to verify if a function is a sucient statistic. The sample mean, variance and standard deviations are commonly used sucient statistics. The next theorem, the factorization theorem, can also be used to determine a sucient statistic T(X(u)) for . Theorem 7. The Factorization Theorem. (Cassella and Berger) Let f(xj) de- note the joint pdf or pmf of a sample X(u). A statistic T(X(u)) is a sucient statistic for if and only if there exist functions g(T(x)j) and h(X) such that, for all sample points u2U and all parameter points f(xj) =g(T(x)j)h(x): (E.4.1) The proof of the Factorization theorem can be found in Cassalla and Berger. Clearly, the sucient statistic is contained in the function g(). There are a class of pdf's that belong to what is called an exponential class or family. The Factorization theorem naturally apply to members of the exponential family and thus the following theorem is applicable to members of the exponential family. Theorem 8. The exponential family. (Cassella and Berger) Let x 1 ;x 2 ;:::;x n be iid observations from a pdf or pmf f(xj) that belongs to an exponential family given by f(xj) =h(x)c()exp k X i=1 w i ()t i (x)) ! (E.4.2) where = ( 1 ; 2 ;:::; d ); d<k. Then, T(X(u)) = 0 @ n X j=1 t 1 (X j (u));:::; n X j=1 t k (X j (u)) 1 A (E.4.3) is a sucient statistic for . The Gaussian pdf is a member of the exponential family and by either the Factoriza- tion theorem or writing the Gaussian pdf in the form of (E.4.2), the vector parameter = (; 2 ) can be jointly estimated by T(X(u)) = (T 1 (X(u));T 2 (X(u))) (E.4.4) = (X; S 2 ) (E.4.5) 226 when both parameters are not known. Application of the Factorization theorem to the normal pdf leads to g(tj) =g(t 1 ;t 2 j; 2 ) = 1 (2 2 ) n=2 e [n(t 1 ) 2 + (n 1)t 2 ] 2 2 ; (E.4.6) where t 1 =x = 1 n n X k=1 x k ; and t 2 =s 2 = 1 (n 1) n X i=1 (x 1 x) (E.4.7) and h(x) = 1: (E.4.8) Denition 34. The arithmetic mean of the random sample is called the sample mean denoted as X(u) = 1 n n X k=1 X(u;k): (E.4.9) Denition 35. The sample variance of the complex random sample is the statistic S 2 (u) = 1 (n 1) n X k=1 jX(u;k)X(u)j 2 : (E.4.10) Denition 36. The sample standard deviation S(u) of the complex random sample is the positive square root of the statistic S 2 (u) and thus S(u) = p S 2 (u): E.5 The 2D Jacobian Consider the transformation of a nite region which is mapped to a region where the points (t;s)2 and (;)2 then ZZ f(t;s)dtds = ZZ f(h(;);g(;))jJ(;)jd d; (E.5.1) where t =h(;) s =g(;); =h 1 (t;s) =g 1 (t;s); and the absolute value of the Jacobian is jJ(;)j = abs @t @ @s @ @t @ @s @ ; (E.5.2) 227 where the notation abs x =jxj. The reciprocal of the Jacobian is useful and can be computed from J 1 (;) = @ @t @ @s @ @t @ @s : (E.5.3) E.6 Appendix: Double integrals ZZ f(x;y)dxdy = Z b a Z 2 (x) 1 (x) f(x;y)dydx (E.6.1) = Z d c Z 2 (y) 1 (y) f(x;y)dxdy (E.6.2) E.7 Convergence of a Geometric Sequence Consider the geometric sequence given by S = 1 +x +x 2 + +x n +x n+1 + (E.7.1) = 1 X k=0 x k : (E.7.2) Theorem 9. The geometric sequence S = P 1 k=0 x k is said to be convergent ifjxj < 1 and is divergent otherwise. Proof: Let s n = 1 +x +x 2 + +x n : (E.7.3) Then xs n =x +x 2 + +x n+1 : (E.7.4) Forming the dierence between (E.7.3) and (E.7.4) yields (1x)s n = 1x n+1 (E.7.5) which shows that the partial sum can be written as s n = 1x n+1 1x : (E.7.6) To verify convergence of the geometric series, there are three cases that must be consid- ered: 1. jxj< 1, 2. x = 1, 3. jxj> 1. 228 For case (1), substitutingjxj< 1 into (E.7.6) and taking the limit ofx n+1 shows that the limit of the n + 1 term clearly converges to zero and therefore, lim n!1 s n = 1 1x : (E.7.7) For case (2), substituting x = 1 into (E.7.6) yields the indeterminate form of 0=0 and thus L'Hospital's rule must be used which demonstrates that lim n!1 s n = lim n!1 (n+1) which is clearly divergent. For case (3), using the partial sum in E.7.1 and looking at the x n+1 term in the series shows clearly that in the limit, the partial sum increases without bound sincejxj> 1: In summary, forjxj< 1, the geometric series S = 1 +x +x 2 + +x n +! 