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 supervision 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 (see Page 12) 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 universities 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 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. However, an overview of this work is provided here on page 12.1. His work in astrophysics from 2013 to 2018 has not been published due to its nascent nature. In addition, unpublished past work on radio-frequency source location and imaging may begin to appear on this website by 2022. Finally, his recreational work on revolutionary wine openers/closures compatible with natural cork for still wine during the period 2000-2005 is reported on here. Further work ensued on cork-free closures with ecological advantages compared with aluminum screw caps in 2009 -2012, and is reported on here. Ongoing work on safe sparkling wine openers that avoid explosive exit of corks from bottles with contents under pressure was reactivated in 2022 (patents pending).
William’s life Partner — Without the love and moral support of my wife Nancy of 56 years (as of June 2022), throughout my entire professional career, the level and duration of my professional commitment would not have been possible. As Professor Thomas Kailath warned me early in my career, “it’s always hard to go against the established order”. My ideas about the importance of developing the theory of cyclostationarity beginning in the 1970s and developing the Fraction-of-Time probability theory, in contrast to the stochastic process theory, beginning in the 1980s met with great resistance, some of which has not subsided 35 to 45 years later. This was especially difficult before I wrote a substantial book in 1987 making the case for the utility of my forward-looking theoretical work much better than was possible in a few research papers. Without Nancy, I doubt I would have fared as well as I have with such a long-term uphill struggle. Everyone needs at least one believer.
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.
An Unlikely Childhood — My adult life is incongruous with my childhood. One could say my adult life could not have been predicted from my childhood or, looking backwards, one could say that someone knowing my adult life would never guess what my childhood was like. The reasons for this were only slowly revealed to me over time. I was not a “good student” in grammar school or in high school, and I entered the military service soon after graduation from high school in 1960. I was discharged early for “defective work attitudes” (one might say “persistent immaturity”, like receiving traffic violations for off-base car racing on public roads out of town), and I received a General Discharge Under Honorable Conditions after about two years. This qualified me for GI benefits, which I used to offset expenses for attending junior college back at home in California. Given my total lack of effort in high school, I did not qualify to enter any 4-year colleges, let alone a university.
So, what happened to that kid one might ask.
Was his stepfather correct? He predicted “this kid will be in the penitentiary before he is 21.” This was his reaction to a high school prank in which I was caught removing hubcaps from a vehicle.
Well, the first thing that happened after early discharge from the military is that I recognized that I felt shame from having been unilaterally discharged. So, I enrolled in Foothill Junior College in Los Altos Hills with the first plan I ever had to achieve something beyond customizing my car. All I knew about myself at that point was that I had received training in the Air Force as an aircraft radio repairman, and the battery of tests I took during the recruitment process indicated I was well suited to this subject. I had no idea why I was well suited, but I decided to build on that by entering the two-year junior college program in radio and television repair.
I was, to my surprise, doing quite well in all my courses, but I was studying very hard—I was determined to not once again be judged unworthy; however, I did get off to a difficult start in the remedial math course I was required to take. This was not surprising to me, because I never studied math—or anything else for that matter—in high school. I did take a few math courses in high school, namely algebra and geometry. Algebra was just OK. I didn’t study, but I got a passing grade. Geometry seemed intuitive to me, and I did better in that course, although I do not recall studying. My interests consisted of only two things: girls and cars. I was not motivated to study and, although I did not know it at the time, I was unable to read with comprehension because of dyslexia. So, I just faked it all the way through high school. I wrote meaningless book reports using the blurbs on the book jackets and skimming chapter titles and inspecting a few paragraphs here and there. I never read a single book. In all my other courses, I just picked up what I could in the classroom and got by with grades of C and some B’s and an occasional A in courses like applied arts, wood shop, and metal shop.
My mother did not receive education past high school and although she was loving, which was of course extremely important and deeply appreciated, she did not seem to be into teaching or engaging me one way or another in learning, although she did arrange for piano lessons, and I took to that fairly well. I learned to really enjoy feeling and expressing emotion through music, and I quickly learned how to memorize pieces so I could enjoy playing them, but I was unable to read and play at the same time—possibly due to dyslexia or some other mental limitation. Even when I became quite good at playing a repertoire of memorized pieces, especially popular music with a swing base when I was in high school, if I attempted to talk while playing, my fingers would stop. Once in a while I tried playing with a friend who played a different instrument, but I could not do it. I seemed to have a severe limitation that prevented me from thinking about more than one thing at a time. Nevertheless, one of my claims to fame among family and friends was the speed with which I could play memorized boogie woogie music—not quite as skillful as these performers though: https://www.youtube.com/watch?v=J9b3ZZywQvg. Also, I wasn’t as young as this dynamo either: https://www.wimp.com/old-school-talent/.
