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# A Brief Autobiographical History of Professor Gardner’s Research Work on Cyclostationarity

This brief history of an uphill struggle is typical—it’s just one more example of how difficult it often is for engineers and scientists who foster change to survive professionally. But even being typical, I consider the communication of histories like this, which end well, to be important especially for those just entering the profession. To conclude this little preface, I quote here a few excerpts from Richard Greenberg’s review, “Rock Star,” of the book The Rock from Mars: A Detective Story on two Planets, by Kathy Sawyer, from the July-August 2007 issue of American Scientist. Greenberg concludes his review with the remarks “Apparently, contempt is viewed [in our professions] as a perfectly normal and appropriate response to anyone who thinks outside the box…But should creativity and innovation be rewarded with contempt? Scientists may smugly believe that any worthy player will tough it out…The problem is that we do not know what science is losing in the process. What does a young, talented person make of a profession that rewards initiative, care and creativity with contempt?” Greenberg concludes with the advice “Anyone who is contemplating a career in science, who wants to understand what scientists do, who is curious about where scientific knowledge comes from, or who is interested in life, the universe, and everything should read this book.”

In 1972, when I completed my PhD dissertation “Representation and estimation of cyclostationary processes,” I realized that I had only scratched the surface of what I believed was a deep subject of fundamental importance to many fields involving statistical analysis and processing of time-series data associated with cyclic phenomena. I was convinced that there was a substantial void that needed to be filled through the development of a new theory for the class of cyclostationary (including polycyclostationary) processes, and that the availability of such a theory would, through the improvements in modeling fidelity it would enable, spawn many new methods with improved results for analysis, processing, and inference. Although I was alone in this conviction at that time, I forged ahead and embarked on what turned out to be a twenty-five year project of developing the foundation and much of the framework of such a theory, and of applying this theory to a variety of practical problems. Although I engaged in research and supervised thesis projects on diverse topics in statistical signal processing for communications by electrical transmission, I maintained a constant program of research in the particular subject of cyclostationary processes.

About ten years into this program (around 1982), in the face of considerable difficulties encountered in obtaining funding for, and in publishing the results of my work on, the foundations of cyclostationary processes, and also in meeting some of the technical challenges (like resolving the “cycloergodic hypothesis” by creating a counterpart for polycyclostationary processes to the Birkhoff ergodic theorem for stationary processes which resolved the “ergodic hypothesis” back in the 1930’s), I formed the alternative hypothesis that the abstraction inherent in the mathematics of stochastic processes, which was clearly a conceptual stumbling block for many engineers, might not even be necessary for many applications. Although the theory of stationary stochastic processes had become palatable for many analytically-oriented engineers, the deeper level of abstraction required for polycyclostationary processes seemed to be holding me back not only in making progress but also in gaining enough acceptance of the progress I had made to enable me to publish my results and obtain research funding (both requirements for maintaining the privilege of a research career).

To test my hypothesis, I set the goal of proving by construction that the present body of knowledge concerning spectral analysis of time-series data, in the absence or irrelevance of ensembles of statistically identical time series, could be explained at least as well using an empirically-oriented theory based on time averaging, rather than the orthodox abstract theory based on mathematical expectation and probability for ensembles. This goal was met two years later in 1984 with the completion of the first draft of what became Part I of my book *Statistical Spectral Analysis: A non-probabilistic theory,* 1987. A precursor of this alternative theory 1987 appeared in my 1985 book Introduction to Random Processes with Applications to Signals and Systems. A major theme of this earlier book is the duality between the well-known stochastic theory based on expectation and the non-stochastic theory I was developing, which was based on time averaging.

Upon reaching my goal in 1984, I set the new goal of reformulating all my research progress to date on cyclostationary stochastic processes in a non-stochastic framework. Two years later with the completion of the manuscript for Part II of my 1987 book, this goal too had been met. Moreover, as I had originally hypothesized, this re-conceptualization had indeed removed the conceptual stumbling blocks. The rate of my progress in developing the theory of cyclostationarity and its applications in communications and signals intelligence had accelerated considerably. The appearance of my books had a significant impact on acceptance of my work, and both funding and publication finally began to flow, fifteen years after initiation of my research program on cyclostationarity.

In contrast to the difficulties I had encountered during the first 15 years, my work was not only being accepted for publication, but also winning awards, such as the international IEEE Stephen O. Rice Prize Paper award in communication theory (1988), and the international EURASIP Best Paper of the Year Award (1987). My thesis students were also winning awards on their joint research with me, including university department (1990), campus-wide (1995), and Sigma Xi (1994) dissertation awards. Some of my colleagues, who recognized the potential of the theory of cyclostationarity, also began to win awards for their publications.

