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 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 21st 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.

  • 9.1 Seminal Contributions of William A. Gardner
    9.1.1 Non-Stochastic Cyclostationarity Theorems

    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 (Definitions and theorem statements are being prepared for posting.) 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.

    Gardner’s Definitions:  

    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:

    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:  

    • Re: Theorems 1 – 18 — Originator of the Fraction-of-Time (FOT) Theory of ACS Time-Series, which is dual to the Theory of ACS Stochastic Processes and therefore comprises the Gardner Isomorphism which is the ACS counterpart of the Wold Isomorphism for stationary stochastic processes, p. 62 in [B2] for CS (for ACS, see [JP34] and pp. 519-520 in [Bk2]).
    • Re: Theorem 1 — Originator of the Gardner Theorem of Temporal Expectation for ACS time-series based on Gardner’s Sine-Wave-Component-Extraction Operator (this is the FOT dual to the standard Fundamental Theorem of Expectation applied to ACS stochastic processes), pp. 43-50, 137-138 in [B2] (see also pp 517-519 in [Bk2]).  
    • Re: Theorem 5 — Originator of the Gardner Relation between the Cyclic Autocorrelation Function and the Spectral Correlation Function (originally dubbed Cyclic Wiener Relation by Gardner because it is an extension and generalization of the Wiener Relation, a term Gardner introduced to distinguish this relation from the Wiener-Khinchin Relation for stochastic processes), pp. 10, 20, 56, 57, 139 in [B2].
    • Re: Theorem 6 — Originator of the Higher-Order Gardner Relation between Temporal and Spectral Higher-than-2nd-Order FOT-moments of time-series, p. 138,139 in [B2].
    • Re: Theorem 7 — Originator of the Gardner Cyclic Periodogram/Correlogram Relation for time-series, p. 57 in [B2], and it’s nth-order counterpart (see p. 3419 in [JP56]). 
    • Re: Theorem 8 — Originator of the FOT Cyclic Moment/Cumulant Relation for times-series (a generalization of the non-stochastic FOT counterpart of the Leonov/Shiryaev Relation between moments and cumulants of stochastic processes, pp. 147-148 in [B2].
    • Re: Theorem 9 — Originator of the Gardner Synchronized Averaging Theorem, for functions containing an additive almost period component, and the decomposition of that component into a sum of periodic components, pp.485-486 in [B2] (see pp. 362-365 and 511-515 in [Bk2] and pp. 332-334 in [Bk3]).
    • Re: Theorems 4, 10, 11 — Original discovery of the Spectral Correlation Characteristics of basic signal processing operations (time-sampling & aliasing, multiplication, convolution, and band limitation), pp. 82-109 in [B2] and [JP15], and Generalization to Higher-Order Moments/Cumulants, pp. 133- 149 in [B2] and Chapt. 2 in [Bk5].
    • Re: Theorem 13 — Invention of Cyclic Wiener Filtering Theory, also called FRESH Filtering, and proof that Fractionally-Spaced-Equalizers are Cyclic Filters with Subsampled Outputs, pp. 330-333 in [B2].
    • Re: Theorem 14 — Invention of the Single-Cycle Detector and proof that it is a Maximum-Signal-to-Noise-Ratio sine-wave generator, and originator of the decomposition of the maximum-likelihood detector for weak ACS signals into a coherent sum of Max-SNR Cycle Detectors, pp.286-290 in [B2]. 
    • Re: Theorem 14 — Original discovery that Nth-order spectral correlation plays a central role in the operation of Nth-Order Nonlinear Synchronizers for ACS signals, pp. 333-335 in [B2]. 
    • Re: Theorem 16 — Original discovery of Blind Phase-Sensitive Channel Identification/Equalization, which is made possible with only 2nd-order statistics by exploiting the cyclostationarity of channel-input signals, p.343 in [B2]. 
    • Re: Theorems 10, 16 — Original discovery of Input/Output-Corruption-Tolerant Linear System Identification, which is made possible by exploiting cyclostationarity of an input-signal component, pp. 335-336 in [B2]; and original extension and generalization of Volterra Nonlinear System Identification methods by exploitation of cyclostationary excitation, p. 344 in [B2]. 

     

    Early Quotations Addressing Gardner’s Seminal Contributions (late1980 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.”

    9.1.2 Stochastic Cyclostationarity Theorems

    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 Definitions:  

    Exhibition of Cyclostationarity: 

    Cyclic Components of Expectation:

    Cyclic Components of Probability: 

    Cyclic Cumulative Distributions:

    Cyclic Temporal Moments:

    Cyclic Temporal Cumulants:

    Spectral Moments:

    Spectral Cumulants:

    Cyclic Periodogram:

    Cyclic Correlogram:

    Synchronized Average:

    Gardner’s Theorems:

    CATEGORY 1: Probabilistic and Statistical Functions of Time and Frequency

    CATEGORY 2: Transformation of Probabilistic and Statistical Functions by Signal Processing Operations 

    CATEGORY 3: Optimum Statistical Inference

    Bibliographical References for Theorems:  

    .
    .
    .

    The early quotations addressing Gardner’s seminal contributions (late 1980 to late 1990s) on Page 9.1.1 also apply here on Page 9.1.2, and are not repeated here.

  • 9.2 Seminal Contributions of Antonio Napolitano

    –     List in preparation using [B2] ; sources being taken from Page 8.2     –

  • 9.3 Seminal Contributions of Others

    –     List in preparation using [B2]; sources being taken from Page 8.3     –