My father vanished so to speak soon after my mother divorced him when I was a year or two old. My stepfather who arrived when I was about three years old was not formally educated and he took no interest in me and never attempted to teach me anything or even offer guidance of any type. Our relationship primarily consisted of my being afraid of him in reaction to his stern behavior and occasional moderate physical abuse. As far as growing up goes, I was on my own and I had limitations I did not understand, which I will get into below. So, all my schooling up through high school amounted to nothing. This is also true for the couple of years I spent at the Palo Alto Military Academy boarding school during the 7th and 8th grades. Picture shown below.
Apparently, my mother thought I would be better off away from my stepfather. This may have been true and, as far as I could tell, it did no harm. It did provide some structure in my life.
Besides piano, I did have one love and this was a love of nature: observing it, inspecting it, wondering about it, admiring its beauty, enjoying just being in it—running wild along animal paths through the woods, picking wild flowers, and building childhood forts—more like nests in the thickets—collecting interesting rocks and pieces of twisted vines, sea shells from the beach, polliwogs from the creek (which grew into frogs that took over the brick planter boxes around the perimeter of our house). Later, in my early teens, I enjoyed growing and harvesting vegetables in a small home garden. I still react to the smell of a tomato plant like cats do to catnip.
With my newfound motivation in junior college to achieve something that would make me feel worthy of people’s respect, after having flunked out of the Air Force, I was pleasantly surprised to find that I did appear to have some abilities besides playing piano by ear—abilities that were never before given a chance to be exercised.
Yet, I experienced anxiety in my math class the first semester, because it seemed there were things I was supposed to already know in order to understand what I was supposed to be learning but, whatever that was, I didn’t know it. Because of my determination to achieve something I could be proud of, I took advantage of all my first math teacher’s office hours seeking answers to my many questions and, later, I did the same with my first physics teacher. I think it’s fair to say that I made quite a pest of myself. Also, instead of doing just the assigned exercises and problems in the math textbook, I did all the exercises in the book. Moreover, to make them seem as perfect as they could be, I typed them all (on an old manual typewriter—this was the early1960s). Despite my teacher’s attempt to comfort me, reduce my anxiety, early in the semester by assuring me that some people simply can’t to math and by urging me to just accept this and move on, my perseverance and his generosity with his office hours resulted in my ultimately receiving an A+ in the course. This was a real eye opener for me. It cemented my understanding, which I had a glimpse of from my piano playing, of the power of hard work and perseverance. And this had a major impact on the rest of my life. It was the first time I tackled something that at first seemed very difficult and yet I mastered it. The piano earlier on did feel like a success, but I learned a different lesson there. Given how difficult and frustrating it was trying to play while reading music, I simply gave up on that standard approach and I happily accepted the workaround of memorizing the music, in little bits at a time. This taught me to seek creative solutions for getting over hurdles.
My success in the remedial math class gave me confidence that I could succeed if I worked hard enough. And that is exactly what I did. I worked very hard on every course for two years at the junior college, and I ended up with nearly all A’s and A+’s. I rose to the top of my two-semester calculus and analytic geometry class, and absolutely loved it. We had a quiz at the beginning of every class meeting, five days a week, and my objective, which I met, was to race through the quiz every day and be the first to put my pencil down and look up to see that everyone else was still working.
After the first year of the two-year radio and television technician program, having discovered that I really liked physics and math and was getting very good at it, I changed my major to an engineering preparatory program. After completion of my second year, I took a big chance and applied for a transfer to the electrical engineering program at Stanford University and, to my great surprise, I was admitted. In two years, I had been transformed from an aimless do-nothing to a highly motivated and successful college student.
