# On Detection of Radio Frequency Interference in Spaceborne Microwave Radiometers using Negentropy Approximations and Complex Signal Kurtosis as Test Statistics

код для вставкиСкачатьCURRICULUM VITAE Name: Damon C. Bradley. Secondary Education: George Washington Carver High School of Engineering and Science, Philadelphia, Pennsylvania. June 1997. Collegiate institutions attended: University of Maryland, Baltimore County, Ph.D., Electrical Engineering, 2014. The University of Maryland, College Park, Master of Engineering, 2005. The Pennsylvania State University, B.S., Electrical Engineering, 2002. Major: Electrical Engineering. Professional publications: D. Bradley, J. M. Morris, T. Adali, J. T. Johnson, and A. Mustafa, “On the detection of RFI using the complex signal kurtosis in microwave radiometry,” in to appear in 13th Specialist Meeting on Microwave Radiometry and Remote Sensing of the Environment (MicroRad) 2014, Pasadena, CA, USA, March 2014 D. Bradley and J. Morris, “On the performance of negentropy approximations as test statistics for detecting sinusoidal RFI in microwave radiometers,” IEEE Transactions on Geoscience and Remote Sensing, vol. 51, no. 10, pp. 4945–4951, October 2013 J. Piepmeier, J. Johnson, P. Mohammed, D. Bradley, C. Ruf, M. Aksoy, R. Garcia, D. Hudson, L. Miles, and M. Wong, “Radio-frequency interference mitigation for the soil moisture active passive microwave radiometer,” IEEE Transactions on Geoscience and Remote Sensing, vol. 52, no. 1, pp. 761–775, January 2014 S. Misra, J. Johnson, M. Aksoy, J. Peng, D. Bradley, I. O’Dwyer, S. Padmanabhan, and R. Denning, “SMAP RFI mitigation algorithm performance characterization using airborne high-rate direct-sampled SMAPVEX 2012 data,” in Geoscience and Remote Sensing Symposium (IGARSS), 2013 IEEE International, Melbourne, Australia, July 2013 S. Misra, J. Johnson, M. Aksoy, D. Bradley, H. Li, J. Mederios, J. Piepmeier, and I. O’Dwyer, “Performance characterization of the SMAP RFI mitigation algorithm using direct-sampled SMAPVEX 2012 data,” in Radio Science Meeting (USNCURSI NRSM), 2013 US National Committee of URSI National, Orlando, FL, USA, July 2013, pp. 1–1 D. Bradley, C. Brambora, A. Feizi, R. Garcia, L. Miles, P. Mohammed, J. Peng, J. Piepmeier, K. Shakoorzadeh, and M. Wong, “Preliminary results from the Soil Moisture Active/Aassive (SMAP) Radiometer Digital Electronics Engineering Test Unit (ETU),” in Geoscience and Remote Sensing Symposium (IGARSS), 2012 IEEE International, Munich, Germany, July 2012, pp. 1077 –1080 D. Bradley, C. Brambora, M. Wong, L. Miles, D. Durachka, B. Farmer, P. Mohammed, J. Piepmier, J. Medeiros, N. Martin, and R. Garcia, “Radiofrequency interference (RFI) mitigation for the soil moisture active/passive (SMAP) radiometer,” in Geoscience and Remote Sensing Symposium (IGARSS), 2010 IEEE International, Honolulu, HI, USA, July 2010, pp. 2015 –2018 Professional positions held: Consultant/Digital Technical Lead, Center of Excellence for Tactical & Advanced Communication Technologies, Morgan State University (October, 2011 – Present). Lead Computer Engineer, Instrument Electronics Development Branch, NASA Goddard Space Flight Center (September, 2012 – Present). DSP Technology Group Founder, Leader, Microelectronics and Signal Processing Branch, NASA Goddard Space Flight Center (August, 2008 – August 2012). Electronics Engineer, Microelectronics and Signal Processing Branch, NASA Goddard Space Flight Center (December, 2001 – August 2008). Engineering Intern, Microelectronics and Signal Processing Branch, NASA Goddard Space Flight Center (May, 2001 – August 2001). Engineering Intern, General Electric Transportation Systems (May 2000, August 2000). Engineering Co-Op, The Boeing Company, Expendable Launch Vehicles (May, 1999 – December 1999). Website Designer, The Pennsylvania State University, Eberly College of Science (September 1997 – May 2002). Information Technology Support Staff, Philadelphia College of Osteopathic Medicine (May 1998 – August 1998). Information Technology Support Staff, The School District of Philadelphia (September 1996 – June 1997). ABSTRACT Title of Dissertation: On Detection of Radio Frequency Interference in Spaceborne Microwave Radiometers using Negentropy Approximations and Complex Signal Kurtosis Name of Candidate: Damon C. Bradley, Doctor of Philosophy, 2014 Dissertation directed by: Dr. Joel M. Morris, Professor Department of Computer Science and Electrical Engineering Microwave radiometers are passive, sensitive radio receivers used as spacecraft instruments for Earth remote sensing. Microwave radiometers work by measuring average power of naturally occurring thermal emission from Earth. Radiometer data are used by researchers to monitor Earth’s hydrosphere and a number of important geophysical processes critical to our understanding of and survival on the planet. Besides measuring natural thermal noise, radiometers also measure inadvertent signal emissions from human-made sources, such as surveillance radar, communications systems, reﬂected signals from broadcast satellites, and signals from a wide array of wireless technologies. Signals of this nature are called radio-frequency interference (RFI) and corrupt radiometric measurements, leading to erroneous geophysical retrievals. RFI threatens the utility of radiometer data as a result of this corruption. This research investigates two new approaches for detecting RFI. The ﬁrst approach investigates the suitability of several different negentropy approximations as test-statistics for detecting RFI in the radiometric signal. The second approach combines polarimetric and baseband quadrature signals into a complex signal model and employs a complex signal kurtosis test-statistic to improve RFI detection, relative to the current state of the art that uses only real-valued signal kurtosis. It is shown that various negentropy approximations do not outperform detection performance of the real-valued kurtosis, but does provide insight as to why kurtosis works well. It is also shown that RFI detection in the complex domain using the complex signal kurtosis offers improved detection performance and less data rate, leading to new digital signal processing systems for future spaceborne microwave radiometers. On Detection of Radio Frequency Interference in Spaceborne Microwave Radiometers using Negentropy Approximations and Complex Signal Kurtosis as Test Statistics by Damon C. Bradley Dissertation submitted to the Faculty of the Graduate School of the University of Maryland Baltimore County in partial fulﬁllment of the requirements for the degree of Doctor of Philosophy 2014 UMI Number: 3624329 All rights reserved INFORMATION TO ALL USERS The quality of this reproduction is dependent upon the quality of the copy submitted. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if material had to be removed, a note will indicate the deletion. UMI 3624329 Published by ProQuest LLC (2014). Copyright in the Dissertation held by the Author. Microform Edition © ProQuest LLC. All rights reserved. This work is protected against unauthorized copying under Title 17, United States Code ProQuest LLC. 789 East Eisenhower Parkway P.O. Box 1346 Ann Arbor, MI 48106 - 1346 c Copyright Damon C. Bradley 2014 This work is dedicated to my little nephew, Derrick Robinson (23), whose life was tragically taken during the time of this writing. iii ACKNOWLEDGEMENTS With great humility and gratitude, there are many people and entities that I would like to thank for making the achievement of this work possible. I’d like to thank my mother, Sandra, who single-handedly raised me with some help from my 2 elder brothers and 2 elder sisters. I owe my integrity and work-ethic to their love and teaching, as well as observations of their diligence and tenacity while growing up in difﬁcult situations in South Philadelphia. I would like to thank my research advisor and mentor, Professor Joel M. Morris, and my distinguished committee members, Dr. Edward Kim, Dr. Tulay Adali, Dr. Paul Racette, Dr. Janet Rutledge and Dr. E.F. Charles Laberge. Your technical wisdom and teaching over the years is reﬂected in this dissertation. In addition, I’d like to thank each of you for your ﬂexibility in working with my hectic schedule, and being available to help whenever I needed you. I truly appreciate the job you do every day. I’d like to thank fellow researchers Professor Joel Johnson, Professor Chris Ruf, and Dr. Siddarth Misra for their help and advice throughout this research, and help with providing the real radiometer data to work with. I’d like to thank Mrs. Denise Atkinson and her relentless support of helping me schedule times with Dr. Rutledge via Google Chat. I can’t express how much easier our correspondence over Google Chat made my life. I’d also like to thank Professor Chein-I-Chang, for our arguments on rate-distortion and maximum-entropy proofs, for his hospitality in giving me a comfortable lab cubicle to work in, and for his friendship and advice. I’d like to give thanks to and show humble appreciation for Dr. Jeff Piepmeier from Goddard Space Flight Center, who, through many discussions on how to design the SMAP iv radiometer, motivated my interest in the ﬁeld of RFI detection. It was Jeff who gave me the opportunity to implement the SMAP radiometer signal processing system, and has also gave me opportunities to participate in IGARSS and MicroRad conferences. Jeff, I really appreciate your teamwork and dedication to the ﬁeld of microwave radiometery, and your support throughout my research at work. I would like to thank the NASA Goddard Space Flight Center and their part-timegraduate study program and their Academic Investment for Mission Success program for funding my part-time pursuit of this goal. In particular, I’d like to thank Marsha-DuboseWilliams and Pamela Barrett for making the paperwork for applying to this program every year as painless as possible, while being cheerleaders for my continued success. Also at NASA, I owe many thanks to my former and current supervisors, Bob Kasa, David Sohl, Wes Powell, Phyllis Hestnes, Lavida Cooper, Renee Reynolds, and Jack McCabe, all of whom were supportive for allowing me the time from work to pursue this degree. In addition, I’d like to thank the many great science and math teachers that I’ve had who guided my inherent interest in the ﬁeld. In particular, I’d like to thank Dorothy Sloan from the Carver High School of Engineering of Science who gave me a calculus book that allowed me to teach myself, long before I needed the subject. I’d like to thank Robert Welsh, a.k.a. callsign N3RW, who taught me the concept of Fourier analysis, which got me a ﬁrst place math award in high school, and led me to linear systems and signals, taught at Penn State by Jeff Schiano. I’d very much like to thank Dr. Schiano, for one of the most challenging, important and interesting courses in linear systems theory. I’ve learned to emulate the example Dr. Schiano, of teaching extemporaneously, rigorously proving theorems and deriving transforms, and showing up at the EE-West lab at 3 o’clock in the morning to conduct diligent research when everyone else was too tired to continue. These teachers helped me build my foundation as a practicing engineer, and I thank them for it. v I have had a considerable number of friends also supporting me throughout this academic journey. My classmates, who became dear friends, provided me the company that misery loves, and the encouragement that I often needed when times were difﬁcult. To my dear friends, who I consider my brothers, Dr. Mark Wong and Dr. Pedro Rodriguez, Dr. Albert Kir, Dr. Ganesh Saiprasad and my sister, Dr. Haleh Safavi, have all shared one or more classes with me, forced me to stay on my “A-game”, but more importantly, provided me with their rich, diverse experience that ultimately enriched my academic career at UMBC and enriched my life overall. To these dear friends, I thank you so much for studying qualifying exams with me, copying my homework solutions, sharing Matlab code, and just simply listening to me vent for the last several years. I am humbled by your friendship. In 2009, I was diagnosed with having a 4.5cm benign meningioma in my frontal lobe, which was removed in 2010 by my friend and internationally famous neurosurgeon, Dr. Alfredo Quininones-Hinojosa. Dr. Q, I simply thank you for saving my life, or as you would put it, using your hands as tools of God to save my life. Your humility, perseverance and strength have been a great motivator and a reminder of my own background, and the importance to keep on pushing, especially in the face of adversity and insurmountable odds. But the person who truly saved my life, who made me make the call to the doctor after I bumped my head in that fateful basketball game at Johns Hopkins APL basketball court, and who has been my most ardent supporter for the last 15 years, my dear wife, Ella. Sweetheart, this degree would not be possible without you. This work is as much yours as it is mine. It is impossible for me to express my thanks and my sheer gratitude for your countless days and nights watching our beautiful girls, taking care of our house, our paperwork, and all of the many functions of life outside of academia. I thank you for waiting up for me to return home after many 14+ hour days at work and at school. I thank vi you for simply spending time with me as I worked, well into the night, while falling asleep. You have been my rock, you have kept me sane, you have made this work possible, and you have picked up the slack in building our family when duties forced me to be there less than 100% of the time. I am eternally thankful for absolutely everything you have done to support my pursuit of this research. I love you. I’d also like to thank my mother-in-law, Kathy, who helped babysit my kids and helped keep my wife sane during the many hours I spent away from home working at NASA and at UMBC. Last but not least, I wish to acknowledge my three brilliant, beautiful, and loving little monsters, Olivia, Alexis and Evelyn. You have always brought me joy and happiness since I started this degree. Olivia, you were born while Dr. Morris was joking about how difﬁcult it would be for me while raising you and taking his Estimation and Detection theory class. Alexis, you were born right after I passed my PhD qualiﬁers, so thank you for being patient and arriving after I passed these exams. Evelyn, thank you for sleeping through the night early-on, after falling asleep in my arms as I ﬁnished my dissertation work, and the last milestone of this degree. To my girls, I am blessed to be your father. I am truly proud of you, and I hope that I have made you proud of me as a result of this dissertation. vii Contents LIST OF TABLES xiii LIST OF FIGURES xv LIST OF ABBREVIATIONS xxiii 1 INTRODUCTION 1 1.1 Research Goals and Motivation . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Main Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 Organization of Dissertation . . . . . . . . . . . . . . . . . . . . . . . . . 6 Chapter 2 BACKGROUND 2.1 2.2 2.3 9 Microwave Radiometer Physics . . . . . . . . . . . . . . . . . . . . . . . . 9 2.1.1 Thermal Emission . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.1.2 Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.1.3 Statistical Characterization of Thermal Noise . . . . . . . . . . . . 15 2.1.4 Brightness Temperature and Antenna Temperature Relationships . . 17 Microwave Radiometers . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.2.1 Total Power Radiometers . . . . . . . . . . . . . . . . . . . . . . . 19 2.2.2 Polarimetric Radiometers . . . . . . . . . . . . . . . . . . . . . . . 20 Radiometer Power and Resolution . . . . . . . . . . . . . . . . . . . . . . 21 viii 2.4 2.5 Digital Radiometers and Signal Processing Considerations . . . . . . . . . 22 2.4.1 Signal Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.4.2 Digital Signal Processing . . . . . . . . . . . . . . . . . . . . . . . 24 2.4.3 Super Heterodyne Digital Radiometers . . . . . . . . . . . . . . . . 25 2.4.4 Radiometer Operation . . . . . . . . . . . . . . . . . . . . . . . . 26 2.4.5 Radiometer Example: The Soil-Moisture Active-Passive Instrument 28 RFI and Mitigation Approaches . . . . . . . . . . . . . . . . . . . . . . . . 33 2.5.1 Prevention . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.5.2 Statistical Detection . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2.5.3 Time-Frequency Detection . . . . . . . . . . . . . . . . . . . . . . 40 2.5.4 Polarization Detection . . . . . . . . . . . . . . . . . . . . . . . . 41 Chapter 3 HISTORICAL RFI DATA AND DEVELOPMENT OF A COMPLEXVALUED RFI SIGNAL MODEL 48 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.2 RFI Model from Microwave Radiometer Frequency Allocations . . . . . . 50 3.3 Historical Examples of RFI . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.4 General RFI Signal Model for L-band . . . . . . . . . . . . . . . . . . . . 53 3.5 RFI Signals from Complex Digital Modulation Model . . . . . . . . . . . . 57 3.6 3.5.1 S, C, X - Bands . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 3.5.2 Ku , K-band, Ka Bands . . . . . . . . . . . . . . . . . . . . . . . . 59 Development of Polarized RFI Signal Model . . . . . . . . . . . . . . . . . 59 3.6.1 SMAP Validation Experiment (SMAPVEX12) Data Set . . . . . . . 60 3.6.2 Canton Data Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 3.6.3 GREX Data Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 ix 3.6.4 3.7 Polarized Complex RFI Signal Model . . . . . . . . . . . . . . . . 68 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . 71 Chapter 4 DETECTION OF SINUSOIDAL RFI USING NEGENTROPY APPROXIMATIONS 73 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 4.2 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 4.3 Review of Gaussianity Tests . . . . . . . . . . . . . . . . . . . . . . . . . 77 4.4 Negentropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 4.4.1 Negentropy Approximations . . . . . . . . . . . . . . . . . . . . . 80 4.4.2 Histogram-Based Approximation . . . . . . . . . . . . . . . . . . . 82 4.4.3 Edgeworth Approximation . . . . . . . . . . . . . . . . . . . . . . 82 4.4.4 Non-polynomial Function-Based Approximations . . . . . . . . . . 85 4.5 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 4.6 Large-Sample Behavior of Negentropy Approximations . . . . . . . . . . . 88 4.7 Performance of Negentropy Approximations for// Detection of Single Sinusoidal-Source RFI . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 4.8 4.7.1 Performance Under CW RFI Sinusoid Signal Model . . . . . . . . 91 4.7.2 Performance Under Pulsed RFI Sinusoid Signal Model . . . . . . . 93 4.7.3 Inﬂuence of the Number of Samples . . . . . . . . . . . . . . . . . 96 Performance for the Multi-PCW Case . . . . . . . . . . . . . . . . . . . . 98 4.8.1 Convergence of Multiple-PCW RFI to a Gaussian Probability Density Function . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 4.8.2 4.9 ROC Performance for Multiple PCW Interference Case . . . . . . . 108 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . 113 x Chapter 5 DETECTION OF RFI USING COMPLEX SIGNAL KURTOSIS COEFFICIENTS 117 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 5.2 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 5.3 Complex Gaussian Detector . . . . . . . . . . . . . . . . . . . . . . . . . . 121 5.4 Baseband Quadrature RFI Detection . . . . . . . . . . . . . . . . . . . . . 122 5.4.1 RFI Signal Models Considered for Complex Baseband RFI Detection123 5.4.2 RFI Detection Results Using Kurtosis of the Complex Quadrature Signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 5.5 5.6 Polarimetric RFI Detection . . . . . . . . . . . . . . . . . . . . . . . . . . 137 5.5.1 Bivariate and Complex Gaussian Noise Relationship . . . . . . . . 137 5.5.2 Polarimetric Detection Performance Results . . . . . . . . . . . . . 140 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . 144 Chapter 6 CONCLUSIONS 146 6.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 6.2 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 6.3 6.2.1 RFI Signal Models and Central-Limit Effects . . . . . . . . . . . . 152 6.2.2 Negentropy-based Test-Statistics for Detection of RFI . . . . . . . . 154 6.2.3 Complex-Valued Kurtosis - Based Test-Statistics for Detection of RFI155 6.2.4 Discussion of Implementation Costs vs. Performance . . . . . . . . 156 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 6.3.1 Additional Polarized RFI Study . . . . . . . . . . . . . . . . . . . 159 6.3.2 Higher-Order Spectral Analysis for Radiometric RFI Detection . . . 160 6.3.3 Sliding-Window Kurtosis Detection . . . . . . . . . . . . . . . . . 162 xi Appendices Appendix A: 164 Proof that the Continuous Gaussian Distribution Maximizes Differential Entropy using Lagrange Multipliers Appendix B: 164 ROC Performance Tables for Negentropy Approximations and Complex Signal Kurtosis Statistics 167 B.1 Negentropy-Based Test Statistics . . . . . . . . . . . . . . . . . . . . . . . 168 B.1.1 Single PCW RFI Signal Model . . . . . . . . . . . . . . . . . . . . 168 B.1.2 Multiple PCW RFI Signal Model . . . . . . . . . . . . . . . . . . . 169 B.2 Complex Signal Kurtosis-Based Test Statistics . . . . . . . . . . . . . . . . 170 B.2.1 Complex Baseband Signal Model . . . . . . . . . . . . . . . . . . 170 B.2.2 Complex Polarized Signal Model . . . . . . . . . . . . . . . . . . . 171 Appendix C: Environmental Data Records and Passive Frequency Allocations 172 REFERENCES 174 xii LIST OF TABLES 2.1 Summary of data products for RFI mitigating radiometer. . . . . . . . . . . 31 3.1 Potential sources of X and K-band RFI from various Geostationary DirectBroadcast Services (DBS) Satellites. Because of their wide transmit swaths, transmit power, and operating frequency, transmissions from these satellites can reﬂect off of the ocean surface and reach the input of a K-band microwave radiometer. These data are available at http://www.remss.com/about/projects/radio-frequency-interference. . . . . . 56 3.2 Parameters of Multi-Pulsed Sinusoid RFI Model. The notation U means uniform distribution, wk is the pulse-width k, T is the radiometer integration period, and 0 ≤ ξ ≤ 1. . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.3 Properties of Canton Data Set . . . . . . . . . . . . . . . . . . . . . . . . . 67 3.4 Properties of GREX Data Set . . . . . . . . . . . . . . . . . . . . . . . . . 67 4.1 Summary of Negentropy-Based Test Statistics Studied . . . . . . . . . . . . 76 4.2 Negentropy Approximations Studied . . . . . . . . . . . . . . . . . . . . . 88 4.3 Parameter set values Λ for RFI signal model s(n, Λ) . . . . . . . . . . . . . 92 4.4 Summary of best and worst PCW RFI detectors in terms of PD (PF = 0.05) 114 4.5 Summary of best and worst Multiple PCW RFI detectors in terms of PD (PF = 0.05) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 xiii 5.1 Signal and parameters Λ chosen for the baseband complex RFI detection simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 5.2 Summary of best and worst complex baseband RFI test-statistics in terms of PD (PF = 0.05). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 5.3 Summary of best and worst polarized complex RFI test-statistics in terms of PD (PF = 0.05) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 6.1 Negentropy Approximations Studied . . . . . . . . . . . . . . . . . . . . . 148 B.1 Single PCW RFI Source Receiver Operating Charcteristic Results: AUC and PD for PF = 0.05, PF = 0.10. . . . . . . . . . . . . . . . . . . . . . . 168 B.2 Multiple PCW RFI Source Receiver Operating Charcteristic Results: AUC and PD for PF = 0.05, PF = 0.10. . . . . . . . . . . . . . . . . . . . . . . 169 B.3 Complex Baseband RFI Detection ROC Performance Analysis: AUC and PD for PF = 0.05, PF = 0.10. . . . . . . . . . . . . . . . . . . . . . . . . 170 B.4 Receiver Operating Charcteristic Results for Polarimetric RFI test-statistics: AUC and PD for PF = 0.05, PF = 0.10. . . . . . . . . . . . . . . . . . . . 171 C.1 Example Environmental Data Records and their acronyms resulting from retrievals in microwave radiometry. . . . . . . . . . . . . . . . . . . . . . . 172 C.2 Frequency allocations for passive remote sensing according to the National Academic Press [8] and the International Telecommunication Union Recommendation ITU-R RS.1029-2. Allocations in bold highlight protected frequency bands exclusive to passive remote sensing . . . . . . . . . . . . . 173 xiv LIST OF FIGURES 2.3 Modern Spaceborne Radiometers as of April 2014 . . . . . . . . . . . . . . 19 2.1 Three-dimensional schematic of an electromagnetic wave and its projection on a plane intersecting its direction of propagation. The wave here is circularly polarized, which is a special case of the more general elliptical polarization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 2.2 Polarization Ellipse. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 2.4 Total-power radiometer. This is also known as a switching radiometer [9]. The antenna receives the total electromagnetic ﬁeld radiated from Earth E(r, t). Radiometric measurements consist of alternating antenna-only measurements (which we focus on in this dissertation), and reference load measurements. The resulting signal is band-limited using an RF bandpass ﬁlter HB (f ). The resulting signal is called the pre-detected signal x(t). The average power P of x(t) is periodically computed over T seconds. . . . . . 44 2.5 Polarimetric Radiometer. . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 2.6 Digital superheterodyne total-power radiometer. . . . . . . . . . . . . . . . 45 2.7 Superheterodyne polarimetric radiometer . . . . . . . . . . . . . . . . . . . 45 2.8 Example Radiometer Timing Diagram. . . . . . . . . . . . . . . . . . . . . 45 2.9 Relative brightness sensitivities versus frequency for atmospheric and ocean environmental data records. National Academic Press [8]. . . . . . . 46 xv 2.10 Relative brightness sensitivities versus frequency for land related environmental data records. National Academic Press [8]. . . . . . . . . . . . . . . 47 2.11 Graph of the various approaches taken to detect and mitigate RFI. . . . . . 47 3.1 Soil-Moisture retrievals from the Soil Moisture Ocean Salinity (SMOS) L-band radiometer, launched in November 2009. RFI is so severe that most of Europe appears as a blind spot where soil-moisture cannot be measured. 3.2 52 Possible C-band RFI reported by the AMSR-E radiometer reported by JAXA [10]. Red corresponds to retrieved brightness temperatures that are abnormally high, well above the average of 300K which is expected for Earth. 53 3.3 K-band RFI observed by SSM/I, WindSat and AMSR-E radiometers in January 2009. SSM/I (top panel), WindSat (middle panel), and AMSR-E (Bottom panel) [11] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 3.4 On the very ﬁrst orbit of the Aquarius L-band radiometer, in which the data system was switched on, RFI was observed over China. Aquarius was launched on June 10, 2011. . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.5 SMAPVEX Flight Path Over Denver, CO, USA. The colorbar indicates the relative brightness temperature observed. Red areas indicate high intensity RFI sources. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 3.6 SMAPVEX12 Combined Sources 1. Signal contains a narrowband CW, wideband CW and a narrowband pulse sources. . . . . . . . . . . . . . . . 63 3.7 SMAPVEX12 Combined Sources 2. Signal contains a narrowband pulse, wideband pulse and a narrowband CW sources. . . . . . . . . . . . . . . . 64 xvi 3.8 SMAPVEX12 Narrowband CW. The H-Pol and V-Pol signal amplitudes are close to each other on average, but generally not equal, resulting in approximate Gaussian pdf s and a circular scatter plot. The CW signal present is not obvious in the time or statistical domains, but is obvious in the joint time-frequency domain plot. Furthermore, the RFI occurs at the same frequency in both polarization channels. . . . . . . . . . . . . . . . . 65 3.9 Air-Route Surveillance Radar in Canton, MI. . . . . . . . . . . . . . . . . 66 3.10 A single radar pulse of the Canton data set. . . . . . . . . . . . . . . . . . 68 3.11 Flight Path of GREX October 6, 2012 experiment. . . . . . . . . . . . . . . 69 3.12 GREX data set, showing self-imposed RFI at every 25 MHz due to an onboard clock from an ethernet controller. This RFI is most apparent in the two power spectral density subplots in the ﬁgure. . . . . . . . . . . . . 70 4.1 Total-power radiometer. RFI detection depends on a ﬁnite N -sample window of x(n). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 4.2 Behavior of the various Gaussian test statistics as N → ∞. The sample means and standard deviations of all negentropy approximations tend to zero and trend similarly, except Jh , which converges slower than the other negentropy-based test-statistics . . . . . . . . . . . . . . . . . . . . . . . . 89 4.3 ROC performance of Ji (x̃) for parameter sets Λ1 , . . . , Λ4 . The dash-dotted lines indicate PF = 0.05 and PF = 0.10. . . . . . . . . . . . . . . . . . . . 93 4.4 Detection Probabilities for the false alarm cases PF = 0.05 and PF = 0.10 and AUC for parameter sets Λ1 . . . Λ4 xvii . . . . . . . . . . . . . . . . . . . . 94 4.5 ROC performance of Ji (x̃) for parameter sets Λ5 , . . . , Λ8 . The AndersonDarling (AD) and Shapiro-Wilk (SW) tests are included for comparison for Λ5 , where d = 50%. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 4.6 Detection Probabilities for the false alarm cases PF = 0.05 and PF = 0.10 and AUC for parameter sets Λ5 . . . Λ8 4.7 . . . . . . . . . . . . . . . . . . . . 96 ROC performance of Ji (x̃) for Λ9 , . . . , Λ12 . Je and JqB perform better than all other Ji for d ≤ 1%. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 4.8 Detection Probabilities for the false alarm cases PF = 0.05 and PF = 0.10 and AUC for parameter sets Λ9 . . . Λ12 . . . . . . . . . . . . . . . . . . . . 98 4.9 ROC performance of Ji (x̃) for N = {3k, 10k, 30k, 100k} samples. Kurtosis outperforms all Ji , Jh has the worst performance, and for 100k, all detectors except Jh perform generally well. . . . . . . . . . . . . . . . . 99 4.10 Detection Probabilities for the false alarm cases PF = 0.05 and PF = 0.10 and AUC for parameter sets Λ13 . . . Λ16 . . . . . . . . . . . . . . . . . . . . 100 4.11 Clockwise from the top left: time, joint time-frequency, amplitude histogrambased pmf,. and power spectral density plots of the RFI model signal x(n), with no noise present. M = 2. . . . . . . . . . . . . . . . . . . . . . . . . 103 4.12 Clockwise from the top left: time, joint time-frequency, amplitude histogrambased pmf,. and power spectral density plots of the RFI model signal x(n), with no noise present. M = 20. . . . . . . . . . . . . . . . . . . . . . . . . 104 4.13 Clockwise from the top left: time, joint time-frequency, amplitude histogrambased pmf,. and power spectral density plots of the RFI model signal x(n), with no noise present. M = 100. . . . . . . . . . . . . . . . . . . . . . . . 105 xviii 4.14 Clockwise from the top left: time, joint time-frequency, amplitude histogrambased pmf, and power spectral density plots of the RFI model signal x(n), with no noise present. M = 1000 . . . . . . . . . . . . . . . . . . . . . . . 106 4.15 Mean-Squared Error between s(n) and Gaussian ﬁt to s(n) versus M . Each data point corresponds to a single Monte-Carlo trial with M interfering signals present. As M increases, the MSE decreases rapidly, particularly between 10 and 20 interfering signals. However, MSE rate of decrease slows beyond 20 interfering signals, suggesting that the pdf of s(n) converges slowly to Gaussian. . . . . . . . . . . . . . . . . . . . . . . . . 107 4.16 Convergence of test statistic values to their Gaussian pdf equivalent values in the absence of radiometric noise. . . . . . . . . . . . . . . . . . . . . . . 108 4.17 Log-Log-scale plot of the convergence of test statistic values to their Gaussian pdf equivalent values in the absence of radiometric noise. . . . . . 109 4.18 ROC performance curves for the six negnetropy-based detectors and kurtosis for 1,2,5, and 10 sinusoidal interferers in Gaussian noise. In all cases, the kurtosis has the highest detection probability given any false alarm probability, and thus the best detection performance. All of the negentropy-based detectors except Jh tend to cluster in their performance, still with a higher detection probability than false alarm probability, but not as good as kurtosis. The histogram-based approximation of negentropy suffered the worst detection performance, having near equal detection and false alarm probabilities for the entire ROC curve. . . . . . . . . . . . . . . 