3. Learning About Cyclostationarity

  • 3.1 Basic Cyclostationarity

    IEEE Signal Processing Society defines the field of Signal Processing as follows:

    Signal processing is the enabling technology for the generation, transformation, extraction, and interpretation of information. It comprises the theory, algorithms with associated architectures and implementations, and applications related to processing information contained in many different formats broadly designated as signals. Signal processing uses mathematical, statistical, computational, heuristic, and/or linguistic representations, formalisms, modeling techniques and algorithms for generating, transforming, transmitting, and learning from signals.

    The most widely cited single article introducing the subject of cyclostationarity is entitled “Exploitation of Spectral Redundancy in Cyclostationary Signals” and, as of this writing (2018), was published almost three decades ago (1991) in the IEEE Signal Processing Society’s IEEE Signal Processing Magazine vol. 8 (2), pp. 14-36; according to Google Scholar as of 1 July 2018, this tutorial article has been cited in 1,217 research papers and, as of this update on 8 May 2020, 1321 papers—this represents a current growth rate of more than one new citation every week in its 29th year. On the basis of this evidence that this introductory article on this topic has been perhaps the most popular among researchers, visitors to this website are referred to this article [JP36] for the first recommended reading.

    Fifteen years later, in 2006, the most comprehensive survey of cyclostationarity at that time, entitled “Cyclostationarity: Half a Century of Research” was published in the European Association for Signal Processing Journal Signal Processing vol. 86 (4), pp. 639-697; according to Google Scholar as of 1 July 2018, this survey had been cited in 740 research papers and, as of this update on 8 May 2020, 930 papers—this represents a current growth rate of more than two new citations every week in its 14th year. This survey paper received from the Publisher (Elsevier) the “Most Cited Paper Award” in 2008; and, each year since its first appearance online up through 2011, it was the most cited paper among those published in Signal Processing in the previous five years, and among the top 10 most downloaded papers from Signal Processing. On the basis of this evidence that this comprehensive survey paper on this topic has been perhaps the most popular among researchers, visitors to this website are referred to this paper [JP64] for the second recommended reading. However, for new students of this subject, it is recommended that this survey paper not be read thoroughly at this stage; it should just be perused to widen one’s perspective on the scope of this subject as of 2006.

    For visitors to this website looking for an introduction to the 2nd order (or wide-sense) theory of cyclostationarity at an intermediate level—more technical than the magazine article cited above but less technical and considerably less comprehensive than the survey paper also cited above—the journal paper entitled “The Spectral Correlation Theory of Cyclostationary Time-Series” [JP15] is recommended. This paper was published in 1986 in the Journal Signal Processing, Vol. 11, pp. 13-36. An indication that this paper was well received is the fact that it was awarded the best paper of the year by the European Association for Signal Processing. In contrast to the 1,217 citations of the magazine article recommended above, this journal paper had been cited in only 351 research papers but, as of this update on 8 May 2020, its citations have grown to 399, a rate of more than two new citations every month. It is suggested that this lesser popularity is more a reflection of differences in the readerships of this magazine and this journal than it is a reflection of the utility of the paper.

    The textbooks/reference-books on the subject of cyclostationarity that have been the most frequently cited in research papers as of this writing (2018) and again as of this update on 8 May 2020 are the following three books which, together, comprise over 1600 pages and have been cited in 2,518 research papers over the last three decades:

        • Introduction to Random Processes with Applications to Signals and Systems, 2nd ed., McGraw-Hill Publishing Company, New York, 546 pages, 1990 (1st ed. Macmillan, 1985). [Bk3] Read book reviews.
        • Statistical Spectral Analysis: A Nonprobabilistic Theory, Prentice-Hall, Englewood Cliffs, NJ, 565 pages, 1987. [Bk2] Author’s Comments, SIAM Review, Vol. 36, No. 3, pp. 528-530, 1994. Read book reviews.
        • Cyclostationarity in Communications and Signal Processing, IEEE Press, Piscataway, NJ, sponsored by IEEE Communications Society, 1994, 504 pages. [Bk5]

    By this update (8 May 2020), the citations of these books have grown to 2806, a growth rate of more than 3 new citations every week.

