Dr. William A. Gardner’s postsecondary education: Certificate of Completion, 1961, Aircraft Radio Repair School, Keesler Air Force Base, Biloxi, MS; course work in Associate of Arts Program—Electronics, 1962-1964, Foothill College, Los Altos, CA; course work in Undergraduate Program—Electrical Engineering, 1964-1966, and Master of Science degree in Electrical Engineering, 1966-1967, Stanford University; graduate course work, 1968-1969, Massachusetts Institute of Technology and Bell Telephone Laboratories; Doctor of Philosophy in Electrical Engineering, 1969-1972, University of Massachusetts, Amherst.
Professor (Emeritus) Gardner has been in the profession of statistical signal processing education, research and development for communications and reconnaissance since 1972 when he joined the faculty of the Department of Electrical and Computer Engineering, School of Engineering, University of California, Davis.
The primary focus of his research and master’s/doctoral thesis advising at UCD was on the development of theory and methodology for cyclostationary signal processing. After rising through the ranks from assistant professor II to professor VII, he transitioned in 2001 to professor emeritus but continued operating his private research firm, Statistical Signal Processing, Inc (SSPI), founded in 1986, until 2011. The focus of research at SSPI also was on the development of signal processing algorithms for cyclostationary signals, primarily for application to radio reconnaissance and signals intelligence, but also including cellular communications and the sale of intellectual property (to Apple Computers Inc. and Lockheed Martin Corporation).
Upon the firm’s 25th anniversary, Dr. Gardner dissolved the corporation but continued his research in statistical signal processing and further developed his collaboration with Professor Antonio Napolitano, University of Naples, Italy—who has made seminal contributions to the theory and methodology for extensions and generalizations of cyclostationarity. Their most recent joint contribution at the time of creation of this website is development of methodology for exploiting irregular statistical cyclicity in scientific data.
Other activities since retirement from university and industry include 1) investigation into electromagnetic modeling of current (ion flow) in cosmic plasma, in support of what some say is the essence of future astrophysics–see Plasma Universe for an overview of the original and ongoing movement and also see Thunderbolts Project, which is an offshoot of the Plasma Universe that is popularizing the revolutionary character of the movement that is intended to create a new paradigm in astrophysics presently called the Electric Universe; and 2) electromagnetic modeling of laboratory-confined/controlled plasma in voluntary support of the SAFIRE Project, funded by International Science Foundation (ISF)1, which aims to better understand the physics of stars and the possibility of energy production with what has been humorously dubbed a “star in a jar”; and voluntary support of the fledgling Institute for Venture Science (IVS). Dr. Gardner began a five-year collaboration in 2013 with the late Dr. James T Ryder (retired Vice President of Lockheed Martin Space Systems Company and head of the Advanced Technology Center in Palo Alto CA—an R&D organization covering a diverse range of technologies including solar and space sciences) during which time Dr. Ryder founded the ISF and served as chairman of the board of the ISF and also the IVS which was founded by Professor Gerald Pollack of the University of Washington.
Professor Gardner has authored three books, Introduction to Random Processes with Applications to Signals and Systems, Macmillan, 1985 (second edition, McGraw-Hill, 1989), Statistical Spectral Analysis: A Nonprobabilistic Theory, Prentice-Hall 1987, and The Random Processes Tutor: A Comprehensive Solutions Manual for Independent Study, McGraw-Hill, 1989, coauthored with Dr. Chih Kang Chen (Re-release, 2014). He is the editor of Cyclostationarity in Communications and Signal Processing, IEEE Press, 1994, and he is the author or co-author of chapters in four books. He is the inventor in thirteen patents, is the author of over 100 peer-reviewed research papers, and he has given many invited lectures at university and industrial research laboratories.
Dr. Gardner received the international Best Paper of the Year award from the European Association for Signal Processing in 1986 for the paper, “The spectral correlation theory of cyclostationary time-series,” the 1987 Distinguished Engineering Alumnus Award from the University of Massachusetts, and the international Stephen O. Rice Prize Paper Award in the Field of Communication Theory from the IEEE Communications Society in 1988 for the paper entitled “Signal interception: A unifying theoretical framework for feature detection”. He was nominated for the 1998 IEEE Graduate Teaching Award and the 1999 IEEE Medal for Pioneering Contributions in Signal Processing. Professor Gardner was elected to Fellow grade in the institute of Electrical and Electronics Engineers in 1991 “For contributions to the development of time-series analysis and stochastic processes with applications to statistical signal processing and communication, and for contributions to engineering education.” He is a biographee in Who’s Who in the World and other biographical sources.