110 xix 4.19 ROC performance curves for the six negnetropy-based detectors and kurtosis for 20, 50, 100, and 200 sinusoidal interferers in Gaussian noise. Kurtosis still outperforms all other detectors in this case except for the case where M = 200. However, all of the detectors have poor performance here because their detection and false alarm probabilities are nearly equal for every ROC curve. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 4.20 ROC performance curves for the six negnetropy-based detectors and kurtosis for 500 and 100 sinusoidal interferers in Gaussian noise. Again, kurtosis appears to perform slightly better in terms of detection vs. false-alarm performance. However, in this case, we expect all detectors to approach the PF − PD line because of the convergence of pdf of s(n) to Gaussian. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 5.1 Polarimetric Radiometer. The polarimetric radiometer consists of two identical signal processing channels for the horizontal and vertical polarization component signals. The pre-detected and digitized radiometer signals are xH (n), xV (n), and their corresponding complex baseband representations are IH (n) + jQH (n) and IV (n) + jQV (n). The complex signals z1 (n) = I(n) + jQ(n) and z2 (n) = xH + jxV (n) are studied for RFI detection in the complex domain. H-Pol and V-Pol subscripts are dropped on I(n) and Q(n) since the same sort of processing is applied to either baseband channel. 119 5.2 Total power radiometer with quadrature downconversion. . . . . . . . . . . 123 xx 5.3 Radiometer signal model for complex baseband RFI detection. We assume an L-band radiometer with input noise N (0, σ 2 ) with IF bandpass ﬁlter centered at 24 MHz with 12 MHz on each side. The ﬁltered signal x(n) is mixed down to complex baseband, and subsequently ﬁltered by a pair of identical image-rejection ﬁlters, producing the signal z1 (n) = I(n) + jQ(n).124 5.4 CW-Model RFI performance. CW signal amplitudes are (a) 0.75, (b) 0.60, (c) 0.45, and (d) 0.30. Noise variance is always unity. In all cases, Γ outperforms all other test-statistics, but only slightly outperforming β in terms of the ROC. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 5.5 CW-Model RFI performance ROC analysis results. The AUC and PF (PD ) values for Γ are higher than that for β, but are very close to each other in all cases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 5.6 Pulsed-CW RFI performance. Duty cycles of 50%, 25%, 10%, and 1% as shown for (a) – (d). In (a), we note that Γ has a detection blind-spot similar to the real kurtosis. In (b) - (d), Γ has a slightly better ROC performance than the other RFI detectors. . . . . . . . . . . . . . . . . . . . . . . . . . 129 5.7 ROC analysis results for the pulsed-CW model case. Again, the AUC and PF (PD ) values for Γ are higher than that for β, but are very close to each other in all cases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 5.8 ROC performance for complex detector vs. real kurtosis and sum of kurtosis for real and imaginary component signals for digital modulation models. (a) BPSK RFI, (b) QPSK, (c) 16-QAM (circular), (d) 16-QAM (rectangular). Noise variance is unity in all cases. . . . . . . . . . . . . . . 131 xxi 5.9 ROC analysis results for the digital modulation model RFI case. The AUC and PF (PD ) values for Γ are higher than that for β, but are very close to each other in all cases. It is interesting to note that for rectangular 16QAM, AUC=1.000, and PF (PD ) = 1, but for circular 16-QAM, detection performance is poor. for all detectors. This is due to the implementation of the circular 16-QAM signal having a much lower SNR than the rectangular case in our implementation. . . . . . . . . . . . . . . . . . . . . . . . . . . 132 5.10 ROC performance for complex detector vs. real kurtosis and sum of kurtosis test-statistics for real and imaginary component signals for the pulsed-CW case, with increasing number of samples N : (a) N=10K, (b) N=20K, (a) N=50K, and (d) N=100K. . . . . . . . . . . . . . . . . . . . . 133 5.11 ROC analysis for increasing number of samples N : (a) N=10K, (b) N=20K, (a) N=50K, and (d) N=100K. As we expect, ROC performance improves as we increase N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 5.12 Signal model for polarimetric RFI. The signal s(n) is corrupted with complex Gaussian noise with parameters μ = 0, σ, and ρ. The complex signal is then separated into its real and imaginary parts, and then fed into the radiometer RFI processor in the form of z(n) = xH (n) + xV (n). . . . . 141 5.13 Polarized RFI detection results. (a) All detectors perform similarly, with the Γ statistic outperforming all others. (b), the real kurtosis of the H-Pol channel has a slightly better ROC than Γ, since the amplitude of the RFI is higher in that channel than the V-Pol. (c) Same situation in (b), swapping H-Pol and V-Pol channels (d) RFI signal amplitude for both channels are equal, and approximately half of that in (a) results in a ROC where RFI is undetectable. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 xxii 5.14 ROC analysis for the polarimetric RFI signal model (a) AH = AV = 0.71. Γ has the best AUC of 0.999. (b) AH > AV . The AUC κre is slightly better than Γ (c) AH < AV , The AUC κim is slightly better than Γand (d) AH = AV = 0.18. RFI is nearly undetectable all with an AUC close to 0.50. 143 6.1 Variance-normalization circuit. The signal x(n) streams in at a clock rate fclk and is split into three parallel paths. The ﬁrst path delays the signal by N samples, the second computes the mean and then squares the mean, and the third computes the square of the samples, and then averages the squares of the samples. The difference between the squared mean and mean squares is computed, yielding the signal variance. The variance is inverted, and then used as a constant factor to multiply every sample of the delayed x(n) for the current N samples. The process repeats every N samples. . . . . . . . . 158 6.2 Radiometer timing diagram. . . . . . . . . . . . . . . . . . . . . . . . . . 162 6.3 Sliding-window kurtosis concept. . . . . . . . . . . . . . . . . . . . . . . . 163 xxiii LIST OF ABBREVIATIONS AD . . . . . . . Anderson-Darling Test APSK . . . . . Amplitude and Phase-Shift Keying AUC . . . . . . Area under the curve BPSK . . . . . Binary Phase-Shift Keying CASPER . . . Collaboration for Astronomy Signal Processing and Electronics Research CW . . . . . . Continuous-Wave DBS . . . . . . Direct-Broadcast Satellite DSP . . . . . . Digital Signal Processor, Digital Signal Processing DVB-S . . . . . Direct Video Broadcast Standard DVB-S2 . . . . Second Direct Video Broadcast Standard EDR . . . . . . Environmental Data Record ESA . . . . . . European Space Agency FFT . . . . . . Fast Fourier Transform FPGA . . . . . Field-Programmable Gate Array GREX . . . . . Goddard Radio-Frequency Explorer I . . . . . . . . In-Phase IBOB . . . . . Integrated Break-Out Box ICA . . . . . . Independent-Component Analysis IF . . . . . . . Intermediate Frequency ITU . . . . . . International Telecommunications Union JAXA . . . . . Japan Aerospace Exploration Agency JqB . . . . . . Jarque-Bera Test MAC . . . . . Multiply-and-Accumulate i MPCW . . . . Multiple Pulsed Continuous-Wave NASA . . . . . National Aeronautics and Space Administration PCW . . . . . Pulsed Continuous Wave PRI . . . . . . Pulse Repetition Interval PSD . . . . . . Power Spectral Density Q . . . . . . . Quadrature QAM . . . . . Quadrature Amplitude Modulaton RFI . . . . . . Radio Frequency Interference ROC . . . . . . Receiver Operating Characteristic rQAM . . . . . Rectangular Quadrature Amplitude Modulation RTE . . . . . . Radio-Thermal Emission SK . . . . . . . Spectral Kurtosis SMAP . . . . . Soil-Moisture Active Passive SMAPVEX12 . SMAP Valadation Experiment 2012 SMOS . . . . . Soil-Moisture Ocean Salinity SNR . . . . . . Signal-to-Noise Ratio SW . . . . . . Shapiro-Wilk Test VCM . . . . . Varible Coding and Modulation ii Chapter 1 INTRODUCTION 1.1 Research Goals and Motivation Microwave radiometers are passive microwave receivers used aboard spacecraft for remotely sensing the Earth. The task of a radiometer is to sense naturally occurring thermal noise from the Earth, as the average power of this noise is used to infer various geophysical properties involving the hydrosphere. A threat to these measurements is inadvertent interference from signals that are also sensed by the radiometer and result from human-made sources. Radio frequency interference (RFI) corrupts the thermal noise signal received by the radiometer, and leads to erroneous measurements of the Earth’s hydrosphere that may be unknowingly used by the geoscience community. RFI degrades the utility of radiometric measurements. Use of corrupted data leads to erroneous geophysical parameter retrieval, which has negative consequences for climate and weather modeling, water and drought monitoring, and prediction of the Earth’s water cycle. The primary goal of this dissertation is to improve the quality of remote sensing data delivered by spaceborne microwave radiometers by helping to detect and mitigate RFI that they receive. This goal is achieved by applying new test statistics to the RFI 1 2 detection problem for different microwave radiometer types. These test statistics include approximations of negentropy and the complex signal kurtosis. Though these statistics are are commonplace in the signal processing community, they are unfamiliar in the geoscience and remote sensing community and have never been applied to RFI detection in microwave radiometers. This dissertation was inspired by the author’s lead role in developing the Soil-MoistureActive-Passive (SMAP) radiometer signal processing electronics subsystem for NASA Goddard Space Flight Center (GSFC) and NASA Jet Propulsion Laboratory (JPL) since August 2008. The SMAP observatory contains an L-band passive microwave radiometer, and an L-band synthetic aperture radar. The project is led by NASA-JPL. SMAP is NASA’s latest spaceborne radiometer scheduled to launch in 2014. This work involved the complete development, including modeling, simulation, implementation and testing of the signal processing on the radiometer digital electronics (RDE) subsystem. The SMAP-RDE will be the most advanced microwave radiometer ever launched. Despite the SMAP radiometer being so advanced, it was identiﬁed that there were places in the RDE signal processing ﬂow where complex-valued signal processing could be employed, where it was not currently being used. Since future polarimetric radiometers will likely be designed similar to SMAP, this research was pursued because radiometric signals naturally admit to complex-valued signal processing and could be advantageous on these future instruments. In addition, since the entire principle of radiometric RFI detection is based on detection of non-Gaussianity, the author realized that negentropy, which is used in information theory, could be used for RFI detection as well. The ideas of using complex-valued signal processing – a statistic that’s based on the complex signal kurtosis signals in particular, and negentropy-based test statistics were then proposed to be the foundation of this dissertation work. 3 A secondary goal of this dissertation is to extend the ﬁeld of statistical and complexvalued signal processing to RFI detection for future instruments. Currently, the SMAP observatory contains the most advanced spaceborne radiometer digital receiver that will ever be ﬂown. It is advanced because it digitally processes the pre-detected radiometer signal – essentially a frequency shifted and bandlimited version of the radiometer signal, as opposed to the post-detected signal – the average power waveform that results after applying a square-law detector to the pre-detected signal. Since the sample rate of the pre-detected signal is orders of magnitude higher than the post-detected signal, it must use FPGA-based DSP to perform the signal processing in-orbit. In concert with the in-orbit processing, ground-based signal processing uses the telemetered data from the SMAP observatory to perform RFI mitigation. SMAP is essentially a spectrum analyzer that computes the ﬁrst four raw sample moments on all of the subband signals. During the time of its design, no literature existed that treated the fact that complex-valued digital signals would be generated in the radiometer. The design stood upon the work of DeRoo, Ruf, and Misra, [12], that only treated the pre-detected radiometer signal as a real-valued signal and did not consider the issue of quadrature downconversion to baseband. Downconversion and decimation were necessary for practical engineering reasons, but this led to complex-valued signals on the spacecraft that could carry RFI, because they originated from their real-valued counterparts at the input of the receiver. In addition, the SMAP design included two identical digital receivers, one for each polarization component signal. It was this situation that lent itself to the idea of considering how the tools, developed by Professor Tulay Adali and colleagues in the complex-valued signal processing community [13–15], could improve the processing of the downconverted radiometer signal and the polarimetric radiometer signal. The ﬁnal goal for this dissertation research was to establish a foundation for treating RFI 4 detection in polarimetric and interferometric radiometers in future research. In particular, interferometric radiometers are generalizations of polarimetric radiometers because they many identical polarimetric receivers. They achieve spatial and polarization-diverse measurements of the Earth, and they’re also subject to RFI. Little research exists on mitigation of RFI for interferometric radiometers, but it is believed that this idea can be generalized to multiple interferometric channels. We contribute scalar complex-valued signal processing in this dissertation, but vector complex-valued extensions of the results presented here can be applied to the interferometric case would use vector formulations, for which complex-valued vector signal processing models currently exist [16]. In addition, other generalizations of this contribution could extend to quaternion-valued signal processing, because the four component signals of two downconverted polarimetric signals can be thought of as a single quaternion-valued signal [17]. 1.2 Main Contributions This dissertation makes two contributions to the area of radiometric RFI detection in the ﬁeld of geoscience and remote sensing. The ﬁrst contribution introduces several negentropy-based test statistics negentropy-based test statistics to the list of statistical RFI detectors, but shows that these are generally unsuitable for RFI detection when compared to the kurtosis test statistic. Second, this dissertation extends complex-valued signal processing to microwave radiometery in the baseband and polarimetric signals of a radiometer, to improve RFI detectability. Once RFI is detected, its effect on resulting radiometric brightness temperature data can be mitigated via excision. The common method for mitigating RFI is to simply discard signal samples thought to contain RFI. 5 The ﬁrst contribution explores the use of negentropy, a quantity originating from information theory, as a test statistic for detecting RFI using the radiometer signal. Negentropy is used in a class of radiometers called total-power radiometers, which are the simplest type of radiometer that do not measure the polarization state of the received electromagnetic radiation. A negative result is shown, however, that although negentropy has attractive theoretical properties, it has limited utility when compared to another test statistic, kurtosis, that is currently used on aircraft and and a new spaceborne radiometer currently under development by NASA. It is shown that negentropy-based approximations have a detection performance that is sub-par to kurtosis, with a few minor exceptions, but these approximations are more complicated and more difﬁcult to implement on spaceborne radiometers than kurtosis. Nonetheless, insight is gained from this research because of the relationship between kurtosis and negentropy. The second contribution of this dissertation interprets polarimetric and baseband signals that appear in the radiometer as complex signals, and applies the complex signal kurtosis to each of these cases to detect RFI. Currently, kurtosis-based detection is only used for real-valued radiometric signals and, thus, does not take advantage of the full information embedded in the radiometric signal. In addition, this dissertation formulates the two channels of a polarimetric radiometer as a single complex-valued signal, and characterizes the kurtosis test statistic associated with it, which has never been done before. We show that for total-power radiometers with a baseband downconverted pre-detected signal, performing kurtosis-based detection in the complex domain as opposed to real-valued kurtosis-based detection for the separate in-phase and quadrature signals, we can improve RFI detection for all types of RFI that a spaceborne radiometer may experience. Another revelation is that the probability distribution function (pdf) of the RFI-free radiometric signal is second-order circular, which simpliﬁes analysis of downconverted radiometric 6 signals in the complex domain. Related to the second contribution is the application of complex-valued signal processing to the channels of a polarimetric radiometer. The horizontal and vertical channel signals are treated as a single complex-valued signal. This new formulation has associated with it a kurtosis, just like the basedband quadrature case. However, since the RFI-free polarimetric signal is shown to be non-circular, detection of RFI is not as straightforward as it is for the circular case. We show that for pulsed and continuous wave RFI sources, the complex signal kurtosis generally performs better than the sum of the real-valued kurtosis of the separate radiometer polarization channels. There are a few esoteric situations where it does not, and this is when the RFI-free signal pdf ’s an extreme form of noncircular, degenerating into a line in the complex plane. This also happens when the RFI-to-noise power ratio in one polarized channel is stronger than the other. 1.3 Organization of Dissertation This dissertation begins with a thorough background in microwave radiometery and RFI detection in chapter 2. The intent is to familiarize the signal processing practitioner with radiometric terminology, how radiometers work, and how geopyhsical parameters are derived from their measurements. The background also introduces two basic radiometer architectures studied in this research, the total-power radiometer, and the polarimetric radiometer, and explains their various signals and how these radiometers are used in operation. Chapter 3 discusses the problem of RFI and develops a signal model for polarized RFI. Historical examples are given from various spacecraft missions. The difference between pre-detection and post-detection of RFI is also explained. A detailed overview 7 of the approaches currently used to deal with RFI is given. Lastly, data was taken from three real radiometers – two on aircraft and one ground-based, and analyzed to strengthen underlying assumptions about RFI and develop a polarimetric RFI signal model. This model is used for developing an RFI detection method for the entire polarimetric signal, not just individual polarized component signals. Time, histogram, spectrogram, and scatter analysis are performed on these data to justify the assumptions of RFI made in this research and to provide additional examples. The analysis of this data is a minor contribution of this dissertation. Chapter 4 is the ﬁrst major technical contribution of this dissertation. A new family of RFI detectors based on negentropy, or negentropy-based test-statistics for total-power radiometers is introduced and their detection vs. false-alarm performance is evaluated for signal models presented in chapter 3. The chapter begins with a literature review of RFI detection methods that use the pre-detected radiometric signal and shows that negentropy was never used before for this purpose. The theory behind using negentropy for RFI detection is developed, along with a formulation of the binary hypothesis-testing problem that is evaluated using the Neyman-Pearson decision rule. These negentropy-based test statistics are compared against those used in the current literature, namely the kurtosis test statistic and the Jarque-Bera test statistic. A number of different RFI source signal models for the alternate hypothesis are developed based on the RFI properties reported in chapter three. These models are used to evaluate the receiver operating characteristic (ROC) performance as a function of the various parameters of the RFI signal models. It is shown that the negentropy-based tests do detect RFI, but do not outperform kurtosis-based test statistics in therms of their ROC performance curves. Chapter 5 connects recent results in complex-valued signal processing theory to RFI detection. Currently, total-power radiometers that downconvert their signal to complex 8 baseband quadrature format have no methods deﬁned that use the both the in-phase and quadrature component signals together to process the radiometric signal and detect RFI. In addition, there are no methods that combine the polarization component signals into a single complex signal to perform RFI detection. The results of chapter 4 extend the ﬁeld of radiometric RFI detection by interpreting the baseband signal and the polarimetric channel signals as a single complex-valued signal, and then applying a test statistic based on the kurtosis coefﬁcients to this signal. It is shown that by treating these signal cases as complex-valued, as opposed to real-valued, gains can be made in detection performance over the kurtosis performance of any of the individual real component signals. Chapter 6 summarizes the results of the prior chapters. It connects the fact that kurtosis-based RFI detectors seem to have the best performance for real-valued signals, which motivated the research into complex-valued generalizations of radiometer signals and how the associated complex signal kurtosis could be used to detect RFI. We elaborate on the implication of the results on the design of future radiometers with improved RFI detection capability. The chapter ends with suggestions for future research, such as generalizing the complex-valued formulation to interferometric radiometers and investigating the use of the trispectral density as an approach that naturally combines joint time-frequency and statistical detection approaches into a concise and theoretically-rich framework. Lastly, this chapter points out the need to establish a global user database of radiometer signals corrupted by RFI, so that more researchers can study the characteristics of RFI and develop better mitigation algorithms. Chapter 2 BACKGROUND 2.1 2.1.1 Microwave Radiometer Physics Thermal Emission All matter, at a ﬁnite temperature above absolute-zero Kelvin, radiates electromagnetic energy [9, 18, 19]. Temperature is a measure of the average thermal energy of a substance due to random collisions and accelerations of its constituent atoms. These atoms contain charged particles. Random motion of these particles also induces random electromagnetic ﬁelds according to Faraday’s law of induction. The electromagnetic radiation that results from the intrinsic thermal kinetic energy of matter is thus called radio-thermal emission (RTE) or simply thermal emission. It is more commonly known in electrical engineering as thermal noise. The spectral radiance density (spectral brightness density) Lbb,f of an object is described by Plank’s law of radiation Lbb,f = 2hf 3 c2 (ehf /kB T0 − 1) 9 W/Hz (2.1) 10 where h = 6.626×10−34 J·s, is Plank’s constant, f is the frequency of the radiating electric ﬁeld, kB = 1.381 × 10−23 J/K is Boltzmann’s constant, c = 3 × 108 m/s is the speed of light, and T0 is the physical temperature of the object in units of Kelvin. Spectral radiance density is given in units of brightness per unit frequency, or radiated power per unit unit solid angle per unit area normal to the direction deﬁned by solid angle Ω [20]. This law states that an object at a physical temperature of T0 radiates electromagnetic energy with spectral brightness density Lbb . Thus, the overall brightness over some ﬁnite bandwidth Δf is given by Lbb Δf watts. At microwave frequencies, in the range of 300 MHz to 300 GHz, the argument in the exponential of Planks’s law becomes very small, much less than unity, since hf << kB T0 . As a result, the exponential term can be replaced by its ﬁrst-order Taylor series approximation exp hf kB T 0 ≈1+ hf . kB T 0 (2.2) Simplifying the resulting expression, we arrive at the Rayleigh-Jeans law by expanding the denominator exponential using the ﬁrst two terms of its Taylor series [19], resulting in Lbb,f = 2kB T0 f 2 c2 (2.3) The spectral brightness Lbb of a blackbody (as opposed to spectral brightness density), in a narrow bandwidth Δf centered at frequency f , is then Lbb = Lbb,f Δf (2.4) where the bandwidth Δf << f . Substituting in (2.3) into (2.4), spectral brightness of a 11 blackbody is given by Lbb = 2kB T0 f 2 Δf c2 (2.5) The units of spectral brightness (or simply brightness) are radiated power per unit area per unit solid angle [20]. The radiation laws described in (2.1) and (2.3) are idealized in the sense that they apply to objects that are perfect blackbodies. A blackbody is a theoretical object at thermodynamic equilibrium that absorbs and re-radiates all incident electromagnetic energy [20]. Contrary to blackbodies, objects called greybodies emit only a fraction of the electromagnetic energy that would otherwise be emitted if it were a black body. In addition, the spectral brightness density of a greybody generally has a directional dependence on the solid angle through which it is observed Ω = (θ, φ) [19]. As a result, the spectral brightness density of a greybody can be written as Lf (θ, φ), to emphasize the angular dependence. In the context of microwave radiometry, the greybody refers to the Earth’s surface. The ratio of the spectral brightness (or brightness density) of an object (or Earth’s surface) at a physical temperature, T0 , and the same object at the same temperature if it were a perfect blackbody is an intrinsic material property called emissivity, e(θ, φ). Emissivity is deﬁned as the ratio between greybody and blackbody brightness [20], e(θ, φ) = Lf (θ, φ, T0 ) , Lbb (T0 ) (2.6) and is a fraction between zero and one that determines the amount of radiation emitted by a greybody relative to that emitted by a blackbody at the same temperature. Greybodies are representative of real matter. An e(θ, φ) = 1 means that an object is a perfect blackbody. If an expression for the spectral brightness of a greybody Lf (θ, φ), whose physical temperature is T0 , is to be written in similar form to 2.5, then we require a blackbody- 12 equivalent temperature. This temperature is called the brightness temperature TB (θ, φ) and is the key quantity sought in the ﬁeld of microwave radiometry. The brightness temperature allows greybody brightness to be deﬁned as [19] Lf (θ, φ) = 2kB TB (θ, φ)f 2 Δf. c2 (2.7) Starting with the deﬁnition of emissivity in 2.6, Lf (θ, φ) e(θ, φ) = = Lbb 2kB f 2 TB (θ, φ) Δf c2 1 c2 2 2kB T0 f Δf = TB (θ, φ) , T0 (2.8) therefore, TB (θ, φ) = e(θ, φ)T0 (2.9) The units of brightness temperature are in Kelvin. The fact that emissivity is between zero and one suggests that a greybody has a “cooler” brightness temperature than its physical temperature [19]. If physical temperature and brightness temperature are known, then emissivity can be determined. Very many variables determine the emissivity of a substance. In the ﬁeld of Earth remote-sensing, substances that we are interested in are geophysical in nature – water, in particular, and its solid, liquid, and gaseous states. Through careful empirical experimentation over the last several decades, relationships between geophysical variables, emissivities, and brightness temperatures have been found [21]. These relationships are the key for using brightness temperatures for performing remote sensing using microwave radiometers. 13 2.1.2 Polarization The electric ﬁeld vector received by an antenna can be written as a time-varying uniform plane wave (wavefront) that has horizontal and vertical (x̂ and ŷ) component vectors, as described in [19, 22]. Denoting the orthogonal unit vectors x̂ and ŷ, the ﬁeld E(r, t) traveling in the z direction mutually orthogonal to x̂ and ŷ can be written as a superposition of two component waves EH (z, t) and EV (z, t) E(r, t) = x̂EH (z, t) + ŷEV (z, t) = x̂EH,0 (z, t)cos(ωt + βz) + ŷEV,0 (z, t)cos(ωt + βz + θ) = Re x̂EH,0 (z, t)ej(ωt+βz) + ŷEV,0 (z, t)ej(ωt+βz+θ) (2.10) (2.11) (2.12) where the horizontal and vertical components of the wavefront are given by EH (z, t) and EV (z, t), ω is the angular frequency of the wavefront, t is time, β is the wave number, and θ is the phase difference between the component waves. The magnitudes of the horizontal and vertical component waves are given by EH,0 and EV,0 . As an electromagnetic ﬁeld propagates through space, the vector normal to its direction of propagation traces an ellipse on the plane that is also normal to the direction of the wave’s propagation. The vector can be projected to two orthogonal polarization components horizontal and vertical. The magnitudes of these components, EH and EV , form a pair of time-varying voltage signals if measured by a dual-polarized antenna. The polarization state of the wave is the shape traced by the ﬁeld vector of E(r, t) along the plane deﬁned by the x̂ and ŷ direction vectors. Anthropogenic radio emissions generally have some form of elliptical polarization, such has right or left-hand circular, or degenerate forms such as horizontal only, vertical only, or slant linear polarizations having a 45◦ angle 14 with respect to horizontal polarization [23, 24]. On the other hand, unlike anthropogenic signals, thermal emission signals are generally partially polarized quasimonocromatic random ﬁelds [9, 22, 25]. This means that the resulting brightness temperatures, TH and TV , from orthogonal horizontal and vertical electromagnetic ﬁeld components, are generally different. The received signal and corresponding brightness temperature measured by a total power radiometer as a result represents only the brightness corresponding to the polarization of the radiometer’s antenna. The polarization state of this ﬁeld is completely described by the Stokes vector, a vector consisting of the four Stokes parameters. The Stokes vector is deﬁned as (dropping subscripts 0 from the ﬁeld amplitudes) ⎞ ⎛ ⎛ ⎞ 2 2 + E E S H ⎟ ⎜ V ⎜ 1⎟ ⎟ ⎜ ⎜ ⎟ ⎜ E 2 − E 2 ⎟ ⎜S 2 ⎟ 1 V H ⎟ ⎜ ⎟ ⎜ =⎜ ⎟= ⎜ S ⎟, ⎜S ⎟ Z ⎜2 Re E · E ∗ ⎟ ⎜ ⎜ 3⎟ V H ⎟ ⎠ ⎝ ⎝ ⎠ ∗ S4 2 Im EV · EH (2.13) where the parameters S1 , . . . , S4 are called the Stokes parameters [9]. The ﬁrst two Stokes parameters represent the total power and difference in power between vertical and horizontal polarization components, respectively. The third and fourth Stokes parameters represent the real and imaginary parts of the complex correlation between the electric ﬁeld polarization components. The variable Z is the impedance of the medium through which the electromagnetic ﬁeld propagates. In microwave radiometry, where the Rayleigh-Jeans approximation holds, the Stokes vector can be converted to one containing brightnesses by means of a scaling factor c2 , kB f 2 noting the temperature-brightness correspondence between power and brightness 15 temperature ⎞ ⎛ 2 2 + E E V H ⎟ ⎜ TV + TH ⎜ ⎜ ⎟ ⎜ ⎜ E 2 − E 2 ⎟ ⎜ TV − TH 2 c V H ⎜ ⎟ ⎜ = S ⎜ ⎟=⎜ 2 kB Zf ⎜ 2 Re E · E ∗ ⎟ ⎜ T ◦ − T ◦ ⎜ ⎜ 45 V −45 H ⎟ ⎝ ⎠ ⎝ ∗ Tlhcp + Trhcp 2 Im EV · EH ⎛ ⎞ ⎛ ⎞ T ⎟ ⎜ 1 ⎟ ⎟ ⎜ ⎟ ⎟ ⎜ T2 ⎟ ⎟ ⎜ ⎟ ⎟=⎜ ⎟ ⎟ ⎜ T ⎟ ⎟ ⎜ U ⎟ ⎠ ⎝ ⎠ TV (2.14) In equation 2.14, TH and TV are the brightness temperatures for the horizontal and vertical polarization components of the radiated electric ﬁeld, respectively. The brightness temperatures T45◦ and T−45◦ are skewed linear polarizations. Lastly, Trhcp and Tlhcp are the right-hand and left-hand circularly polarized component brightness temperatures of the radiated electric ﬁeld. 2.1.3 Statistical Characterization of Thermal Noise Thermal emission from a material is described by its electric ﬁeld E(r, t), where the position vector r and time index t summarizes the four-dimensional space-time evolution of the ﬁeld [26]. The amplitude of this ﬁeld is commonly modeled as a Gaussian random process and has been justiﬁed in several bodies of work [19, 27, 28]. The basic principle is that charge carriers in a material exhibit Brownian motion, which leads to Johnson-Nyquist electrical noise. Each charge carrier randomly contributes a random ﬁeld component to the total ﬁeld. By application of the central limit theorem, due to the large number of charge carriers present in a material, all of the random ﬁelds add incoherently. As a result, the total ﬁeld, when measured by an antenna, will result in an output voltage signal v(t) that has a zero-mean Gaussian probability density function (pdf ). In addition, Ulaby and Dicke state in [18, 19] that this behavior is exhibited across all microwave frequencies, justiﬁed 16 by the results of Ragazzini and Chang [29] and S.O. Rice [30]. Therefore, it is widely accepted that the thermal noise signal resulting from RTE, measured by the antenna of a total power radiometer, has a random voltage signal v(t) characterized by a zero-mean Gaussian probability density function with variance σ 2 , fv = √ v2 1 e− 2σ2 , 2πσ (2.15) and the power spectral density measured over a ﬁnite bandwidth B of this noise is given by Svv (f ) = σ 2 (2.16) The pre-detected polarization signals xH (t) and xV (t) are modeled as a bivariate Gaussian [31, 32] random signal with correlation coefﬁcient ρ and zero-mean horizontal 2 and σV2 , respectively and vertical component variances σH fxH (t),xV (t) (xH , xV ) = 2πσH σV 1 2 xH 1 yV2 2ρxH yV . exp − + 2 − 2 2(1 − ρ2 ) σH σV σH σV (1 − ρ2 ) (2.17) We assume that the assumptions for the signal pdf s in (2.15) and (2.17) hold during a short interval of time, T , in the order of milliseconds, for a microwave radiometer. In Section 2.2, we will further deﬁne T as the integration time, or integration period, which corresponds to the amount of time a microwave radiometer is considered to be stable [20] and is used to compute a time average of v(t) for computation of signal power [9]. 