    As of this 8 May 2020 update, there has been another step forward in the publication of comprehensive book treatments of cyclostationarity that can be highly recommended for serious students of the subject: Cyclostationary Processes and Time Series: Theory, Applications, and Generalizations, by Professor Antonio Napolitano, the most prolific contributor to this field for two-to three-decades now. Besides providing the most comprehensive treatment of the subject, including both the FOT probability theory and the stochastic probability theory of almost cyclostationary time series or processes and all established generalizations of these (discussed in the following section of this page), this book also is the most scholarly treatment since the seminal book [Bk2]. Being a historian of time-series analysis, I can say with confidence that no other treatment of the history of contributions to the theory of cyclostationarity can compete with this exemplary book. It will be *the* definitive treatment covering the period from the inception of this subject of study to the end of 2019 (more than half a century) for the foreseeable future.

  • 3.2 Extensions and Generalizations of Cyclostationarity

    Relative to the aforementioned three introductory but thorough treatments of cyclostationarity, there is one textbook/reference-book that is highly complementary and can be strongly recommended for advanced study:Generalizations of Cyclostationary Signal Processing: Spectral Analysis and Applications, John Wiley & Sons, West Sussex, UK, 480 pages, 2012. The generalizations of cyclostationarity introduced in this unique book are summarized on Page 7.An even more recent development of the cyclostationarity paradigm is its extension to signals that exhibit irregular cyclicity, rather than the regular cyclicity we call cyclostationarity. This extension enables application of cyclostationarity theory and method to time-series data originating in many fields of science where there are cyclic influences on data (observations and measurements on natural systems), but for which the cyclicity is irregular as it typically is in nature. This extension originates in the work presented in the article “Statistically Inferred Time Warping: Extending the Cyclostationarity Paradigm from Regular to Irregular Statistical Cyclicity in Scientific Data,” written in 2016 and published in 2018 [JP65].

    However, from an educational perspective, visitors to this website whose objective is to develop a firm command of not only the mathematical principles and models of cyclostationarity but also the conceptual link between the mathematics and empirical data—a critically important link that enables the user to insightfully design or even just correctly use algorithms for signal processing (time-series data analysis), are strongly urged to take a temporary detour away from cyclostationarity per se and toward the fundamental question:

    “What should be the role of the theory of probability and stochastic processes in the conceptualization of cyclostationarity and even stationarity?”

    As discussed in considerable detail on Page 4, one can argue quite convincingly that, from a scientific and engineering perspective, a wrong step was taken back around the middle of the 20th Century in the nascent field of time-series analysis (more frequently referred to as signal processing today) when the temporal counterpart referred to here—introduced by Norbert Wiener in his 1949 book, Extrapolation, Interpolation, and Smoothing of Stationary Time Series, with Engineering Applications—was rejected by mathematicians in favor of Ensemble Statistics, Probability, and Stochastic Processes. This step away from the more concrete conceptualization of statistical signal processing that was emerging and toward a more abstract mathematical model, called a stochastic process, is now so ingrained in what university students are taught today, that few STEM (Science, Technology, Engineering, and Mathematics) professors and practitioners are even aware of the alternative that is, on this website, argued to be superior for the great majority of real-world applications—the only advantage of stochastic processes being their amenability to mathematical proof-making, despite the fact that it is typically impossible to verify that real-world data satisfies the axiomatic assumptions upon which the stochastic process model is based! In essence, the assumptions pave the way for constructing mathematical proofs in the theory of stochastic processes, not—as they should in science—pave the way for validating applicability of theory to real-world applications.

