Table Of Contents

9. Seminal Contributions to Basic Theory and Methodology of Cyclostationarity

On this page, the core seminal contributions to the theory of cyclostationarity are listed and concisely described. There are two salient contributors who are the sources of the great majority of these contributions, and the contributions of each occur in two distinct periods of time, with one exception which is a recent contribution. The first period is the mid-1980s to the mid-1990s, which is when William Gardner’s contributions were made, and the second period is the mid-1990s to 2020, which is when Antonio Napolitano’s contributions were made. Contributions from others occurred primarily in the later period.

The most comprehensive treatise on the subject of cyclostationarity written to date is the just-published 2020 book entitled Cyclostationary Processes and Time Series, Theory, Applications, and Generalizations, written by the most prolific contributor to this subject, Professor Antonio Napolitano. Because this book is the most current and authoritative and scholarly and encyclopedic treatment of this subject, it has been used as the sole source for identifying the core seminal contributions made by the several contributors included below in Sections 9.1, 9.2, and 9.3.

The use on this page of the term seminal contribution is based on the standard definition: 

Seminal def: adjective

1. (of a work, event, moment, or figure) strongly influencing later developments. Similar terms: influential, formative, groundbreaking, pioneering, original

It is noted that this definition favors the WCM’s contributions over those of others over the last 40 years because he “got there first” and because refinements and deeper mathematization and translation from FOT-probability versions to stochastic-process versions of already-formulated concepts do not qualify as seminal, though they certainly can be fundamentally important contributions to the theory.

  • 9.1 Seminal Contributions of William A. Gardner

    Because the earliest period of concentrated work would naturally be the period during which the most seminal contributions would likely be made (the earliest bird gets the most worms), this first section starts off with a collection of quotations about the first major contributor’s work that were extracted from letters submitted to the University of California, Davis, during the latter part of that unusually productive period: the mid-1980s to the mid-1990s. The authors, some of whom have passed on, of these letters are or were well-known leaders in their various fields, all of which relate to statistical signal processing. 

    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 M. 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.

    Professor Akiva M. 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 L. 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.

        –     List coming soon     

  • 9.2 Seminal Contributions of Antonio Napolitano

        –     List coming soon     

  • 9.3 Seminal Contributions of Others

        –     List coming soon