Dr. Gardner received the international Best Paper of the Year award from the European Association for Signal Processing in 1986 for the paper, “The spectral correlation theory of cyclostationary time-series,” the 1987 Distinguished Engineering Alumnus Award from the University of Massachusetts, and the international Stephen O. Rice Prize Paper Award in the Field of Communication Theory from the IEEE Communications Society in 1988 for the paper entitled “Signal interception: A unifying theoretical framework for feature detection”. He was nominated for the 1998 IEEE Graduate Teaching Award and the 1999 IEEE Medal for Pioneering Contributions in Signal Processing. Professor Gardner was elected to Fellow grade in the institute of Electrical and Electronics Engineers in 1991 “For contributions to the development of time-series analysis and stochastic processes with applications to statistical signal processing and communication, and for contributions to engineering education.” He is listed in Who’s Who in the World and his biography is posted in Wikipedia’s Biographies of Living Persons.
Dr. Gardner has been principal investigator for over fifty research awards for theoretical work totaling in excess of $23 million between 1980 and 2010 from industry and from numerous federal research offices, including the Defense Advanced Research Projects Agency, the U. S. Office of Research and Development, the National Security Agency, the National Reconnaissance Office, the National Science Foundation, the Office of Naval Research, the Air Force Office of Scientific Research, the Army Research Office, the U.S. Army Communications Electronics Command Center, the U. S. Space and Naval Warfare Systems Command, the USAF Rome Laboratory, the USAF Wright Laboratory, the Republic of Singapore Center for Strategic Infocomm Technologies, Swiss Army TEMPEST/EMSEC Research Program, and the French National Institute of Applied Sciences. Dr. Gardner is considered to be the father of and (as of the time of initial posting of this website, 2018) the world’s leading authority on the theory and application of cyclostationary signals, and he organized and chaired the first international Workshop on Cyclostationary Signals (in 1992) invited by the National Science Foundation and cosponsored by NSF and the Offices of Research of the U. S. Army, Navy and Air Force. Dr. Gardner has been Principal Investigator on several programs to develop advanced signal processing technology for cellular communications.
After transitioning to Professor Emeritus in 2001, Dr. Gardner increased his employment with Statistical Signal Processing, Inc (SSPI), which he founded 15 years earlier, from part time to full time for 10 more years. SSPI, a theoretical research and algorithms development firm, employed many of Dr. Gardner’s current and previous graduate students, supporting their thesis research and postdoctoral work. From 2011 to 2013, Dr. Gardner was with Lockheed Martin Space Systems Company’s Advanced Technology Center as Research Scientist Senior Principal, assisting with the transition of research contracts from SSPI to LMC.
Dr. Gardner expects his latest research publication, “Statistically Inferred Time Warping: Extending the Cyclostationarity Paradigm from Regular to Irregular Statistical Cyclicity in Scientific Data”, to appear late-2018, to be his last publication in a research journal. Any new research results that he might produce are expected to be communicated exclusively on this website.
Most of Dr. Gardner’s research from 2000 to 2013 has not been published in the open literature because of its proprietary nature and in some cases its sensitive nature from a national security perspective. That work, despite its relevance to cyclostationarity, will not appear on this website in the foreseeable future. His work in astrophysics from 2013 to 2018 has not been published due to its nascent nature. But notes on this work may begin to appear on this website by 2019. In addition, unpublished past work on radio-frequency source location and imaging may begin to appear on this website by 2019.
1 The International Science Foundation funded the SAFIRE project for several years and then closed down at the time of its Founder/COB, Dr. James T. Ryder’s, death in May 2018.
William A Gardner (born Allen William Mclean, November 4, 1942) is a theoretically inclined electrical engineer specializing in advancement of the theory of statistical time-series analysis with emphasis on signal processing algorithm design and performance analysis. He is also an entrepreneur, a professor emeritus with the University of California, Davis, founder of the R&D firm Statistical Signal Processing, Inc. (SSPI), and former president, CEO, and chief scientist of this firm for 25 years (1986 to 2011) prior to sale of its IP to Lockheed Martin.