    This is an extremely serious mis-step for this important field of study and it parallels a similar egregious mis-step taken early on in the 20th Century when astrophysics and cosmology became dominated by mathematicians who were bent on developing a theory that was particularly mathematically viable, rather than being most consistent with the Scientific Method. This led to wildly abstract models and associated theory (such as black holes, dark matter, dark energy, and the like) that are dominated by the role of the force of Gravity, whereas Electromagnetism has been scientifically demonstrated to play the true central role in the workings of the Universe. As in the case of the firmly established but mistaken belief that stochastic process models for stationary and cyclostationary time-series are the only viable models, the gravity-centric model of the universe, upon which all mainstream astrophysics is based, is so ingrained in what university students have been taught since early in the 20th century, that few professors and mainstream astrophysics practitioners can bring themselves to recognize the alternative electromagnetism-centric model that is strongly argued to be superior in terms of agreeing with empirical data. Interested readers are referred to the major website www.thunderbolts.info, where the page “Beginner’s Guide” is a good place to start.

    With this hindsight, this website would be remiss to simply present the subject of cyclostationarity within the framework of stochastic processes, which has unfortunately become the norm. This would be the path of least resistance considering the impact of over half a century of using and teaching the stochastic process theory as if it was the only viable theory, not even mentioning the fact that an alternative exists and actually preceded the stochastic process concept before it was buried by mathematicians behaving as if the scientific method was irrelevant.

  • 3.3 A Narrative Description of the Full Content of the Original Source of the Nonstochastic Theory of Statistical Spectral Analysis [Bk2]

    This part of Page 3 consists of the narrative introductions and summaries of each of the 15 Chapters comprising Parts I and II of Statistical Spectral Analysis: A Nonprobabilistic Theory, written by the WCM and published in 1987 [Bk2] . Part I treats stationary time series and Part II treats cyclostationary time series.Before proceeding, it is important to clarify the relationship between the subtitle “A Nonprobabilistic Theory”, the term “Nonstochastic Theory” in this Section 3.2 heading, and the fact that the book presents a theory that comprises a “Nonprobabilistic Foundation”, then develops further theory for giving this foundation “Probabilistic Interpretations”, but in no case relies on the abstraction of stochastic process concepts or theory. Thus the complete theory is nonstochastic and its foundation is nonprobabilistic, but the complete theory includes a probabilistic mathematical interpretation that facilitates quantitative analysis and modeling, and the book establishes that this theory is dual to the more abstract stochastic process theory—that is, the fraction-of-time definition of probability upon which the probabilistic interpretation is built in this book is dual to the more common and considerably more abstract stochastic definition of probability, which is a poor fit for many physical applications involving empirical time-series data.

    In producing the narratives that follow, minor editing of the book’s content has been used—primarily the excerpting of only portions of the book’s introductions and summaries, and the occasional use of boldface font and broken-out text for added emphasis.


    A good deal of our statistical theory, although it is mathematical in nature, originated not in mathematics but in problems of astronomy, geomagnetism and meteorology: examples of fruitful problems in these subjects have included the clustering of stars, also galaxies, on the celestial sphere, tidal analysis, the correlation of fluctuations of the Earth’s magnetic field with other solar-terrestrial effects, and the determination of seasonal variations and climatic trends from weather data. All three of these fields are observational. Great figures of the past, such as C. F. Gauss (1777—1855) (who worked with both astronomical and geomagnetic data, and discovered the method of least square fitting of data, the normal error distribution, and the Fast Fourier Transform algorithm), have worked on observational data analysis and have contributed much to our body of knowledge on time series and randomness.

    Much other theory has come from gambling, gunnery, and agricultural research, fields that are experimental. Measurements of the fall of shot on a firing range will reveal a pattern that can be regarded as a sample from a normal distribution in two dimensions, together with whatever bias is imposed by pointing and aiming, the wind, air temperature, atmospheric pressure and Earth rotation. The deterministic part of any one of these influences may be characterized with further precision by further firing tests. In the experimental sciences, as well as in the observational, great names associated with the foundations of statistics and probability also come to mind.