To be sure, Stanford was considerably more competitive than the junior college had been, but I just kept working as hard as I could, so I could continue experiencing the rewarding feeling of succeeding at something that is a real challenge. Little by little, I began to realize there were hints early on, even way back in grammar school, that I had some facility for math and physics, despite my lackluster performance. For example, in the 4th or 5th grade, I discovered what is called the Diagonal Paradox while walking home from school in Redwood City, CA. There was an empty lot on a corner where two perpendicular streets intersect. It seemed clear to me that the diagonal path across the lot was a shorter walk than walking along the sidewalk to the corner, turning the corner, and then walking along the sidewalk of the perpendicular street. Yet, I reasoned that if I took just one step parallel to the first street and then one step parallel to the cross street, and repeated this, I would be doing very nearly the same thing as walking along the diagonal path across the empty lot, but the sum of the lengths of all the steps would surely be the same as if I had walked along the sidewalks only. And, as my steps were made smaller and smaller, I would come closer and closer to walking the diagonal path, yet the total path length would not get any smaller and would therefore remain longer than the diagonal path. This bothered me a lot. Many years later, in my calculus and analytic geometry course, I found I had stumbled across a mathematical paradox all on my own with no background knowledge—just a curiosity. And, through many other instances over the years I came to recognize that I am an unusually curious person who is frequently prone to giving considerable attention to all kinds of thoughts that provoke my curiosity. I found that the questions that captured my interest were the “why” questions, not the “how” questions. I sometimes wonder if settling into engineering in academia was more a result of circumstances than a good match to my natural talents and drive. Nevertheless, I found a way to concentrate on the former rather than the latter questions because I was in academia, not in industry, and I therefore had great freedom in where I put my energies.
Even at Stanford, I was still discovering unusual capabilities I had, while I also was still struggling with impediments. For example, I spent what seemed like a lot of money to take a course called Evelyn Wood Reading Dynamics (also called speed-reading) because I was unable to keep up with the huge amount of reading assignments in Stanford’s three-quarter sequence on the history of western civilization. But I found it was impossible for me to do what I was being instructed to do in this speed-reading course, so I had to drop out and forfeit the tuition. As a drastic measure, I had to resort to purchasing some Schaum’s Outlines on history from an off-campus bookstore. I doubt that I learned much about history this way, but I had become very averse to failure of any kind. I was amazed that I was able to get passing grades on essay tests in the year-long history course, but I remained uncomfortable as I continued to become more aware of how little reading I was able to do with comprehension. Fortunately, in some ways, the more technical the writing, the better my comprehension. And, the more math, the better my comprehension (up to a point). But this says nothing about speed. The speed of my reading has remained painfully slow throughout my life, and this has greatly restricted the amount of reading I have been able to do. In order to read, I must imagine pronouncing every word, and sometimes I do actually pronounce the words in a whisper. Thank goodness for the growth in documentary programs on television and more recently on the internet. Yet, the very slow rate at which I read also results in part from the effort I put forth to comprehend every little bit of insight that can be gained from words on a page. Perhaps this is just one of many reflections of obsessive-compulsive behavior which I exhibit to some degree.
Other experiences at Stanford that revealed to me more about who I was and what I am good at, in contrast to the drawbacks of my dyslexia, being easily distracted, and my “one-track mind”, included chemistry and the electromagnetics portion of the three-quarter sequence in physics. I was selected to participate in the honors lab in chemistry. I didn’t take any further courses in chemistry beyond what was required since my major was electrical engineering and, as a result, I never found out what was seen in my performance that led to this honor, except to recognize that it must be something of an analytical nature.
In my physics course on electromagnetics, I do recall that the subject seemed easy for me because I was able to geometrically envision many of the mathematical properties of electromagnetic fields. I was still playing the game, when I could, at Stanford that I played in my calculus courses at the junior college: racing though exams to see how fast I could complete them. In the final exam for electromagnetics, the allotted time was something like 90 minutes, and I finished it in about 30 minutes. The teaching assistant asked me on my way out of the room if I had given up. I don’t recall if I responded or just shook my head (being bashful), but I did get an A in the course.
I also recall questioning some of the standard theory of electromagnetics that was being taught, because I found that some of the concepts were not supported with foolproof arguments (in the sources we used) like, for example, exactly where the energy in a capacitor is stored. Was it really in the electric field between the parallel conducting plates or was it more directly associated with the electrical charges accumulated on the plates? This led one of the assistant professors to urge me to stay on for PhD studies in electromagnetics, but my major was circuit theory, and I felt the need to get out into industry so I would better understand how I would use what I was studying so hard to learn.
