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

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CURRICULUM 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
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