    Experimental subjects are traditionally distinguished from observational ones by the property that conditions are under the control of the experimenter. The design of experiments leads the experimenter to the idea of an ensemble, or random process, an abstract probabilistic creation illustrated by the bottom less barrel of well-mixed marbles that is introduced in elementary probability courses. A characteristic feature of the contents of such a barrel is that we know in advance how many marbles there are of each color, because it is we who put them in; thus, a sample set that is withdrawn after stirring must be compatible with the known mix.

    The observational situation is quite unlike this. Our knowledge of what is in the barrel, or of what Nature has in store for us, is to be deduced from what has been observed to come out of the barrel, to date. The probability distribution, rather than being a given, is in fact to be intuited from experience. The vital stage of connecting the world of experience to the different world of conventional probability theory may be glossed over when foreknowledge of the barrel and its contents — a probabilistic model — are posited as a point of departure. Many experimental situations are like this observational one.

    The theory of signal processing, as it has developed in electrical and electronics engineering, leans heavily toward the random process, defined in terms of probability distributions applicable to ensembles of sample signal waveforms. But many students who are adept at the useful mathematical techniques of the probabilistic approach and quite at home with joint probability distributions are unable to make even a rough drawing of the underlying sample waveforms. The idea that the sample waveforms are the deterministic quantities being modeled somehow seems to get lost.

    When we examine the pattern of fall of shot from a gun, or the pattern of bullet holes in a target made by firing from a rifle clamped in a vise, the distribution can be characterized by its measurable centroid and second moments or other spread parameters. While such a pattern is necessarily discrete, and never much like a normal distribution, we have been taught to picture the pattern as a sample from an infinite ensemble of such patterns; from this point of view the pattern will of course be compatible with the adopted parent population, as with the marbles. In this probabilistic approach, to simplify mathematical discussion, one begins with a model, or specification of the continuous probability distribution from which each sample is supposed to be drawn. Although this probability distribution is not known, one is comforted by the assurance that it is potentially approachable by expenditure of more ammunition. But in fact, it is not.

    The assumption of randomness is an expression of ignorance. Progress means the identification of systematic effects which, taken as a whole. may initially give the appearance of randomness or unpredictability. Continuing to fire at the target on a rifle range will not refine the probability distribution currently in use but will reveal, to a sufficiently astute planner of experiments, that air temperature, for example, has a determinate effect which was always present but was previously accepted as stochastic. After measurement, to appropriate precision, temperature may be allowed for. Then a new probability model may be constructed to cover the effects that remain unpredictable.

    Many authors have been troubled by the standard information theory approach via the random process or probability distribution because it seems to put the cart before the horse.

    Some sample parameters such as mean amplitudes or powers, mean durations and variances may be known, to precision of measurement, but if we are to go beyond pure mathematical deduction and make advances in the realm of phenomena, theory should start from the data. To do otherwise risks failure to discover that which is not built into the model.

    Estimating the magnitude of an earthquake from seismograms, assessing a stress-test cardiogram, or the pollutant in a storm water drain, are typical exercises where noise, systematic or random, is to be fought against.

    Problems on the forefront of development are often ones where the probability distributions of neither signal nor noise is known; and such distributions may be essentially unknowable because repetition is impossible.

    Thus, any account of measurement, data processing, and interpretation of data that is restricted to probabilistic models leaves something to be desired.

    The techniques used in actual research with real data do not loom large in courses in probability. Professor Gardner’s book demonstrates a consistent approach from data, those things which in fact are given, and shows that analysis need not proceed from assumed probability distributions or random processes. This is a healthy approach and one that can be recommended to any reader.