Gardner has authored four advanced-level engineering books on statistical signal processing theory including Statistical Spectral Analysis: A Nonprobabilistic Theory, 1987, which remains the most referenced book on the statistical theory of cyclostationarity, with about one thousand citations in peer-reviewed journal articles. Gardner’s approach in this book is considered to be in keeping with the work of Norbert Wiener in his classic treatise Generalized Harmonic Analysis first published in 1930.
In the literature, Gardner is referred to as an influential pioneer of cyclostationarity theory and methodology, on the basis of his being a prolific contributor of seminal advances spanning nearly half a century.Gardner has written more than 100 peer-reviewed original-research articles, a number of which received most-cited-paper and best-research-paper awards. His research papers and books have been cited in over ten thousand peer-reviewed journal articles.
Gardner married Nancy Susan Lenhart in June 1966 and the following year completed his M.S. in Electrical Engineering from Stanford University, attended Massachusetts Institute of Technology while employed as a member of technical staff at Bell Telephone Laboratories from 1967 to 1969, and completed his Ph.D. in Electrical Engineering from University of Massachusetts under the supervision of Lewis E. Franks in 1972, at which time he joined the University of California, Davis as an Assistant Professor.
Gardner performed research and teaching there for nearly 30 years, becoming Professor Emeritus in 2001. In 1982, while at University of California, Gardner founded the R&D firm Statistical Signal Processing, Inc. (SSPI), an engineering research services company serving primarily the national security sector but also the cellular RF communications industry. He served as the president, CEO, and chief scientist of SSPI for 25 years. He also founded several entrepreneurial ventures during the latter 15 years of that period, including Gardner Technologies in 2001 for which he served as IP inventor and chief technology officer for five years.
After completing his Ph.D. dissertation entitled “Representation and Estimation of Cyclostationary Processes,” in 1972, Gardner began working on developing a new theory for the class of cyclostationary and polycyclostationary random processes.
In 1985, he wrote his first book, Introduction to Random Processes with Applications to Signals and Systems, which focused on the duality between the stochastic theory based on mathematical expectation and the nonstochastic theory based on time averaging, which theory he was developing. Amir Atiya wrote “The book is an excellent introduction to the theory of random processes… I recommend everyone working in the areas of signal processing and communications to own a copy.” Lawrence Marple wrote “The depth of coverage and the ease of readability can be compared to classic texts such as [A Papoulis’s book which emphasizes theory and J Bendat & A Piersol’s book which emphasizes estimation in practice]. The two chapters on stochastic calculus and the theory of duality and ergodicity are two of the most accessible and easy to understand presentations of these topics in a textbook… The book covers all key aspects of second order statistics of random processes using many examples and without the theoretical trappings of other introductory texts on this subject.
Gardner completed the fundamentals of his nonstochastic theory for stationary processes in 1984 and then reformulated all his research progress to date on cyclostationary stochastic processes within a nonstochastic framework: he developed the novel theory of Fraction-of-Time (FOT) Probability for Poly-Cyclostationary time-series data.
Gardner’s 1987 book Statistical Spectral Analysis: A Non-probabilistic Theory presented his FOT theory of both stationary and poly-cyclostationary processes and/or time-series in Part I and Part II, respectively. After publication of this book, recognition of his work, together with the cornucopia of practical applications it spawned, initiated a long period of growth of this new field of study, including approximately 50 research grants and contracts awarded to Gardner over the following 30 years, garnering nearly $25M in awards of research and development funding from approximately 25 government agencies and industrial research laboratories.
Reviewing the book Statistical Spectral Analysis, Enders A Robinson wrote “In this work Professor Gardner has made a significant contribution to statistical spectral analysis, one that would please the early pioneers of spectral theory and especially Norbert Wiener.” James Massey wrote “I admire the scholarship of this book and its radical departure from the stochastic process bandwagon of the past 40 years.” Akiva Yaglom wrote “It is important . . . that until Gardner’s . . . book was published there was no attempt to present the modern spectral analysis of random processes consistently in language that uses only time-averaging rather than averaging over the statistical ensemble of realizations [of a stochastic process] . . . Professor Gardner’s book is a valuable addition to the literature”.
Gardner’s contributions throughout the literature of the last 50 years are identified by Antonio Napolitano in Cyclostationary Processes and Time Series. In Napolitano’s book, there are the 720 pages containing citations of about 1400 distinct research publications on cyclostationarity (primarily theory), including 582 citations to publications by Gardner. Gardner provided the original definition and mathematical characterization of almost cyclostationary (ACS) stochastic processes, including poly-CS stochastic processes. He further gave the original definition and mathematical characterization of non-stochastic fraction-of-time (FOT) probabilistic models of CS, ACS, and poly-CS time-series. He also originated the extensions and generalizations of the core theorems and relations comprising the second order and higher-order theories of stationary stochastic processes and stationary non-stochastic time-series to CS, poly-CS, and ACS processes and times-series.