17 2.1.4 Brightness Temperature and Antenna Temperature Relationships In general, the objective of radiometery is to recover brightness temperatures TB from antenna temperatures TA . If only one polarization of the received electromagnetic wavefront is measured, then TA is a scalar. Otherwise, for polarimetric radiometers T A is a vector. In both cases, solving for the brightness temperature is done in ground processing, and the data product delivered by the radiometer is a continuous series of antenna temperatures. For a total-power radiometer, the antenna temperature, TA and brightness temperature TB (θ, φ) are directly related, according to [25], as Ae TA = 2 λ Fn (θ, φ)TB (θ, φ) dθ dφ (2.18) 4π For a polarimetric radiometer, T A is a vector of brightness components, corresponding to four Stokes parameters, and is given by η0 TA = 4Z0 λ2 −1 4π T M C LP T M C T B (θ, φ) dθ dφ, (2.19) where λ is the wavelength of the received electromagnetic wavefront, the series of −1 operators T M C LP T M C convert the a coherency vector into a modiﬁed stokes operator, and T B (θ, φ) is the modiﬁed Stokes vector, containing the stokes parameters of the radiometric measurement. The idea is that in orbit, telemetry that corresponds to antenna temperature via power measurement is sent to the ground, where the antenna temperature is computed. Next, the relationships in (2.18) and (2.19) are inverted to ﬁnd the corresponding brightness 18 temperatures TB or modiﬁed Stokes vector T B (θ, φ) of the corresponding radiometer. Each ground pixel – an area of Earth with an associated TB , corresponds to one of these measurements. This pixel is determined by the radiometer antenna footprint, which corresponds to tens of square kilometers, depending on the orbit and antenna geometry of the radiometer. As the entire Earth is scanned, a constant stream of this telemetry is sent over the life of the mission, and the corresponding processed data is archived and processed further to extract the geophysical parameter of interest. 2.2 Microwave Radiometers A microwave radiometer is a very sensitive radio receiver, typically with an antenna input, that is used to measure radiated electromagnetic power from some direction in space [9,19, 20,33]. Two recently launched and currently orbiting radiometers include the Soil-Moisture Ocean Salinity (SMOS) [34], managed by the European Space Agency, and the Aquarius Mission [35], managed by the National Aeronautics and Space Administration (NASA). A radiometer that is currently under its ﬁnal stages of testing, and scheduled to be launched in 2014, is the Soil-Moisture Active-Passive radiometer (SMAP) [36]. The author led the development of the signal processing system as part of the SMAP radiometer. Illustrations of the three radiometers are shown in Figure 2.3. 19 (a) SMOS (b) Aquarius (c) SMAP Figure 2.3: Modern Spaceborne Radiometers as of April 2014 2.2.1 Total Power Radiometers The simplest possible description of a microwave radiometer is given by the block diagram in Figure 2.4. This kind of radiometer is called the total-power radiometer. At a minimum, a total-power radiometer will consist of an antenna connected by an ampliﬁer with gain G, followed by an analog microwave bandpass ﬁlter with frequency response HB (f ), center frequency fc , and bandwidth B. The output signal from this assembly is called the predetected signal x(t). The pre-detected signal connects to a square-law detector, which is simply a specialized diode that outputs a voltage proportional to the square of x(t). The reasoning behind this is the square of the signal is proportional the power of the signal, which is the key quantity that is measured by the radiometer. Lastly, the squared predetected signal is averaged over a ﬁnite time T , using an integrator circuit (lowpass ﬁlter). The ﬁnal output is called a“video” signal, or the post-detected signal. This signal is of prime importance in microwave radiometry. In this conceptual block diagram, the antenna illuminates a ﬁnite area of Earth and receives broadband radio emission within its passband. Given the levels of emissivity, atmospheric attenuation, and distance from the scene, the received electromagnetic ﬁeld E(r, t) is relatively weak. The antenna signal, v(t), is ampliﬁed with a gain of G. 20 The broadband signal is bandlimited by a bandpass ﬁlter with bandwidth B and center frequency fc determined by the radiometer’s application. This crucial ﬁlter shapes the overall bandpass signal used for subsequent radiometer operation and is the ﬁrst major defense against RFI. The instantaneous power of this bandlimited signal is computed by a device called a square law detector that outputs a voltage signal that is proportional to the power within the radiometer bandwidth. The average power of this signal is computed by a lowpass ﬁlter or integrator. To improve radiometric sensitivity and to account for gain ﬂuctuations of the receiver, a reference load or calibration source with roughly the same expected brightness temperate of the scene under observation is often used near the antenna input. A switch periodically switches between one or more reference loads and the antenna input. This technique was developed by R. H. Dicke in [18] and is a standard technique in spaceborne microwave radiometry. The resulting power measurements are collected and telemetered to Earth, along with the switch state, and calibration algorithms are used to estimate the resulting scene brightness temperature T̂B . 2.2.2 Polarimetric Radiometers A microwave radiometer that measures all four modiﬁed Stokes parameters is called a polarimetric radiometer [20]. The block diagram of a radiometer of this type is shown in ﬁgure 2.5. An example of such radiometer is the Soil Moisture Active Passive radiometer, currently being built by NASA Goddard Space Flight Center and scheduled for launch in 2015 [7]. Without loss of generality, a polarimetric radiometer consists of two identical radiometer receivers, which can each be thought of as total power radiometers. Each radiometer takes one of each horizontal and vertical polarization component signal from an antenna assembly, which is capable of receiving both polarizations. The pre-detected 21 output polarization signals xH (t) and xV (t) are also cross-correlated by a correlation circuit that outputs the real T3 and imaginary T4 parts of the cross-correlation result. The quantities T3 and T4 are referred to as the third and fourth Stokes brightness temperatures, and contain information about the polarization sense of the electromagnetic ﬁeld received by the radiometer [9, 20]. 2.3 Radiometer Power and Resolution In microwave radiometry, the terms “Brightness”, and “power” are often used interchangeably. Power P is linearly related to sum of the receiving antenna temperature TA and the receiver noise temperature TSY S through the relationship by [9] P = kB G (TA + TSY S ) B, (2.20) where kB is Boltzmann’s constant, G is the gain of the receiver, and B is the observation bandwidth of the radiometer. This power, since it is a measure of random noise-like ﬂuctuations resulting from the antenna and receiver brightness temperatures, is a random variable itself. A radiometer measures RTE over several repetition intervals (called pulse repetition intervals - PRI) of time T . These measurements are transmitted to Earth, and additional processing yields a thermal brightness TB for the observed area. The standard deviation of the power measurement, in terms of a temperature equivalent quantity, is called the noiseequivalent change in temperature, radiometric resolution, or NEΔT , NEDT, or simply, 22 ΔT . Radiometric resolution was derived in [33], and is given by ΔT = (TA + TSY S ) √ , Bτ (2.21) where TA is the antenna temperature, TSY S is the noise temperature of the receiver itself, B is the spectral bandwidth observed by the radiometer, and T is the ﬁxed integration time of the receiver. Equation 2.21 is known as the ideal radiometer equation [9] and is a measure of the minimum detectable brightness change that can be detected by a radiometer. In all radiometry applications (as well as radio astronomy), this is a key performance speciﬁcation. For radiometers to be useful for Earth remote sensing, they must meet or exceed the radiometric resolution for the speciﬁc application. For example, for measuring soil moisture, ΔT is required to be known to a resolution about 1 Kelvin. For sea surface salinity measurement, the requirement is even higher, a few tenths of a Kelvin [9]. 2.4 Digital Radiometers and Signal Processing Considerations Regarding the two radiometer types presented thus far, total power and polarimetric, the post-detected data is collected by the spacecraft, and sent to Earth for further processing to recover brightness temperatures. Of interest in this research are radiometers in which the pre-detected radiometer signal x(t) is sampled, and the remainder of the radiometer signal processing, i.e., the squaring, integration, and cross-correlation functions, are handled digitally [7,32]. Though these types of radiometers have been in development and in use in airborne remote sensing and laboratory experiments for the last decade, the Soil Moisture 23 Active Passive (SMAP) radiometer, part of a new NASA mission under development, will be the ﬁrst radiometer that uses an entirely digital backend processor. A digital radiometer digitizes the pre-detected signal x(t) in the total-power (single channel) case and both of the polarization component signals, xH (t) and xV (t), in the polarimetric (two-channel) case. Squaring and integration functions, traditionally carried out by the square-law detector diode and analog integrator circuit, are replaced by digital squaring and averaging circuits and typically implemented in Field Programmable Gate Array (FPGA) [37] logic that is interfaced directly to the output of the analog-to-digital converter. The digital counterparts to the analog total-power radiometer and analog polarimetric radiometer are shown in ﬁgures 2.6 and 2.7. In a digital radiometer, the pre-detected voltage signal x(t) of bandwidth B is sampled by an analog-to-digital converter with sample rate FS greater than 2B with b-bit resolution, resulting in a digital signal x(n). This signal is processed by a digital processor, (normally a FPGA since B is on the order of MHz or greater). There are two main advantages of digital radiometer signal processing [38]. Digital radiometers, relative to their analog counterparts, have improved stability and independence from receiver temperature variations, thus simplifying calibration. In addition, using digital signal processing allows expansion of capabilities far beyond what is feasible or implementable on analog radiometers, including the ability to mitigate RFI on-board. Effects of digitization on the radiometer resolution NEΔT and correlation have been studied by Fischmann, Piepmier, Bosch-Lluis [31, 32, 39]. Basic processing, without RFI mitigation, consists of simply computing the sample mean and variance of the x(n) for the total power case and the mean, sample variance, and cross-correlation of xH (n) and xV (n) for the polarimetric case. Assuming wide-sense stationarity of the input signal in 24 both cases, the total number of samples used to estimate these statistics are given by N = T FS (2.22) where FS is the analog-to-digital converter sample rate, T is the integration time, and (·) denotes the ceiling function (round towards greater integer). 2.4.1 Signal Assumptions The thermal noise random process, observed by a microwave radiometer, is inherently non-stationary [40]. A key assumption made is that during the N -sample interval, the observed signal, x(n) is a discrete-time random process (sequence) consisting of N i.i.d. random variables. We further assume that in the RFI-free case, these random variables are Gaussian distributed, with zero-mean, and variance to be estimated by radiometer signal processing. The key problem is that this variance estimate is biased by RFI, which may exist as part of x(n) upon observation. Lastly, we assume that for this short time interval T , that the discrete-time random signal x(n) is wide-sense stationary. As T increases, the wide-sense stationary assumption breaks down. 2.4.2 Digital Signal Processing Signal processing for the total power radiometer starts with the antenna voltage signal v(t). This signal is a wide-sense stationary Gaussian process with variance σ 2 over the integration time T . The signal is ampliﬁed by a low noise ampliﬁer and then ﬁltered by an analog bandpass ﬁlter designed to isolate the radiometer observation band B as ideally as possible. The resulting signal x(t) is sampled, producing x(n). The sample mean μx and 25 variance σx2 are then computed by the time averages in (2.23) and (2.24) and σx2 N −1 1 x(n), μx = x(n) = N n=0 (2.23) N −1 1 2 = x (n) = x (n) − μ2x N n=0 (2.24) 2 In addition, for the polarimetric processing, without RFI detection, the cross-correlation (lag-zero) is computed by RHV (n) = xH (n)x∗V (n) N −1 1 = xH (n)x∗V (n) N n=0 (2.25) The real part of RHV (n) is proportional to the the third Stokes parameter, and the imaginary part is proportional to the fourth Stokes parameter. In remote sensing literature, they are referred to as the modiﬁed Stokes parameters T3 and T4 [9, 19, 20]. 2.4.3 Super Heterodyne Digital Radiometers Often, in practice, the total power and polarimetric radiometers downconvert the radiometer bandwidth B to a convenient intermediate frequency (IF) range. Downconversion allows the radiometer receiver to be designed in stages, which can help isolate the radiometer band and achieve greater stability [9]. For digitizing the radiometer bandwidth B, downconversion is a practical necessity. The lowest frequency that spaceborne microwave radiometers operate at is the 1.4 GHz L-band, where the bandwidth for a passive allocation at this band is 27 MHz. This band is also shared by radio telescopes used for radio astronomy and astrophysics. If Nyquist sampling were directly applied to this band, a minimum sample rate, FS , of 2.854 GHz would be required (twice the L-band upper 26 frequency of 1.427 GHz), which is extremely challenging to accomplish with current space digitizer systems. However, the actual required sample rate for this band is a minimum of 54 MHz, which is much more reasonable to achieve in space using analog mixing and ﬁltering prior to sampling. Given that the center frequency for this band is in L-band, downconversion of the radio frequency (RF) band should occur in stages so that a lower-rate analog-to-digital converter can be used. There are numerous practical advantages of downconverting the radiometer band to a lower frequency, including reducing the spacecraft data rate, reducing system power dissipation, and reducing component cost. The functional block diagram of a super heterodyne radiometer is shown in ﬁgure 2.6 In a hetrodyne microwave radiometer, the antenna signal v(t) is ampliﬁed and bandlimited by an RF bandpass ﬁlter centered at the radiometer’s operational band. The resulting bandlimited signal is then mixed to a lower IF frequency using a local oscillator with angular frequency ωLO . The resulting signal is ﬁltered again with an IF bandpass ﬁlter with the same bandwidth B as the ﬁrst RF ﬁlter. The second ﬁlter HIF (f ) ﬁlters out harmonics resulting from mixing and reinforces the radiometer observation bandwidth, B. The resultant pre-detected signal x(t) is then squared and integrated to estimate the power of v(t) in the integration period of T . The polarimetric superheterodyne radiometer is analogous to the total power super heterodyne radiometer in ﬁgure 2.6 and is shown in ﬁgure 2.7. 2.4.4 Radiometer Operation The discussion of total power and polarimetric radiometers thus far has considered the ideal theoretical models on which they are based. The key assumption is that the receiver parameters – gain G, receiver system noise TSY S , and bandwidth B do not vary during an 27 integration period T . In reality, they do vary, and are accounted for in calibration [9], [19]. In addition to gain and receiver noise variation, radiometers must be sensitive to changes in brightness temperature that are fractions of the absolute brightness temperature measured by the antenna. For example, a radiometer may measure an Earth brightness temperature of 300 Kelvin, but in order to make measurements of soil moisture, its radiometric resolution needs to be on the order of 1 Kelvin [9]. To account for this variation and to achieve this measurement resolution, most spaceborne microwave radiometers include calibration targets (reference loads), or loads connected to the front-end of the radiometer receiver that emit a known speciﬁc brightness temperature. The radiometer will periodically switch between measuring the brightness temperature and physical temperature of one or more of these loads, before switching back to measure the antenna voltage signal. A general timing diagram of this process is shown in Figure 2.8. The time that a radiometer spends on the antenna is called an antenna look. In this case, this signal is represented as xi (n), n ∈ [0, . . . , N − 1], with the subscript i corresponding to the radiometer state. Since the DC component of the the signal can be removed by AC coupling the radiometer analog signal to the analog-to-digital converter of its digital processor, the signal power is the same as its sample variance Pi = N −1 x2i (n), (2.26) n=0 where N samples correspond to the integration time T of a single PRI. A radiometer footprint will consist of several of these power measurements, Pi , that consist of the periodic power measurements of the antenna or calibration sources. At the bare minimum, these power measurements are the key telemetry transmitted to Earth to perform subsequent 28 estimation of TA , followed by estimation of TB . Many power measurements are taken per footprint, and are averaged to form a pixel of the Earth scene, corresponding to the geophysical parameter measured. A summary of the kinds of geophysical parameters, also called environmental data records (EDRs) [8] that spaceborne microwave radiometers are used to produce is summarized in Appendix C, Table C.1. This telemetry is collected by receiving stations on the ground and processed. Processing consists of using calibration references to correct for radiometer gain variation, and the Dicke principle [18] is applied to compute the sensitive antenna temperature measurements, from which the surface brightness measurements are computed. In the past, spaceborne microwave radiometers have largely been analog-only systems that measure power only [9, 19, 39]. As we will see in this dissertation, radiometers can now measure additional quantities corresponding to higher-order statistics (besides power, which is second-order) that aid in detection of RFI [7]. In addition, on-board digital signal processing has allowed for additional manipulation, such as spectrogram analysis [41], of the pre-detected radiometer signal that was impossible or too costly to perform using analog-only hardware. The Soil Moisture Active Passive (SMAP) radiometer is an example of all-digital radiometric processing utilizing higher-order statistics and spectrogram analysis [3]. This system discussed in the next section since it is the primary motivator for this dissertation research. 2.4.5 Radiometer Example: The Soil-Moisture Active-Passive Instrument The SMAP radiometer is an example of the polarimetric hetrodyne type radiometer [3, 7]. It performs computations of the sample moments on the four digital signals IH,V (n) 29 and QH,V (n). Since the basic job of this radiometer is to compute the Stokes vector for every integration period, it must compute all of the second-order sample statistics of the four digital signals. These statistics include the variance of the individual polarization channel signals, in-phase and quadrature components each, as well as the cross correlation between the baseband quadrature signals. The total-power products, namely the secondorder statistics of the independent polarization channels, includes computation of the signal variance. Computation of the variance requires the mean. An efﬁcient method was found by [7] to compute the variance ofﬂine: using the computed mean and second raw moment, in lieu of the variance, on the spacecraft. In addition, the third and fourth raw sample moments were computed on-board the SMAP radiometer. The purpose of doing so was to estimate the skewness and kurtosis from every radiometer integration period, and send that data back to Earth. Finally each digital signal was split into 16 subband signals. The ﬁrst four raw moments are computed on each of the 16 subband signals, and these signals are also individually cross-correlated, channel-by-channel. All sample moments are computed using an integration time corresponding to N samples. The angle brackets denote a sample average of the signal xj (n) as follows N −1 1 j xj (n) = x (n) N n=0 (2.27) The baseband downconverted in-phase and quadrature signals and their statistics are referred to in notation as fullband data because the signal and its associated moments originate from the full radiometer bandwidth, B. Conversely, the signals that were split into 16 channels and their corresponding sample moments are referred to as subband data. The subband data is essentially a spectrogram of the ﬁrst through fourth sample moments 30 of each polarimetric channel. The cross correlation data is also delineated into fullband and subband products. In this manner, SMAP essentially computes a continuous spectrogram of sorts of the ﬁrst through fourth sample moments and their polarization cross-correlations. Fullband data is referred to using capital letters, preceded by an H or V to indicate which polarization channel the data comes from. If the in-phase component signal is used, then the notation will has an I appended. Otherwise, a Q is appended. For example, HM2 I refers to the second fullband raw sample moment of the in-phase signal corresponding to the horizontal polarization channel. Subband data also have the superscript k afﬁxed to their notation, where k corresponds to a frequency channel number. In this case, k = 0, . . . , 15, corresponding to one of the 16 subband channels. Lower case letters denote subband data. [k] For example, rV H corresponds to the subband cross-correlation between horizontal and vertical polarization channels for frequency channel number k. Given this notation, a summary of the computations performed by the SMAP radiometer is shown in Table 2.1, (2.32) and (2.37). RV H = ∗ EV (n)EH (n) N −1 1 ∗ = EV (n)EH (n) N n=0 N −1 1 = (IV (n) + jQV (n)) (IH (n) − jQH (n)) N n=0 = N −1 1 (IH (n)IV (n) + QH (n)QV (n)) N n=0 +j N −1 1 (IH (n)QV (n) − IV (n)QH (n)) N n=0 = S3 + jS4 (2.28) (2.29) (2.30) (2.31) (2.32) 31 Table 2.1: Summary of data products for RFI mitigating radiometer. Moment First, fullband Quadrature In-Phase T 1 −1 M 1H,V = I(n) T 1 −1 M 1H,V = n=0 Second, fullband M 2H,V = T 1 −1 I (n) 2 M 2H,V = T 1 −1 n=0 Third, fullband M 3H,V = T 1 −1 M 4H,V = T 1 −1 I 3 (n) M 3H,V = T 1 −1 mi1H,V = I 4 (n) M 4H,V = T 1 −1 i(n) mq1H,V = n=0 Second, subband mi2H,V = Third, subband mi3H,V = i (n) 2 mq2H,V = mi4H,V = T 2 −1 n=0 q(n) T 2 −1 q 2 (n) n=0 i (n) 3 mq3H,V = n=0 Second, subband T 2 −1 n=0 n=0 T 2 −1 Q4 (n) n=0 T 2 −1 T 2 −1 Q3 (n) n=0 n=0 First, subband Q2 (n) n=0 n=0 Fourth, fullband Q(n) n=0 T 2 −1 q 3 (n) n=0 i4 (n) mq4H,V = T 2 −1 n=0 q 4 (n) 32 and N −1 ∗ ∗ 1 [k] [k] [k] [k] [k] rV H = eV (n) eH (n) eV (n) eH (n) , k = 0, . . . , 15 = N n=0 N −1 1 [k] [k] [k] [k] = i (n) + jqV (n) iH (n) − jqH (n) N n=0 V N −1 1 [k] [k] [k] [k] = i (n)iV (n) + qH (n)qV (n) N n=0 H N −1 1 [k] [k] [k] [k] +j i (n)qV (n) − iV (n)qH (n) N n=0 H [k] [k] = S3 + jS4 (2.33) (2.34) (2.35) (2.36) (2.37) Once telemetered to the ground, the raw moment data are centralized using the moment centralization formula μp = p p j=0 j (−1)p−j xj (n) x(n)p−j , p = 1, 2, 3, 4, (2.38) were μp corresponds to the p−th sample central moment. The statistics of the sample mean μ, variance σ 2 , skewness γ, and kurtosis κ are derived from these moments. In particular, the variance is given by σ 2 = μ2 − μ21 , (2.39) the skewness is given by 3 γ = μ3 /σ 2 , (2.40) 33 and the kurtosis is given by κ = μ4 /σ 4 . (2.41) As will be shown in the next chapter, the skewness and kurtosis are used as auxiliary data products to help detect the presence of RFI within a group of N samples of data from the four digital signals of the polarimetric radiometer. Overall, the radiometer outputs an ongoing time-frequency-statistical diversity data set so that the Stokes brightness temperatures as well as RFI can be continually derived. 2.5 RFI and Mitigation Approaches Radio Frequency Interference (RFI) in microwave radiometers occurs as a result of anthropogenic (man-made) signals that enter the passband of the microwave radiometer receiver and corrupt calibrated brightness temperature measurements of naturally occurring background thermal emission. RFI can be thought of as the additional power that is contributed to a brightness temperature measurement that exceeds the radiometric uncertainty [40]. Use of corrupted brightness temperatures result in erroneous geophysical retrievals, and is a growing and serious problem in microwave radiometery [42–44]. For a total-power radiometer, the quantity of interest is the antenna power. For a polarimetric radiometer, the quantities of interest include the total power from each polarization channel, as well as a complex cross-correlation between the polarization channels. The real and imaginary parts of the complex cross-correlation between radiometer channels are known as the third and fourth Stokes Parameters, respectively. Data products telemetered down to Earth from currently orbiting radiometers will only 34 contain measurements corresponding to the total power in each polarimetric channel and the Stokes Parameters. Because of fundamental limitations to data downlink, on-board storage, and processing, the original pre-detected signal x(n) (or the combination of pre-detected polarization signals xH (n) and xV (n), cannot be downlinked directly, only the post-detected data can. As a result, algorithms that mitigate RFI using the pre-detected radiometer signal have to be performed on the spacecraft itself. In post-detection, RFI is detectable if it contributes enough power to the Earth RTE signal such that the resulting TB exceeds thresholds determined by what is physically possible. Hence, post-detected RFI will appear as “hot” or “bright” spots of area on a map of brightness temperatures corresponding to an environmental data record. The key point is that this is easily detectable RFI, because it is strong enough to corrupt an observation beyond a threshold established by the physics of the corresponding environment that was measured. Since the RFI signal(s) are generally inseparable from the total observation of average power, the resulting detected power value has to be discarded. If the RFI occurs in a small area (small number of pixels in the resulting brightness image), the neighboring pixels (in space and in time) can be used to approximate what the true brightness is. In the case of RFI that contributes additional power to the radiometric observation, but does not cause the resulting TB to exceed physical thresholds, the RFI is said to be “low-level”. This type of RFI is more difﬁcult to detect because it leads to a valid TB given the post-detected signal y(n). To detect low-level RFI, it is advantageous to do so using the pre-detected signal, x(n), which still retains information about the RFI signal(s) interspersed throughout its samples as opposed to being blurred and averaged away in the post-detected power of x(n). Since RFI signals vary over space, time, and frequency, and as the radiometer orbits 35 and scans the surface of the Earth, the actual signal properties of RFI signals are difﬁcult to characterize, even if the presence of RFI is detected in the post-detected radiometer signals. Each observation will last on the order of milliseconds, so long temporal records of pre-detected signals are not yet possible to obtain from the spacecraft. Even if they were, because the area of the antenna footprint will be on the order of hundreds of square kilometers, multiple unknown sources of the same and different types would all contribute to x(n), making complete source separation difﬁcult if not impossible for complete signal characterization [45, 46]. Two approaches are used to gain insight into the offending signal sources. First, international spectrum allocations are used to infer the kind of sources that have contributed to the corrupted post-detected radiometer signal. Since the radiometer observation frequencies are known, transmitters in neighboring frequency allocations are assumed as likely RFI candidates [8, 44, 47]. Second, a variety of radiometers are actually ﬂown on aircraft for short duration ﬂights, over areas of interest that may cause RFI [4, 46]. The aircraft experiments use the same bands as orbiting microwave radiometers, and sometimes store the pre-detected waveform for as many samples as practical. In this case, the waveform signal properties can be analyzed and more insight can be gained into the nature of the offending signals. Understanding of these signals can lead to the development of better RFI detection and mitigation algorithms. The problem of RFI mitigation is dealt with in a variety of different ways, depending on the RFI environment and the requirements of the radiometer in a space mission. In addition, RFI is either mitigated in the pre-detected or post-detected signal. To date, strong RFI is easily detectable in brightness temperature maps because the resulting brightness of a scene on Earth exceeds a pre-determined threshold established by the physics of the environment. 