    Ronald N. Bracewell
    Stanford, California


    This book grew out of an enlightening discovery I made a few years ago, as a result of a long-term attempt to strengthen the tenuous conceptual link between the abstract probabilistic theory of cyclostationary stochastic processes and empirical methods of signal processing that accommodate or exploit periodicity in random data. After a period of unsatisfactory progress toward using the concept of ergodicity1 to strengthen this link, it occurred to me (perhaps wishfully) that the abstraction of the probabilistic framework of the theory might not be necessary. As a first step in pursuing this idea, I set out to clarify for myself the extent to which the probabilistic framework is needed to explain various well-known concepts and methods in the theory of stationary stochastic processes, especially spectral analysis theory. To my surprise, I discovered that all the concepts and methods of empirical spectral analysis can be explained in a more straightforward fashion in terms of a deterministic theory, that is, a theory based on time-averages of a single time-series rather than ensemble-averages of hypothetical random samples from an abstract probabilistic model. To be more specific, I found that the fundamental concepts and methods of empirical spectral analysis can be explained without use of probability calculus or the concept of probability and that probability calculus, which is indeed useful for quantification of the notion of degree of randomness or variability, can be based on time-averages of a single time-series without any use of the concept or theory of a stochastic process defined on an abstract probability space. This seemed to be of such fundamental importance for practicing engineers and scientists and so intuitively satisfying that I felt it must already be in the literature.

    To put my discovery in perspective, I became a student of the history of the subject. I found that the apparent present-day complacence with the abstraction of the probabilistic theory of stochastic processes, introduced by A. N. Kolmogorov in 1941, has been the trend for about 40 years. Nevertheless, I found also that many probabilists throughout this period, including Kolmogorov himself, have felt that the concept of randomness should be defined as directly as possible, and that from this standpoint it seems artificial to conceive of a time-series as a sample of a stochastic process. (The first notable attempt to set up the probability calculus more directly was the theory of Collectives introduced by Von Mises in 1919; the mathematical development of such alternative approaches is traced by P. R. Masani [Masani 1979].) In the engineering literature, I found that in the early 1960s two writers, D. G. Brennan [Brennan 1961] and E. M. Hofstetter [Hofstetter 1964], had made notable efforts to explain that much of the theory of stationary time-series need not be based on the abstract probabilistic theory of stochastic processes and then linked with empirical method only through the abstract concept of ergodicity, but rather that a probabilistic theory based directly on time-averages will suffice; however, they did not pursue the idea that a theory of empirical spectral analysis can be developed without any use of probability. Similarly, the more recent book by D. R. Brillinger on time-series [Brillinger 1975] briefly explains precisely how the probabilistic theory of stationary time-series can be based on time-averages, but it develops the theory of empirical spectral analysis entirely within the probabilistic framework. Likewise, the early engineering book by R. B. Blackman and J. W. Tukey [Blackman and Tukey 1958] on spectral analysis defines an idealized spectrum in terms of time-averages but then carries out all analysis of measurement techniques within the probabilistic framework of stochastic processes. In the face of this 40-year trend, I was perplexed to find that the one most profound and influential work in the entire history of the subject of empirical spectral analysis, Norbert Wiener’s Generalized Harmonic Analysis, written in 1930 [Wiener 1930], was entirely devoid of probability theory; and yet I found only one book written since then for engineers or scientists that provides more than a brief mention of Wiener’s deterministic theory. All other such books that I found emphasize the probabilistic theory of A. N. Kolmogorov usually to the complete exclusion of Wiener’s deterministic theory. This one book was written by a close friend and colleague of Wiener’s, Y. W. Lee, in 1960 [Lee 1960]. Some explanation of this apparent historical anomaly is given by P. R. Masani in his recent commentary on Wiener’s Generalized Harmonic Analysis [Masani 1979]: “The quick appearance of the Birkhoff ergodic theorem and the Kolmogorov theory of stochastic processes after the publication of Wiener’s Generalized Harmonic Analysis created an intellectual climate favoring stochastic analysis rather than generalized harmonic analysis.” But Masani goes on to explain that the current opinion, that Wiener’s 1930 memoir [Wiener 1930] marks the culmination of generalized harmonic analysis and its supersession by the more advanced theories of stochastic processes, is questionable on several counts, and he states that the “integrity and wisdom” in the attitude expressed in the early 1960s by Kolmogorov suggesting a possible return to the ideas of Von Mises “. . . should point the way toward the future. Side by side with the vigorous pursuit of the theory of stochastic processes, must coexist a more direct process-free [deterministic] inquiry of randomness of different classes of functions.”

