On this page, the core seminal contributions to the theory of cyclostationarity are identified 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 almost entirely in two distinct periods of time. 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. The smaller number of contributions from others occurred primarily in the later of these two periods.
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 in this 21^{st} Century, Professor Antonio Napolitano [B2]. Because this encyclopedic book is the most current and authoritative and scholarly treatment of this subject, it has been used here as the primary source for identifying the core seminal contributions made by the several key contributors included below in Sections 9.1, 9.2, and 9.3. This book is comprised of 720 pages, comprehensively containing citations of about 1400 distinct research publications on cyclostationarity (a substantial portion of which are theoretical contributions), including 582 citations of Gardner’s publications (some of which are multiple citations of single works, like books).
The use on this page of the term seminal contribution is based on the standard definition:
Def, Seminal: adjective
(Of a work, event, moment, or figure) strongly influencing later developments;
Similar terms: influential, formative, groundbreaking, pioneering, original
The meaning of seminal favors Gardner’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 Gardner’s FOT-probability versions to stochastic-process versions of already-formulated concepts do not qualify as seminal, though they certainly can be important contributions to the theory.
The conceptual and theoretical foundation and framework of William Gardner’s Fraction-of-Time (FOT) Probabilistic Theory of Cyclostationarity for Time Series is succinctly captured by the set of 18 numbered theorems listed below, following Gardner’s 10 basic Cyclostationarity Definitions. All definitions and theorems were originated by Gardner. The 18 theorems are partitioned into three categories, Probabilistic and Statistical Functions of Time and Frequency, Transformations of Probabilistic and Statistical Functions by Signal processing Operations, and Optimum Statistical Inference.
To assist the reader with achieving an overview of the terminology and theorems, the terms only are given first and only concise word descriptions of the theorems are given next. These are followed by 1) citations of and bibliographical references to these theorems and 2) quotations from commentary on these contributions from experts in the field. Only then are the terminology defined and theorem content stated.
Gardner’s Terminology (terms only):
Exhibition of Cyclostationarity:
Cyclic Expectation:
Cyclic FOT Probability:
Cyclic Cumulative FOT Distributions:
Cyclic Temporal Moments:
Cyclic Temporal Cumulants:
Spectral Moments:
Spectral Cumulants:
Cyclic Periodogram:
Cyclic Correlogram:
Synchronized Average:
Gardner’s Theorems (concise descriptions only):
CATEGORY 1: Probabilistic and Statistical Functions of Time and Frequency
Theorem 1: Gardner’s Fundamental Theorem of Sine-Wave Component Extraction (original definition and proof of validity)
Theorem 2: Gardner’s Cyclic FOT Cumulative Probability Distribution Function and associated Cyclic Moments and Cumulants for Time Series Exhibiting Cyclostationarity (original definitions and proof of validity)
Theorem 3: Gardner’s Approximately Cyclic Finite-Time FOT Cumulative Probability Distribution Function for Time Series Exhibiting Approximate Cyclostationarity (original definition and proof of validity), see Page 3.3 here.
Theorem 4: Gardner’s Sine-Wave Extraction Derivation of the Non-Stochastic temporal Cumulant Function (original definition and derivation)
Theorem 5: Gardner Relation between Spectral Correlation and Cyclic Temporal Autocorrelation [generalization of the Wiener Relation between non-stochastic average power spectral density function and temporal autocorrelation function] (original derivation)
Theorem 6: Higher-Order Gardner relation between non-stochastic temporal and spectral higher-order moments (Original derivation)
Theorem 7: Gardner Cyclic-Periodogram/Correlogram Relation and its Higher-Order Counterpart (original definitions and derivations)
Theorem 8: Gardner Relation between non-stochastic Higher-Order Cyclic moments and Cyclic Cumulants (original derivation of generalization, from stationary to cyclostationary signals, of non-stochastic counterpart of Leonov/Shiryaev Relation between stochastic moments and cumulants)
Theorem 9: Gardner’s Synchronized Averaging Relation for FOT-Probabilistic Functions (original derivation)
CATEGORY 2: Transformation of Probabilistic and Statistical Functions by Signal Processing Operations
Theorem 10: Gardner’s Spectral-Correlation Input/Output Relations for Key Signal Processing Operations (9.1, sampling & aliasing; 9.2. multiplication; 9.3, convolution & band limitation
Theorem 11: Gardner’s Higher-Order Spectral-Moment Input/Output Relations for Key Signal Processing Operations (9.1; 9.2; 9.3)
Theorem 12: Gardner’s Derivation of Signal Selectivity of Cyclic Cumulants
CATEGORY 3: Optimum Statistical Inference
Theorem 13: Gardner Theory of Optimum Polyperiodically Time-Variant Filtering of Polycyclostationary Signals (generalization of the theory of non-causal Wiener filtering)
Theorem 14: Gardner Theory of Optimum Detection of Cyclostationary Signals (Maximum-SNR and Maximum-Likelihood spectral-line regenerators)
Theorem 15: Gardner Theory of Cyclostationary Signal Classification
Theorem 16: Gardner Theory of Time-Invariant Linear System Identification
Theorem 17: Gardner Theory of Time-Invariant Volterra Nonlinear System Identification
Theorem 18: Gardner Theory of Periodic and Poly-Periodic Nonlinear Volterra System Identification
Bibliographical References for Theorems:
Early Quotations from Commentary on Gardner’s Seminal Contributions (late1980s to late 1990s)
Professor Enders A. Robinson, Columbia University and Member of the National Academy of Engineering, states in a letter of reference on behalf of Dr. Gardner:
From time to time it is good to look back and see in perspective the work of those people who have made a difference in the engineering profession. One of the important members of this group is William A. Gardner.