1 1x (E.7.8) and is divergent otherwise. 229 Appendix F Generalized Binomial Theorem F.1 Binomial Theorem For any x;y 2 C and n 0, the well known (see the Mathematical Handbook for Engineers in [30] ) binomial expansion of (x +y) n is (x +y) n = n X k=0 n k x nk y k (F.1.1) with n k = n! k!(nk)! : (F.1.2) F.2 Binomial Series The Binomial Theorem was generalized to the binomial series by 26 year old Sir Issac Newton in 1669. He demonstrated that for every real number r, a function of the form f(x) = (1 +x) r , can be expanded into an innite series (1 +x) r = 1 X k=0 r k x rk y k ; jxj< 1 (F.2.1) =x r +rx r1 + r(r 1) 2! x r2 +: Ifr is a positive integer, the series is convergent and the series terminates at the term x r : Abel, later considered the series for x =z2C andr2C and showed that the expansion as demonstrated by Newton was still valid on the unit disk (see Remmert in [47] ). 230 Appendix G Linear Algebra G.1 The Hadamard Product Let A and B denotes two matrices and each having dimension nm. The Hadamard product for matrix multiplication is given by (A B) i;j =A i;j B i;j : (G.1.1) Then, (A A ) i;j =jA i;j j 2 : (G.1.2) 231 Index Admissible set, 23 Almost Sure Convergence, 217 Average complex power, 54 Average power, 223 Cardinality of a set, 15 Cauchy Criterion, 215 Characteristic impedance Frequency-dependent, 47 Chebychev inequality, 216 Complex sample mean, 119 Condition number, 132 Conjugate transpose of a vector, 127 Consistent estimator, 119 Curl-Curl identity, 65 Denitions Almost sure, 217 Convergence in probability, 216 Cross-correlation, 214 Cross-covariance, 214 Mean square convergence, 217 Random Variable, 210 Sigma algebra, 210 Statistical average, 211 Strict sense stationary, 213 Wide sense stationary, WSS, 214 Delivered load voltage, 37 Device under test (DUT), 114 Dirac delta functions, 2 Directivity, 56 Distortionless lter, 39 Distortionless transmission lines, 39 Divergence of the curl of any vector, 64 Divergence theorem, 58 Divergenceless, 65 DUT, 114 Eective length of an antenna, 84 232 Energy, 223 Equivalence class, 22 Ergodic in the mean, 119 Ergodicity, 219 Euclidean norm, 129 Exponential class, 226 Exponential family, 226 Factorization theorem., 226 FCC, 2 Fourier Transform Time shifting property, 15 Gauss theorem, 58 Generalized functions, 2 Hadamard Matrix Product, 231 Hadamard Product, 231 Hilbert Matrix, 137 Ill-conditioned system, 137 Impulse radio, 1 Incident power, 50 Innite impulse response eect, 37 innite-impulse response lter, 36 Inner product denition, 18 IR, 1 Laplacian, 66 Least upper bound, 129 Lorentz condition or potentials, 69 Lossless transmission line, 43 Low loss transmission line, 44 Magnetic potential Divergence of (i.e.,r A), 70 Magnetic Potential A, 57 Match ltering, 57 Mean estimation, 119 Measurement Interval, 6 Measurement Range, 6 Metric Space, 128 Noise equivalent bandwidth, 116 Non-homogeneous wave equation for the scalar potential, 70 Non-homogeneous wave equation for the vector potential, 70 Norm of a vector, 128 Normed space, 128 Null identity, 60 233 Null Identity 2, 64 Order of a set, 15 p-norm, 128 Parseval's theorem, 224 Poisson equation, 63 Potential Magnetic potential, 57 Power, 223 Average complex power, 54 Instantaneous power, 223 Poynting vector, 55 Real power, 55 Spatial power density, 55 Time averaged power, 223 Power spectral density, 224 Power spectral density denition, 224 Power-waves, 112 Poynting vector, 55 Prolate spheroidal functions, 10 Prolate Spheroidal Wave Functions, 9 PSD, 224 Pulse equalization, 18 Pulse-shaping factor, 18 Radiation Intensity Isotropic radiator, 56 Random Process, 5 Real power, 55 Reciprocity theorem, 125 rms value, 224 Sample mean, 216 sample mean, 227 sample standard deviation, 227 Sample variance, 