In fact, I was getting so anxious to put my accumulating knowledge to work that, when I noticed in the Stanford catalog of programs, curricula, and courses that there was a program called “University Division” in which an undergraduate with well formulated educational objectives could skip some of the senior elective courses and begin taking graduate courses early in the chosen specialty, leading more quickly to a Master of Science degree instead of a Bachelor of Science degree, I applied and was admitted. I received the Master of Science degree in several quarters less than it would have taken had I gone on and earned a bachelor’s degree first. Upon reflection later, I began to see that there was a pattern in my behavior: I was always rushing as if I could not learn fast enough—could not solve problems fast enough—could not erase so much ignorance fast enough—could not start applying what I had been learning fast enough. Yet, I was saddled with an inability to read at anything beyond what I considered to be a snail’s pace, and this discouraged me from reading as much as I wanted to.
There also was something else about myself that I wanted to understand but could not—until many years later. Following a couple of years working in research and development at Bell Telephone Labs and then three years earning my PhD degree at University of Massachusetts, and then about five to ten years on the faculty at University of California, Davis, while doing research one day in the medical library about personal quirks I had, I discovered I had something called Tourette Syndrome—a nervous system disorder known most for the physical tics it causes. At that time few medical professionals were aware of this syndrome but now, decades later, it is understood that this syndrome affects the mind in various ways not all of which are yet understood definitively. This late revelation about 30 years after its onset (which typically occurs around age seven) together with the late discovery (around age twenty-eight) that I was dyslexic explained a lot of mysteries I had been living with and that had shaped my personal life and my career.
As my experience working at Bell Labs grew as did other experiences from returning to school for a PhD education, and then teaching and doing independent research, my interests continued to evolve more toward the theoretical, and my recognition of my own drive became more clear—I needed to know the “why” of subjects I became engaged in much more than the “how”. This is in fact what the PhD degree originally designated: a level of knowledge in a subject area sufficient to philosophize about the subject. Engineering is, I believe, much more about “how” than it is about “why”. And this makes the degree of PhD in engineering a bit of an oddity, though I do not want to oversimplify what a PhD in engineering should represent. From my perspective, there are often multiple answers to questions of “how” but, one typically does not have a good basis for choosing among them without knowing the answer to the question of “why”.
This drive of mine is evidently responsible for my efforts throughout my career to change the way some things are done in engineering. The most significant example of this is my proposed paradigm shift, during my tenth year as a faculty member, from modeling time series data (signals) as mathematically abstract Stochastic Processes—a worldwide standard for decades—to more concrete Fraction-of-time Probability models (as explained in depth on Page 3 at this website).
My striving to answer the question of “why” led me to a persistent search for deeper truths and this, in turn, resulted in my work leading to unifications of various ideas previously considered by others to be unrelated. Specific examples and associated publications are as follows:
Nevertheless, by about my 10th year on the faculty at the University, there were some subjects I had studied very deeply and understood completely and, when asked a question—even a very broad one—I was able to speak at length in a very organized manner without mental blocks or loss for words. It was (and is today) almost as if there are extensive scripts stored in my brain for these subjects. This might not be true; but the complete confidence that I know a subject unusually well gives me the confidence to speak freely about it, without the hang-ups I otherwise experience. With increasing age and corresponding continued studies, the ease with which I could speak—without specific preparation—about my areas of expertise continued to increase.
Another quirk of my personality is bashfulness. My typical quietness in the presence of people has been interpreted by some to be an indication of aloofness when, in fact, it is nearly the opposite. I simply avoid embarrassing myself by keeping my mouth shut. This developed at an early age as a way of not giving others anything to criticize me about or to even just form negative opinions about me—my worth as a person. I believe this to be a direct consequence of Tourette Syndrome, which I did not know about and could not use as an excuse to give to myself for my sometimes-strange behavior, especially the tics. This tendency made public speaking quite painful; and it created mental blocks when I attempted to speak my mind. I had difficulties finding the right words to use, and I was constantly afraid I would lose my train of thought in the middle of an attempt to communicate my ideas. This required the writing of a script for essentially all my classroom lectures and research presentations in industry and at professional conferences, symposia, and seminars. This considerably multiplied the time it took me to prepare for any public or classroom speaking or presentations. In addition to preparing visual aids (slides or viewgraphs), I needed to prepare a script and come as close to memorizing it as I could.
A consequence of this early timidness there was a reluctance to travel to non-English speaking countries, where my fear of being tongue tied was even more severe. Nevertheless, my wife Nancy has a facility with languages, majored in French and comparative literature, and was happy to act as my ambassador during foreign travels. And we enjoyed these travels so much that I began favoring foreign venues for conferences where I could present my research and hopefully interest others in it. Eventually, many years later, after receiving enough positive feedback about my contributions, I began to loosen up and reduce my dependence on prepared scripts. But still, writing soon became my favored means of communication and my skill at this grew considerably over the years. My words typed out would flow like I wish my spoken words would. I am pleased that writing has only continued to get easier as time passes.