Professor Gardner has the ability to impart a fresh approach to many difficult problems. 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. It is this quality that he so well imparts to his students, who have gone forth and solved important and far-reaching problems in their own right.
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. 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.
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.
Recent Quotations Addressing Gardner’s Contributions (2020)
In addition to the early quotations (excerpts from letters) given above, some new quotations—excerpts taken from the preface of the 2020 book [B2]—which include bibliographical notes and references, are provided below, and apply to Theorems 1 – 18.
Preface, p. xxii, “Fundamental noise and interference immunity and signal-separability properties of CS and ACS processes [and time series] discovered in the early 1980s by Gardner (e.g., (Gardner,1985) [Bk3] and (Gardner, 1987f) [Bk2]), and theoretically developed in his seminal work performed primarily during the subsequent decade provided the foundation for the last 30 years of exploitation of almost-cyclostationarity in the design and analysis of signal processing methods and algorithms for communications, telemetry, and radar systems.”
Preface, p. xxiv, “The FOT probability framework for CS and ACS [time series] is introduced in (Gardner, 1985) [Bk3] and treated in depth in (Gardner, 1986c) [JP15], (Gardner, 1987f, Part II ) [Bk2], (Gardner and Brown, 1991) [JP34], and (Gardner, 1991a) [JP36] with reference to continuous-time signals and in (Gardner, 1994) [JP53] for both continuous- and discrete-time signals; the case of complex signals is considered in (Gardner, 1987f, Part II) [Bk2] and (Brown, 1987) [p.8.3].”
Preface, p. xxvi, “The first fundamental treatment on cyclostationary processes in a book can be found in (Gardner, 1985, Chap. 12) [Bk3]. Several books are entirely dedicated to cyclostationarity. The extensive treatment in Part II of (Gardner, 1987f) [Bk2] contains the foundations of cyclic spectral analysis in the FOT approach.”
Definitions of Gardner’s Terms:
The terms defined here are complimented by those defined on the Home Page.
IN PREPARATION
Statements of Gardner’s Theorems:
Each of the theorems stated below have various alternative versions for which the differences lie in the details of assumptions made and the strength of results obtained. This is typical for many if not most theorems in mathematics. Nevertheless, for pragmatic users of the results provided by theorems, the practical essence of each theorem is often unique. For example, the exact nature of the convergence of an infinite series depends on the particular properties of the series. The formulator of a theorem typically has flexibility in the choice of assumptions made about properties. Generally, the more restrictive the assumptions made, the stronger the results that follow. In many applications, what’s important to users is the fact that specific types of series converge in some sense, regardless of the particular sense. Or the fact that other specific types of series diverge in every useful sense.
All the theorems stated below concern statistical functions of signals (time series). In order to not clutter the theorem statements provided here, detailed assumptions are omitted. It is taken for granted that all mathematical quantities referred to in a theorem statement exist. For example, it is assumed that for whatever statistical moments or cumulative distributions or the like that are referred to in a theorem, there exist practical signal models for which these statistical functions do indeed exist. The essence of the theorems lies in the relationships among these functions. These relations are generally expressed in the form of equations. The particular sense in which equality holds in these equations typically depends on detailed assumptions. An equation among time functions may hold for all time, or for almost all time (meaning all time except for a set of times of measure zero), or in temporal mean square, or with fraction-of-time-probability equal to one, etc. More often than not, when empirical quantities are being dealt with, these distinctions become irrelevant. One can sum this up by saying an equation holds for all time of interest. Similar remarks apply to functions of multiple time variables, or frequency variables, or both types of variables, and in some cases, spatial variables.
In summary, existence of statistical functions is not the focus in these theorems; the focus is the relationships among the functions (when they exist)—this is what matters in these theorems. Nevertheless, readers are forewarned that, when the objective is further development of the underlying theory of cyclostationarity, the details of assumptions under which the results of theorems are valid must be stated explicitly. An important case in point is Theorem 2 below. There are technical issues here that will be explained in the material yet to be added to this page.