36 The basic categories of RFI mitigation fall into prevention, detection/excision, and cancellation. Prevention and cancellation are limited to the design of the radiometer and its on-board signal processing system. RFI can be prevented by enforcement of global frequency allocations for passive remote sensing, or by simply cooperative spectrum use between the radiometer and potential interfering source. RFI is also avoided by proper radiometer design, such that any self-imposed harmonics or spurious signals are out of the radiometer bandwidth. Most importantly, RFI is avoided ﬁrst by employing adequate bandpass ﬁltering of the radiometer band, with analog ﬁltering exceeding 50 dB of stopband rejection or more [9]. Cancellation techniques are still an active research topic, and are difﬁcult to perform onboard because they require computationally expensive hardware. In addition, cancellation is limited by knowledge of interfering signals, and the ability to adaptively cancel them [48]. Detection is the most prevalent method of RFI mitigation, and can be performed both on-board the spacecraft and on the ground, with post-detected data [49]. However, as the offending signals are integrated with the desired signal w(n), they become impossible to identify and separate from the original pre-detected signal. Post-detection methods rely on detection of brightness temperature deviations beyond the range of natural brightness temperature variability. For example, consider soil moisture. Njoku in [50] showed that for sensing soil moisture at below 5GHz, water has a dielectric constant of -80, whereas for dry soil, the dielectric constant is -3.5. These correspond to emissivities of 0.6 and 0.9, respectively. If we use (2.6), along with an assumed soil physical temperature T0 = 300 Kelvin, then the natural variability for soil moisture ranges from 180 Kelvin to 270 Kelvin. Since RFI contributions are additive, any brightness temperature of an Earth scene that falls outside of 270 Kelvin (assuming the soil is no hotter than 300 Kelvin) can be ﬂagged as RFI because 37 the natural variability of soil is exceeded. For all other radiometer bands, the thresholds can be computed similarly using the plots in ﬁgures 2.9 and 2.10, and the NEΔT requirement of the radiometer. 2.5.1 Prevention Based on the radiometer sensitivity (2.21), the larger the observation bandwidth, the more sensitive the radiometer is. However, there are many reasons why an arbitrarily large observation bandwidth cannot be used. First is the issue of spectrum occupancy. In the radio spectrum, there exists a multitude of human-made signal sources, such as ground, aircraft, and air trafﬁc control radar, satellite television broadcasts, wireless communication and networking signals, that co-exist with and add to the weak, naturally-occurring RTE signal from Earth. Secondly, the radiometer bandpass ﬁlter(s) cannot be designed to have arbitrarily wide bandwidths, arbitrarily high stopband, rejection and maximally ﬂat passbands. Practical RF design impose constraints on the performance of this ﬁlter. This bandpass ﬁlter is the ﬁrst line of defense against RFI. The ﬁlter response HB (f ) is designed to be as ideal as possible - which means having a narrow transition band, ﬂat passband, and as much stopband rejection as practically possible. Out-of-band rejection requirements often exceed 40 dB, leading to multi-stage designs. Beyond the bandwidth issue, there is also an issue of center frequency selection of operation for microwave radiometers. The geophysical parameter to be observed typically determines the neighborhood of center frequency the radiometer should operate. For example, in soil-moisture radiometers, it is desirable to observe at L-band frequencies since the emissions from the soil penetrate vegetation. For snow or ice cover applications, typically a center frequency in K-band is desirable. In all cases, there are anthropogenic signals with center frequencies occurring throughout most of the radio frequency band. 38 To address the spectrum occupancy issue, regulatory agencies allocate protected frequency bands for what is called Earth Exploration Satellite Services (EESS). Often times, these bands are shared with Radio Astronomy Services (RAS) [8]. For EESS services, ITU regulation of protected passive bands is required because the entire Earth is observed. The result of international regulation sets aside a collection of frequency bands for passive observation, which means transmitting anywhere on Earth within these bands is prohibited. However, there are circumstances and areas of the world where these regulations are not observed. Even in areas where they are observed, RFI still occurs and is reported in [43, 44] The various microwave radiometer conﬁgurations allow for the detection of diverse features in the radiometric signal that is corrupted by anthropogenic RFI. According to Gasiewski in [51], anthropogenic signals will be relatively bandlimited with respect to the radiometer observation bandwidth, have unusual degrees of slant-linear polarization, have high degrees of correlation between polarizations, and have a high degree of directional anisotropy. According to Ruf, et. al. [46], the presence of anthropogenic RFI also perturbs the Gaussian pdf of the radiometric signal, allowing for the development of statistical detectors of RFI. In summary, there exist to date - three basic methodologies for detecting the presence of RFI in the pre-detected radiometer signal [51, 52] include statistical amplitude detection, time-frequency detection, and polarization detection. 2.5.2 Statistical Detection Statistical RFI detectors are based on the underlying assumption that the probability density function (pdf ) of the uncontaminated x(n) is a zero-mean, white, Gaussian process. The presence of RFI in x(n) implies that the pdf would no longer be Gaussian. As a result, detection algorithms rely on tests for signal non-Gaussianity. One such RFI detection 39 algorithm was proposed by Ruf, DeRoo, and Misra [46]. The algorithm uses the sample kurtosis of x(n) to detect the presence of RFI in x(n). The sample kurtosis of a signal x(n) ∈ RN with mean μx and standard deviation σ is deﬁned by (x(n) − μx )4 kurt(x(n)) = σx4 N −1 1 = (x(n) − μx )4 4 N σx n=0 (2.42) If x(n) does not contain RFI, then it should have a Gaussian pdf. Therefore the the kurtosis should have a value at or near 3. Otherwise, for RFI that causes the pdf of x(n) to be non-Gaussian, then the kurtosis should deviate from 3. A platykurtic pdf is one in which the kurtosis is less than 3, and hence the pdf is spread wide around the mean of the distribution. A leptokurtic pdf is one in which the kurtosis is greater than three, thus having a distribution concentrated around the mean. The uniform distribution is the most platykurtic distribution and a delta function is the most leptokurtic distribution. An equivalent kurtosis deﬁnition is known as the kurtosis excess, which simply subtracts the value of 3 from (2.42) such that if x(n) has a Gaussian pdf, then the kurtosis excess would evaluate to zero. For L-band Earth-Observing radiometers, sources of RFI with the highest probability of occurrence include air trafﬁc control and military radars [24]. Signals of this class are usually continuous-wave (CW) or pulsed-CW signals. It was found in [53] that CW signals in noise exhibit a platykurtic pdf and pulsed-CW signals in noise exhibit a leptokurtic pdf. As a result, the kurtosis detector can also act as a classiﬁer between CW and pulsed-CW RFI. One drawback to using the kurtosis detector is that it cannot detect the presence of RFI from pulsed-CW sources with a 50% duty cycle. However, this problem is addressed two 40 ways. First, nearly all terrestrial pulsed-CW sources will have a duty cycle less than 50%. Secondly, this detection blind spot is addressed in [54] using the 6th moment. Another drawback is that the kurtosis detector in principle cannot detect RFI present in x(t) when the composite signal pdf appears Gaussian [55]. This can also be a problem in cases where there are multiple RFI sources present in a single observation footprint. Multiple sources of RFI in a single footprint is a reasonable assumption since an Earth-observing radiometer sees footprints on the order of 10’s to 1000’s of square kilometers. This is pointed out by Misra in [55]. It was reported by Misra that given a multi-source pulsed-sinusoid model with N sources, the kurtosis detector misses obvious large RFI signals as N increases. Put another way, as N increases, a Central-Limit Theorem argument can be applied to the pdf of the RFI signal. As the N increases, the pdf looks more Gaussian, leading to RFI detection failures. The premise of RFI detection formulated as a non-Gaussianity detector leads to additional varieties of statistical tests besides kurtosis. In the work by Tarongi [56], ten different normality tests, including kurtosis, were evaluated for pulsed-CW, linear FM chirp, and pseudo-random noise signals. It was concluded in this work that kurtosis performed the best for RFI detection given the class of signals presented. 2.5.3 Time-Frequency Detection RFI can be detected by both time and frequency diversity of the radiometric signal. Tarongi and Camps [41], devise a method that uses the signal spectrogram to identify RFI. The key to this approach is that they deﬁne sufﬁcient time and frequency resolution cells in the spectrogram in order to resolve known RFI signatures. Next, they treat the spectrogram as an image, and threshold the image to determine the expected brightness temperature. The premise of this approach assumes that RFI will appear as distinct features of sufﬁciently 41 high brightness in the spectrogram in order to be thresholded. Many other researchers have also used time-frequency diversity in the form of the spectrogram [57–59]. 2.5.4 Polarization Detection It is thought that RFI can also be detected by analyzing the third and fourth Stokes parameters of a polarimetric radiometer [60]. Currently, RFI can be detected in the post-processed brightness temperatures of the radiometric signal. Our research will investigate the use of mutual information between polarization channels to see if RFI can be detected prior to cross-correlation in a polarimetric radiometer. A depiction of the various RFI mitigation techniques are shown in Figure 2.11. In Figure 2.11, the RFI detection methods are grouped into 6 different areas Polarization, Cancellation, Time-Frequency Diversity, Statistical Tests, Excision, and Prevention. All methods on the graph besides Excision and Prevention are considered pre-detection methods. These methods operate directly on x(n) and not the power P of x(n). In this context, this dissertation contributes two new methods for RFI detection - (1) statistical tests based on negentropy and (2) the complex signal kurtosis. In actual operation, many of these approaches can be combined with one another. Research is currently being conducted to ﬁnd optimal ways to combine algorithms [3]. 42 ŷ EV x̂ EH Figure 2.1: Three-dimensional schematic of an electromagnetic wave and its projection on a plane intersecting its direction of propagation. The wave here is circularly polarized, which is a special case of the more general elliptical polarization. 43 ŷ EV φ EH Figure 2.2: Polarization Ellipse. x̂ 44 Pre-Detected Signal x(t) v(t) E(r, t) G TA + TSY S Antenna Low Noise Ampliﬁer Reference Load HB (f ) Bandpass Filter Post-Detected Power Signal P ∝ T̂A τ ( )2 dt Square Law Detector, Integrator Figure 2.4: Total-power radiometer. This is also known as a switching radiometer [9]. The antenna receives the total electromagnetic ﬁeld radiated from Earth E(r, t). Radiometric measurements consist of alternating antenna-only measurements (which we focus on in this dissertation), and reference load measurements. The resulting signal is band-limited using an RF bandpass ﬁlter HB (f ). The resulting signal is called the pre-detected signal x(t). The average power P of x(t) is periodically computed over T seconds. Reference Load Bandpass Filter G TSY S E(r, t) HB (f ) Square Law Detector, Integrator xH (t) τ ( )2 dt Ampliﬁer OMT τ xV (t)x∗H (t) dt Antenna Correlator G HB (f ) Ampliﬁer Bandpass Filter xV (t) Reference Load Figure 2.5: Polarimetric Radiometer. τ ( )2 dt Square Law Detector, Integrator 45 v(t) E(r, t) Low Noise Ampliﬁer TA + TSY S Mixer HB (f ) G × HIF (f ) Bandpass Filter Antenna x(t) ADC x(n) 1 N N −1 n=0 x2 (n) P Square Law Detector, Integrator Bandpass Filter cos(ωLO t) Figure 2.6: Digital superheterodyne total-power radiometer. Reference Load TSY S E(r, t) vH (t) cos(ωc n) G HB (f ) Ampliﬁer Bandpass Filter xH (t) ADC × sin(ωc n) LPF × LPF × LPF cos(ωc n) × LPF IH (n) QH (n) OMT Antenna vV (t) G HB (f ) Ampliﬁer Bandpass Filter Reference Load xV (t) ADC sin(ωc n) Figure 2.7: Superheterodyne polarimetric radiometer Figure 2.8: Example Radiometer Timing Diagram. IV (n) QV (n) 46 Figure 2.9: Relative brightness sensitivities versus frequency for atmospheric and ocean environmental data records. National Academic Press [8]. 47 Figure 2.10: Relative brightness sensitivities versus frequency for land related environmental data records. National Academic Press [8]. Wavelet Denoising CrossFrequency Spectrogram Smoothing Cancellation TimeFrequency Diversity Pre-Detection Methods Post-Detection and Other Methods Prior Research Cited Research Contribution Stokes Parameters Filterbank Peak Detect Negentropy Complex Signal Kurtosis RFI Mitigation Methods Polarization Diversity Statistical Tests Real Signal Kurtosis Other Gaussianity Tests Prevention Excision Flagging ITU-R RS.1029-2 NTIA FCC Discard Data Figure 2.11: Graph of the various approaches taken to detect and mitigate RFI. Chapter 3 HISTORICAL RFI DATA AND DEVELOPMENT OF A COMPLEX-VALUED RFI SIGNAL MODEL 3.1 Introduction In this chapter, we review historical examples of RFI from spaceborne radiometers, connect these examples to the current RFI signals model developed in [55], and use experimental radiometer data to extend this model to the complex domain. We employ the current RFI signal model in [55] for evaluating negentropy-based test-statistics in Chapter 4 and employ the extension of this model, developed in this chapter, to the complex-valued formulation of radiometric RFI detection in Chapter 5. 48 49 In Chapter 5, we re-formulate the RFI detection problem in such a way that allows us to take advantage of complex-valued signal processing in two radiometric situations: 1. When the pre-detected radiometer signal x(n) (or a single channel of a polarimetric radiometer) is downconverted to complex-baseband representation z1 (n) with in-phase I(n) and quadrature Q(n). Therefore z1 (n) = I(n) + jQ(n). 2. In a polarimetric radiometer when the polarimetric channel signals (horizontal xH (n) and vertical xV (n)) are combined to form a single complex-valued signal z2 (n) = xH (n) + jxV (n). For the ﬁrst case, the RFI model developed in [55] is adequate because we can generate real-valued RFI for x(n) and then downconvert it to the complex representation z1 (n). However, for the second case, the signal model is not adequate since it can only treat xH (n) and xV (n) separately, and not as a single complex-valued random signal. In all cases, we assume the digitized RFI signal component is represented by s(n, Λ) with parameter set Λ , the thermal noise signal component is represented by w(n), and the RFI-corrupted signal x(n) observed by a total-power radiometer is represented by the sum of the signal components s(n) and w(n) x(n) = s(n, Λ) + w(n), 0 ≤ n ≤ N − 1, (3.1) for N samples every integration period. For the RFI-free case, s(n, Λ) = 0. We make further assumptions about the noise component w(n) in Chapter 5 for the polarized RFI case. The goal of RFI mitigation is to detect the presence of s(n) given x(n) and minimize the effect of s(n, Λ) on the average power measurement of x(n), leading to an RFI-free brightness measurement, TB . 50 3.2 RFI Model from Microwave Radiometer Frequency Allocations Radiometers operate over one or more speciﬁc narrow microwave spectrum bands depending on the set of EDRs that they are intended to measure. Some of these microwave bands are protected by the International Telecommunications Union (ITU), making it illegal to transmit any signal within the band anywhere in the world. These bands are referred to as Primary Exclusive Allocations [8]. Other microwave frequency bands are designated as shared allocations, so that radiometry can be performed, but on a limited basis since terrestrial and airborne transmitters are allowed to operate within the passband. The three different band allocation types are: • P - Primary Exclusive Allocation. This prohibits any entity from transmitting signals within the bandwidth range allocated • p - Shared Primary Allocation. Transmission is allowed, but cannot interfere with the observing radiometer. • s - Shared. There are no restrictions as to who uses spectrum in the band. This is the most harmful for microwave radiometer operation. A list of the radio frequency bands that different radiometers operate within is summarized in Appendix C in Table C.2, along with the geophysical parameters that can be obtained when operating in the band. Although in cases where the observation takes place in a primary exclusive allocation, there are spurious emissions that can fall within the passband. Furthermore, not all entities around the planet observe ITU regulations, and hence transmit signals directly within the radiometer’s passband [60]. Another, more 51 prevalent and insidious source of RFI can come from spurious emissions from transmitters of center frequency outside of the radiometer’s band. Spectral leakage from neighboring bands, caused by nonlinearities in the transmitters’ power ampliﬁer can spill over into the radiometer passband. An example of a strong RFI from a source thought to be a communications system in L-band is detailed in [47]. Lastly, RFI can originate from the radiometer itself, in the form of harmonics from internal clocking sources and local oscillator (LO) leakage. 3.3 Historical Examples of RFI Strong RFI appears as abnormally-high brightness temperatures in the brightness temperature maps corresponding to the radiometer output data. As this data is collected over time, others construct global maps of the relative frequency of occurrence of large brightness temperature excursions, or RFI ﬂagged by the instrument. Some examples of these are shown for illustration in Figures 3.1 through 3.4 for L, C, X, and K-band radiometers currently in orbit. As can be seen in the plots, RFI occurs globally. Radiometers can expect to encounter more RFI in heavily populated areas, however, because the number of interfering sources increases as there are more users to use devices that cause RFI [8, 23] . In Figure 3.1, for example, Much of Europe has RFI, as the color scale indicates that most of the radiometer data over Europe taken by SMOS is discarded. Essentially, there is a 100% chance of encountering RFI over nearly all areas of Europe, some areas of North America, and many areas of Asia in L-band. About a decade ago, a similar observation was made by the AMSR-E mission, but in C-band, shown in Figure 3.2. In this case, abnormally high brightness temperatures were reported over land masses which had dense populations. 52 Figure 3.1: Soil-Moisture retrievals from the Soil Moisture Ocean Salinity (SMOS) L-band radiometer, launched in November 2009. RFI is so severe that most of Europe appears as a blind spot where soil-moisture cannot be measured. Even at higher frequencies, such as K-band, RFI was reported in [11] by the Special Sensor Microwave Imager (SSM/I) radiometer. Other sources of RFI include ocean-reﬂected signals transmitted from Geostationary Direct Broadcast Satellites. Table 3.1 lists satellites that are potential transmitters of RFI since they transmit near passive microwave frequency bands. In addition, [61] points out several speciﬁc possibilities for RFI that originate L-band radars, mobile and and navigation satellite services, and other satellite services. Skou et al., in [23] elaborate on the polarization characteristics of linear and circular polarized radar, 53 Figure 3.2: Possible C-band RFI reported by the AMSR-E radiometer reported by JAXA [10]. Red corresponds to retrieved brightness temperatures that are abnormally high, well above the average of 300K which is expected for Earth. and their effects on the third and fourth Stokes Parameters. 3.4 General RFI Signal Model for L-band For L-band radiometers, interference is largely expected from terrestrial radar sources, such as air defense radar, and air trafﬁc control radar [60]. Though the center frequencies of the sources are outside of the L-band allocation, these radars transmit at high powers required to cover large distances, and have been shown to cause interference within this allocation. Justiﬁcation for using this model is supported by [12]. Misra, DeRoo, and 54 Figure 3.3: K-band RFI observed by SSM/I, WindSat and AMSR-E radiometers in January 2009. SSM/I (top panel), WindSat (middle panel), and AMSR-E (Bottom panel) [11] 55 Figure 3.4: On the very ﬁrst orbit of the Aquarius L-band radiometer, in which the data system was switched on, RFI was observed over China. Aquarius was launched on June 10, 2011. Ruf [55] developed a RFI-model that assumes a total of L radars existing within an antenna footprint. The continuous-time signal model is given by x(t) = w(t) + s(t, Λ), (3.2) where s(t) = L Ak cos (2πfk t + θk ) Π k=1 t − t0 wk . (3.3) The parameters, given by the notation Λ, of this model are summarized in table 3.2, and the rectangle pulse function is given by Π(t) . 56 Table 3.1: Potential sources of X and K-band RFI from various Geostationary Direct-Broadcast Services (DBS) Satellites. Because of their wide transmit swaths, transmit power, and operating frequency, transmissions from these satellites can reﬂect off of the ocean surface and reach the input of a K-band microwave radiometer. These data are available at http://www.remss.com/about/projects/radio-frequency-interference. Spacecraft DirecTV-10 DirecTV-12 DirecTV-11 Hispasat 1E Intelsat 3R (Sky Brazil) Intelsat 11 (Sky Brazil) Atlantic Bird 4 Eutelsat W3A Hot Bird 7A → Eurobird 9A Eutelsat W2A Hot Bird 6 Hot Bird 8 Astra 1KR Astra 1E Astra 2D Astra 2C Longitude 102.8◦ W (257.2◦ E) 102.8◦ W (257.2◦ E) 99.2◦ W (260.8◦ E) 30.0◦ W (330.0◦ E) 43.0◦ W (317.0◦ E) 43.0◦ W (317.0◦ E) 7.2◦ W (352.8◦ E) 7.0◦ E 13.0 → 9.0◦ E 10.0◦ E 13.0◦ E 13.0◦ E 19.2◦ E 19.2 → 23.5◦ E 28.2◦ E 28.2 → 31.5◦ E Launch date 7/6/2007 12/28/2009 3/19/2008 12/29/2010 1/12/1996 10/5/2007 2/27/1998 3/16/2004 3/11/2006 4/3/2009 8/21/2002 8/5/2006 4/20/2006 10/19/1995 12/19/2000 6/19/2001 Areas Affected North America North America North America Europe Brazil Brazil Middle East Europe Europe Europe Europe Europe Europe Europe Europe Europe Channel 18.7 GHz 18.7 GHz 18.7 GHz 10.730 GHz 10.65 GHz 10.65 GHz 10.65 GHz 10.65 GHz 10.65 GHz 10.65 GHz 10.65 GHz 10.65 GHz 10.65 GHz 10.65 GHz 10.65 GHz 10.65 GHz The pdf was derived for x(t) by means of characteristic functions and is given by fx(t) (t) = ∞ −∞ e − σu 2 2 N (dk J0 (Ak u) + (1 − dk )) ejut du (3.4) k=1 Where J0 is a zeroth order Bessel function. A conclusion drawn from this model in [54] was that although the kurtosis-based detector performs well for low duty-cycle radar interfering sources, the performance degrades rapidly when a large number of high-duty cycle RFI sources are present. The reasoning behind this is that the resulting probability distribution function pdf in (3.4) experiences central-limit theorem conditions due to the additive effect of many high duty cycle signals. This model assumes that for 57 Table 3.2: Parameters of Multi-Pulsed Sinusoid RFI Model. The notation U means uniform distribution, wk is the pulse-width k, T is the radiometer integration period, and 0 ≤ ξ ≤ 1. Parameter Deﬁnition Probability density assumption Ak Exponentially distributed random amplitude f (Ak ) = ν1 eAk /ν fk Uniformly distributed random frequency U (0, 2πB) θk Uniformly distributed random phase U (0, 2π) t0 Center of “on” pulse in the duty cycle, uniformly distributed over radiometer integration period U (0, T ) dk = wk /T Randomly distributed duty cycle ξ dk bw e − d2 k 2b2 w + (1 − ξ) 1 − 1 − νk e 1−d νk communications signals, they behave more like CW sources as opposed to radar, hence kurtosis would not perform well in the presence of many signals of this type. 3.5 RFI Signals from Complex Digital Modulation Model RFI from complex digital modulated signals are considered in this section because they represent a type of RFI that all radiometer types can experience [11, 60, 62]. Using the DVB-S2 standard as a guide for what types of modulation a spaceborne radiometer is likely to experience, we consider the following digital modulation types. 1. Binary Phase-Shift Keying (BPSK) 2. Quadrature Phase-Shift Keying (QPSK) 3. M-ary Phase Shift Keying (M-PSK) 4. M-ary Quadrature Amplitude Modulation (M-QAM) 58 where, the M is simply the number of symbols transmitted in the modulation type. The signal model for RFI resulting from digitally-modulated sources should be represented by s(t) = Am(t)ej(2πfc t+Θc ) (3.5) where A is the amplitude of the waveform, m(t) is the linearly modulated signal, fc is the carrier frequency of the waveform, and Θc is carrier phase. The linearly modulated signal is of the form m(t) = K pk h(t − kτsym ) (3.6) k=0 where pk is the k-th constellation point in an M -order digital modulation scheme, K is the number of symbols present in the interfering signal during the radiometer observation time τ , h(t) is the signal shaping ﬁlter, and τsym is the symbol period. Any digital modulation can be represented by this model, and it takes into account both amplitude and phase variability of an RFI source. Study of the various RFI types outside of L-band shows that this is an adequate model to describe RFI originating from terrestrial and broadcast satellite signal types [11, 44, 62]. 3.5.1 S, C, X - Bands No S-band spaceborne radiometers have been found either in use or planned for use. However, several cases of airborne S-band radiometers have been found. According to Table C.2 in the appendix, there exists a 10 MHz protected passive frequency allocation, but apparently it is not used in spaceborne microwave radiometry. In C-band, there is no passive frequency allocation. As a result, large amounts of RFI from terrestrial sources are expected to be present in a standard radiometer observation as conﬁrmed in [44]. In X-band, RFI is expected because most of the band is shared 59 with terrestrial ﬁxed communications links. Based on a literature search, it was found that likely RFI sources include ﬁxed and mobile communications links that primarily include terrestrial communications stations. Common signal modulations found in these potential RFI sources include M-ary quadrature modulation (QAM) and quadrature phase shift keying (QPSK) [62]. 3.5.2 Ku , K-band, Ka Bands K-band interference has been found to originate from reﬂections of direct broadcast satellites [11]. A majority of these satellites employ a form of quadrature modulation in the form of the digital video broadcast satellite (DVB-S) standard, or the second direct video broadcast standard (DVB-S2) [63] . DVB employs binary phase-shift-keying (PSK) and quadrature phase shift keying (QPSK) modulation in its transmissions. DVB-S2 uses one of many constellations in a variable coding and modulation (VCM). The modulations include QPSK, 8-PSK, 16 amplitude and phase shift keying (APSK), and 32-APSK. Both DVB and DVB-S2 use digital modulation employing a constellation of points in the in-phase and quadrature (complex plane), so they share a similar signal model. In general, the model in 3.5 can be used to describe both modulation types. 3.6 Development of Polarized RFI Signal Model Given the numerous examples of RFI observed in the brightness temperature maps of actual radiometers currently in orbit, the pulsed-CW and complex digital modulation models of these RFI sources cover a wide range of radar, communications system, and broadcast satellite RFI signal types identiﬁed to be potential sources of anthropogenic RFI [11, 24, 47]. These models are useful for analytical study, simulation of RFI, and performance 60 characterization of RFI detectors. However, what’s missing from the literature is an RFI signal model that describes the joint signal received from both polarization channels of a polarimetric radiometer that was described in 2.2.2. In this section, we use real radiometer data from two different research radiometers to develop and justify a simple complex-valued signal model for polarimetric RFI. We develop the model as follows. For every radiometric data set, analyze the time, frequency, joint time-frequency, and statistical behavior of the RFI to get a sense of the general amplitude, frequency, and phase behavior of the RFI. We then use this behavior, combined with the characterization of polarized electromagnetic waves [16, 22] to determine this signal model. We wish to construct an RFI model as simple as the PCW model in (3.3), for L = 1 (single interferer model) for polarimetric radiometers, and use it later, in Chapter 5, to evaluate the performance of a test statistic that is based upon a complex signal kurtosis coefﬁcient as deﬁned in [14] for detecting polarimetric RFI. Two of the three data sets used have known, strong, RFI, which is simply detected using second-order statistics, such as the spectrogram, or by inspection of the amplitude of the time-domain plot. The third data set, from the Goddard RF Explorer (GREX) would have been used to aid in the development of this model, but it was found that the data was corrupted by self-imposed RFI, originating from a 25 MHz oscillator the digital back-end of GREX. 3.6.1 SMAP Validation Experiment (SMAPVEX12) Data Set The integrated break-out board (IBOB) is a digitizer system developed by the University of California Berkely’s CASPER Lab. The IBOB system was originally intended for use 61 as a digital receiver radio astronomy applications. NASA Jet Propulsion Laboratory (JPL) used this system as a digital receiver back-end for a polarimetric L-band radiometer, and ﬂew a test ﬂight campaign in 2012 to characterize the RFI environment a small portion of the United States, as a means to help calibrate and validate the upcoming SMAP mission. This test campaign was called the SMAP Validation Experiment (SMAPVEX12) [4]. The SMAPVEX12 data originates from a ﬂight over Denver, Colorado, USA, on July 22, 2012. The ﬂight path of the SMAPVEX12 campaign is shown in Figure 3.5. The amplitude vs. time, power spectral density (PSD), joint time-frequency domain, and amplitude histogram plots of this data were studied. Scatter plots of the horizontal polarization versus vertical polarization data samples were also studied to gain some insight into the circularity/ellipticity of the data. The various plots are shown in Figures 3.6 through 3.8. The analysis of the SMAPVEX12 data was used to help develop a general complex-valued signal model for RFI as experienced by a polarimetric radiometer. The objective of this analysis was to support the use of complex-valued signal processing on polarimetric radiometer data in Chapter 5. From the resulting plots, it was learned that the polarimetric channels have similar spectrograms. When strong RFI appears in the H-Pol channel, it usually appears in the V-pol channels as well. This suggests that the frequencies of the RFI are the same for each polarization. Moreover, the scatter plot data is generally not circular. This is to be expected since the the amplitudes of an RFI source would generally be different in each polarization receiver. Lastly, the histogram analysis shows that most of the data is near Gaussian, unless the power of RFI source is signiﬁcantly higher than the thermal noise. DeRoo presents a model for this scenario in [12], which describes the pdf of a pulsed sinusoid in noise. As the RFI power to noise power ratio increases, this distribution has a sharper peak and smaller variance. This behavior is consistent across polarization channels, but not exactly 62 Figure 3.5: SMAPVEX Flight Path Over Denver, CO, USA. The colorbar indicates the relative brightness temperature observed. Red areas indicate high intensity RFI sources. the same in terms of variance. The mean of the data is approximately zero in all cases. Even if the mean is not zero for radiometric data, the mean can be computed for every integration period, and subtracted out. Moreover, proper design of a microwave radiometer tries to eliminate any DC bias in the front-end analog and back-end digital circuitry [6, 9]. 