216 sample variance, 227 Sampling Distribution, 225 Scalar potential denition, 61 Scattering matrix, 112 Single antenna method, 122 SLLNs, 217 Spatial pulse ltering, 57 Statistic denition, 225 Stochastic integral, 218, 219 Stokes theorem, 58 Strong Law of Large Numbers, 217 234 Strong Law of Large Numbers (SLLN), 119 Sucient statistic, 225 sup, 129 supremum, 129 Three antenna method, 124 Time-averaged power, 223 Transmission coecient, 37 Transmission line transfer function, 36 Transmission lines, 27 Transmission lines:Initially relaxed con- ditions, 27 Transverse electromagnetic waves, 79 Triangle inequality, 128 Two antenna method, 123 UWB, 1 Vector space, 128 Weak approximation model, 86 Weak Law of Large Numbers, 216 Weak Law of Large Numbers (WLLN), 119 Wide-Sense Stationary noise, 115 Wide-Sense Stationary Process, 5 Wiener-Khintchine theorem, 224 WLLNs, 216 WSS, 5, 115 235
Abstract (if available)
Abstract
Sets of ultra-wideband signals are designed and generated using a mathematical model of a zero-order hold, digital-to-analog converter. An efficiency measure of filling the FCC spectral mask is developed and optimized for various choices of digital-to-analog converter parameters and waveform time duration. The proposed optimization technique takes into account filtering effects in the digital-to-analog converter as well as a measured antenna transfer function characteristic. The research presented here demonstrates that a simple ultra-wideband transmitter can be designed which consists only of a high speed digital-to-analog converter and a suitable ultra-wideband antenna. The waveform design problem is explicitly dependent on the transmitter's equivalent transfer function, the FCC spectral mask constraint, and the ultra-wideband antenna that is included as a factor in the equivalent transfer function of the system.
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University of Southern California Dissertations and Theses
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Digital to radio frequency conversion techniques
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Creator
Lewis, Terry Pernell
(author)
Core Title
An ultrawideband digital signal design with power spectral density constraints
School
Viterbi School of Engineering
Degree
Doctor of Philosophy
Degree Program
Electrical Engineering
Publication Date
07/27/2012
Defense Date
08/27/2011
Publisher
University of Southern California
(original),
University of Southern California. Libraries
(digital)
Tag
carrierless modulation,digital signal synthesis,FCC,FCC mask,impulse radio,mask,OAI-PMH Harvest,ultrawideband,UWB,UWB communications,UWB mask,widebandwidth communications
Language
English
Contributor
Electronically uploaded by the author
(provenance)
Advisor
Scholtz, Robert Arno (
committee chair
), Welch, Lloyd Richard (
committee chair
), Golomb, Solomon Wolf (
committee member
), Haydn, Nicolai T. A. (
committee member
), Weber, Charles (Chuck) L (
committee member
)
Creator Email
terry.lewis@gmail.com,terry.lewisevo@gmail.com
Permanent Link (DOI)
https://doi.org/10.25549/usctheses-c3-73293
Unique identifier
UC11289364
Identifier
usctheses-c3-73293 (legacy record id)
Legacy Identifier
etd-LewisTerry-1042.pdf
Dmrecord
73293
Document Type
Dissertation
Rights
Lewis, Terry Pernell
Type
texts
Source
University of Southern California
(contributing entity),
University of Southern California Dissertations and Theses
(collection)
Access Conditions
The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the a...
Repository Name
University of Southern California Digital Library
Repository Location
USC Digital Library, University of Southern California, University Park Campus MC 2810, 3434 South Grand Avenue, 2nd Floor, Los Angeles, California 90089-2810, USA
Tags
carrierless modulation
digital signal synthesis
FCC
FCC mask
impulse radio
ultrawideband
UWB
UWB communications
UWB mask
widebandwidth communications