In summary, my better-late-than-never arrival at an understanding of the origins of my afflictions enabled me to identify and label and compartmentalize these afflictions from dyslexia and Tourette syndrome and this in turn enabled me to understand the difficulty and aimlessness of the first twenty years of my life. My continuing experiences deriving from my never-ending commitment to learn and to help others learn what I have learned gave me a better conscious understanding of both the limitations I was working within and, of no lesser importance, the special talents I have: great perseverance, undying curiosity, and strong independence of thought which gave me freedom from suppression by authoritarians, of which I found no shortage, and eventually a finely honed skill for writing. Nevertheless, I have been criticized for using long sentences in my writing, but I prefer the nuances than can be expressed this way better than if I used short sentences. This style was also illustrated in my classroom lectures. One of my student admirers once stated in a course review that I spoke in paragraphs. This was, no doubt, a result of my having prepared scripts.
I believe my hesitancy to blindly trust authority has facilitated my creativity of thought. I came to trust very little of anything I had not thought about deeply and drawn my own conclusions about. I realized that the afflictions I have are a double-edged sword: obsessive-compulsive behavior can produce unusual perseverance and a striving for perfectionism; being forced to read slowly can actually increase comprehension. The propensity to be easily distracted from thinking by noises or stimulation of any of my senses eventually led to an ability to concentrate very deeply for many hours at a time, with no sense of how much time passes. These characteristics have benefitted my intellectual growth and productivity. It has been a long road from an aimless youth to a deeply committed educator, and I have enjoyed every bit of it, including the creation of this website and the opportunity it offered me to talk a bit here and there about autobiographical topics.
This mix of highly technical concepts, theories, and methodologies together with autobiographical tid bits spread throughout this website is, I think, unusual. But it is my hope that it will personalize the material to some extent and reveal its connections to a living, thinking person, exposing strengths and weaknesses and even some emotion—in engineering literature no less! Some would say this is blasphemy!
This biographical sketch, posted on 1 September 2022, of the incongruity of my childhood and adult life is complemented by further discussion of the ongoing impact of my mental impediments on my professional life in the material titled “Notes added in May 2022” in the next section, 10.2. These added remarks underscore the common truth that such impediments usually do not fade away, but the severity of their impact can often be reduced and in some cases, essentially erased, through adaptation. A great deal of brain research over the years has revealed the amazing degree to which the brain can adapt to both inborn abnormalities and major injuries.
[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, rather than to the university teaching community, as evidenced by [Google Scholar] over 1,000 citations in leading 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], as well as nearly 16,000 citations of my journal papers and books.
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 13) star ranging systems using IBI (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 7), 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 7, 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).
Notes added in May 2022
The following paraphrased abstract from an article in Journal of Market Research stimulated some thought about my experience with creativity since my college days in the 1960s, and this led to the following autobiographical observations about my experiences in learning throughout my life.
“Open contests for innovation that display all submitted ideas to contestants are sometimes used to generate ideas. But studies have shown that having access to numerous competitive ideas from others dulls rather than stimulates creative performance. Such competitive ideas interfere with idea generation, perhaps because one’s own ideas need to be differentiated from others’ ideas in order to be considered original. Exposure to an increasing number of prior ideas therefore heightens one’s perceived constraints on expressing ideas and apparently thereby reduces creativity performance. It has also been observed that creativity performance decreases with an increasing number of others’ ideas that one has access to. Restricting access to only a few ideas of others and grouping others’ ideas both appear to reduce the harmful influence on the creation of new ideas.”
Because of the centrality of creativity and problem solving in my career, I’m entering here my reactions to the above findings.
First of all, I mention that the above findings suggest to me two of possibly more mental mechanisms responsible for the findings:
In my case, I believe my lifelong accommodation of dyslexia has played an interesting role in my creativity. The mental strain and resultant exhaustion produced by reading while being hindered by dyslexia has severely limited the volume of written material that I have been able to read. Frankly, I have been surprised that I received grades as good as they were in college courses with assignments based on lots of reading and then writing about what was read. On the other hand, I understood my successes in science, mathematics, and engineering courses to be at least in part a result of how slowly I had to read the textbooks. There has always been an obsession/compulsion to achieve a full understanding of every sentence of text and every equation before moving on to the next.