    As a result of my discovery and my newly gained historical perspective, I felt compelled to write a book that would have the same goals, in principle, as many existing books on spectral analysis—to present a general theory and methodology for empirical spectral analysis—but that would present a more relevant and palatable (for many applications) deterministic theory following Wiener’s original approach rather than the conventional probabilistic theory. As the book developed, I continued to wonder about the apparent fact that no one in the 50 years since Wiener’s memoir had considered such a project worthy enough to pursue. However, as I continued to search the literature, I found that one writer, J. Kampé de Fériet. did make some progress along these lines in a tutorial paper [Kampé de Fériet 1954], and other authors have contributed to development of deterministic theories of related subjects in time-series analysis, such as linear prediction and extrapolation [Wold 1948], [Finch 19691, [Fine 1970]. Furthermore, as the book progressed and I observed the favorable reactions of my students and colleagues, my conviction grew to the point that I am now convinced that it is generally beneficial for students of the subject of empirical spectral analysis to study the deterministic theory before studying the more abstract probabilistic theory.

    When I had completed most of the development for a book on a deterministic theory of empirical spectral analysis of stationary time-series, I was then able to return to the original project of presenting the results of my research work on cyclostationary time-series but within a nonprobabilistic framework. Once I started, it quickly became apparent that I was able to conceptualize intuitions, hunches, conjectures, and so forth far more clearly than before when I was laboring within the probabilistic framework. The original relatively fragmented research results on cyclostationary stochastic processes rapidly grew into a comprehensive theory of random time-series from periodic phenomena that is every bit as satisfying as the theory of random time-series from constant phenomena (stationary time-series). This theory, which brings to light the fundamental role played by spectral correlation in the study of periodic phenomena, is presented in Part II.

    The focus in this book is on fundamental concepts, analytical techniques, and basic empirical methods. In order to maintain a smooth flow of thought in the development and presentation of concepts that steadily build on one another, various derivations and proofs are omitted from the text proper, and are put into the exercises, which include detailed hints and outlines of solution approaches. Depending on the students’ background, the instructor can either assign these as homework exercises, or present them in the lectures. Because the treatment of experimental design and applications is brief and is also relegated to the exercises and concise appendices, some readers might desire supplements on these topics.


    1 Ergodicity is the property of a mathematical model for an infinite set of time-series that guarantees that an ensemble average over the infinite set will equal an infinite time average over one member of the set.


    BLACKMAN, R. B. and J, W. TUKEY. 1958. The Measurement of Power Spectra. New York: American Telephone and Telegraph Co.
    BRENNAN, D. G. 1961. Probability theory in communication system engineering. Chapter 2 in Communication System Theory. Ed. E. J. Baghdady, New York: McGraw-Hill.
    BRILLINGER, D. R. 1975. Time Series. New York: Holt, Rinehart and Winston.
    FINCH, P. D. 1969. Linear least squares prediction in non-stochastic time-series. Advances in Applied Prob. 1:111—22.
    FINE, T. L. 1970. Extrapolation when very little is known about the source. Information and Control. 16:33 1—359.
    HOFSTETTER, E. M. 19M. Random processes. Chapter 3 in The Mathematics of Physics and Chemistry, vol. 11. Ed. H.
    Margenau and G. M. Murphy. Princeton, N.J.: D. Van Nostrand Co.
    KAMPÉ DE FÉRIET, J. 1954. Introduction to the statistical theory of turbulence. I and 11. J. Soc. Indust. Appl. Math. 2, Nos. I and 3:1—9 and 143—74.
    LEE, Y. W. 1960. Statistical Theory of Communication. New York: John Wiley & Sons.
    MASANI, P. R. 1979. “Commentary on the memoir on generalized harmonic analysis.” pp. 333—379 in Norbert Wiener:
    Collected Works,
    Volume II. Cambridge. Mass.: Massachusetts Institute of Technology.
    WIENER, N. 1930. Generalized harmonic analysis. Acta Mathematika. 55:117—258.
    WOLD, H. O. A. 1948. On prediction in stationary time-series. Annals of Math Stat. 19:558—67.