In 1987, Professor Gardner was invited by the Editor of IEEE Signal Processing Magazine to write an introduction, for the signal processing community, to the recently discovered 1914 contribution of Albert Einstein to time-series analysis. This introduction reveals the central role played by Professor Gardner’s just-published time-average theory in understanding the relationship between Einstein’s and Norbert Wiener’s (1930) contributions to statistical spectral analysis.
Gardner won the international IEEE Stephen O. Rice Prize Paper award in communication theory in 1988 and the International EURASIP Best Paper of the Year Award in 1987; both papers treated his theory of cyclostationarity. Gardner and his students went on to further prove the uses of his theory of cyclostationarity in applications in communications and signals intelligence. Together with his doctoral student Chi Kang Chen, he wrote the book of mathematical problem solving, The Random Processes Tutor: A Comprehensive Solutions Manual for Independent Study in 1989.
Gardner, with the assistance of his doctoral student Chad Spooner, also generalized his theory from second-order to higher-order cyclostationarity in the early 1990s, and provided novel insight into the statistical quantity called the cumulant. Later, he worked on cyclostationarity exploitation in the areas of enhanced radio reception for wireless communications and, more extensively, advanced RF signals intelligence. He was the editor and contributing author of the 1994 book, Cyclostationarity in Communications and Signal Processing. Douglas Cochran wrote “this book is a timely contribution that should be a valuable reference for academic and industrial R&D engineers in signal processing and communication systems.” This book was an outgrowth of the first international Workshop on Cyclostationary Signals in 1992, which was funded jointly by the National Science Foundation and the Offices of Research of the US Army, Navy, and Air Force. Gardner served, by invitation of the NSF, as organizer and chair. His 2006 review paper, “Cyclostationarity: Half a Century of Research” received the Elsevier Most Cited Paper Award for multiple years.
Applications of Gardner’s theory include his discovery and development of the fundamental operational principles of cyclostationarity—Insensitivity to Noise and Interference, and Selectivity/Separability of spectral correlation measurements and the signals themselves—as well as demonstration of applicability to design and analysis of signal processing methods and algorithms for communications, telemetry, and radar systems. This body of work has demonstrated that substantial improvements in system performance can be obtained in various signal processing applications, such as detection, estimation, and classification of signals, by exploiting cyclostationarity—that is, by recognizing and modeling the properties CS and ACS instead of using the stationary-process models which were the standard before Gardner. Major applications include cellular telephone, spectrum sensing–for cognitive radio–and signals intelligence for national security. Chapters 9 and 10 of the book survey fields of application of the cyclostationarity paradigm and identify on the order of 100 distinct areas of application and cite about 500 published papers addressing these applications. Gardner in 2016 developed the ad hoc concept of time de-warping into the basic theory of converting irregular cyclostationarity into regular cyclostationarity as a means for rendering the extensive and powerful tools of cyclostationary signal processing technology applicable to natural data exhibiting irregular cyclicity, which pervades essentially all fields of science as well as engineering.
Gardner founded Gardner Technologies, Inc. and served as president and chief technical officer until 2006. Through Gardner Technologies, he ventured into more functional wine-packaging with patented wine bottle openers and closures. Upon terminating his brief cellular-telephone-technology venture with partner Stephen Schell, PureWave Technologies in 2001, he sold the IP to Apple.