In 1987, I was invited by* IEEE Signal Processing Magazine* to write an introduction, for the signal processing community, to the recently discovered early contribution of Albert Einstein to time-series analysis. This introduction reveals the central role played by my time-average theory in understanding the relationship between Einstein’s and Norbert Wiener’s contributions to statistical spectral analysis.

In the ensuing ten years (1987-1997), my thesis students and I went on to prove the utility of my theory of cyclostationarity in numerous and diverse applications in communications.

In the early 1990’s, I also generalized the theory from second-order to higher-order cyclostationarity and, in the process, provided completely novel insight into the statistical quantity known as the cumulant. My student’s and my derivation of the cumulant as the solution to a practical engineering problem concerning generation of sine waves by nonlinear transformation of time series appears to be one of the most fundamental advances in the cumulant since its inception nearly a century ago. It seems unlikely that this discovery would have been made without the non-stochastic conceptual framework I developed. Strangely enough, the problem to which we found the cumulant to be the solution is a non-stochastic problem! The concept of probability does not even arise in the problem statement.

The application areas in which my students and I have pioneered the invention of new signal processing algorithms with improved performance based on cyclostationarity exploitation include: blind-channel identification and equalization using second-order statistical quantities, blind adaptive spatial filtering using second-order statistical quantities, signal-selective high-resolution direction finding (and new reduced Cramer-Rao performance bounds), signal selective time-difference estimation (and new reduced Cramer-Rao performance bounds), separation of spectrally overlapping signals with linear processing (cyclic Wiener filtering, also called frequency-shift filtering or FRESH filtering), weak-signal presence-detection, classification of multiple spectrally-overlapping signals, blind despreading of direct-sequence spread-spectrum signals, and nonlinear Volterra-system identification.

My most recent work on cyclostationarity exploitation is in the area of enhanced radio reception for wireless communications and includes patented technology, some of which was purchased by Apple Computer, inc.

A few selected quotations from colleagues, during the period from the late 1980’s to the late 1990’s, regarding the work discussed above follow:

Professor Enders A. Robinson of Columbia University (Member of the National Academy of Engineering) states in a letter of reference on behalf of Dr. Gardner:

*William is one of those few people who can effectively do both the analytic and the practical work required for the introduction and acceptance of a new engineering **method. His general approach is to go back to the basic foundations, and lay a new framework. This gives him a way to circumvent many of the stumbling blocks confronted by other workers . . . I am particularly impressed by the fundamental work in spectral analysis done by Professor Gardner. Whereas most theoretical developments make use of ensemble averages, he has gone back and reformulated the whole problem in terms of time-averages. In so doing he has discovered many avenues of approach which were either not known or neglected in the past. In this way his work more resembles some of the outstanding mathematicians and engineers of the past. This approach took some courage, because generally people tend to assume that the basic work has been done, and **that no new results can come from re-examining avenues that had been tried in the past and then dropped. William’s success in the approach shows the strength of his engineering insight. He has been able to solve problems that others have left as being too difficult.*

Professor Bernard C. Levy, Chairman of the Department of Electrical & Computer Engineering at the University of California, Davis, states in a nomination letter:

*Dr. Gardner’s random processes textbook has several original features which make it **stand out among all other textbooks in the same general area. First, it contains a chapter on cyclostationary processes, which have been one of the main topics of research for Dr. Gardner throughout his research career. These processes play a key role in the study of digital communications systems, and virtually all recent digital communications textbooks refer to Dr. Gardner’s random processes book as well as to his research papers on cyclostationary signal processing. Another original feature of Dr. Gardner’s random processes book is its detailed development of the time-average approach for evaluating the statistics of random signals. This approach provides the theoretical underpinning for the textbook Statistical Spectral Analysis: A Nonprobabilistic Theory which was written by Dr. Gardner for his Spectral Analysis course (ECE 262). Because of its revolutionary **time-average approach (which can be traced back in part to the pioneering work of Norbert Wiener on generalized harmonic analysis), this textbook has been the subject of entertaining exchanges in the Signal Processing Magazine of the IEEE Signal Processing Society. As a consequence of Bill Gardner’s courage and vision in pursuing a radically new path, based on the eminently sensible view that the analysis of random signals should be based on statistics extracted from the observed data, this book has had a huge impact on modern spectrum analysis practitioners.*

Professor Lewis E. Franks, previous NSF program director and previous chairman of the Department of Electrical & Computer Engineering at the University of Massachusetts, Amherst, states:

*I believe I have read a major portion of Gardner’s papers and textbooks. I feel that a unique feature of all these publications, compared to other engineering documents of a similar nature, is the presence of a strong scholarly style. Previous contributions to the topic are meticulously sought out and referenced. It’s not just a matter of being polite to colleagues or avoiding confrontations over omitted citations; but a genuine attempt to establish an important historical context for new results or interpretations. The relevance of prior contributions to the topic is carefully laid out and unified… On the **topic of cyclostationary processes, I feel that he has, almost single-handedly, developed the theoretical and applied engineering aspects of the topic to the point of today’s widespread recognition of its utility.*

Dr. Bart F. Rice, of Lockheed Research, past chairman of the Santa Clara chapter of the IEEE Signal Processing Society, in 1992 letters of reference on behalf of Dr. Gardner states:

*Gardner’s crowning achievement is the development of the theory of spectral correlation and cyclostationary signal processing and analysis. It is hard to overstate the **importance of this work. Like many important theoretical developments, the theory is ‘unifying’ in that it brings into a common, cohesive framework results that previously seemed unrelated, or whose relation was not completely appreciated or understood. The consequences have been new insights and new results. In the not-too-distant future, it will constitute part of the core graduate curriculum in signal processing and in communications . . . And, as with much seminal work of a profound nature, Gardner’s theories have spawned a large amount of activity and new ideas and applications by others.*

Dr. Nelson Blachman, well known communication systems author, writes:

*My interest in Dr. Gardner’s research is concerned with the advances in cyclostationary signal processing that has been his greatest contribution to electrical engineering research. In fact, Professor Gardner is “Mr. Cyclostationary”, the promoter and leading international researcher in this important signal processing area, with two textbooks, numerous papers, and a federal government subsidized workshop to his credit. As a scientist involved with Department of Defense signal processing research aimed at threat analysis of signals related to national security interests, I can indicate to you that Dr. Gardner’s work has had profound impact on the analysis of these signals, but **classification of the analysis has kept the importance of his work from being known to the general public and others in academia. There is another attribute of Dr. Gardner’s research and tutorial material that makes **him stand out among so many of my other academic colleagues, and that is his depth of research (especially historical and mathematical detail) and his attention to precision and detail in his writings. I have always found it difficult to find errors and to take issue with any of Professor Gardner’s papers because he has meticulously done his research; this is in contrast to so many other academics who tend to be more sloppy in their mathematical precision and who do not always thoroughly check the technical literature in depth. I believe this high degree of professional research has contributed greatly to the widespread acceptance of Dr. Gardner’s technical writings as being the preeminent authority on cyclostationary signal processes and their exploitation.*

Dr. Akiva Yaglom, Mathematician and Physicist, USSR Academy of Sciences, wrote in a book review published in *Theory of Probability and Its Applications*:

*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**”*

Professor James Massey, information theorist and cryptographer, Professor of Digital Technology at ETH Zurich, member of the National Academy of Engineering, wrote in a prepublication book review in 1986:

*I admire the scholarship of this book and its radical departure from the stochastic process bandwagon of the past 40 years.*

Around the turn of the century, now 15 years ago, the recognition in a variety of fields of study of the utility of the theory of cyclostationary time-series began growing at an accelerating pace. A comprehensive literature survey was published in 2006 (see Tutorial Publications herein), and a major extension and generalization of the theory that accommodates the effects of rapid motion between radio-frequency transmitters and receivers appeared in 2012 (ISBN: 978-1-119-97335-5, written by a close colleague, Professor Antonio Napolitano). Numerous research publications in the professional journals of various fields of science and engineering have appeared during this recent period.

**My more recent work illustrating Professor Robinson’s observations, quoted above, about going back to basic foundations and building a new framework began around 2007, with a several-year focus on research leading to a new theoretical framework for RF position finding by adaptive aperture synthesis. The results of this work have been waiting to be communicated in book form since 2011, due to distractions from my several years of employment at Lockheed Martin’s Advanced Technology Center, followed by my entry into a substantially different field of study: The Electric Universe (see Research Areas herein). Many scientists are now predicting that a major scientific revolution will occur this century, as the failed theory of the cosmos–developed over the previous century and based primarily on mathematics, a lack of observational data, and the mistaken belief that gravity is the dominant cosmic **

**force—gives**

**way to the developing empirically-based theory in which electricity and magnetism are the dominant cosmic forces, as is being confirmed at a startling rate as data from deep space probes and extremely powerful telescopes is collected and analyzed.**