    William A. Gardner



    The subject of Part I is the statistical spectral analysis of empirical time-series. The term empirical indicates that the time-series represents data from a physical phenomenon; the term spectral analysis denotes decomposition of the time-series into sine wave components; and the term statistical indicates that the squared magnitude of each measured or computed sine wave component, or the product of pairs of such components, is averaged to reduce random effects in the data that mask the spectral characteristics of the phenomenon under study.

    The purpose of Part I is to present a comprehensive deterministic theory of statistical spectral analysis and thereby to show that contrary to popular belief, the theoretical foundations of this subject need not be based on probabilistic concepts.

    The motivation for Part I is that for many applications the conceptual gap between practice and the deterministic theory presented herein is narrower and thus easier to bridge than is the conceptual gap between practice and the more abstract probabilistic theory.

    Nevertheless, probabilistic concepts are not ignored. A means for obtaining probabilistic interpretations of the deterministic theory is developed in terms of fraction-of-time distributions, and ensemble averages are occasionally discussed.

    A few words about the terminology used are in order. Although the terms statistical and probabilistic are used by many as if they were synonymous, their meanings are quite distinct. According to the Oxford English Dictionary, statistical means nothing more than “consisting of or founded on collections of numerical facts”. Therefore, an average of a collection of spectra is a statistical spectrum. And this has nothing to do with probability. Thus, there is nothing contradictory in the notion of a deterministic or nonprobabilistic theory of statistical spectral analysis. (An interesting discussion of variations in usage of the term statistical is given in Comparative Statistical Inference by V. Barnett [Barnett 1973]). The term deterministic is used here as it is commonly used, as a synonym for non-probabilistic. Nevertheless, the reader should be forewarned that the elements of the nonprobabilistic theory presented herein are defined by infinite limits of time averages and are therefore no more deterministic in practice than are the elements of the probabilistic theory. (In mathematics, the deterministic and probabilistic theories referred to herein are sometimes called the functional and stochastic theories, respectively.) The term random is often taken as an implication of an underlying probabilistic model. But in this book, the term is used in its broader sense to denote nothing more than the vague notion of erratic unpredictable behavior.


    This introductory chapter sets the stage for the in-depth study of spectral analysis taken up in the following chapters by explaining objectives and motives, answering some basic questions about the nature and uses of spectral analysis, and establishing a historical perspective on the subject.


    A premise of this book is that the way engineers and scientists are commonly taught to think about empirical statistical spectral analysis of time-series data is fundamentally inappropriate for many applications. The subject is not really as abstruse as it appears to be from the conventional point of view. The problem is that the subject has been imbedded in the abstract probabilistic framework of stochastic processes, and this abstraction impedes conceptualization of the fundamental principles of empirical statistical spectral analysis. Hence, the probabilistic theory of statistical spectral analysis should be taught to engineers and scientists only after they have earned the fundamental deterministic principles—both qualitative and quantitative. For example, one should first learn 1) when and why sine wave analysis of time-series is appropriate, 2) how and why temporal and spectral resolution interact, 3) why statistical (averaged) spectra are of interest, and 4) what the various methods for measuring and computing statistical spectra are and how they are related. One should also learn how simultaneously to control the spectral and temporal resolution and the degree of randomness (reliability) of a statistical spectrum. All this can be accomplished in a non-superficial way without reference to the probabilistic theory of stochastic processes.

    The concept of a deterministic theory of statistical spectral analysis is not new. Much deterministic theory was developed prior to and after the infusion, beginning in the 1930s, of probabilistic concepts into the field of time-series analysis. The most fundamental concept underlying present-day theory of statistical spectral analysis is the concept of an ideal spectrum, and the primary objective of statistical spectral analysis is to estimate the ideal spectrum using a finite amount of data. The first theory to introduce the concept of an ideal spectrum is Norbert Wiener’s theory of generalized harmonic analysis [Wiener 1930], and this theory is deterministic*. Later, Joseph Kampé de Fériet presented a deterministic theory of statistical spectral analysis that ties Wiener’s theory more closely to the empirical reality of finite-length time-series [Kampé de Fériet 1954]. But the very great majority of treatments in the ensuing 30 years consider only a probabilistic theory of statistical spectral analysis, although a few authors do briefly mention the dual deterministic theory (e.g., [Koopmans 1974; Brillinger 1976]).