3.6.2 Canton Data Set In June 2005, ground-based radiometric observations were made in an experiment to characterize L-band RFI and the performance of three digital back-end receivers for mitigating this RFI. The instrument conﬁguration consisted of a common L-band radiometer that fed its signal to three distinct digital back-end receivers [64]. The observations were made approximately 200 meters away from the antenna of an air-route surveillance radar (ARSR) 63 Figure 3.6: SMAPVEX12 Combined Sources 1. Signal contains a narrowband CW, wideband CW and a narrowband pulse sources. in Canton, MI Figure 3.9. The characteristics of the the digital radiometer signal are summarized in Table 3.3. Similar analysis in terms of the time, joint time-frequency domain, and statistical study to the SMAPVEX12 data was performed on the Canton data set. However, the Canton data set, according to [64] was downconverted to complex baseband as a matter of convenience. As a result, the two polarization channel signals themselves were complex-valued. Only the H-Pol channel was studied in this case simply to verify the radar pulse in the data, and 64 Figure 3.7: SMAPVEX12 Combined Sources 2. Signal contains a narrowband pulse, wideband pulse and a narrowband CW sources. to see its effect in the joint time-frequency domain. An example plot of this data is shown in Figure 3.10. 3.6.3 GREX Data Set Data was used from the Goddard RF Explorer (GREX), the most recently implemented radiometer digital back-end, on October 6, 2012. GREX is by far the largest data set in 65 Figure 3.8: SMAPVEX12 Narrowband CW. The H-Pol and V-Pol signal amplitudes are close to each other on average, but generally not equal, resulting in approximate Gaussian pdf s and a circular scatter plot. The CW signal present is not obvious in the time or statistical domains, but is obvious in the joint time-frequency domain plot. Furthermore, the RFI occurs at the same frequency in both polarization channels. terms of the number of bytes, since both polarimetric channels were sampled at 350 MHz, with 14-bit resolution, over the course of a 6-hour ﬂight. A summary of the GREX data set properties is listed below. The GREX instrument was attached to a P3 aircraft and ﬂown along the DelMarVa Peninsula and the Atlantic Ocean. The ﬂight path is indicated in Figure 3.11. The intent of GREX was to sample L-band directly at IF, for horizontal and vertical polarizations. The 66 Figure 3.9: Air-Route Surveillance Radar in Canton, MI. ﬂight path was designed to capture L-band RFI by ﬂying over densely populated areas, as well as RFI-free data by ﬂying over open ocean. A total of 23 Terabytes of data was collected. Time, frequency and histogram analyses were performed on a subset of the data, and is shown in Figure (3.12). It was quickly discovered that the GREX data set would be problematic. No data was recovered that was RFI free. It was found that the actual data system caused self-imposed RFI as described in Section 3.2, in the form of harmonics of an on-board 25 MHz clock circuit. This circuit was responsible for controlling the gigabit ethernet controller on the system, and was afﬁxed to the motherboard. It was found later that this clock could not be disabled. The problem with this clock is and its harmonics is 67 Table 3.3: Properties of Canton Data Set Property Value Polarizations V,H Intermediate Frequency 25 MHz Sample Rate 100 MHz Sample Resolution 14 bits Downconverted to complex baseband? Observation Base Yes Ground-based Observation General Location Canton, Michigan Table 3.4: Properties of GREX Data Set Property Value Polarizations V,H Intermediate Frequency 120 MHz Sample Rate 350 MHz Sample Resolution 14 bits Downconverted to complex baseband? Observation Base No Observation General Location P3 Aircraft-based Atlantic Ocean and DelMarVa Peninsula that they appeared every multiple of 25 MHz, leading to signiﬁcant RFI directly within the radiometer IF band, at 125 MHz. 68 Figure 3.10: A single radar pulse of the Canton data set. 3.6.4 Polarized Complex RFI Signal Model Given the historical observations of RFI and the radio frequency bands that they occur, as well as observations of the SMAPVEX12 and Canton data sets, we formulated the following digital signal model for complex polarimetric RFI s(n) = AH cos 2πfc n FS + jAV sin 2πfc n FS Π(n), (3.7) 69 Figure 3.11: Flight Path of GREX October 6, 2012 experiment. where the digital signal has real and imaginary component amplitudes AH and AV , common frequency fc , sample rate FS , duty cycle d, and a 90◦ phase difference between real and imaginary components (due to orthogonal polarization). The function Π(n) is simply a rectangular window function that zeroes-out a percentage of the signal over total N samples, similar to the pulse-CW model in 3.3 for L = 1. The parameter set Λ is thus represented by Λ = {AH , AV , fc , dFS , N }. The received polarimetric signal x(n) = xH (n) + jxV (n) can therefore be represented 70 Figure 3.12: GREX data set, showing self-imposed RFI at every 25 MHz due to an onboard clock from an ethernet controller. This RFI is most apparent in the two power spectral density subplots in the ﬁgure. by x(n) = s(n, Λ) + w(n), (3.8) assuming that we have complex radiometric noise w(n) = wH (n) + jwV (n). Therefore, 71 combining 3.7 with 3.8, we have the general received complex RFI model x(n) = xH (n) + jxV (n) 2πfc n 2πfc n = AH cos + wH (n) + jAV sin + jwV (n) Π(n), FS FS (3.9) (3.10) Where the signals wH (n) and wV (n) represent the horizontal and vertical polarized radiometric noise components, respectively. According to [31, 32], wH (n) and wV (n) cannot be considered uncorrelated, and have correlation coefﬁcient ρ. 3.7 Summary and Conclusions In this chapter, RFI data from a number of historical and currently-orbiting radiometers were reviewed. These radiometers included SMOS, AQUARIUS, AMSR-E, SSM/I, and WindSat. The data revealed that RFI occurs globally, as shown in Figures 3.1 — 3.4. We also saw that RFI occurs more frequently over heavily populated areas, but also occurs as a result of reﬂected signals from direct broadcast video satellites. We also discussed two signal models that cover a large representative set of RFI as seen by a radiometer. The ﬁrst signal model developed in [55] considered the multiple PCW case. In this model, all of the random-parameter sinusoids are additive, contributing their individual powers to the power of the thermal noise signal w(n). The parameters themselves come from various pdf s according to observational data reported in [55]. We consider this model to represent most of the kinds of signals that make up RFI, as supported by [23, 44, 47, 65]. The second RFI signal model discussed was a digital modulation model for linearly modulated communications systems. This generic digital modulation model is a slight extension of the multiple PCW case since it considers phase modulation, as well as 72 the sinusoidal nature of the envelope of the RFI signal. Although the two aforementioned signal models cover a wide variety of radiometric RFI signal types, they are real-valued only, and correspond to a single total-power channel of a spaceborne microwave radiometer. In Chapter 5, the horizontal and vertical polarization channel signals are interpreted as a single complex-valued signal. To demonstrate complexvalued RFI signal detection in this case, a complex-valued RFI model is required. We developed such a model based on analysis of SMAPVEX12 data, and presented in 3.7. This new RFI model is a complex-valued extension of the PCW model used in prior work. Chapter 4 DETECTION OF SINUSOIDAL RFI USING NEGENTROPY APPROXIMATIONS FOR TOTAL-POWER RADIOMETERS 4.1 Introduction Many methods exist for detecting RFI and subsequently mitigating its impact. These methods can be grouped into two categories: non-statistical methods and statistical methods. Non-statistical methods tend to exploit time, frequency, or wavelet-domain characteristics of x(n) signal [41, 49, 66–68]. Statistical methods primarily depend on 73 74 detecting the Gaussianity of the probability density function (pdf ) of the pre-detected waveform x(t) [12,56]. In the absence of RFI, this signal is considered to have a zero-mean Gaussian pdf with variance σ 2 (notation given by N (0, σ 2 )), during a single radiometer integration period T [19]. In the presence of RFI, the pdf of x(t) deviates from N (0, σ 2 ). Detection of this deviation is the principle upon which statistical detection methods are based [49]. Seemingly missing from the RFI detection literature is a discussion of negentropy, which is often used for Gaussianity detection in source-separation problems involving Independent Component Analysis (ICA) [69, 70]. Negentropy was introduced as a method for detecting RFI in [2]. Though negentropy is a function of the pdf of a continuous random variable, several negentropy approximations exist for use with discrete random variables. We need these approximations for application to digital microwave radiometers. As a result, we show that negentropy can be utilized for microwave radiometric RFI detection. The results of this chapter add to the current literature by introducing and evaluating negentropy and its various approximations for RFI detection. 4.2 Overview In this chapter, we consider RFI detection in the case of a digital total-power radiometer model, as depicted in Figure 4.1. Of particular interest is the the pre-detected radiometer signal x(t), with a corresponding pdf fX (x), and the pre-detected digitized radiometer signal, x(n), with corresponding probability mass function (pmf ) fˆX (x). All test statistics in this chapter depend a ﬁnite, N -sample segment of x(n) which corresponds to an integration time T in the analog-radiometer case. RFI detection happens in parallel to the usual operation of power estimation as shown in the ﬁgure. 75 v(t) E(r, t) Low Noise Ampliﬁer TA + TSY S Antenna G Mixer HB (f ) × Bandpass Filter HIF (f ) x(t) ADC Bandpass Filter x(n) 1 N N −1 n=0 x2 (n) P Square Law Detector, Integrator cos(ωLO t) Figure 4.1: Total-power radiometer. RFI detection depends on a ﬁnite N -sample window of x(n). This chapter begins with a thorough review of the current literature on RFI detection methods that test for the Gaussianity of the pdf of the digitized, total-power pre-detected radiometer signal x(n) in section 4.3. Each method reviewed utilizes some test statistic of a single integration period of x(n). We point out that none of the current methods utilize negentropy and that it is of interest in this dissertation because it is based on the same principle as current methods — namely Gaussianity detection. In section 4.4, we introduce negentropy and show why we are limited to using approximations of it for RFI detection since we are operating on the digitized radiometric signal, x(n). In section 4.5 the formal binary hypothesis testing problem we wish to solve using negentropy as an RFI test statistic is presented. We focus on six speciﬁc approximations of the true negentropy: one based on the Edgeworth series approximation in [71], one that directly uses the histogram approximation of the pdf, two based on non-polynomial approximations of negentropy [69], and two others derived from the non-polynomial approximations of negentropy that exploit known properties of the radiometric signal in the RFI-free case. We then analyze how these negentropy approximations perform with increasing numbers of samples under the Gaussian noise-only scenario in section 4.6. RFI is then modeled in section 4.7 as a simple single pulsed-sinusoidal signal, with constant amplitude, frequency, phase, and duty cycle. We deﬁne duty cycle as the amount 76 Table 4.1: Summary of Negentropy-Based Test Statistics Studied Name Symbol Notes Histogram Jh Uses histogram approximation of the pdf of x(t) to compute negentropy directly Edgeworth Je Edgeworth Series approximation Hyvärinen approximation A Ja Non-polynomial series approximation Hyvärinen approximation B Jb Non-polynomial series approximation Secondary Hyvärinen approximation A Jaa Same as Ja , dropping ﬁrst term Secondary Hyvärinen approximation B Jbb Same as Jb , dropping ﬁrst term of time within a single integration period in which the sinusoidal signal is present. The sinusoidal signal is absent otherwise. We add constant-variance Gaussian noise to this pulsed-sinusoidal signal and then apply all of our detectors to it, varying the signal parameters to explore the limits of detectability for this case. Subsequently, in section 4.8, we test negentropy using a more elaborate and realistic RFI model, one which consists of multiple pulsed-sinusoidal sources present in Gaussian noise, constant variance. In this experiment, we generate a new signal that is composed of M component interfering signals, each with parameters randomly chosen from an experimental model, developed by Misra [55]. We then add Gaussian noise of unit variance to this signal, forming a RFI model that is more representative of what a spaceborne microwave radiometer would actually experience. In all cases, we apply the Neyman-Pearson decision rule to derive receiver operating characteristic (ROC) performance curves [56]. All six negentropy-based tests are compared 77 in ROC performance to the kurtosis test for RFI [12, 56]. In addition, we quantify and compare the various ROC performance curves using the area-under-the-curve (AUC) metric, as well as two detection probabilities PD as a function of the false-alarm probabilities PF = {0.05, 0.10}. The AUC metric and two PD (PF ) points are plotted as bar plots in subsequent ﬁgures throughout the discussion. All ROC performance data can be found in Appendix B, sections B.1.1 and B.1.2. We ﬁnally summarize our results of this chapter in section 4.9. However, none of the negentropy-based detectors outperform the kurtosis-based detector ROC performance, with the exception of the kurtosis blind-spot case. The kurtosis is currently currently used on airborne platforms [46], and will soon be used in space [3] for the ﬁrst time. 4.3 Review of Gaussianity Tests Current RFI detection methods depend on detecting non-Gaussianity of the digitized radiometric signal prior to integration. Tests for Gaussianity are numerous. For example, Tarongi and Camps [56] evaluate ten different Gaussianity tests, including the Jarque-Bera (JqB), Shapiro-Wilk (SW), Chi-square(CHI2), Anderson-Darling(AD), Lilliefors-Smirnov-Kolmogorov (L), Lin-Muldhokar (LM), Agostino-Pearson (K2), Cramer-von Mises (CM), kurtosis (k), and skewness (S) tests. It was concluded that the kurtosis provides the best receiver-operating-characteristic (ROC) performance for a wide class of RFI signal models [56]. These models included continuous-wave (CW), pulsed continuous-wave (PCW), chirp (CH), and pseudo random noise (PRN) signals. In addition, it was shown that the combination of the kurtosis and Anderson-Darling (AD) tests were able to detect most types of RFI. In [52], Guner explores the SW test in more depth for PCW signals, in particular, and 78 shows comparable performance to kurtosis. However, there are two exceptions to this comparable performance. The SW test outperforms kurtosis when the signal has a duty cycle of 50%. In this case, the kurtosis cannot detect the PCW signal because its value is the same as if the PCW signal were Gaussian noise instead. The case where RFI is present in x(n) but is not detected at all by kurtosis is deﬁned as the kurtosis blind spot. Kurtosis has this blind spot for pulsed signals having a 50% duty cycle, which a number of tests do not have. As the RFI signal-power-to-Gaussian-noise ratio decreases, the performance of the SW test decreases as well, relative to kurtosis. It was DeRoo, Misra, and Ruf [12], who ﬁrst introduced kurtosis as a Gaussianity teststatistic with application to radiometric RFI detection. Computation of kurtosis depends on the ﬁrst through fourth central moments of x(n). The kurtosis blind spot was addressed in [40] considering the 6th -order moment, but the method suffers from a large variance relative to kurtosis, as well as two detection blind spots. Nevertheless, kurtosis has been shown to be a reliable test-statistic for detecting CW and PCW signals buried in radiometric noise. For this reason, we use kurtosis as a baseline for comparison of other RFI test-statistics and seek those test-statistics that have a better receiver-operating-characteristic performance. Negentropy is of interest in this dissertation because it is a Gaussianity test-statistic and because of its connections to kurtosis [69, 71]. If the kurtosis outperforms all other test-statistics that depend on the pdf of the pre-detected radiometer signal x(n), then how does Negentropy compare to it? The hypothesis of this chapter is that negentropy should outperform kurtosis in terms of detection because it depends on the entire pdf of the radiometric signals, as opposed to a single test statistic of the pre-detected radiometer signal. We ﬁnd that surprisingly, this is not the case, and we explain why in this chapter. 79 4.4 Negentropy The differential entropy of a continuous random variable X with pdf fX (x), is deﬁned as h(X) = − ∞ −∞ fx (x) loge fx (x) dx, (4.1) using the natural logarithm [72]. Of all continuous random variables with the same mean μX and variance σ 2 , the Gaussian random variable maximizes differential entropy [72] (refer to Appendix A for detailed proof). Moreover, the differential entropy of a Gaussian random variable XG with pdf f (x) is given by h(XG ) = − ∞ f (x) log f (x)dx, 1 =− f (x) log √ 2πσ 2 −∞ ∞ 1 = log 2πσ 2 f (x) dx + 2 −∞ ∞ 1 2 = log 2πσ f (x) dx + 2 −∞ 1 1 = log 2πσ 2 + 2 σ 2 , 2 2σ 1 1 = log 2πσ 2 + log e, 2 2 1 = log 2πeσ 2 . 2 −∞ ∞ (4.2) + 1 2σ 2 1 2σ 2 (x − μ)2 − dx, 2σ 2 ∞ f (x)(x − μ)2 dx, −∞ ∞ f (x)(x − μ)2 dx, (4.3) (4.4) (4.5) −∞ (4.6) (4.7) (4.8) The negentropy J(X) of X is deﬁned as the difference in differential entropies of X and a Gaussian random variable XG with the same mean and variance as X [69]: J(X) = h(XG ) − h(X) = 1 loge (2πeσ 2 ) − h(X). 2 (4.9) 80 Since the differential entropy is maximized for XG , the negentropy will equal zero iff X is also Gaussian and will deviate positively away from zero otherwise. Negentropy is nonnegative, invariant for invertible linear transformations, and is considered to be an optimal estimator of non-Gaussianity in a statistical sense [69]. These desirable properties suggest that negentropy can be used as a distance measure of the pdf of x(n) from Gaussianity and, hence, as a test-statistic for detecting RFI. The challenge with using negentropy is that one requires the underlying pdf of X, which can only be estimated from the data in practice. As a result, approximations of negentropy are required in this case. These approximations are frequently utilized in signal processing, in particular as a cost function used in ICA applications [69]. Another challenge with using the negentropy, as deﬁned in (4.9), is the fact that in digital radiometers, x(n) is discrete-time and discrete-amplitude. Consequently, only the digital signal would be available, and only negentropy approximations would be applied in the digital processor portion of the radiometer. Since we have only a ﬁnite, N -sample x(n) to work with, we can only apply approximations of negentropy in lieu of the true negentropy. Thus we expect as N → ∞, we expect the approximations to improve and hence the ROC performance of the corresponding detector to improve along with it. 4.4.1 Negentropy Approximations Considering an integration period of x(n), we employ a discrete-amplitude version of (4.9), J(x) = 1 log(2πeσ 2 ) − H(x), 2 (4.10) 81 where h(X) ≈ H(x) = − fˆX (x) log fˆX (x), (4.11) x∈S σ 2 is the sample variance of x(n), and fˆX (x) is the probability mass function — (pmf ) that approximates fX (x) with region of support S. The pmf fˆ(x) can be the histogram-based estimate, Parzen estimate [73], or other suitable estimate of the pdf of x(n). The histogram-based approximation of negentropy uses fˆX (x) directly. Two other negentropy approximations that involve sample statistics were developed by Hyvärinen [69] and Edgeworth [71]. In particular, Edgeworth presents negentropy as a polynomial series of cumulants. In addition, Hyvarı̈nen developed a family of approximations that use non-polynomial functions to estimate non-Gaussianity that were less sensitive to outliers — sparse, large values in the data, than kurtosis. In all negentropy computations, the signal-sample standard-deviation, σx , is required. The various negentropy approximations, except for the histogram-based approximation, require x(n) to be standardized (zero-mean and unit variance) x̃(n) = x(n) − x(n) , ∀n. σx (4.12) Standardizing x(n) before computing negentropy eliminates the negentropy statistic’s dependence on sample variance (brightness). The variance estimate can easily be computed on-board a digital radiometer for negentropy-based detection in the same manner as it would be for kurtosis [7]. 82 4.4.2 Histogram-Based Approximation The most direct method for computing J(x) uses fˆ(x) as an estimator of the true probability density, fX (x). We denote this negentropy approximation as Jh (x). The histogram estimate requires a bin size and a ﬁnite number of bins. There are many different rules for choosing a bin size, but they fall into two categories [74–76]. One category uses a function of the number of elements in the data set (samples) N , and the other uses the number of elements and the values of these elements together to estimate bin size. In this √ work, we choose N as the number of histogram bins because of simplicity and so that the histogram bin size is independent of the data sample values, making it consistent for all histograms, regardless of the data, x(n). The estimate, fˆX (x), is computed by forming a histogram of the samples of x(n), and then normalizing each bin by the histogram bin width so that the total area under the normalized histogram sums to one. We also use the convention 0 log 0 = 0. 4.4.3 Edgeworth Approximation Consider the pdf fX (x). Its r-th moment (also known as moment about the origin or raw moment) μr is deﬁned in [77] as μr = ∞ −∞ xr fX (x) dx. (4.13) The r-th central moment μr (also known as moment about the mean), in terms of μr , is deﬁned by μr = ∞ −∞ (x − μ1 )r fX (x) dx. (4.14) 83 We can write central moments in (4.14) in terms of the raw moments in (4.13). In terms of raw moments, the r-th central moment is given by μr = n r j j=0 (−1)r−j μj μr−j , (4.15) with μ = μ1 = μ1 . The inverse Fourier transform of fX (x) is the characteristic function φ(t) and is given by φ(t) = ∞ −∞ fX (x)ejtx dx, (4.16) and the r-th cumulant κr is deﬁned as the r-th coefﬁcient of the Maclaurin series expansion of the natural log of φ(t) ∞ (jt)r log φ(t) = . κr r! r=1 (4.17) In terms of central moments, the r-th cumulant is given by the recursive formula κr = μr − r−1 r−1 m=1 m−1 κm μr−m , (4.18) and the ﬁrst four cumulants evaluate to κ1 = μ1 , (4.19) κ 2 = μ2 , (4.20) κ 3 = μ3 , (4.21) κ4 = μ4 − 3μ22 (4.22) 84 Furthermore, the standardized cumulant [71] of the standardized random variable x̃ is given by the ratio r/2 ρr = κr /κ2 . (4.23) The Edgeworth approximation [71] of negentropy Je is given by a series of standardized cumulants of x̃(n), ρr , with terms that diminish as o(N −2 ): Je (x̃) ≈ 1 2 1 7 1 ρ3 + ρ24 + ρ43 − ρ23 ρ4 + o(N −2 ). 12 48 48 8 (4.24) The skewness and kurtosis are deﬁned as ρ3 and ρ4 , respectively. We can rewrite (4.24) as Je (x̃) ≈ 1 2 1 7 1 ρ3 + ρ24 + ρ43 − ρ23 ρ4 , 12 48 48 8 (4.25) neglecting the vanishing o(N −2 ) terms. The Edgeworth approximation is thus a function of skew and kurtosis f (ρ3 , ρ4 ). Note that for zero-skew pdfs, ρ3 = 0 and (4.25) reduces to ρ24 /48. For comparison to the Edgeworth approximation, the Jarque-Bera test-statistic is considered. The Jarque-Bera test is a goodness-of-ﬁt test of whether sample data has the skewness and kurtosis that matches a Gaussian pdf [78]. The test is named after Carlos Jarque and Anil K. Bera. The test statistic JB is deﬁned as N JqB(x̃) = 6 ρ23 1 2 + (ρ4 − 3) , 4 (4.26) which was evaluated by Tarongi in [56] for RFI detection suitability. Like the Edgeworth 85 approximation, the JqB test statistic also follows the form f (ρ3 , ρ4 ). The performance of the JqB statistic is compared with the Edgeworth and other approximations in this dissertation. We expect JqB and Je to perform similarly because they both are of the form of f (ρ3 , ρ4 ). 4.4.4 Non-polynomial Function-Based Negentropy Approximations Citing that cumulant-based (polynomial) approximations of negentropy are sensitive to outliers, Hyvärinen developed more statistically robust approximations based on non-polynomial functions [69] for ICA and projection pursuit using the maximum entropy method. These new approximations are statistically robust in the sense that they are relativity insensitive to sparse, large data values, unlike the kurtosis. The form of these approximations follows 1 i 2 J(x̃) ≈ E G (x̃) , 2 i=1 n (4.27) where the measuring functions Gi (x) form an orthonormal basis set, do not grow faster than quadratically as a function of |x|, and capture the properties of the pdf of x̃ that are pertinent in entropy measurement [69]. The expectation operator E is the expected value. Two special cases of (4.27) are 2 Ja (x̃) ≈ k1 G1 (x̃)2 + ka Ga (x̃) − 2/π (4.28) 2 Jb (x̃) ≈ k1 G1 (x̃)2 + kb Gb (x̃) − 1/2 , (4.29) 86 where the non-polynomial approximating functions and coefﬁcients are given by G1 (x) = x exp(−x2 /2), Ga (x) = |x|, Gb (x) = exp(−x2 /2), √ k1 = 36/(8 3 − 9) ka = 1/(2 − 6/π) √ kb = 24/(16 3 − 27) (4.30) (4.31) (4.32) with representing a time average over N data samples. The coefﬁcients k1 , ka , and kb originate from orthonormalizing the functions G1 , Ga , and Gb . A lengthy derivation of these coefﬁcients is given in chapter 5 of [69]. In (4.28) and (4.29), G1 (x) is an odd function intended to measure asymmetry of fˆ(x), whereas even functions Ga (x) and Gb (x) measure sparsity and bimodality of fˆ(x). Since we expect the pdf to be symmetric in practice, G1 (x) will go to zero as N increases. Therefore, we can set k1 = 0, dropping the ﬁrst term in both approximations. Two complimentary approximations, Jaa (x̃) and Jbb (x̃), result from this simpliﬁcation, and are also evaluated herein. 4.5 Problem Formulation We model RFI detection as a binary hypothesis test for one integration period of the received radiometric signal x(n). Under the null hypothesis H0 , x(n) is modeled as a zero-mean Gaussian signal w(n). Under the alternate hypothesis H1 , x(n) is the sum of w(n) and RFI signal s(n, Λ) with parameter set Λ. Formulated mathematically, we then 87 have H0 : x(n) = w(n) (4.33) H1 : x(n) = w(n) + s(n, Λ), (4.34) and the goal is to use the negentropy-based test statistics to decide the correct hypothesis given a single integration period of the radiometer. Since we do not have prior probabilities, we rely on the Neyman-Pearson decision rule for RFI detection. We assume that N corresponds to the number of discrete-time samples at a sample rate of FS , where N = T FS . We further assume that FS is chosen such that each sample of w(n) can be thought of as an independent, identically-distributed (iid) random variable for the corresponding hypothesis. Since x̃ is a random variable, all of the negentropy approximations, which are functions of x̃, are a random variables also. We deﬁne the conditional negentropies Ji (x̃|Hk ) : Negentropy of x̃(n) under Hk , k = 0, 1 (4.35) where the subscript i denotes one of the negentropy approximations studied. Summarizing, we consider the six negentropy approximations in table 4.2. In all cases in this chapter, the six negentropy approximations are compared with the kurtosis and the Jarque-Bera test statistics for their performance in detecting various forms of RFI signals. In a speciﬁc case, where the kurtosis has a detection blind-spot, the negentropy approximations are also compared to the Anderson-Darling and Shapiro-Wilk tests statistics. Given the RFI signal parameter set Λ, multiple binary-hypothesis tests using the 88 Table 4.2: Negentropy Approximations Studied Subscript, i Negentropy Notation Description h Jh Historgram-based approximation e Je Edgeworth approximation a Ja Non-polynomial Hyvärinen (a) aa Jaa Same as Ja but with k1 = 0 b Jb Non-polynomial Hyvärinen (b) bb Jbb Same as Jb but with k1 = 0 approximation approximation used by used by negentropy test-statistic are evaluated in terms of the receiver-operating-characteristic (ROC) performance. However, we ﬁrst study how Ji (x̃|H0 ) behaves as a function of x̃ N. 4.6 Large-Sample Behavior of Negentropy Approximations Under H0 , the pdf of x(t) is Gaussian — N (0, σ 2 ). Negentropy as deﬁned in (4.9) is zero in this case. We therefore expect any approximation to approach zero as N → ∞. To verify this claim, we performed a set of Monte-Carlo simulations that evaluated Ji (x|H0 ) as a function of N increasing from 10 to 1M samples by powers of 10, repeating each case 100 times. For each case, the mean and variance of Ji (x|H0 ) was plotted in linear and logarithmic scales, respectively, in Figure 4.2. A noise variance of unity was used in this experiment. 89 0 Sample Mean 10 Ja Jb Je Jh Jaa Jbb −2 10 −4 10 −6 10 10 100 1k 10k 100k 1M 0 Sample Standard Deviation 10 Ja Jb Je Jh Jaa Jbb −2 10 −4 10 −6 10 10 100 1k 10k Number of Gaussian Samples 100k 1M Figure 4.2: Behavior of the various Gaussian test statistics as N → ∞. The sample means and standard deviations of all negentropy approximations tend to zero and trend similarly, except Jh , which converges slower than the other negentropy-based test-statistics Figure 4.2 shows that the means and variances of Ji , i = a, aa, b, bb, e, trend similarly and converge to zero. Though the Jh evaluates to zero under H0 , the variance decreases slower than the other negentropy approximations. Thus we have shown numerically that as N increases, Ji (x̃|H0 ) approaches zero for every negentropy approximation considered. Not shown in the ﬁgure is kurtosis, which has an asymptotic mean of 3, and the Jarque-Bera statistic, which asymptotically converges to a χ2 pdf with 2 degrees of freedom, and hence, the average value of 2 for x(n) ∼ N (0, σ 2 ). 90 4.7 Performance of Negentropy Approximations for Detection of Single Sinusoidal-Source RFI The performance of all test statistics was computed by simulating H0 and H1 for every parameter set Λ in Table 4.3 using the Monte-Carlo method for 10,000 trials perΛ. The parameters pertain to CW and PCW signal models used for H1 : s(n, Λ) = A sin(2πfc n/FS )Π(n) (4.36) where Π(n) is the unit-pulse function Π(n) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ 1, 0 ≤ n ≤ K − 1 ⎪ ⎪ ⎪ ⎪ ⎩ 0, K ≤ n ≤ N. (4.37) The case K = N corresponds to the CW model and the cases where K < N imply the PCW model with duty cycle d = K/N × 100%. We assumed signal parameters that were commensurate with a hypothetical L-band radiometer with x(n) downconverted to an intermediate frequency between DC and 27 MHz. Under H0 , x(n) consisted of N samples generated from N (0, 1), and Ji (x̃|H0 ) was evaluated. Under H1 , we used FS = 100 MHz, fc = 13 MHz, amplitude A, and duty-cycle d. ROC performance curves were computed for everyΛ simulated. In every case, the noise variance was ﬁxed at unity. Only the parameters A, d, and N were varied. The SNR is given 91 by dA2 /2, and is related to the RFI-power-to-radiometric resolution-ratio R = TRF I /NEΔT. This ratio evaluates to dA2 (4.38) N/8 assuming Nyquist sampling. Probability mass functions (pmfs) for every Ji (x̃|H0 ) and Ji (x̃|H1 , Λ) were computed from the Monte-Carlo trials. ROC performance curves were then computed using these functions. It was assumed that only one-pulse-per-integration-period is necessary for the PCW model, ignoring the pulse-repetition frequency. Since we are concerned with the pdf of x, the time that a pulse occurs does not matter. Only the fact that it does occur during the integration time matters. As a result, the effect of having multiple pulses spread across an integration time has the same effect on the pdf as if these pulses were all adjacent to each other in time. This effect is captured by the choice of d in the model, so that pulse repetition rate and/or time spacing need not be considered. 