CATEGORY 1: Probabilistic and Statistical Functions of Time and Frequency
Theorem 1: Gardner’s Fundamental Theorem of Sine-Wave Component Extraction (original definition and proof of validity)
IN PREPARATION
Theorem 2: Gardner’s Cyclic FOT Cumulative Probability Distribution Function and associated Cyclic Moments and Cumulants for Time Series Exhibiting Cyclostationarity (original definitions and proof of validity)
IN PREPARATION
Theorem 3: Gardner’s Approximately Cyclic Finite-Time FOT Cumulative Probability Distribution Function for Time Series Exhibiting Approximate Cyclostationarity (original definition and proof of validity)
IN PREPARATION
Theorem 4: Gardner’s Sine-Wave Extraction Derivation of the Non-Stochastic temporal Cumulant Function (original definition and derivation)
IN PREPARATION
Theorem 5: Gardner Relation between Spectral Correlation and Cyclic Temporal Autocorrelation [generalization of the Wiener Relation between non-stochastic average power spectral density function and temporal autocorrelation function] (original derivation)
IN PREPARATION
Theorem 6: Higher-Order Gardner relation between non-stochastic temporal and spectral higher-order moments (Original derivation)
IN PREPARATION
Theorem 7: Gardner Cyclic-Periodogram/Correlogram Relation and its Higher-Order Counterpart (original definitions and derivations)
IN PREPARATION
Theorem 8: Gardner Relation between non-stochastic Higher-Order Cyclic moments and Cyclic Cumulants (original derivation of generalization, from stationary to cyclostationary signals, of non-stochastic counterpart of Leonov/Shiryaev Relation between stochastic moments and cumulants)
IN PREPARATION
Theorem 9: Gardner’s Synchronized Averaging Relation for FOT-Probabilistic Functions (original derivation)
IN PREPARATION
CATEGORY 2: Transformation of Probabilistic and Statistical Functions by Signal Processing Operations
Theorem 10: Gardner’s Spectral-Correlation Input/Output Relations for Key Signal Processing Operations (10.1, sampling & aliasing; 10.2. multiplication; 10.3, convolution & band limitation
IN PREPARATION
Theorem 11: Gardner’s Higher-Order Spectral-Moment Input/Output Relations for Key Signal Processing Operations (11.1; 11.2; 11.3)
IN PREPARATION
Theorem 12: Gardner’s Derivation of Signal Selectivity of Cyclic Cumulants
IN PREPARATION
CATEGORY 3: Optimum Statistical Inference
Theorem 13: Gardner Theory of Optimum Polyperiodically Time-Variant Filtering of Polycyclostationary Signals (generalization of the theory of non-causal Wiener filtering)
IN PREPARATION
Theorem 14: Gardner Theory of Optimum Detection of Cyclostationary Signals (Maximum-SNR and Maximum-Likelihood spectral-line regenerators)
IN PREPARATION
Theorem 15: Gardner Theory of Cyclostationary Signal Classification
IN PREPARATION
Theorem 16: Gardner Theory of Time-Invariant Linear System Identification
IN PREPARATION
Theorem 17: Gardner Theory of Time-Invariant Volterra Nonlinear System Identification
IN PREPARATION
Theorem 18: Gardner Theory of Periodic and Poly-Periodic Nonlinear Volterra System Identification
IN PREPARATION
The conceptual and theoretical foundation and framework of William Gardner’s Theory of Cyclostationarity for Stochastic Processes is succinctly captured by the set of numbered theorems listed below, following Gardner’s 11 basic Cyclostationarity Definitions. Definitions and theorems designated with the symbol * were originated by Gardner. The theorems are partitioned into three categories, Probabilistic and Statistical Functions of Time and Frequency, Transformations of Probabilistic and Statistical Functions by Signal Processing Operations, and Optimum Statistical Inference.
Gardner’s Terminology:
Exhibition of Cyclostationarity: IN PREPARATION
Cyclic Components of Expectation: IN PREPARATION
Cyclic Components of Probability: IN PREPARATION
Cyclic Cumulative Distributions: IN PREPARATION
Cyclic Temporal Moments: IN PREPARATION
Cyclic Temporal Cumulants: IN PREPARATION
Spectral Moments: IN PREPARATION
Spectral Cumulants: IN PREPARATION
Cyclic Periodogram: IN PREPARATION
Cyclic Correlogram: IN PREPARATION
Synchronized Average: IN PREPARATION
Gardner’s Theorems (concise descriptions only):
CATEGORY 1: Probabilistic and Statistical Functions of Time and Frequency
IN PREPARATION
CATEGORY 2: Transformation of Probabilistic and Statistical Functions by Signal Processing Operations
IN PREPARATION
CATEGORY 3: Optimum Statistical Inference
IN PREPARATION
Bibliographical References for Theorems:
IN PREPARATION
The early quotations addressing Gardner’s seminal contributions (late 1980s to late 1990s) on Page 9.1.1 also apply here on Page 9.1.2, and are not repeated here.
Definitions of Gardner’s Terms:
IN PREPARATION
Statements of Gardner’s Theorems:
IN PREPARATION