1 William Gardner – Google Scholar
2 Positions Held
3 Statistical spectral analysis : a nonprobabilistic theory
4 Statistical Spectral Analysis—A Nonprobabilistic Theory
5 Hyperbolic-tangent-function-based cyclic correlation: Definition and theory
6 Hyperbolic-tangent-function-based cyclic correlation: Definition and theory
7 Short Overview of Cyclostationary Signal Processing
8 Cyclostationary Processes and Time Series
9 William A. Gardner – Google Scholar
10 A Brief Autobiographical History of Professor Gardner’s Research Work on Cyclostationarity
12 INTRODUCTION TO RANDOM PROCESSES (McGraw-Hill, 1989)” (PDF)
13 Introduction to Random Signal Processes With Application to Signals & Systems (2nd Edition)
14 Fraction-of-time probability for time-series that exhibit cyclostationarity
15 William Gardner
16 Book reviews “Statistical Spectral Analysis–A Nonprobabilistic Theory”
17 Excerpts from Reviews of Professor Gardner’s Books”(PDF)
18 Cyclostationarity: Half a century of research
19 The Random Processes Tutor A Comprehensive Solutions Manual For Independent Study PDF, ePub eBook
20 The cumulant theory of cyclostationary time-series. I. Foundation
21 Signal interception: a unifying theoretical framework for feature detection
22 Signal interception: performance advantages of cyclic-feature detectors
23 NATIONAL SCIENCE FOUNDATION (GRANT # MIP-91-12800)” (PDF)
24 Statistically inferred time warping: extending the cyclostationarity paradigm from regular to irregular statistical cyclicity in scientific data
25 Industrial Research and Entrepreneurial Experience
26 Suppression of Cochannel Interference in GSM by Pre-demodulation Signal Processing
[This statement was written around the time of my retirement from the University of California at the turn of the century, when it was posted on my University webpage.]
The written word is my preferred mode of communication of ideas to others. Two things I have always placed high value on in my writing are–
I’ve always tried to bridge the gap between mathematicians, too many of whom seem not to understand or even care about the essence of real-world problems to which mathematics can be applied, and engineers, too many of whom seem not to understand very well the mathematical models and methods they attempt to use to solve their real-world problems. This continuing effort has often put me in the middle between two largely non-communicating groups, often seeming to be misunderstood by both; that is, not succeeding at communicating well with either group, each of which speaks a different dialect. Fortunately, I have had enough success over the long run to have gathered the support of enough reputable professionals to obtain some degree of satisfaction that the effort has been worthwhile.
Two examples of this effort are the books Random Processes [Bk1, Bk3] and Statistical Spectral Analysis [Bk2] in which I tried to do something substantial about the fundamentally important duality between the stochastic expectation operation and the time-average expectation operation that, with extremely few exceptions, has simply not been written about or taught since the middle of the last century when stochastic processes began being adopted as what was then considered to be the preferred approach. This duality has, therefore, not been understood by the great majority of those who would benefit, which I estimate to be the great majority of practicing engineers and scientists. Furthermore, a solid understanding of time-average expectation is a prerequisite for learning and effectively using the newer subject of cyclostationary signal processing theory and method, which is based on the novel sine-waves-extraction operation.
Notes added in June 2020:
With 20 more years of hindsight since the above was written, the impact of my effort described above appears to be restricted to the research community, as evidenced by approximately 1000 citations [Google Scholar] in recognized research journals of each of the two aforementioned books three decades since their publication, and almost a 1000 more citations of my third book, Cyclostationarity in Communications and Signal Processing [Bk5].
A more recent example of my effort described above is a forthcoming monograph I am writing on the novel concept of generalized (4-dimensional) adapted-antenna patterns–or, equivalently, 3-dimension patterns of contours of constant gain–which graphically depict antenna gain vs. 3-dim position of radiating-sources after data-directed adaptation of the array to sources in the near and very-near fields of the widely spaced array. This is to be contrasted with the traditional 2-dimensional antenna patterns for direction of sources in the far field. This new tool is useful for studying passive synthetic aperture theory and method, including statistically optimum aperture synthesis and Bayesian probabilistic performance metrics, such as posterior probability density functions for source location and for percent-containment regions. The theory I have developed provides a more mathematically sound framework for developing and understanding the statistical signal processing techniques of radio-frequency astronomy such as VLBI (Very-Long-Baseline Interferometry), and satellite-based radio-frequency surveillance of Earth’s surface and atmosphere, and proposed (see Page 10.1) star ranging systems using IPBI (Interplanetary Baseline Interferometry).
Upon reflection on four decades of dedication to the professional objectives described above, and considering what I judge to be a very limited impact of my effort on university curricula in the area of statistical signal processing, together with the results of my research into the history of science (Page 11), I have come to recognize the universality of the nature of the challenge I had been focusing on only within the field of statistical signal processing. This challenge, described on Page 11, has motivated one of the objectives of my present path forward:
To promote better recognition of the challenge science faces—the human condition—with the hope that our human shortcomings can be more effectively compensated for in the practice of science.
The philosophically minded reader will have noticed that a worthy path forward for all of humanity is described by replacing the word “science” in two places in the above objective with the word “living” (See Human Condition).