4.7.1 Performance Under CW RFI Sinusoid Signal Model The ROC performance curves are shown in Figure 4.3 for parameter sets Λ1 , . . . , Λ4 , N = 10k samples. The SNR was decreased from 0.00 dB to -6.11 dB in 4 steps. In the case of strong CW RFI, corresponding to parameter set Λ1 , all detectors achieved a nearly-perfect ROC performance. RFI with this SNR is effectively detected with nearly 100% detection probability and nearly zero false-alarm probability. For Λ2 , the situation is similar except the ROC performance of Jh begins to suffer. The SNR is degraded by about 2dB in this case, and this difference is enough for the ROC performance of the histogram-based approximation of negentropy to degrade relative to the other methods. For Λ2 , . . . , Λ4 , all detectors outperform Jh , with kurtosis having the best performance in 92 Table 4.3: Parameter set values Λ for RFI signal model s(n, Λ) Param. Signal Set Type N SNR SNR (dB) R d% Λ1 CW 10k 0.99 -0.03 70.29 100 Λ2 CW 10k 0.63 -2.03 44.35 100 Λ3 CW 10k 0.40 -4.02 28.00 100 Λ4 CW 10k 0.24 -6.11 17.32 100 Λ5 PCW 10k 1.00 0.00 70.71 50 Λ6 PCW 10k 0.50 -3.01 35.36 25 Λ7 PCW 10k 0.20 -6.99 14.14 10 Λ8 PCW 10k 0.02 -16.99 1.41 1 Λ9 PCW 10k 0.04 -13.47 3.18 1 Λ10 PCW 10k 0.02 -16.48 1.59 0.5 Λ11 PCW 10k 0.01 -20.46 0.64 0.2 Λ12 PCW 10k 0.00 -23.47 0.32 0.1 Λ13 CW 3k 0.24 -6.11 9.49 100 Λ14 CW 10k 0.24 -6.11 17.32 100 Λ15 CW 30k 0.24 -6.11 30.01 100 Λ16 CW 100k 0.24 -6.11 54.78 100 every case. Although Jbb outperforms all other negentropy-based detectors here, it lags behind the kurtosis performance as SNR decreases. For every ROC curve generated, the false alarm probabilities PF = 0.05 and PF = 0.10 were chosen to ﬁnd corresponding detection probabilities PD . Bar plots of the two PD for the chosen PF ’s were generated for each test-statistic. In addition, the area-under-the-curve Detection Probability, PD Detection Probability, PD 93 1 1 0.8 0.8 Ja Jb Je Jh k JqB Jaa Jbb 0.6 0.4 Λ : (CW) 1 d: 100%, A=1.41 N: 10k SNR: 0.99 (−0.03 dB) R: 70.29 0.2 0 0 0.2 0.4 0.6 0.8 0.4 1 0 0 0.8 0.8 Ja Jb Je Jh k JqB Jaa Jbb Λ3: (CW) d: 100%, A=0.89 N: 10k SNR: 0.4 (−4.02 dB) R: 28.00 0.2 0 0 0.2 0.4 0.6 0.8 False−Alarm Probability, PF d: 100%, A=1.12 N: 10k SNR: 0.63 (−2.03 dB) R: 44.35 0.2 1 0.4 Λ : (CW) 2 1 0.6 Ja Jb Je Jh k JqB Jaa Jbb 0.6 0.2 0.4 0.6 0.8 Ja Jb Je Jh k JqB Jaa Jbb 0.6 0.4 Λ4: (CW) d: 100%, A=0.70 N: 10k SNR: 0.24 (−6.11 dB) R: 17.32 0.2 1 0 0 0.2 0.4 0.6 1 0.8 1 False−Alarm Probability, PF Figure 4.3: ROC performance of Ji (x̃) for parameter sets Λ1 , . . . , Λ4 . The dash-dotted lines indicate PF = 0.05 and PF = 0.10. (AUC) for every ROC curve was computed and plotted for every test statistic. For the CW RFI case, these bar plots are given in Figure 4.4 corresponding also to the parameter sets Λ1 , . . . , Λ4 . 4.7.2 Performance Under Pulsed RFI Sinusoid Signal Model Similar to the CW-RFI model test, every detector was tested for pulsed signals given by (4.36), with d taking on values from the set {50%, 25%, 10%, 1%}, and A = 2. This experiment was repeated for shorter duty cycles d = {1%, 0.5%, 0.2%, 0.1%} and A = 3. 94 Figure 4.4: Detection Probabilities for the false alarm cases PF = 0.05 and PF = 0.10 and AUC for parameter sets Λ1 . . . Λ4 The ROC results of each experiment are plotted in Figures 4.5 and 4.7. For d = 50% the AD and SW tests were included in the ROC comparison since they performed well in [56] for this case. For Λ5 , the kurtosis blind-spot is apparent because PF is approximately equal to PD despite the high SNR. In addition, all Ji except Je outperform the kurtosis in this case. Je and JqB suffer a mild detection blind-spot in this case because they also include kurtosis directly in their computation. However, these statistics depend also on skewness, so they are also sensitive to the slight skew present in x(n), due to the ﬁnite number of samples. As a result, they perform slightly better than kurtosis, although there seems to be a reversal 95 of PD and PF . For other cases ofΛ, all detectors perform well, with kurtosis outperforming all others and the histogram-based negentropy suffering the worst performance among the Detection Probability, PD Detection Probability, PD set. The AD and SW tests outperform negentropy in this case, for comparison. 1 1 0.8 0.6 0.4 Λ : (PCW) 5 d: 50%, A=2.00 N: 10k SNR: 1 (0.00 dB) R: 70.71 0.2 0 0 0.2 0.4 0.6 0.8 Ja Jb Je Jh k JqB Jaa Jbb SW AD 0.8 0.4 6 1 0 0 1 0.8 0.8 0.4 Ja Jb Je Jh k JqB Jaa Jbb Λ7: (PCW) d: 10%, A=2.00 N: 10k SNR: 0.2 (−6.99 dB) R: 14.14 0.2 0 0 0.2 0.4 0.6 0.8 False−Alarm Probability, PF Λ : (PCW) d: 25%, A=2.00 N: 10k SNR: 0.5 (−3.01 dB) R: 35.36 0.2 1 0.6 Ja Jb Je Jh k JqB Jaa Jbb 0.6 0.2 0.4 0.6 0.8 Ja Jb Je Jh k JqB Jaa Jbb 0.6 0.4 Λ8: (PCW) d: 1%, A=2.00 N: 10k SNR: 0.02 (−16.99 dB) R: 1.41 0.2 1 0 0 0.2 0.4 0.6 1 0.8 1 False−Alarm Probability, PF Figure 4.5: ROC performance of Ji (x̃) for parameter sets Λ5 , . . . , Λ8 . The AndersonDarling (AD) and Shapiro-Wilk (SW) tests are included for comparison for Λ5 , where d = 50%. As d is reduced, Je and JqB swap performance with the other negentropy-based detectors. Performance improves with increasing SNR. This is apparent in Figure 4.5, southeast panel, and in all panels in Figure 4.7. It is conceivable that the sensitivity of kurtosis to outliers makes it, and detectors that depend directly on it, the central reason 96 Figure 4.6: Detection Probabilities for the false alarm cases PF = 0.05 and PF = 0.10 and AUC for parameter sets Λ5 . . . Λ8 behind this performance behavior. This seems to imply that the non-polynomial based approximations are unsuitable for short duty-cycle RFI detection. Their robustness to outliers is actually a disadvantage for detecting short pulses, which can be interpreted as statistical outliers in an otherwise Gaussian signal. 4.7.3 Inﬂuence of the Number of Samples To investigate the performance of negentropy as a function of N , the CW case was chosen with a SNR of -6.11dB and N = {3k, 10k, 30k, 100k} samples. The results are shown in Figure 4.9. Although detection performance improves for all negentropy-based Detection Probability, PD Detection Probability, PD 97 1 1 0.8 0.8 0.6 0.4 Λ : (PCW) 9 d: 1%, A=3.00 N: 10k SNR: 0.045 (−13.47 dB) R: 3.18 0.2 0 0 0.2 0.4 0.6 Ja Jb Je Jh k JqB Jaa Jbb 0.8 0.6 0.4 10 0.2 1 0 0 1 1 0.8 0.8 0.6 0.4 Λ11: (PCW) d: 0.2%, A=3.00 N: 10k SNR: 0.009 (−20.46 dB) R: 0.64 0.2 0 0 0.2 0.4 0.6 Ja Jb Je Jh k JqB Jaa Jbb 0.8 False−Alarm Probability, PF Λ : (PCW) d: 0.5%, A=3.00 N: 10k SNR: 0.022 (−16.48 dB) R: 1.59 0.2 0.4 0.6 0.8 0.6 0.4 Λ12: (PCW) d: 0.1%, A=3.00 N: 10k SNR: 0.0045 (−23.47 dB) R: 0.32 0.2 1 0 0 0.2 0.4 0.6 Ja Jb Je Jh k JqB Jaa Jbb 1 Ja Jb Je Jh k JqB Jaa Jbb 0.8 1 False−Alarm Probability, PF Figure 4.7: ROC performance of Ji (x̃) for Λ9 , . . . , Λ12 . Je and JqB perform better than all other Ji for d ≤ 1%. approximations, kurtosis outperforms negentropy approximations for all values of N . Noteworthy is the fact that the histogram approximation suffered poor performance, even for 100k samples. All other tests perform well for 100k samples, which corresponds to a 1 ms integration time. Increasing N implies that we either increase FS on the spacecraft, which is costly, or we increase T by integrating longer, which compromises spatial accuracy of TB . 98 Figure 4.8: Detection Probabilities for the false alarm cases PF = 0.05 and PF = 0.10 and AUC for parameter sets Λ9 . . . Λ12 4.8 Performance for the Multi-PCW Case In this section, we characterize the performance of negentropy-based test statistics for a more realistic radiometric RFI signal environment, one in which multiple PCW interfering signals with randomly chosen amplitudes, phases, frequencies, and duty cycles are present within an integration period of x(n). The performance of the kurtosis test-statistic was explored in this case by Misra [39], where it was shown that as we increase the number of interfering signals present in x(n), the pdf of the composite signal converges asymptotically to a zero-mean Gaussian pdf. As a result, the ROC performance of the Detection Probability, PD Detection Probability, PD 99 1 1 0.8 0.8 Ja Jb Je Jh k JqB Jaa Jbb 0.6 0.4 Λ : (CW) 13 d: 100%, A=0.70 N: 3k SNR: 0.24 (−6.11 dB) R: 9.49 0.2 0 0 0.2 0.4 0.6 0.8 0.4 1 0 0 0.8 0.8 Ja Jb Je Jh k JqB Jaa Jbb Λ15: (CW) d: 100%, A=0.70 N: 30k SNR: 0.24 (−6.11 dB) R: 30.01 0.2 0 0 0.2 0.4 0.6 0.8 d: 100%, A=0.70 N: 10k SNR: 0.24 (−6.11 dB) R: 17.32 0.2 1 0.4 Λ : (CW) 14 1 0.6 Ja Jb Je Jh k JqB Jaa Jbb 0.6 0.2 0.4 0.6 0.8 Ja Jb Je Jh k JqB Jaa Jbb 0.6 0.4 Λ16: (CW) d: 100%, A=0.70 N: 100k SNR: 0.24 (−6.11 dB) R: 54.78 0.2 1 False−Alarm Probability, PF 0 0 0.2 0.4 0.6 1 0.8 1 False−Alarm Probability, PF Figure 4.9: ROC performance of Ji (x̃) for N = {3k, 10k, 30k, 100k} samples. Kurtosis outperforms all Ji , Jh has the worst performance, and for 100k, all detectors except Jh perform generally well. kurtosis test statistic would degrade, i.e., the detection and false alarm probabilities would approach each other, leading to an unreliable RFI detector for a large number of interferers. The pdf approach to Gaussianity for an increasing number of random PCW interferers is essentially a result of the central limit theorem. We show in this section that all negentropy-based test statistics lead to detectors that suffer from this same central-limit phenomenon as kurtosis. We take an additional step beyond characterizing the pdf and compute the value of every test statistic as a function 100 Figure 4.10: Detection Probabilities for the false alarm cases PF = 0.05 and PF = 0.10 and AUC for parameter sets Λ13 . . . Λ16 of the number interferers in different radiometric noise cases. We ﬁnd as a result that all these statistics converge to their asymptotic value for a Gaussian pdf. By plotting the various test statistic value versus the number of interferers, we further show that all of the negentropy-based test statistics converge to their asymptotic values faster than kurtosis, leading to the realization that kurtosis can outperform negentropy in the multi-signal RFI case as well. Finally, we show ROC performance curves for all test statistics for different numbers of interferers and relate their performance to the asymptotic behavior of the test statistic values. 101 4.8.1 Convergence of Multiple-PCW RFI to a Gaussian Probability Density Function An experiment was conducted in which M PCW signals were generated according to the model in (3.3), repeated here for clarity x(t) = w(t) + s(t, Λ), (4.39) where s(t) = M Ak cos (2πfk t + θk ) Π k=1 t − t0 wk (4.40) and the signal parameters are all grouped into a single the vector Λ = {M, Ak , fk , θk , t0 , dk }. The corresponding discrete-time signal model used to represent the RFI embedded in x(n) in this section is given by x(n) = w(n) + s(n, Λ), (4.41) where s(n) = M k=1 Ak cos (2πfk n + θk ) Π n − n0 wk . (4.42) The parameters of this model are listed in Table 3.2. We chose M = {1, 2, 5, 10, 20, 50, 100, 200, 500, 1000}. For each M , we generated 100 random realizations of s(n). For each realization, we computed all of the negentropy-based test statistics as well as the kurtosis of s(n). The average kurtosis excess (kurtosis minus 3) was plotted, rather than the kurtosis, 102 with negentropy, so that the curves could be compared more easily when they all converge to zero. The 100 test-statistic values corresponding to the 100 random realizations of s(n) were then averaged for every value of M . The average values were then plotted versus M on a logarithmic x-axis scale in Figure 4.16 and a log-log scale in Figure 4.17. The mean-squared error between the pdf of s(n) and its Gaussian ﬁt for one trial of each value of M was computed and plotted in Figure 4.15. To illustrate the properties of this RFI model, plots were generated for a single trial of M interferers. The amplitude vs. time, spectrogram, normalized histogram, and power spectral density were computed and shown in these plots. Four of these cases are shown in Figures 4.11 through 4.14 for M = 2, 20, 100, and 1000. For this multiple-signal (M > 1) interference model, several features of the plots are noteworthy. As we can see in Figures 4.11 through 4.14, the waveform plots resemble a noise signal as M is increased. For the case where M = 2, the spectrogram has a sparse distribution of two sinusoidal terms, but when M = 1000, this spectrogram becomes more densely packed with sinusoidal signals. The histograms in Figures 4.11 through4.13 have large peaks at the mean, corresponding to a signal amplitude of zero. This implies that there is a signiﬁcant number of zero values in the time waveform, relative to other amplitude values in the signal. The pmf is plotted with an overlay of a ﬁtted Gaussian. In all cases, the pmf has a higher peak than the ﬁtted Gaussian, and is also narrower than the Gaussian. The pmf of s(n) approaches the shape of a Gaussian as we increase M . Quantitatively, we can see that the MSE between the pmf and the ﬁtted Gaussian decreases from 95.17% to 0.69% as M increases from 2 to 1000. Lastly, we note that the power spectral density, which can be viewed as a spectrogram ﬂattened in time, has minima and maxima that approach each other as M increases, implying that the spectrum becomes more white as M increases. Comparing power spectral densities, the difference between the maximum and 103 Figure 4.11: Clockwise from the top left: time, joint time-frequency, amplitude histogrambased pmf,. and power spectral density plots of the RFI model signal x(n), with no noise present. M = 2. minimum values decreases slightly from about 30 dB/Hz to 20 dB/Hz. Worthy of note in Figure 4.16 is that beyond M = 100, the rate of convergence of kurtosis seems to slow, and reach a small asymptotic value of 0.2392, while the other statistics reach zero. To gain additional insight into this behavior, this data is plotted in a log-log scale plot in Figure 4.17. As seen in Figure 4.17, the convergence rate of every test statistic slows with more than 104 Figure 4.12: Clockwise from the top left: time, joint time-frequency, amplitude histogrambased pmf,. and power spectral density plots of the RFI model signal x(n), with no noise present. M = 20. M = 100 sources. The kurtosis excess and Jh test statistics reach zero slower than the remaining test statistics. The remaining test statistics all behave similarly for M > 100. The Edgeworth negentropy approximation reaches values close to the kurtosis excess for M < 10, but then trends along with the remaining negentropy approximations for M > 10. We see that for 10 ≤ M ≤ 20, the Edgeworth approximation transitions from kurtosis convergence behavior to negentropy convergence behavior. 105 Figure 4.13: Clockwise from the top left: time, joint time-frequency, amplitude histogrambased pmf,. and power spectral density plots of the RFI model signal x(n), with no noise present. M = 100. If we take a second look at Figure 4.15, although the data points in the plot represent only a single realization of the the multiple-RFI signal, convergence to a zero percent MSE still appears to be slow. This suggests that although increasing M causes the pdf of s(n) to approach Gaussian, it is not actually Gaussian. In fact, the error between the pdf of s(n) and a Gaussian with the same mean and variance does not reach signiﬁcantly below 10%, even for M = 1000. Figures 4.11 through 4.14 suggest that the underlying pdf is 106 Figure 4.14: Clockwise from the top left: time, joint time-frequency, amplitude histogrambased pmf, and power spectral density plots of the RFI model signal x(n), with no noise present. M = 1000 supergaussian, therefore the kurtosis value is always above 3. In Figure 4.15, the slight bias of Kurtosis may be due to the fact that there is still a notable difference between the pdf of s(n) and a Gaussian pdf. The results of analyzing the multiple-PCW interference model show that the ROC performance all RFI detectors will degrade as M increases since the resulting pdf function in (3.3) experiences central-limit theorem conditions due to the additive effect of many 107 Figure 4.15: Mean-Squared Error between s(n) and Gaussian ﬁt to s(n) versus M . Each data point corresponds to a single Monte-Carlo trial with M interfering signals present. As M increases, the MSE decreases rapidly, particularly between 10 and 20 interfering signals. However, MSE rate of decrease slows beyond 20 interfering signals, suggesting that the pdf of s(n) converges slowly to Gaussian. random-parameter pulsed-sinusoidal signals. In addition, based on the the convergence plots in Figures 4.17 and 4.17, the negentropy-based test-statistics would also not be able to detect RFI for large M . Since the pdf of s(n) approaches Gaussian, however slowly, any statistic that measures non-Gaussianity would erroneously report that the there is no RFI despite the fact that there may be large numbers of interfering signals present in x(n). 108 Figure 4.16: Convergence of test statistic values to their Gaussian pdf equivalent values in the absence of radiometric noise. 4.8.2 ROC Performance for Multiple PCW Interference Case Another experiment was conducted in which M PCW signals were generated according to the model in Chapter 3, and then the RFI model signal, s(n) was added to zero-mean, unit-variance Gaussian noise. Again, we choose M = {1, 2, 5, 10, 20, 50, 100, 200, 500, 1000} signals. The goal of the experiment was to investigate the detectability of RFI in the multiple PCW plus noise case, for the various negentropy-based test statistics Ji . We 109 Figure 4.17: Log-Log-scale plot of the convergence of test statistic values to their Gaussian pdf equivalent values in the absence of radiometric noise. expect that for increasing M , as we saw in the previous section, the RFI-only signal model converges to a Gaussian pdf. However, if s(n) is added to Gaussian noise, the pdf is more Gaussian and we should observe quickly degrading detectability for all of the detectors based on negentropy. In Figures 4.18 through 4.20 we verify that this is indeed the case. For M > 5, all ROC performance curves approach the PF = PD line, implying that RFI becomes harder to detect for all detectors based on negentropy as M increases. In addition, the kurtosis 110 suffers as well, but its ROC performance is better than those for negentropy, implying that kurtosis is superior to negentropy for detecting multiple RFI sources as well. Detection Probability, PD Number of Interferers = 1 Number of Interferers = 2 1 1 0.8 0.8 Ja Jb Je Jh k JqB Jaa Jbb 0.6 0.4 SNR: 0.09 (−10.69 dB) # of Inteferers, M: 1 0.2 0 0 0.2 0.4 0.6 0.8 0.4 1 0 0 Number of Interferers = 5 Detection Probability, PD SNR: 0.08 (−10.98 dB) # of Inteferers, M: 2 0.2 0.2 0.4 0.6 0.8 1 Number of Interferers = 10 1 1 0.8 0.8 Ja Jb Je Jh k JqB Jaa Jbb 0.6 0.4 SNR: 0.01 (−19.86 dB) # of Inteferers, M: 5 0.2 0 0 Ja Jb Je Jh k JqB Jaa Jbb 0.6 0.2 0.4 0.6 0.8 False−Alarm Probability, PF Ja Jb Je Jh k JqB Jaa Jbb 0.6 0.4 SNR: 0.05 (−12.89 dB) # of Inteferers, M: 10 0.2 1 0 0 0.2 0.4 0.6 0.8 1 False−Alarm Probability, PF Figure 4.18: ROC performance curves for the six negnetropy-based detectors and kurtosis for 1,2,5, and 10 sinusoidal interferers in Gaussian noise. In all cases, the kurtosis has the highest detection probability given any false alarm probability, and thus the best detection performance. All of the negentropy-based detectors except Jh tend to cluster in their performance, still with a higher detection probability than false alarm probability, but not as good as kurtosis. The histogram-based approximation of negentropy suffered the worst detection performance, having near equal detection and false alarm probabilities for the entire ROC curve. 111 Figure 4.19: ROC performance curves for the six negnetropy-based detectors and kurtosis for 20, 50, 100, and 200 sinusoidal interferers in Gaussian noise. Kurtosis still outperforms all other detectors in this case except for the case where M = 200. However, all of the detectors have poor performance here because their detection and false alarm probabilities are nearly equal for every ROC curve. 112 Detection Probability, P D Number of Interferers = 500 Number of Interferers = 1000 1 1 0.8 0.6 0.4 SNR: 0.07 (−11.58 dB) # of Inteferers, M: 500 0.2 0 0 0.2 0.4 0.6 0.8 Ja Jb Je Jh k JqB Jaa Jbb 0.8 0.6 0.4 F SNR: 0.06 (−12.11 dB) # of Inteferers, M: 1000 0.2 1 False−Alarm Probability, P Ja Jb Je Jh k JqB Jaa Jbb 0 0 0.2 0.4 0.6 0.8 1 False−Alarm Probability, P F Figure 4.20: ROC performance curves for the six negnetropy-based detectors and kurtosis for 500 and 100 sinusoidal interferers in Gaussian noise. Again, kurtosis appears to perform slightly better in terms of detection vs. false-alarm performance. However, in this case, we expect all detectors to approach the PF − PD line because of the convergence of pdf of s(n) to Gaussian. 113 4.9 Summary and Conclusions Six approximations of negentropy were introduced as test-statistics for detecting radiometric RFI. These include the histogram-based and Edgeworth aproximations, and four non-polynomial negentropy approximations. All negentropy-based test-statistics were compared in ROC performance to kurtosis and the Jarque-Bera statistics. For the kurtosis blind-spot case for a single PCW interferer, all of the negentropy-based test-statistics were compared to the Shapiro-Wilk and Anderson-Darling test statistics as well. Simulations were performed to characterize the ROC performance of these tests under CW and PCW RFI model assumptions as well as under the multiple PCW interference case in (3.3). Additionally, the behavior of these approximations as a function of the number of samples of x(n) was characterized in the RFI-free case. Tables 4.4 and 4.5 summarize the quantitative ROC performance results for the best and worst-performing test statistics for PD (PF = 0.05). Complete performance results are given in Tables B.1 and B.2. It was shown that negentropy of the pre-detected radiometer signal can be used for sinusoidal RFI detection, but the ROC performance is inferior to kurtosis except for the kurtosis blind-spot case. The approximations Ja , Jaa , Jb , and Jbb outperformed all others in terms of the ROC for duty cycles greater than 1%, whereas Je performed better for d = 1%. Negentropy-based tests generally performed better for CW as opposed to PCW interference. The histogram-based test, Jh , suffered sub-par performance compared with all other tests and RFI signal models. Lastly, the Je and JqB tests were found to perform similarly, as expected. Comparing our results herein to the work of [56] in the speciﬁc case of d = 50%, all negentropy-based tests performed better than kurtosis-based detectors. In [56], the D’Agostino, K-squared, SW, and AD tests performed better than kurtosis as well. The 114 Table 4.4: Summary of best and worst PCW RFI detectors in terms of PD (PF = 0.05) Param. Set Λ 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Test Maximum Statistic PD (PF = 0.05) Jbb 1.000 κ 0.999 κ 0.834 κ 0.333 AD 0.600 Jb 1.000 κ 1.000 κ 0.317 κ 0.995 κ 0.777 κ 0.319 κ 0.140 κ 0.161 κ 0.354 κ 0.657 κ 0.986 Test Minimum Statistic PD (PF = 0.05) Jh 0.999 Jh 0.524 Jh 0.118 Jh 0.058 κ 0.026 Jh 0.866 Jh 0.637 Jh 0.052 Jh 0.323 Jh 0.024 Ja 0.067 Ja 0.057 Jh 0.057 Jh 0.048 Jh 0.061 Jh 0.150 Table 4.5: Summary of best and worst Multiple PCW RFI detectors in terms of PD (PF = 0.05) # of Signals M 1 2 3 4 5 6 7 8 9 10 Test Maximum Statistic PD (PF = 0.05) κ 1.000 κ 0.551 κ 0.998 κ 0.853 κ 0.112 κ 0.105 κ 0.085 κ 0.074 κ 0.094 Jh 0.062 Test Minimum Statistic PD (PF = 0.05) Jh 0.718 Jh 0.045 Jh 0.428 Jh 0.132 Jh 0.065 Jh 0.053 Je 0.038 Jaa 0.039 Jh 0.044 Jb 0.046 115 negentropy-based tests all outperformed JqB, but the AD and SW tests outperformed all Ji . The AD test had the best ROC performance, followed by SW, which closely follows Jaa . The R values deﬁned in 4.38 that correspond to the best-performing ROC curves for negentropy are rather high for practical RFI detection, since RFI detection at interference levels near the radiometric sensitivity are generally desired. Increasing N improves performance of all tests, but this increases R as well. For these large values of R, non-statistical detectors would probably perform better than statistical detectors since they are mostly transform-based and do not require large sample sizes. In the case that N → ∞, we would expect the negentropy-based test statistics to approach the true negentropy. If one had access to the pdf of x(n), for N → ∞, we would also expect that the true negentropy would perform as well or better than the kurtosis in terms of the ROC performance metrics. The theoretical proof of this was not pursued, since in a real radiometer we do not have access to an inﬁnite number of samples N , and cannot access the true pdf of x(n). Although kurtosis appears in the terms of various series expansions of negentropy, it appears that including terms other than kurtosis for detection seems to degrade ROC performance. Omitting terms that are sensitive to skewness of the pdf from 4.28 and 4.29 improved ROC performance. This suggests that for pdfs that are approximately zero-skew, the terms that account for this skew actually degrade detection performance. If the pdf is zero-skew, then these terms are zero and do not impact detection performance. This phenomenon was also consistent with the Je performance, as well as results for the skewness-dependent tests in [56]. It was ﬁnally shown that for more realistic multiple-source RFI signals, all negentropybased statistics do not perform well with more than 20 interfering signals present in the same integration period. The kurtosis outperforms all other test-statistics, followed by the 116 Edgeworth expansion of negentropy. As the number of sources increases, the composite RFI-only signal converges to a Gaussian pdf. When added to Gaussian noise, the RFI becomes indistinguishable from the Gaussian noise in which we are interested. We can conclude that while negentropy-based test statistics can be used to detect RFI in microwave radiometers, the test-statistics considered are all outperformed in ROC performance by the kurtosis in almost all cases. In addition, kurtosis is much simpler to calculate than negentropy. As a result, negentropy offers no signiﬁcant advantage over kurtosis for detecting RFI in total-power microwave radiometers, where x(n) is real-valued. It is this reason why, in the next chapter, we consider extending the application of kurtosis to RFI detection in cases where the pre-detected radiometer signal is complex. Chapter 5 DETECTION OF RFI USING COMPLEX SIGNAL KURTOSIS COEFFICIENTS 5.1 Introduction In the previous chapter, it was shown that the kurtosis statistic outperformed all negentropybased test-statistics and performed consistently well in terms of the ROC for a wide variety of RFI signal types. real-valued signals. However, the literature to date only uses kurtosis for This implies that there is no current analytical formulation for using kurtosis for radiometric signals interpreted as complex-valued signals that naturally appear in microwave radiometers. Such an interpretation is fundamentally important, because complex-valued signals naturally arise in two distinct scenarios in the context of total-power and polarimetric radiometers. In the ﬁrst scenario, the pre-detected radiometric signal, x(n), which originates from 117 118 a total-power radiometer, is downconverted to a complex baseband representation with in-phase and quadrature signal components. The complex baseband representation of x(n) is practically advantageous to use over the non-downconverted case because it allows decimation of the signal and reduction of the overall signal sample rate; hence, leading to reductions in overall power consumption in the spacecraft. In addition, working with the complex-valued signal as opposed to two real component signals leads one to possibly use a single test-statistic for RFI, as opposed to two (possibly redundant) test-statistics for the in-phase and quadrature component signals thus, possibly reducing the amount of data required to be telemetered to the ground for processing. This advantage is highly desirable since the communications bandwidth from the spacecraft down to Earth is fundamentally limited. In the second scenario, the two channels of a polarimetric radiometer can be interpreted as a single, complex-valued signal. This interpretation is natural due to the fact that the horizontal and vertical polarization component signals originate from the same electromagnetic wavefront, which itself is represented as a complex-valued function of space and time as described in Section (2.1.2). Considering the polarization component signals jointly as a complex-valued signal would preserve more information about the original electromagnetic signal than would any of the polarization component signals alone. It is therefore reasonable to argue that, for baseband downconverted total-power radiometric signals as well as polarimetric radiometer signals, the complex-valued representation of these signals carries more information than any of the individual real components. As a result, one should expect the kurtosis, deﬁned for complex-valued signals, should perform as well as or better than the real-valued kurtosis of any of the individual real component signals. We show in this chapter that this is indeed the case. In this chapter, we focus our attention on the polarimetric radiometer, shown in Figure 119 5.1. Reference Load TSY S E(r, t) cos(ωc n) G HB (f ) Ampliﬁer Bandpass Filter xH (t) ADC × sin(ωc n) LPF × LPF × LPF cos(ωc n) × LPF IH (n) QH (n) OMT Antenna G VB (f ) Ampliﬁer Bandpass Filter Reference Load xV (t) ADC IV (n) QV (n) sin(ωc n) Figure 5.1: Polarimetric Radiometer. The polarimetric radiometer consists of two identical signal processing channels for the horizontal and vertical polarization component signals. The pre-detected and digitized radiometer signals are xH (n), xV (n), and their corresponding complex baseband representations are IH (n)+jQH (n) and IV (n)+jQV (n). The complex signals z1 (n) = I(n) + jQ(n) and z2 (n) = xH + jxV (n) are studied for RFI detection in the complex domain. H-Pol and V-Pol subscripts are dropped on I(n) and Q(n) since the same sort of processing is applied to either baseband channel. 5.2 Overview We begin in section 5.3 with an introduction of the test-statistic Γ, that is developed in [14] for complex Gaussian detection. In section 5.4, we formulate the RFI detection problem for the downconverted baseband quadrature signal in the radiometer. We then interpret it as a complex random signal and formulate the detection problem. Since we are beginning with the baseband quadrature signal case, our RFI signal model that we use is a complex baseband representation of the PCW signal model used in Chapter 4. Since we consider 120 the in-phase and quadrature component signals in this chapter, we also use signals from a generalized digital modulation model that was discussed in Chapter 3, developed in [79], as an RFI model. Complex digital modulation is another type of RFI that is experienced in L-band as well as K-band [11,23]. Of particular interest is K-band RFI that originates from reﬂected signals from direct broadcast satellites such as the Direct-TV 10 satellite [11]. Satellites such as these transmit signals according to the Direct Video Broadcast Standards (DVB-S and DVB-S2) [63]. These signals were discussed in section 3.5. In section 5.5, we then consider our second complex signal scenario, which is the combination of the polarized component signals into a single complex signal. We describe how we use the RFI model we developed in Chapter 3 to simulate the polarized RFI case, varying the model parameters AH and AV to model real RFI with different polarization component amplitudes. Overall, we compare the test-statistic Γ against the individual real kurtosis of the individual polarization channel signals kre and kim , and the statistic β = kre + kim . We show that in general, Γ outperforms all other test-statistics for RFI detection. For the downconverted case, it clear that it is better to use Γ than to sum the kurtoses of the individual channel signals. For the polarized case, Γ has a better ROC performance than β when AH = AV , but otherwise the real kurtosis corresponding to the polarized channel with the strongest RFI dominates. However, the real kurtosis does not have a signiﬁcantly better ROC performance than Γ in this case. We summarize and conclude these points in section 5.6 and also in [1]. All ROC performance data can be found in Appendix B, sections B.2.1 and B.2.2. 121 5.3 Complex Gaussian Detector Consider the complex discrete-time random signal z(n), n = 0, . . . , N − 1. Dropping the time index n, the p(= + m)th-order sample central moment of this random variable is deﬁned by # " α,m = E (z − E[z]) (z − E[z])∗m , , m ∈ Z≥0 , where E is the expectation operator and ∗ (5.1) is the complex conjugate [14]. Standardized moments are deﬁned by ;m = α,m , σ +m (5.2) where σ 2 = α1,1 [13]. We note that the pseudovariance is deﬁned by 2;0 [13,16,80]. Since ∗ there are p + 1 different moments, and some moments are redundant in the sense that α,m = αm, , there are actually three different kurtosis deﬁnitions for the complex case. These kurtosis coefﬁcients γ4;0 , γ3;1 , and γ4;0 , as deﬁned in [14], are given by γ4;0 = 4;0 − 322;0 (5.3) γ3;1 = 3;1 − 322;0 (5.4) γ2;2 = 2;2 − 2 − |2;0 |2 (5.5) In this chapter, we are particularly interested in (5.5) since the quantity itself is real-valued, although it measures complex signal kurtosis. 