So, dyslexia has greatly limited the amount of reading I could do throughout my formal education and the career that followed; yet, I came to believe that this relative ignorance/innocence, coupled with an insatiable curiosity, enabled me to be more creative. At the same time, it would seem that not having as much knowledge to build on would interfere with my intellectual growth. . . and maybe it has!
My approach, upon recognizing a challenge or a problem needing a solution has always been to come up with an idea on my own (in a “vacuum” so to speak) and then dig through the literature in a very directed manner and with great perseverance to find out what competing solutions might already exist and/or what might have already been done with “my” idea. This strong direction made such searches more productive than if they had been conducted before I had my own idea for a solution. The ramifications of the search results had more of a personal impact than they would otherwise have had.
Besides always having been very curious, I also have always had a mistrust in others’ knowledge. As a child of about 4 to 5 years of age, I was known to have chronically stretched my parent’s patience with the questions “why” and “what for”. Often one answer after another to the question “why” or “what for” was followed by another “why” or “what for”. And this was not just a game—I really wanted to know.
I did not continue active pursuit of this demanding curiosity in grammar school or high school. The social ramifications of my Tourette’s Syndrome led to social withdrawal and non-participation in the classroom, and my dyslexia prevented me from doing homework or otherwise engaging in the program of formal education. My types of departure from normality were typically not addressed in that era.
Once I reached college, there was a sudden strong desire to succeed at something—for a change. I started in junior college because I did not qualify for admission to a 4-year college due to my lackluster high school grades, and my dyslexia affected my performance on college entrance exams. My newfound motivation together with quirks of Tourette’s syndrome resulted in an incessant need to know the origin or derivation of every bit of knowledge coming my way. In my first engineering jobs, I feared that, when engaged in problem solving, if I accepted offered knowledge without question, I would waste my time diving into the middle of a problem by taking previous results for that problem on faith. I always needed to go back to the beginning and proceed step-by-step until I either understood and accepted previous work or rejected it as erroneous or shortsighted or otherwise inferior to alternatives that had occurred to me along the way.
I have never been reluctant to pose “What if” questions to myself and to invest time in pursuing potential answers or new thoughts before I had any evidence that those thoughts merited investment of time.
I have always been in a rush to catch a first glimpse of a solution I was seeking, and I frequently skipped steps and made not-yet-justified assumptions to see how far I could get and how fast I could get there. If I was encouraged by this initial approach, I then was much more patient about taking the time to verify every step I had made. Sometimes, I found a fatal flaw. Other times, I was able to verify everything needed, or find a way of circumventing detected flaws.
I have always been oriented toward recognizing problems needing solutions, and I have always been drawn toward seeking solutions, be it in my professional field of endeavor or in everyday life. This is what led me to patent inventions in diverse fields. Eventually, I stopped the patenting effort as an independent inventor because I had learned by trial and error that commercialization of an invention is typically a daunting task for an individual, requiring significant investment of time and money and that the efficacy of an invention is not the sole determinant of its commercial success. The power of large corporations is often put to work either squelching an invention’s success or getting around the patent(s). And markets can be quite fickle and often unpredictable. Commercial success with inventions is often not a scientific endeavor. I had one commercial success with a cell phone technology invention, one near success with a new-tech wine packaging bottle opening/closing invention, and I had numerous failures. The net effect over many years was a financial loss and a poor use of my time and energy. This is, I believe, a fairly common outcome for independent inventors, though there are indeed some very prolific inventors. Yet patenting is especially expensive with today’s world markets because international patenting is essential and typically much more expensive in countries outside the USA. Prolific patenting requires huge sums of money. For employees, their employers are typically the sources of such funds, but most employers—those that do not have diverse business areas—do not have an appetite for diverse patenting activity. Not many employers would be interested in patenting inventions in all of dentistry, wine packaging, and cellular telephone, for example (areas in which I obtained patents).
In summary, my career has been shaped by uncommon behaviors caused by biological abnormalities creating both atypical challenges and atypical talents. Creativity has been a common thread and I have immensely enjoyed creating new theories and methodologies in my fields of endeavor and learning where they fit into the historical development of the fields involved. However, as discussed at length on Page 7, except under special circumstances, innovators have always faced an uphill battle to achieve acceptance of their novel contributions and that acceptance often does not come during their lifetimes. Hence the wisdom of the advice: seek enjoyment of life in the process, not in the end results. I have found this a constant challenge because of the intensity of my drive to “find the solution”.
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
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