122 The test-statistic Γ using (5.2) was derived in [14] to test for complex Gaussianity Γ= γ2;2 . 1 + 12 |2;0 |2 (5.6) If x(n) is complex Gaussian, then Γ = 0. Otherwise, Γ deviates from zero. We apply the test-statistic of (5.6) for the cases of the complex baseband signal model and, later, the polarimetric RFI signal model in this chapter. 5.4 Baseband Quadrature RFI Detection A polarimetric radiometer, such as in Figure 5.1, consists of two identical receiver channels, each of which can beneﬁt from quadrature baseband downconversion due to sample rate reduction after downconversion without loss of radiometric information [7]. Due to the fact that spacecraft instrument cost is driven by mass and power, downconversion can be a step to lower the effective sample rate of the pre-detected radiometer signal at the cost of longer integration times. If we consider FPGA-based signal processing of the radiometer signal, as in [7], lowering the signal sample rate reduces the subsequent required clocking rate of the digital logic that processes the radiometric signal, thereby reducing the amount of digital logic power consumption. In addition, processing at the lower sample rate allows time-interleaving of logic functions, so that digital logic can be re-used on few devices as opposed to being fanned-out across multiple devices. If we consider a total-power digital radiometer, such as in Figure 5.2, the antenna signal is downconverted at least twice prior to detection. The ﬁrst frequency translation shifts the radiometer observation bandwidth B to an intermediate frequency (IF), and the second stage of frequency translation shifts this bandwidth down to baseband, in the form of pair of signals, I(n) and Q(n), called 123 cos(ωc n) TSY S cos(ωIF n) v(t) E(r, t) Antenna G HRF (f ) Ampliﬁer RF Bandpass Filter × × HIF (f ) x(t) ADC IF Bandpass Filter Reference Load I(n) LPF DSP x(n) × LPF Q(n) sin(ωc n) Figure 5.2: Total power radiometer with quadrature downconversion. quadrature component signals. We assume that the intermediate-frequency signal, x(n), is bandlimited with bandwidth, B. After downconversion, the complex baseband signal is given by z1 (n) = I(n) + jQ(n). (5.7) In section 5.4.1 we generate z1 (n) using the pulsed-CW model and the complex digital modulation RFI models for s(n, Λ) discussed in Chapter 3. 5.4.1 RFI Signal Models Considered for Complex Baseband RFI Detection As in Chapter 4, we ﬁrst consider the simple signal model for the input signal x1 (n), s(n, Λ) = A sin(2πfc n/FS )Π(n), (5.8) 124 cos(ωc n) x1 (n) = s(n, λ) + N (0, σ 2 ) HIF (ejω ) × LPF × LPF I(n) x(n) IF Bandpass Filter Q(n) sin(ωc n) Figure 5.3: Radiometer signal model for complex baseband RFI detection. We assume an L-band radiometer with input noise N (0, σ 2 ) with IF bandpass ﬁlter centered at 24 MHz with 12 MHz on each side. The ﬁltered signal x(n) is mixed down to complex baseband, and subsequently ﬁltered by a pair of identical image-rejection ﬁlters, producing the signal z1 (n) = I(n) + jQ(n). where Π(n) is the unit-pulse function ⎧ ⎪ ⎨ 1, 0 ≤ n ≤ K − 1 Π(n) = ⎪ ⎩ 0, K ≤ n ≤ N. (5.9) as our source of RFI. As depicted in Figure 5.3, this signal is passed through a digital bandpass ﬁlter that emulates the passband of a radiometer, and then downconverted. RFI detection is then performed on z1 (n) = I(n) + jQ(n). We wish to detect RFI in z1 (n) The literature to date only applies statistical RFI detection techniques to either the real signal, x(n) [6, 12, 56], to the individual component signals of z1 (n) [2], but not to the complex signal z1 (n) as a whole. We formulate the binary detection problem as follows. Under the null hypothesis H0 , z1 (n) is RFI-free and, therefore, bivariate or complex Gaussian noise at baseband. Under hypothesis H1 , z(n) contains one or more sources of RFI, which consists of the 125 downconverterd signal s(n). In our general additive RFI model, where the received signal x(n) = s(n, Λ) + w(n), w(n) is real-valued Gaussian noise, but has a complex representation with in-phase and quadrature components after it is downconverted and embedded in z(n) [77]. Formally, we state the detection problem H0 : z1 (n) = w̃(n) (5.10) H1 : z1 (n) = w̃(n) + s̃(n, Λ), (5.11) where we designate s̃(n, Λ) as the downconverted version of the RFI signal component s(n, Λ) and w̃(n) is the downconverted noise component of x(n). Here, both s̃(n) and w̃(n) are complex-valued signals. Following the diagram in Figure (5.2), the in-phase and quadrature component signals are given by I(n) = x(n) cos(ωc n) ∗ hLP F (n) (5.12) Q(n) = −x(n) sin(ωc n) ∗ hLP F (n), (5.13) where hLP F is an ideal lowpass ﬁlter with bandwidth B/2 and the convolution operator is given by ∗. Therefore, combining (5.12) with (5.13), we have z1 (n) = x(n) cos(ωc n) ∗ hLP F (n) + jx(n) sin(ωc n) ∗ hLP F (n) = x(n) [cos(ωc n) ∗ hLP F (n) + j sin(ωc n) ∗ hLP F (n)] (5.14) (5.15) 126 5.4.2 RFI Detection Results Using Kurtosis of the Complex Quadrature Signal A series of Monte-Carlo simulations were performed using signal models generated in (4.36). In each simulation, zero-mean, unit-variance noise was generated for H0 , and one of the RFI cases was generated for s(n, Λ), for 16 different parameter sets for Λ, which are summarized in Table 5.1. The ROC results of these simulations are included in Figures 5.4, 5.6, 5.8, and 5.10. Similar to the analysis in Chapter 4 the ROC performance curves for each RFI test-statistic in each RFI case were quantiﬁed in terms of their AUC, and detection probability PD for false-alarm probabilities PF = 0.05 and PF = 0.10. The AUC and PD (PF ) results are given in Figures 5.5, 5.7, 5.9, and 5.11. In Table B.3, the test-statistics associated with the largest PD (PF = 0.05) (best) and smallest PD (PF = 0.05) are summarized. Table 5.1: Signal and parameters Λ chosen for the baseband complex RFI detection simulation Parameter set Λ A d N Λ1 . . . Λ4 0.75, 0.60, 0.45, 0.30 100% in all 4 cases 20k Λ5 . . . Λ8 1.00 in all 4 cases 50%, 25%,10%, 1% 20k Λ9 . . . Λ12 Λ13 . . . Λ16 BPSK, QPSK, 16-QAM, and 16-rQAM — All normalized to A = 1 2 in all 4 cases 0.5% in all 4 cases 10k, 20k, 50k, 100k 127 Figure 5.4: CW-Model RFI performance. CW signal amplitudes are (a) 0.75, (b) 0.60, (c) 0.45, and (d) 0.30. Noise variance is always unity. In all cases, Γ outperforms all other test-statistics, but only slightly outperforming β in terms of the ROC. 128 Figure 5.5: CW-Model RFI performance ROC analysis results. The AUC and PF (PD ) values for Γ are higher than that for β, but are very close to each other in all cases. Detection Probability, P D 129 1 1 0.8 0.8 0.6 0.4 Λ : (PCW) 0 0 d: 50%, A=1.00 SNR: −6.02 dB N: 20k 0.2 0.4 0.6 κ 0.2 0.4 0.6 1 0 0 0.2 0.4 Detection Probability, P D (a) 1 0.8 0.8 0.6 Λ : (PCW) 0 0 d: 10%, A=1.00 SNR: −13.01 dB N: 20k 0.2 0.4 Γ β κ 0.6 κ 0.2 0.4 im 0.8 1 0.6 re 8 d: 1%, A=1.00 SNR: −23.01 dB N: 20k im 0.8 Γ β κ Λ : (PCW) re 7 0.2 0.6 κ (b) 1 0.4 re 6 d: 25%, A=1.00 SNR: −9.03 dB N: 20k im 0.8 Γ β κ Λ : (PCW) re 5 0.2 Γ β κ 1 0 0 0.2 0.4 0.6 κ im 0.8 1 False−Alarm Probability, P False−Alarm Probability, P (c) (d) F F Figure 5.6: Pulsed-CW RFI performance. Duty cycles of 50%, 25%, 10%, and 1% as shown for (a) – (d). In (a), we note that Γ has a detection blind-spot similar to the real kurtosis. In (b) - (d), Γ has a slightly better ROC performance than the other RFI detectors. 130 Figure 5.7: ROC analysis results for the pulsed-CW model case. Again, the AUC and PF (PD ) values for Γ are higher than that for β, but are very close to each other in all cases. Detection Probability, P D 131 1 1 0.8 0.8 0.6 0.4 Λ : (CXDM) 0 0 bpsk N: 20k 0.2 0.6 κ 0.2 0.4 re 9 0.2 Γ β κ 0.6 0.8 qpsk N: 20k κ 1 0 0 0.2 D Detection Probability, P 1 0.8 0.8 0.6 Λ : (CXDM) 16qam N: 20k 0.2 Γ β κ 0.6 κ 0.2 0.4 re 11 0 0 0.8 1 0.6 0.8 Λ : (CXDM) Γ β κ 16rqam N: 20k κ re 12 im 0.4 0.6 (b) 1 0.2 im 0.4 (a) 0.4 re 10 im 0.4 Λ : (CXDM) Γ β κ 1 0 0 0.2 0.4 im 0.6 0.8 1 False−Alarm Probability, P False−Alarm Probability, P (c) (d) F F Figure 5.8: ROC performance for complex detector vs. real kurtosis and sum of kurtosis for real and imaginary component signals for digital modulation models. (a) BPSK RFI, (b) QPSK, (c) 16-QAM (circular), (d) 16-QAM (rectangular). Noise variance is unity in all cases. 132 Figure 5.9: ROC analysis results for the digital modulation model RFI case. The AUC and PF (PD ) values for Γ are higher than that for β, but are very close to each other in all cases. It is interesting to note that for rectangular 16-QAM, AUC=1.000, and PF (PD ) = 1, but for circular 16-QAM, detection performance is poor. for all detectors. This is due to the implementation of the circular 16-QAM signal having a much lower SNR than the rectangular case in our implementation. 133 Figure 5.10: ROC performance for complex detector vs. real kurtosis and sum of kurtosis test-statistics for real and imaginary component signals for the pulsed-CW case, with increasing number of samples N : (a) N=10K, (b) N=20K, (a) N=50K, and (d) N=100K. 134 Figure 5.11: ROC analysis for increasing number of samples N : (a) N=10K, (b) N=20K, (a) N=50K, and (d) N=100K. As we expect, ROC performance improves as we increase N . 135 Table 5.2: Summary of best and worst complex baseband RFI test-statistics in terms of PD (PF = 0.05). Param. Set Λ 1 2 3 4 5 6 7 . 8 9 10 11 12 13 14 15 16 Test Maximum Statistic PD (PF = 0.05) Γ 1.000 Γ 0.955 Γ 0.381 Γ 0.090 β 0.087 Γ 0.918 Γ 0.804 Γ 0.101 Γ 0.405 Γ 0.386 Γ 1.000 Γ 0.104 Γ 0.655 Γ 0.951 Γ 1.000 Γ 1.000 Test Minimum Statistic PD (PF = 0.05). κim 0.989 κim 0.656 κre 0.211 κim 0.069 Γ 0.073 κim 0.550 κim 0.456 κim 0.073 κre 0.199 κim 0.215 κre 1.000 κim 0.080 κim 0.398 κim 0.694 κim 0.957 κim 0.998 In 5.4(a), all detectors have near-perfect detection performance in terms of the ROC due to the high SNR of the CW RFI used. As the SNR decreases, from Figures 5.4(b) through 5.4(d), the performance of every test-statistic degrades accordingly. Inspection of the summary Table 5.2, the test-statistic with the largest PD is primarily Γ, except for parameter sets 1, 5, and 8. For Λ8 , according to the complete Table B.3 in Appendix B, both β and Γ share a PD (0.05) of 1.000. In this case, more than three signiﬁcant ﬁgures are needed to distinguish the PD performance of β from Γ — the performance of the two teststatistics for this point on the ROC curve are roughly equivalent. Even the AUC results for both, according to Table 5.5 indicate an AUC=1.000. The performance difference between β and Γ is negligible, and they both outperform κre and κim . For the other three parameter cases, Λ2 . . . Λ3 , Γ clearly outperforms all other test-statistics. For Λ4 it is not clear by the 136 ROC curves which test-statistics preform the best and worst, but according to Figure 5.5, Γ dominates the performance, followed by β and then κre and κim . A blind-spot, similar to that discovered in [12], was discovered in the ROC performance curve in Figure 5.4(a). PCW RFI was used with a duty cycle of d = 50%. Despite the relatively high SNR of this RFI -6.02dB PF is approximately equal to PD for nearly the entire ROC curve. Figure 5.7(a) supports this fact, since the AUC is approximately 0.5 for every test-statistic. Table 5.2 indicates that β is the best test-statistic, but 5.7(a) only shows a difference of 0.002 in the ROC. This difference is considered negligible from the standpoint of RFI detection, since we are near the line PD = PF for every point on the ROC. In Figures 5.7(b) and 5.7(c), Γ outperforms the other test-statistics in terms of the AUC and PD (PF ) performance. For Figure 5.7(d), the RFI power is low, hence bringing down all of the corresponding ROC curves to the PD = PF line. Seemingly β also slightly outperforms Γ here, but again, this is another case where the AUC of all test-statistics are so low that the difference between β and Γ does not matter. BPSK, QPSK, 16-QAM, and 16 rectangular QAM communications signals were generated for the RFI s(n, Λ) in Figure 5.9. The signals were normalized as to all have an amplitude of unity. In every case, in Figures 5.9(a) through 5.9(d), the Γ test-statistic outperforms all others, as substantiated by the ROC analysis in Figure 5.9 and summary in Table 5.2. Lastly, ROC performance for the Γ, β, κre , and κim test-statistics were measured by increasing the number of samples N from 20K to 200K for a PCW signal with amplitude A = 2 and small duty cycle d = 0.5%. As we expected, the AUC, and PD (PF ) results improved with increasing N . In nearly all cases, the test-statistic Γ outperformed all other cases, with the sum of the kurtosis of the real components not lagging far behind. In the cases where β outperformed 137 Γ, either all test-statistics were in an undetectability region of the ROC curve, where PF = PD , or PD = 1. In these cases, Γ performed so similar to Γ, that the differences in AUC performance is considered to be negligible. Therefore, the claim that performance can be improved using the complex formulation has been justiﬁed by these test cases. 5.5 Polarimetric RFI Detection In this section, we consider the last RFI detection case where we interpret the polarimetric radiometer signal as a complex signal. We then show that the pdf of the RFI-free signal, where the joint pdf of the real and imaginary components of this signal is considered to be a bivariate Gaussian random process [32], has an alternate interpretation as a complex Gaussian process given in [16]. We generate this complex Gaussian according to [14], and apply this to the complex polarized signal model proposed in (3.7). We ﬁx the noise variance at unity and the pseudovariance at zero, considering only the circular noise case as a result. We then vary the amplitudes AH and AV of the signal model to observe the ROC performance of our four detectors, Γ, β, kre , and kim . 5.5.1 Bivariate and Complex Gaussian Noise Relationship Consider two discrete-time, zero-mean, real random signals xH (n) and xV (n). The signals can be combined into a complex random signal x(n) = xH (n) + jxV (n). (5.16) If N samples are collected during an integration period T , we can then write x(n) as a vector x, formed by simply stacking the component signal vectors into a single column 138 vector called the real composite 2N -dimensional vector [13] x = xTH , xTV . (5.17) In [31, 32], the joint probability density function pdf of the RFI-free polarimetric signal set {xH (n), xV (n)} is given by the bivariate Gaussian pdf fxH ,xV (xH , xV ) = 2 xH x2V 1 1 2ρxH xV , exp − + 2 − 2 2(1 − ρ2 ) σH σV σH σV 2πσH σV (1 − ρ2 ) (5.18) with the following deﬁnitions using the expectation operator E 2 σH = E(x2H ), variance of horizontal polarized radiometer channel (5.19) 2 σH = E(x2V ), variance of vertical polarized radiometer channel (5.20) RHV = E(xH xV ), horizontal and vertical polarized signal correlation ρ= RHV , correlation coefﬁcient. σH σV (5.21) (5.22) We wish to relate the bivariate pdf in (5.18) to the pdf of the complex signal x(n) in (5.16). First, let the variables u and v correspond to the radiometric signals xH (n) and xV (n). Furthermore, form the complex variable x as done in [80] x = u + jv. (5.23) 139 The bivariate Gaussian pdf given in [80] by $ −1 fu,v (u, v) = 1 exp 2Ruu (1 − ρ2uv ) (2πRuu (1 − ρ2uv )) 2 ' & −1 1 . × 1 exp 2πRvv (2πRvv ) 2 1 √ 2 % Ruu u− √ ρuv v Rvv (5.24) Collecting coefﬁcients and arguments of the exponential, we have fu,v (u, v) = 1 1 × 2π (Ruu Rvv (1 − ρ2uv )) 2 $ √ ' 2 −1 RU U 1 u− √ . ρuv v − exp 2Ruu (1 − ρ2uv ) 2πRvv RV V (5.25) (5.26) The variances Ruu and Rvv , and correlation coefﬁcient ρuv in (5.26), are connected to the 2 and σV2 , and correlation coefﬁcient ρHV via polarimetric radiometer signal variances σH the following relations 2 Ruu = E{u} = σH (5.27) Rvv = E{v} = σV2 2 2 Ruv = E{uv} = Ruu Rvv ρuv = σH σV ρHV (5.28) ρHV = RHV . σH σV (5.29) (5.30) Therefore, we can rewrite (5.26) in terms of the radiometer signal variances and correlation 140 coefﬁcient fu,v (u, v) = 1 1 × 2πσH σV (1 − ρ2uv ) 2 $ √ ' 2 RU U 1 −1 . u− √ ρHV v − exp 2 2σH (1 − ρ2HV ) 2πσV2 RV V (5.31) (5.32) After expanding the squared term in the exponential of (5.32), and some algebraic manipulation, we obtain fu,v (u, v) = & 1 1 2πσH σV (1 − ρ2uv ) 2 exp −1 2 (1 − ρ2HV ) u2 2ρHV uv v2 − + 2 σH σH σV σV2 ' . (5.33) Equation (5.33) is precisely the form of the bivariate pdf given in (5.18). This is also the pdf of the complex signal x(n). Without loss of generality, the sampled signal set is assumed to follow a similar discretized version of the bivariate Gaussian distribution given enough bits for quantization [31]. According to [14], the joint pdf in equation (5.18) is identiﬁed with the distribution of x. Therefore we can interpret the polarimetric radiometer signal set {xH (n), xV (n)} as a single complex signal x(n) as in (5.16) with the underlying bivariate pdf given by (5.18). This is a new interpretation of the pre-detected polarimetric radiometer signal set. Over a single radiometer integration time, T , corresponding to N samples, we can represent x(n) alternatively by the real composite 2N -dimensional vector x of (5.17). 5.5.2 Polarimetric Detection Performance Results For the purposes of this section, we assume the signal model for x2 (n) given in 3.7. In this case, we obtain the horizontal and vertical polarization noise component signals from 141 Re{ } HB (ejω ) xH (n) Bandpass Filter x2 (n) = s2 (n, λ) + CN (0, σ 2 , ρ) Im{ } HB (ejω ) xV (n) Bandpass Filter Figure 5.12: Signal model for polarimetric RFI. The signal s(n) is corrupted with complex Gaussian noise with parameters μ = 0, σ, and ρ. The complex signal is then separated into its real and imaginary parts, and then fed into the radiometer RFI processor in the form of z(n) = xH (n) + xV (n). a complex elliptically symmetric Gaussian distribution (CES) [14] with psuedovariance parameter 2;0 = 0. The complex noise is added to the complex signal s2 (n), and the polarization components are extracted as real and imaginary parts of this composite signal, respectively. A series of four Monte-Carlo simulations were conducted using the RFI signal model 3.7. The ROC performance results are shown in Figure 5.13 and summarized in terms of the ROC performance in Figure 5.14. Inspection of Figures 5.14(a) through Figure 5.14(d) shows that only in the cases where the RFI component amplitudes AH = AV , that the test-statistic Γ has the best ROC performance. This observation is quantiﬁed in Table 5.3. In the cases where AH is not equal to AV , the real kurtosis test-statistic corresponding to the polarimetric channel with the dominant RFI component will perform slightly better than Γ. This suggests that Γ is generally a good test-statistic to use for detecting polarimetric RFI, but the real kurtosis of the individual channel signals should not be ignored. Since there is no way to know a-priori which polarization channel has the dominant RFI, Γ, kre and 142 kim should all be telemetered from the spacecraft. Since the performance of Γ is so close to kre and kim , only computing and transmitting Γ on a spaceborne microwave radiometer would yield the advantage of halving the data rate from the spacecraft, which is a deﬁnite advantage. Figure 5.13: Polarized RFI detection results. (a) All detectors perform similarly, with the Γ statistic outperforming all others. (b), the real kurtosis of the H-Pol channel has a slightly better ROC than Γ, since the amplitude of the RFI is higher in that channel than the V-Pol. (c) Same situation in (b), swapping H-Pol and V-Pol channels (d) RFI signal amplitude for both channels are equal, and approximately half of that in (a) results in a ROC where RFI is undetectable. 143 Figure 5.14: ROC analysis for the polarimetric RFI signal model (a) AH = AV = 0.71. Γ has the best AUC of 0.999. (b) AH > AV . The AUC κre is slightly better than Γ (c) AH < AV , The AUC κim is slightly better than Γand (d) AH = AV = 0.18. RFI is nearly undetectable all with an AUC close to 0.50. Table 5.3: Summary of best and worst polarized complex RFI test-statistics in terms of PD (PF = 0.05) Param. Set Λ 1 2 3 4 Test Maximum Statistic PD (PF = 0.05) Γ 0.995 κre 0.965 κim 0.962 β 0.055 Test Minimum Statistic PD (PF = 0.05) κre 0.820 κim 0.053 κre 0.053 κre 0.050 144 5.6 Summary and Conclusions It was noted that for a polarimetric radiometer with downconverted channels, two scenarios were identiﬁed where a complex signal could be deﬁned. The ﬁrst complex signal formed was formed by combining the in-phase and quadrature components of the baseband signal, and the second complex signal was formed by combining the horizontal and vertical polarization component signals. For the ﬁrst case, our null hypothesis consisted of real Gaussian noise, downconverted to baseband circular Gaussian noise, and the alternate hypothesis consisted of a downconverted version of s(n, Λ) that was used throughout Chapter 4. The test-statistics Γ, β, kre , and kim were in terms of their ROC performance, measured by the AUC and PD (PF ) for PF = 0.05 and PF = 0.10, respectively. It was shown that there is an advantage to using the complex-valued formulation of the baseband quadrature signal and the polarimetric radiometer signal for RFI detection using the complex signal kurtosis statistics. Speciﬁcally, the test-statistic Γ in (5.6) was shown to have a better ROC performance for a variety of RFI signal models according to Table 5.2. In the total-power radiometer case, this test-statistic Γ outperformed the individual kurtosis of the real component baseband signals, and the sum of the kurtosis β as well. The β test-statistic is being used on the SMAP radiometer. It was shown that detection performance can be improved, but not signiﬁcantly, given the signal models evaluated. For the polarized case, it was determined that the test-statistic Γ outperformed the others for RFI detection only when the amplitudes of the RFI model signal had equal amplitudes AH = AV . Otherwise, the best detector was the real kurtosis corresponding to the polarized channel signal with the greatest signal amplitude. Since it was shown in [14] that γ2;2 degenerates to the real kurtosis in the degenerate case, that is, when the signal degenerates to a line on the complex plane, this implies that the signal-to-noise-ratio is dominant in one 145 polarimetric channel, the real kurtosis will have the best ROC performance. However, Γ did outperform the β test-statistic in all cases, suggesting that it is better to use a combination of Γ and one of the real kurtosis test-statistics to detect RFI in the polarized case, rather than to use the β test-statistic. It was also shown that Γ has a detection blind-spot, similar to the real kurtosis. This is not an issue because the blind-spot case is very unlikely to happen on a real spaceborne radiometer. Chapter 6 CONCLUSIONS 6.1 Summary Background of the physics of thermal noise emission and the ﬁeld of spaceborne microwave radiometry was introduced in Chapter 2. In addition, two types of microwave radiometers were introduced — the single-channel total-power radiometer and the dual-channel polarimetric radiometer. Examples of these spaceborne microwave radiometers, such as the Soil-Moisture Active Passive (SMAP) [3], Soil-Moisture Ocean Salinity (SMOS) [34], and AQUARIUS [35] radiometers were provided. The signal processing of radiometer signals by these two radiometer types was also introduced. Finally, the main problem of RFI and RFI detection was described. In general, the assumption of RFI as it impacts the thermal noise signal yields an additive signal model given by x(t) = w(t) + s(t, Λ), (6.1) where w(t) is the white Gaussian thermal noise signal that is desired to be measured by the radiometer, and s(t, Λ) is the undesired RFI signal with a set of parameters Λ determined 146 147 by the underlying signal model assumed for s(t). In Chapter 3, more detail was given to the description of RFI. Historical mission data was presented that showed how RFI is a global problem, and that it occurs across all passive microwave frequency bands such as L-, C-, and K- bands. In addition, it was pointed out that not only can RFI be generated by anthropogenic sources on the ground, but it can also be caused by signals reﬂected off of the Earth by powerful direct broadcast satellites, mostly operating in K-band. Focus was directed to L-band interference, where the RFI problem is the worst. Chapter 3 also presented a general RFI model signal developed by Dr. Sid Misra, et al., that encompassed multiple additive RFI sources seen by a microwave radiometer. The general model for communications signals as RFI sources was also presented, and used later in Chapter 5 in the case of RFI detection for complex baseband signals. Chapter 4 contributed six new approximations of negentropy (Table 6.1) , based on a ﬁnite number of samples N , and used them as test statistics for RFI detection. This was based on the principle that under the RFI-free condition, the radiometer signal would observe only a signal with a Gaussian pdf, and only have to measure its variance (power), which is the objective of microwave radiometry. Under the RFI-present condition, the pdf of this signal would deviate from Gaussian. The six approximations included: (1) a direct computation of negentropy using the histogram of the radiometer signal; (2) the Edgeworth approximation that could be thought of as cumulant series approximation of negentropy; (3/4) two approximations based on nonlinear functions used by Hyvärinen for ICA-based blind-source-separation; (5/6) and slight modiﬁcations of these nonlinear functions based on the fact that skewness is usually zero for radiometric signals. Since the six negentropy approximations are based on a ﬁnite number of samples, an analysis of their behavior under the null hypothesis — the Gaussian noise-only case 148 Table 6.1: Negentropy Approximations Studied Subscript, i h e a Negentropy Notation Jh Je Ja aa b Jaa Jb bb Jbb Description Historgram-based approximation Edgeworth approximation Non-polynomial approximation Hyvärinen (a) Same as Ja but with k1 = 0 Non-polynomial approximation Hyvärinen (b) Same as Jb but with k1 = 0 used by used by — was performed for increasing number of samples N . It is shown that all of the approximations behaved as expected, that is, approach zero as N increased, implying that if spaceborne radiometers used these approximations on their RFI signal processors, detection performance would improve for increasing N , or a higher sample rate. The large-sample behavior was explored in some detail. After the six negentropy approximations of negentropy were introduced, their performance as test-statistics for RFI detection was researched for two distinct RFI signal model types — the general pulsed-CW signal model s(n, Λ) = A sin(2πfc n/FS )Π(n), (6.2) where Π(n) is the unit-pulse function ⎧ ⎪ ⎨ 1, 0 ≤ n ≤ K − 1 Π(n) = , ⎪ ⎩ 0, K ≤ n ≤ N, (6.3) 149 and the multiple pulsed-CW signal model s(n) = M Ak cos (2πfk n + θk ) Π k=1 n − n0 wk . (6.4) with component signal parameters of amplitude Ak , phase φk , center frequency fk , duty cycles dk = ωk /N , and number of samples N . For the multiple-PCW RFI signal model, the behavior of the six negentropy approximations was explored by ﬁxing N , but increasing the number of signal components M in the model. The goal was to show the limitations of detection for a large number of interferers M . We expected that adding a large number of pulsed-CW signals with random phases, frequencies, amplitudes, and duty cycles would yield an RFI signal with a pdf that approached Gaussian as M increased. This signal would then be added to true Gaussian noise, making this RFI signal undetectable. Since kurtosis was shown to have a detection limit with a large number of interfering signals present, we expected negentropy to have the same limitation, which we found in Chapter 4, subsection 4.8.1. The detection performance of all six negentropy approximations was evaluated via computer simulation and analysis via ROC curves. The ROC curve performance for every approximation was further quantiﬁed by: (1) evaluating detection probability PD versus two separate false-alarm probabilities PF = 0.05 and PF = 0.10, and (2) computing the area under the ROC curve (AUC) metric. In all cases, the approximations were compared with the kurtosis and Jarque-Bera test statistics. For the single pulsed-CW RFI signal model with a pulse duty cycle of 50%, (known as the kurtosis blind-spot) [12], all test-statistics were compared with the performance of the Anderson-Darling and Shapiro-Wilk test statistics as well. Plots of the ROC, AUC, and PD (PF ) performance measures were generated throughout Chapter 4. 150 By Chapter 5, it was clear that the detection performance all of the negentropy-based test statistics did not surpass that of the kurtosis for all types of RFI, with the exception of the kurtosis-blind spot case. The kurtosis had the best AUC and ROC performance of all other detectors, regardless of RFI signal model, in general. However, as was addressed in Chapter 5, there was still more to be achieved with kurtosis. Prior literature on the performance of kurtosis made the fundamental assumption that the signal being operated upon for RFI detection was real-valued and single-channel. In the case of polarimetric radiometers, or any radiometer that downconverts the received signal to complex quadrature baseband representation, there are at least two signals to use for RFI detection. This issue was largely unexplored in the literature. The only literature on RFI detection found that mentioned use of the complex signal kurtosis was spectral kurtosis (SK) estimator presented in [81]. In the case of the SMAP radiometer, it was shown in Chapter 2 that a total of four kurtosis statistics were computed simultaneously for detecting RFI, corresponding to the two sets of in-phase and quadrature component signals from the horizontal and vertical polarimetric channels. In Chapter 5, we extended the use of kurtosis to the complex domain, and employed the test statistic Γ Γ= γ2;2 , 1 + 12 |2;0 |2 (6.5) which depends on the complex signal kurtosis γ2;2 normalized by the complex signal variance 2;0 to detect RFI in the complex baseband radiometer signal of a total-power radiometer channel. We also applied (6.5) to the two bandpass signals (prior to baseband downconversion) of the polarimetric radiometer signal channels combined as a single complex-valued signal. For the baseband quadrature case, we used the following RFI signal models: 151 • Pulsed-CW, varying parameters A and d • Complex digital modulation: BPSK, QPSK, 16-Circular-QAM, and 16-RectangularQAM • Pulsed-CW, ﬁxing A = 2, d = 0.5% and varying N = 10k, 20k, 50k, 100k samples We computed Γ for them, after adding standard Gaussian noise (zero mean, unit variance). For the complex polarized case, we ﬁrst developed a polarized RFI signal model in (3.7) s(n) = AH cos 2πfc n FS + jAV sin 2πfc n FS Π(n), n = 0, . . . , N − 1, (6.6) and applied the same procedure to detect this RFI in complex circular noise. The only parameters that were varied in this case were the amplitudes AH and AV of the polarized channel RFI signals. The other model parameters were ﬁxed at the same values used in Chapter 4, with the exeption of d, which was always set to 100%. All three models, the multiple PCW, digital modulation model, and polarized complex models covered a broad range of RFI signal types that current and future spaceborne microwave radiometers are likely to experience. For comparison, the complex-kurtosis-based test-statistic in (6.5) was compared to another statistic called β, which was a test-statistic formed from summing the individual kurtoses of in-phase and quadrature channel signals for the complex baseband case, or the bandpass polarimetric channel signals in the polarized signal case. In addition, the individual channel kurtosis, denoted kre and kim , were included for comparison. The performance of (6.5), β, kre , and kim were measured via ROC performance curves. The performances were also quantiﬁed by analyzing the PD (PF ) for false-alarm probabilities of 0.05 and 0.01. The AUC metric was also computed. Graphs of the PD (PF ) 152 and AUC measurements were plotted in Figures 5.4 through 5.10, and 5.13, for each RFI signal model case. The ROC analysis graphs (AUC, PD (PF ) values) corresponding to the plots are indicated in Figures 5.5 through 5.11, and 5.14. Tables of the ROC performance via PD (PF ) and the AUC for Chapters 4 and 5 are included in Appendix B. For Chapter 4, all of the ROC performance plots and tables indicate that negentropybased test-statistics are useful for detecting RFI, but do not outperform the kurtosis ROC, except for the blind-spot case. For Chapter 5, the ROC plots show that the test-statistic Γ is a good detector of RFI. For the baseband complex case, this test-statistic always outperforms the real kurtosis of the in-phase and quadrature channels, and also outperforms β, but with a more comparable AUC performance than the real kurtosis cases. For the polarized case, the ROC plots and analysis show that when the polarized signal has equal RFI amplitudes in the real and imaginary parts, Γ deﬁnes the best detector of RFI. Otherwise, the better test-statistic is determined by the polarized channel that has the strongest RFI. 6.2 6.2.1 Conclusions RFI Signal Models and Central-Limit Effects Two of the three RFI signal models that were used to characterize the ROC performance of all test-statistics in this research were based on prior research using spaceborne and airbore radiometers [44, 47, 55, 56, 82]. The consensus is that RFI is caused by communications systems, military and air route surveillance radar, and out-of-band harmonics from other sources. Other RFI can be self-imposed, via poor design of the radiometer, allowing higherorder intermodulation products enter the radiometer’s passband. All of these sources can 153 be modeled as sinusoidal sources, the pulsed-continuous wave model presented in Chapter 3 is a broad signal model that encompasses these RFI types. The third model, also developed in this research in Chapter 3 is the ﬁrst model that captures the joint behavior of RFI that appears simultaneously on both polarization channels of a polarimetric radiometer. It was formed as a modest extension of the PCW model for a single total-power channel, and supported by analysis of measured pre-detected radiometer data from the SMAPVEX12 [4] campaign. Given the multiple PCW model, we observed the known behavior that as the number of interferers increase, the net effect on the pdf of the radiometer signal x(n) is to make it more Gaussian. Since these signals are assumed to add to each other, and they are all random amplitude, frequency, and phase sinusoids, relative to the radiometer, the addition of an increasing number of these has a central-limit theorem type of behavior. Misra, et al., found this in [55], and this was shown again in Chapter 4, subsection 4.8.1. An additional step was taken to measure how close the pdf of the RFI approached Gaussian by measuring the mean-squared-error between the signal pdf and a Gaussian pdf with the same mean and variance as the multiple PCW signal s(n), for an increasing number of randomly-generated RFI signals M . We then measured the same difference using our negentropy approximations, averaging over many trials of randomly-generated RFI signals s(n). The results of both of these measurements are shown in Figures 4.15 – 4.17. We concluded that although the resulting pdf of the multiple-PCW is not exactly Gaussian, it is sufﬁciently close such that when added to Gaussian noise w(n), becomes indistinguishable from pure thermal noise from the perspective of the detector. This effect suggests that as the number of RFI sources M → ∞, any RFI detector that is based on a Gaussianity vs. non-Gaussianity test will fail. 154 6.2.2 Negentropy-based Test-Statistics for Detection of RFI It was shown that although negentropy has attractive analytical properties that suggest that it is an ideal candidate for RFI detection, the fact that it can only be approximated in practice leads to sub-optimal results with respect to ROC performance when compared to kurtosis. Due to its simplicity and superior performance ROC, the kurtosis was again shown to be the best test-statistic of RFI for almost every signal type. Some insight was gained from the fact that negentropy approximations that used terms that are sensitive to skewness performed worse that those that did not. This suggests that the best approximations of negentropy should use as few terms as possible, and only should be sensitive to bimodality for the CW and PCW RFI models. The results also suggest that any new approximations that result by choosing new non-polynomial functions [69] should only consider properties of even moments of the amplitude probability distribution, as it would seem that including terms that were based on odd-order moments of the pdf introduced error to the detector and degraded performance. The negentropy results strengthened the argument for using kurtosis as a basis for RFI detection. However, for digital radiometers that use baseband quadrature downconversion, and for polarimetric radiometers, the kurtosis of a real random variable limited the possibilities for detection. The current state of the art using kurtosis applies it either to the in-phase and quadrature channel signals separately, to the horizontal and vertical polarization signals only, or to sums of the kurtoses for the in-phase and quadrature or horizontal and vertical polarization signal sets, respectively. 155 6.2.3 Complex-Valued Kurtosis - Based Test-Statistics for Detection of RFI Treating the in-phase and quadrature component signals as a single complex signal, and then applying the kurtosis deﬁned for this complex signal, yielded improved ROC detection performance in all cases and signal models considered. It was originally thought that second-order circularity was useful for detecting RFI, but we found later that the quadrature baseband signal is always circular. This is due to the fact that the in-phase and quadrature component signals are always orthogonal, so their cross-correlation is zero. It was also due to the fact that the quadrature component signals always have the same energy, simply because they originated from the same signal, which was split evenly along the quadrature signal paths. The combination of having the same energy and orthogonality for quadrature component signals yields a pseudovariance of zero, regardless of the RFI. However, the fourth-order statistics, namely the normality test-statistic Γ in [14] was shown to produce favorable detection results, compared to the kurtoses of the quadrature component channels and the sum thereof. In addition, treating the polarization channel signals as a single complex-valued signal yielded a better ROC performance than β in terms of the AUC and PD (PF ) measures. In addition, it was shown that circularity of this signal is very unlikely for a number of reasons, including polarization mixing, existence of linearly-polarized RFI, and the simple fact that the variances of the independent polarization RFI-free channel signals generally have different variances. Thus the RFI-free distribution of the complex polarimetric signal is complex elliptical Gaussian. In all cases, ROC performance is largely determined by SNR. Detection performance can always be improved in all cases by increasing N , which implies either increasing the 156 digital radiometer sample rate FS , or integrating for a longer integration time T . 6.2.4 Discussion of Implementation Costs vs. Performance For the next decade or so, it is likely that all future spaceborne microwave radiometers will use FPGAs to perform RFI mitigation functions. FPGAs are highly programmable digital logic devices, very capable of processing large amounts of data very quickly since their array of reconﬁgurable digital logic allows for the design of custom parallel computers that operate at hundreds of MHz [37]. As such, The minimum operational sample rate of an RFI detection system would follow twice Nyquist, i.e., twice the radiometer signal bandwidth, 2B [83]. Since the minimum spectrum allocation for passive microwave radiometery is about 27 MHz, this implies that the minimum sample rate must be 100 MHz, if we are to be conservative and use close to 4X sampling [7]. FPGAs can easily operate at these rates, but conventional microprocessors cannot [37]. This is why, in the last decade or so, FPGAs have become the computational tool of choice for handling radiometer signal processing, particularly for the application of RFI detection [3, 46, 58]. Implementation of processing algorithms on FPGAs is straightforward and easy to do for linear multiply-and-accumulate (MAC) operations [84]. The computation of higher-order moments such as variance, skewness, and kurtosis are also ideal in FPGA digital logic. This is true despite the fact that computing the pth power of a digital signal increases the number of required bits and corresponding registers by a factor of p, using ﬁxed-point operations. On the other hand, operations that are not amenable to MAC operations, such as the computation of nonlinear or transcendental functions (e.g. log or sin) requires a bit more effort [7]. These functions require series approximations, look-up tables, or other algorithms to compute within an FPGA, and are better suited to slower ﬂoating-point processors. This presents a problem for all of the negentropy-based 157 computations, since variance must be normalized prior to the approximation function. For example, a signal ﬂow graph for the simple computation of normalizing the radiometer signal by its standard deviation is shown in Figure 6.1. The standard deviation of the input signal x(n) is given by σ= ( x2 (n) − x(n)2 , (6.7) where the is the sample mean. The delayed version of x(n) is given by x(n − N ). The sample standard deviation corresponds to the prior N samples of x(n), hence x(n − N ) is scaled by this variance instead of x(n). The output of the normalizing circuit is then y(n) = x(N − n)/σ (6.8) The square-root function introduces complexity in the FPGA processing, as does scaling x(n − N ) by σ. Though nonlinear functions can be implemented in an FPGA, implementation of the kurtosis is much simpler, and less costly in terms of digital logic. Since in most digital radiometer processors, the signal second moment is already computed to approximate brightness temperature, not much more complexity is required for the same circuit to compute skewness and kurtosis in parallel on the same data. In fact, this is precisely what is done on the SMAP radiometer signal processor [7]. Given the simple nature of computing kurtosis on-board, and the fact that the kurtosis outperforms negentropy in almost all relevant cases for RFI mitigation, implementation of the negentropy computation on spaceborne microwave radiometer’s RFI processor is not worth the cost to implement as a primary RFI processor, and perhaps has utility only as a secondary RFI processor to cover the rare kurtosis blind-spot. 158 × z −N x(n) x(n − N )/σ √ 1 ( ) ( )2 ( )2 − + Figure 6.1: Variance-normalization circuit. The signal x(n) streams in at a clock rate fclk and is split into three parallel paths. The ﬁrst path delays the signal by N samples, the second computes the mean and then squares the mean, and the third computes the square of the samples, and then averages the squares of the samples. The difference between the squared mean and mean squares is computed, yielding the signal variance. The variance is inverted, and then used as a constant factor to multiply every sample of the delayed x(n) for the current N samples. The process repeats every N samples. For the computation of the complex signal kurtosis, MAC computations are also required. In addition, like the negnetropy case, a variance normalization circuit is also required. The same argument for implementing the real signal kurtosis for total-power radiometers can be made for implementing the complex signal kurtosis-based statistic for the complex signal case. Given its superior performance, the complex signal kurtosis is recommended particularly for quadrature downconverted radiometer signals, as it has better detection performance while reducing the required telemetry sent down to Earth by half. Although using Γ instead of β, κre, and κim achieves modest ROC performance gain, in practical use, there is a signiﬁcant gain from both a data rate and spacecraft data storage perspective. The test-statistic Γ is a single real-valued quantity, that effectively 159 detects RFI using both I/Q signal pairs or horizontal and vertical polarization signal pairs, respectively. This means that for every two test-statistic values that need to be computed by the radiometer digital signal processing system, κ + re and κim , only one value, Γ needs to be computed. This represents a data rate reduction by a factor of two, for all polarimetric radiometers that use kurtosis for RFI detection. This is particularly important for radiometers like SMAP, which calculate separate κ + re and κim test-statistics for each I/Q signal, for each polarization channel. This implies that less on-board data needs to be stored, and that we can recover twice the radiometer data bandwidth needed to operate a radiometer like SMAP in the future. The test statistic β has the same data-rate advantage, only if it is computed on board. But even if β is computed (by adding κ + re and κim ), Γ outperforms β in ROC performance in all relevant test scenarios considered. The cost to implement β on-board is the same as it would be for implementing SMAP, as described in Chapter 2. The cost to implement Γ would be slightly greater, but at a negligible level because only a few additional hardware multipliers and adders would be necessary. This cost is certainly justiﬁable, particularly for radiometers that operate at K-band or higher, where sampling at many hundreds of MHz for these larger radiometer bands will result in the generation and storage of data over constrained telemetry downlink bandwidths. 6.3 6.3.1 Future Work Additional Polarized RFI Study The complex-valued interpretation of polarized component signals in Chapter 5 yielded good results, but there’s additional work that can be done. Detection of RFI using both polarization channels simultaneously is still a relatively new area. Researchers mostly 160 use the third and fourth Stokes parameters to detect RFI [3, 23], which is really using the cross-correlation between horizontal and polarization channel signals for detection. A comparison between the performance of the test-statistic Γ that uses the complex signal kurtosis coefﬁcient γ2;2 against the correlation between Stokes parameter signals should be performed. The noise in the complex-baseband case is always circular, but the noise in the polarized case, though not necessarily circular in the real radiometer case (as seen in Chapter 3), was made circular in simulation by setting the pseudovariance to zero. Future studies will explore the implication of non-circular noise on the detection of polarized RFI using the complex signal model in 3.7. 6.3.2 Higher-Order Spectral Analysis for Radiometric RFI Detection Others have shown that the joint-time-frequency domain decomposition yields signiﬁcant and obvious advantages for RFI detection [41,53,66,68]. The joint time-frequency analysis was combined with kurtosis in [12], showing that if the pre-detected radiometer signal is ﬁrst partitioned into subbands, and the kurtosis of each subband is computed, detection performance is improved and RFI can be excised at greater precision by simply using only those subbands that are ﬂagged as not containing RFI. Essentially, this technique amounts to spectrogram smoothing if the discarded time-frequency cells are replaced with the local power spectrum values [41]. In radio astronomy literature, such as in [81,85], the spectral kurtosis (SK) test-statistic is used to detect RFI leaking into radio telescopes. The spectral kurtosis is deﬁned as the normalized complex signal kurtosis of the k th frequency bin fk of the power-spectral density of x(n). The SK is exactly zero if x(n) ∼ N (0, σ 2 ). Radio telescopes and microwave radiometers are similar in that the their task is to 161 measure average thermal noise power in passive microwave frequency bands. However, they point in opposite directions - radio telescopes point away from Earth and radiometers point towards Earth. The SK has never been applied to radiometric signals, so this is yet another area for applying these test-statistics: in this case, the complex signal kurtosis for RFI detection. However, the Fourier transform of the m-th -order cumulant is deﬁned as the m-th -order polyspectra, or higher-order spectral densities beyond the second-order power spectrum [83, 86]. In particular, for m = 2 the Fourier transform of the autocorrelation function is deﬁned as the power-spectral density. For m = 3, the Fourier transform of the third-order autocorrelation is deﬁned as the bispectrum, or bispectral density. For m = 4, the fourth-order autocorrelation is deﬁned as the trispectrum, or trispectral density. The bispectrum and trispectrum, although applied for RFI detection again in radio astronomy, can be studied in the context of RFI detection for spaceborne microwave radiometers. Moreover, the SK and the trispectral density require a comparative study with regards to utility for microwave radiometers. The SK is essentially the complex signal kurtosis of the Fourier transform, whereas the trispectral density is the Fourier transform of the kurtosis. Some obvious questions to ask include: 1. Do these higher-order statistics represent the same thing? 2. What is the relationship between them? 3. How do they compare with each other in terms of detection performance for different RFI signal types? 4. What is the computational cost of implementation on spacecraft? The SK and the trispectral density are both appealing since they combine the best of 162 all worlds in terms of statistical analysis and joint time-frequency analysis of RFI. Other extensions of this may include study of the cross-bispectral and cross-trispectral densities for RFI detection for multiple polarization channels of polarimetric and interferometric radiometers. It is important to note that in [12], the kurtosis of the frequency bins of the FFT of x(n) was taken, which appears similar to the deﬁnition of the SK statistic; however, only the real kurtosis was taken of the real part of the FFT. Current systems that compute the real kurtosis of real FFT frequency bins can easily be modiﬁed to compute the SK statistic by computing the complex signal kurtosis γ2;2 of the complex FFT output. It is reasonable to expect RFI detection can be improved in this case since the imaginary data are used with the real data, as opposed to real data only, for kurtosis computation. 6.3.3 Sliding-Window Kurtosis Detection To date, current RFI detectors operate only within a single integration period, that is, all pertinent test-statistics are computed on sequential blocks of N samples of x(n). Because of the ﬂexibility of DSP in space applications, the limitation to compute statistics only on sequential blocks of N samples is only restricted by the required radiometric resolution. An alternative approach for computing statistics uses a sliding window of L < N samples, updated for each new sample of x(n). For example, consider the radiometer timing diagram in Figure 6.2. The ﬁrst four Figure 6.2: Radiometer timing diagram. 163 cumulants of x(n) are computed over N samples if the kurtosis is used to detect RFI. The kurtosis is only known for subsequent blocks of N samples. Using what we deem sliding cumulant windows, we can gain better time resolution sample-by-sample of all of the cumulants, as opposed to getting an update every N samples. This measurement scenario is shown in Figure 6.3. Figure 6.3: Sliding-window kurtosis concept. Having ﬁner time-resolution on the statistics would allow more time-localized mitigation of RFI, perhaps reducing the amount of data loss once RFI is detected. This approach would possibly increase detection performance and reduce data loss at the expense of additional on-board signal processing and memory resources. A circular buffer containing N samples would have to be maintained at all times, for every statistic computed in this algorithm. This RFI detection method can be implemented on any future microwave radiometer that uses digital signal processing. Appendix A Proof that the Continuous Gaussian Distribution Maximizes Differential Entropy using Lagrange Multipliers The fact that the Gaussian distribution maximizes the differential entropy was proven in [72, 87]. This proof is repeated here for completeness. The differential entropy of a continuous random variable X with probability density function fX (x) is deﬁned as h(X) = − ∞ −∞ fX (x) log fX (x) dx, (A.1) where the logarithm with with respect to base e. We seek the probability density function fX (x) that maximizes h(X), subject to the following constraints: 164 165 2 1. The mean μX and variance σX are ﬁxed. 2. The total area under fX (x) = 1, hence it is indeed a probability density function. As a result, we consider the constraint involving the mean and variance, simultaneously 2 E (x − μX )2 = σX ∞ −∞ 2 (x − μX )2 fX (x) dx = σX . (A.2) (A.3) We also consider the constraint involving the total area under fX (x), ∞ −∞ fX (x) dx = 1. (A.4) Given the two constraints, we have two Lagrange multipliers, λ1 and λ2 , and the Lagrangian function G(fX (x), λ1 , λ2 ), and constraints from (A.3) and (A.4) we have ∞ G(fX λ1 , λ2 ) = h(X) + λ1 fX (x) dx − 1 −∞ ∞ 2 2 + λ2 (x − μX ) fX (x) dx − σX −∞ ∞ ∞ =− fX (x) log fX (x) + λ1 fX (x) dx − 1 −∞ −∞ ∞ 2 2 + λ2 (x − μX ) fX (x) dx − σX (A.5) (A.6) −∞ To maximize G implies that we maximize all the terms within a single integrand in (A.6). Isolating these terms, and differentiating, we have ∂G fX (x) = − log fX (x) − + λ1 + λ2 (x − μx )2 ∂fX (x) fX (x) = − log fX (x) − 1 + λ1 + λ2 (x − μx )2 = 0 (A.7) (A.8) 166 Therefore, log fX (x) = −1 + λ1 + λ2 (x − μx )2 (A.9) and solving for fX (x), we have fX (x) = e−1+λ1 +λ2 (x−μx ) 2 (A.10) The ﬁnal steps require some algebraic manipulation. Substitution of (A.10) into (A.3) and (A.4), we obtain 1 λ1 = log 2 1 2πσ 2 (A.11) and λ2 = − log 1 2σ 2 . (A.12) Plugging these back into (A.10), we arrive at the Gaussian probability density function fX (x) = √ (x−μ)2 1 e− 2σ2 . 2πσ (A.13) Appendix B ROC Performance Tables for Negentropy Approximations and Complex Signal Kurtosis Statistics 167 168 B.1 Negentropy-Based Test Statistics B.1.1 Single PCW RFI Signal Model Table B.1: Single PCW RFI Source Receiver Operating Charcteristic Results: AUC and PD for PF = 0.05, PF = 0.10. Param. Set Λ1 Λ1 Λ1 Λ1 Λ1 Λ1 Λ1 Λ1 Λ4 Λ4 Λ4 Λ4 Λ4 Λ4 Λ4 Λ4 Λ4 Λ4 Λ7 Λ7 Λ7 Λ7 Λ7 Λ7 Λ7 Λ7 Λ10 Λ10 Λ10 Λ10 Λ10 Λ10 Λ10 Λ10 Λ13 Λ13 Λ13 Λ13 Λ13 Λ13 Λ13 Λ13 Λ16 Λ16 Λ16 Λ16 Λ16 Λ16 Λ16 Λ16 Test Stat. Ja Jb Je Jh k JqB Jaa Jbb Ja Jb Je Jh k JqB Jaa Jbb PD (0.05) 1.000 1.000 1.000 0.999 1.000 1.000 1.000 1.000 0.132 0.166 0.165 0.058 0.333 0.157 0.172 0.215 PD (0.10) 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.224 0.262 0.262 0.110 0.447 0.249 0.267 0.312 AUC 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.618 0.646 0.648 0.529 0.801 0.644 0.651 0.681 Ja Jb Je Jh k JqB Jaa Jbb Ja Jb Je Jh k JqB Jaa Jbb Ja Jb Je Jh k JqB Jaa Jbb Ja Jb Je Jh k JqB Jaa Jbb 0.985 0.996 0.999 0.637 1.000 0.999 0.991 0.997 0.188 0.257 0.610 0.024 0.777 0.614 0.276 0.338 0.078 0.083 0.079 0.057 0.161 0.066 0.085 0.091 0.812 0.903 0.914 0.150 0.986 0.914 0.878 0.939 0.992 0.998 0.999 0.749 1.000 0.999 0.994 0.999 0.293 0.364 0.684 0.064 0.868 0.691 0.369 0.450 0.151 0.157 0.133 0.105 0.277 0.125 0.155 0.172 0.884 0.944 0.958 0.255 0.995 0.958 0.926 0.965 0.996 0.999 1.000 0.912 1.000 1.000 0.997 0.999 0.663 0.711 0.862 0.504 0.955 0.866 0.696 0.747 0.544 0.561 0.560 0.525 0.681 0.557 0.554 0.567 0.957 0.978 0.981 0.672 0.996 0.981 0.969 0.985 Param. Set Λ2 Λ2 Λ2 Λ2 Λ2 Λ2 Λ2 Λ2 Λ5 Λ5 Λ5 Λ5 Λ5 Λ5 Λ5 Λ5 Λ5 Λ5 Λ8 Λ8 Λ8 Λ8 Λ8 Λ8 Λ8 Λ8 Λ11 Λ11 Λ11 Λ11 Λ11 Λ11 Λ11 Λ11 Λ14 Λ14 Λ14 Λ14 Λ14 Λ14 Λ14 Λ14 Test Stat. Ja Jb Je Jh k JqB Jaa Jbb Ja Jb Je Jh k JqB Jaa Jbb SW AD Ja Jb Je Jh k JqB Jaa Jbb Ja Jb Je Jh k JqB Jaa Jbb Ja Jb Je Jh k JqB Jaa Jbb PD (0.05) 0.990 0.998 0.996 0.524 0.999 0.995 0.994 0.999 0.405 0.226 0.083 0.280 0.026 0.085 0.495 0.290 0.532 0.600 0.099 0.112 0.186 0.052 0.317 0.201 0.133 0.154 0.067 0.080 0.200 0.078 0.319 0.208 0.081 0.085 0.151 0.174 0.177 0.048 0.354 0.167 0.176 0.219 PD (0.10) 0.996 0.999 0.998 0.682 1.000 0.998 0.998 0.999 0.533 0.344 0.153 0.426 0.067 0.157 0.618 0.433 0.647 0.735 0.165 0.187 0.283 0.098 0.439 0.293 0.191 0.241 0.126 0.133 0.273 0.159 0.448 0.284 0.138 0.156 0.231 0.265 0.265 0.104 0.481 0.253 0.249 0.309 AUC 0.998 0.999 0.998 0.895 1.000 0.998 0.999 1.000 0.810 0.710 0.548 0.773 0.512 0.550 0.844 0.751 0.872 0.902 0.543 0.571 0.628 0.538 0.780 0.633 0.563 0.594 0.529 0.543 0.637 0.617 0.776 0.639 0.541 0.553 0.605 0.640 0.650 0.528 0.811 0.646 0.632 0.675 Param. Set Λ3 Λ3 Λ3 Λ3 Λ3 Λ3 Λ3 Λ3 Λ6 Λ6 Λ6 Λ6 Λ6 Λ6 Λ6 Λ6 Λ6 Λ6 Λ9 Λ9 Λ9 Λ9 Λ9 Λ9 Λ9 Λ9 Λ12 Λ12 Λ12 Λ12 Λ12 Λ12 Λ12 Λ12 Λ15 Λ15 Λ15 Λ15 Λ15 Λ15 Λ15 Λ15 Test Stat. Ja Jb Je Jh k JqB Jaa Jbb Ja Jb Je Jh k JqB Jaa Jbb PD (0.05) 0.516 0.615 0.577 0.118 0.834 0.574 0.596 0.694 1.000 1.000 1.000 0.866 1.000 1.000 1.000 1.000 PD (0.10) 0.645 0.736 0.729 0.216 0.914 0.720 0.733 0.795 1.000 1.000 1.000 0.923 1.000 1.000 1.000 1.000 AUC 0.863 0.899 0.900 0.641 0.967 0.898 0.893 0.921 1.000 1.000 1.000 0.975 1.000 1.000 1.000 1.000 Ja Jb Je Jh k JqB Jaa Jbb Ja Jb Je Jh k JqB Jaa Jbb Ja Jb Je Jh k JqB Jaa Jbb 0.615 0.771 0.955 0.323 0.995 0.956 0.697 0.825 0.057 0.064 0.100 0.090 0.140 0.104 0.060 0.061 0.332 0.423 0.430 0.061 0.657 0.424 0.421 0.495 0.735 0.845 0.962 0.429 0.998 0.963 0.796 0.886 0.115 0.111 0.153 0.165 0.235 0.157 0.117 0.120 0.474 0.547 0.555 0.129 0.793 0.548 0.543 0.620 0.890 0.939 0.986 0.738 0.999 0.986 0.914 0.953 0.509 0.515 0.548 0.602 0.644 0.551 0.513 0.517 0.773 0.817 0.820 0.568 0.927 0.818 0.808 0.848 169 B.1.2 Multiple PCW RFI Signal Model Table B.2: Multiple PCW RFI Source Receiver Operating Charcteristic Results: AUC and PD for PF = 0.05, PF = 0.10. # of Signals M M =1 M =1 M =1 M =1 M =1 M =1 M =1 M =1 M =5 M =5 M =5 M =5 M =5 M =5 M =5 M =5 M = 20 M = 20 M = 20 M = 20 M = 20 M = 20 M = 20 M = 20 M = 100 M = 100 M = 100 M = 100 M = 100 M = 100 M = 100 M = 100 M = 500 M = 500 M = 500 M = 500 M = 500 M = 500 M = 500 M = 500 Test Stat. Ja Jb Je Jh κ JqB Jaa Jbb Ja Jb Je Jh κ JqB Jaa Jbb Ja Jb Je Jh κ JqB Jaa Jbb Ja Jb Je Jh κ JqB Jaa Jbb Ja Jb Je Jh κ JqB Jaa Jbb PD (0.05) 0.990 0.999 0.999 0.718 1.000 0.999 0.994 0.999 0.912 0.970 0.987 0.428 0.998 0.987 0.940 0.985 0.072 0.070 0.067 0.065 0.112 0.071 0.073 0.078 0.049 0.047 0.038 0.054 0.085 0.064 0.057 0.055 0.053 0.053 0.062 0.044 0.094 0.065 0.051 0.056 PD (0.10) 0.995 0.999 1.000 0.803 1.000 1.000 0.997 1.000 0.945 0.984 0.991 0.557 0.999 0.991 0.965 0.990 0.120 0.125 0.125 0.128 0.189 0.128 0.117 0.120 0.094 0.099 0.088 0.104 0.147 0.117 0.105 0.113 0.100 0.100 0.109 0.105 0.163 0.111 0.105 0.117 AUC 0.998 1.000 1.000 0.933 1.000 1.000 0.998 1.000 0.979 0.993 0.996 0.828 0.999 0.996 0.987 0.996 0.520 0.522 0.531 0.524 0.626 0.533 0.516 0.520 0.502 0.503 0.500 0.504 0.570 0.502 0.506 0.505 0.502 0.501 0.512 0.520 0.562 0.512 0.501 0.503 # of Signals M M =2 M =2 M =2 M =2 M =2 M =2 M =2 M =2 M = 10 M = 10 M = 10 M = 10 M = 10 M = 10 M = 10 M = 10 M = 50 M = 50 M = 50 M = 50 M = 50 M = 50 M = 50 M = 50 M = 200 M = 200 M = 200 M = 200 M = 200 M = 200 M = 200 M = 200 M = 1000 M = 1000 M = 1000 M = 1000 M = 1000 M = 1000 M = 1000 M = 1000 Test Stat. Ja Jb Je Jh κ JqB Jaa Jbb Ja Jb Je Jh κ JqB Jaa Jbb Ja Jb Je Jh κ JqB Jaa Jbb Ja Jb Je Jh κ JqB Jaa Jbb Ja Jb Je Jh κ JqB Jaa Jbb PD (0.05) 0.199 0.240 0.367 0.045 0.551 0.371 0.239 0.307 0.495 0.622 0.699 0.132 0.853 0.707 0.554 0.682 0.058 0.058 0.072 0.053 0.105 0.078 0.064 0.073 0.053 0.059 0.049 0.048 0.074 0.050 0.039 0.041 0.054 0.046 0.062 0.062 0.053 0.061 0.061 0.048 PD (0.10) 0.280 0.334 0.472 0.107 0.677 0.482 0.351 0.434 0.617 0.716 0.779 0.208 0.918 0.785 0.694 0.784 0.109 0.126 0.124 0.103 0.174 0.130 0.126 0.128 0.109 0.106 0.112 0.106 0.134 0.106 0.097 0.097 0.110 0.099 0.102 0.121 0.107 0.105 0.122 0.108 AUC 0.653 0.701 0.764 0.540 0.890 0.768 0.692 0.740 0.840 0.886 0.908 0.600 0.973 0.911 0.872 0.913 0.503 0.511 0.506 0.502 0.612 0.509 0.511 0.518 0.507 0.504 0.504 0.503 0.545 0.503 0.503 0.504 0.502 0.503 0.500 0.518 0.503 0.501 0.504 0.501 170 B.2 Complex Signal Kurtosis-Based Test Statistics B.2.1 Complex Baseband Signal Model Table B.3: Complex Baseband RFI Detection ROC Performance Analysis: AUC and PD for PF = 0.05, PF = 0.10. Param. Set Λ1 Λ1 Λ1 Λ1 Λ3 Λ3 Λ3 Λ3 Λ5 Λ5 Λ5 Λ5 Λ7 Λ7 Λ7 Λ7 Λ9 Λ9 Λ9 Λ9 Λ11 Λ11 Λ11 Λ11 Λ13 Λ13 Λ13 Λ13 Λ15 Λ15 Λ15 Λ15 Test Stat. Γ β κre κim Γ β κre κim Γ β κre κim Γ β κre κim Γ β κre κim Γ β κre κim Γ β κre κim Γ β κre κim PD (0.05) 1.000 1.000 0.989 0.989 0.381 0.322 0.211 0.222 0.073 0.087 0.081 0.078 0.804 0.710 0.477 0.456 0.405 0.338 0.199 0.246 1.000 1.000 1.000 1.000 0.655 0.591 0.399 0.398 1.000 0.999 0.961 0.957 PD (0.10) 1.000 1.000 0.995 0.996 0.523 0.466 0.324 0.345 0.129 0.145 0.141 0.137 0.882 0.827 0.605 0.593 0.556 0.463 0.312 0.382 1.000 1.000 1.000 1.000 0.775 0.703 0.528 0.505 1.000 1.000 0.982 0.978 AUC 1.000 1.000 0.997 0.997 0.823 0.790 0.716 0.716 0.510 0.509 0.504 0.508 0.960 0.939 0.863 0.857 0.839 0.797 0.709 0.733 1.000 1.000 1.000 1.000 0.918 0.894 0.813 0.809 1.000 1.000 0.992 0.991 Param. Set Λ2 Λ2 Λ2 Λ2 Λ4 Λ4 Λ4 Λ4 Λ6 Λ6 Λ6 Λ6 Λ8 Λ8 Λ8 Λ8 Λ10 Λ10 Λ10 Λ10 Λ12 Λ12 Λ12 Λ12 Λ14 Λ14 Λ14 Λ14 Λ16 Λ16 Λ16 Λ16 Test Stat. Γ β κre κim Γ β κre κim Γ β κre κim Γ β κre κim Γ β κre κim Γ β κre κim Γ β κre κim Γ β κre κim PD (0.05) 0.955 0.893 0.682 0.656 0.090 0.086 0.080 0.069 0.918 0.832 0.578 0.550 0.101 0.091 0.076 0.073 0.386 0.339 0.221 0.215 0.104 0.095 0.084 0.080 0.951 0.902 0.719 0.694 1.000 1.000 0.998 0.998 PD (0.10) 0.979 0.941 0.782 0.792 0.166 0.164 0.148 0.142 0.958 0.915 0.730 0.697 0.166 0.166 0.149 0.143 0.537 0.459 0.344 0.346 0.176 0.170 0.156 0.149 0.977 0.951 0.803 0.785 1.000 1.000 1.000 0.999 AUC 0.992 0.980 0.924 0.925 0.582 0.573 0.555 0.549 0.984 0.969 0.909 0.906 0.590 0.578 0.554 0.555 0.831 0.793 0.716 0.718 0.595 0.581 0.551 0.563 0.991 0.982 0.932 0.927 1.000 1.000 1.000 1.000 171 B.2.2 Complex Polarized Signal Model Table B.4: Receiver Operating Charcteristic Results for Polarimetric RFI test-statistics: AUC and PD for PF = 0.05, PF = 0.10. Param. Set Λ1 Λ1 Λ1 Λ1 Λ2 Λ2 Λ2 Λ2 Λ3 Λ3 Λ3 Λ3 Λ4 Λ4 Λ4 Λ4 Test Stat. Γ β κre κim Γ β κre κim Γ β κre κim Γ β κre κim PD (0.05) 0.995 0.976 0.820 0.837 0.911 0.688 0.965 0.053 0.915 0.672 0.053 0.962 0.050 0.055 0.050 0.054 PD (0.10) 0.998 0.990 0.896 0.910 0.962 0.807 0.984 0.108 0.967 0.809 0.103 0.984 0.103 0.102 0.104 0.102 AUC 0.999 0.995 0.963 0.968 0.984 0.937 0.993 0.512 0.984 0.938 0.510 0.992 0.511 0.513 0.510 0.510 Appendix C Environmental Data Records and Passive Frequency Allocations Spaceborne microwave radiometers yield measurements that form the basis for estimating brightness temperatures of various environmental data records (EDR) across the planet. EDRs can be divided into land, sea, and atmospheric parameters. These parameters are summarized in table C.1. Abbreviations start with a letter corresponding to the kind of observation they pertain to. The letter ‘A’ denotes an atmospheric measrurement, ‘S’ denotes a sea or ocean measurement, and ‘L’ denotes a land measurement. Table C.1: Example Environmental Data Records and their acronyms resulting from retrievals in microwave radiometry. Land Soil Moisture Vegetation Biomass Snow Water Equivalent Surface Roughness LSM LVB LSWE LSR Sea Sea Surface Salinity Sea Surface Temperature Sea Surface Wind Speed Sea Ice Concentration Sea Ice Age 172 SSS SST SSW SIC SIA Atmosphere Temperature Proﬁle Moisture Proﬁle Integrated Water Vapor Cloud Liquid Water Cloud Ice Water Precipitation ATP AMP AIWV ACLW ACIW AP Table C.2: Frequency allocations for passive remote sensing according to the National Academic Press [8] and the International Telecommunication Union Recommendation ITU-R RS.1029-2. Allocations in bold highlight protected frequency bands exclusive to passive remote sensing Band L S C X Ku K Ka V W Frequency Allocation (GHz) Range (GHz) 1-2 1.370 - 1.400s 1.400 - 1.427P 2-4 2.640 - 2.655s 2.655 - 2.690s 2.690 - 2.700P 4-8 4.200 - 4.400s 4.950 - 4.990s 6.425 - 7.250s 8 - 12 10.60 - 10.68s 10.68 - 10.70P 15.20 - 15.35s 12 - 18 15.35 - 15.40P 18 - 27 18.60 - 18.80p 21.20 - 21.40p 22.21 - 22.50s 23.60 - 24.00P 27 - 40 31.30 - 31.50P 31.50 - 31.80p 36.00 - 37.00p 40 - 75 50.20 - 50.40P 52.60 - 54.25P 54.25 - 59.30p 75 - 110 86.00 - 92.00P 100.0 - 102.0P 109.5 - 111.8P Bandwidth, B (MHz) EDR LSM, SSS 27.000 NA 10.000 200.000 SST, LSM 200.000 100.000 SSW, LSM 50.000 200.000 100.000 10.000 200.000 200.000 200.000 200.000 100.000 100.000 10.000 10.000 173 NA SSW, SIC, SIA, LSM, ACLW,LSWE SWE, LSR, SIA, AP, ACLW, SIC, SSW ATP, AMP, AP AMP, ACIW, LSR REFERENCES [1] D. Bradley, J. M. Morris, T. Adali, J. T. Johnson, and A. 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