    The primary objective of Part I of this book is to adopt the deterministic viewpoint of Wiener and Kampé de Fériet and show that a comprehensive deterministic theory of statistical spectral analysis, which for many applications relates more directly to empirical reality than does its more popular probabilistic counterpart, can be developed. A secondary objective of Part I is to adopt the empirical viewpoint of Donald G. Brennan [Brennan 1961] and Edward M. Hofstetter [Hofstetter 1964], from which they develop an objective probabilistic theory of stationary random processes based on fraction-of-time distributions, and show that probability theory can be applied to the deterministic theory of statistical spectral analysis without introducing a more abstract mathematical model of empirical reality based on the axiomatic or subjective probabilistic theory of stochastic processes. This can be interpreted as an exploitation of Herman O. A. Wold’s isomorphism between an empirical time-series and a probabilistic model of a stationary stochastic process. This isomorphism is responsible for the duality between probabilistic (ensemble-average) and deterministic (time-average) theories of time-series [Wold 1948] [Gardner 1985].

    There are two motives for Part I of this book. The first is to stimulate a reassessment of the way engineers and scientists are often taught to think about statistical spectral analysis by showing that probability theory need not play a primary role. The second motive is to pave the way for introducing a new theory and methodology for statistical spectral analysis of random data from periodically time-variant phenomena, which is presented in Part II. The fact that this new theory and methodology, which unifies various emerging—as well as long-established—time-series analysis concepts and techniques, is most transparent when built on the foundation of the deterministic theory developed in Part I is additional testimony that probability theory need not play a primary role in statistical spectral analysis.

    The book, although concise, is tutorial and is intended to be comprehensible by graduate students and professionals in engineering, science, mathematics, and statistics. The accomplishments of the book should be appreciated most by those who have studied statistical spectral analysis in terms of the popular probabilistic theory and have struggled to bridge the conceptual gaps between this abstract theory and empirical reality.


    *Other contemporaries of Wiener who contributed to the underlying nonprobabilistic theory of time-series and statistical spectral analysis have been discovered by the diligent scholarship of Professor Antonio Napolitano, et. al. [B2] , who also has moved the theory forward, including deep comparison of the mathematical underpinnings with those of the stochastic theory that supplant edit. It is the WCM’s perspective that this paradigm shift in the 1940s was driven by the esthetics of mathematicians, not the utility that would be found by empiricists. This is not the first time in the history of science that mathematicians have usurped scientists in the development of theories of importance to science for the sake of their sense of beauty which they give precedence to over scientific utility.


    1. What is Spectral Analysis?

    Spectral analysis of functions is used for solving a wide variety of practical problems encountered by engineers and scientists in nearly every field of engineering and science. The functions of primary interest in most fields are temporal or spatial waveforms or discrete data. The most basic purpose of spectral analysis is to represent a function by a sum of weighted sinusoidal functions called spectral components; that is, the purpose is to deompose (analyze) a function into these spectral components. The weighting function in the decomposition is a density of spectral components. This spectral density is also called a spectrum.1 The reason for representing a function by its spectrum is that the spectrum can be an efficient, convenient, and often revealing description of the function.

    As an example of the use of spectral representation of temporal waveforms in the field of signal processing, consider the signal extraction problem of extracting an information-bearing signal from corrupted (noisy) measurements. In many situations, the spectrum of the signal differs substantially from the spectrum of the noise. For example, the noise might have more high-frequency content; hence, the technique of spectral filtering can be used to attenuate the noise while leaving the signal intact. Another example is the data-compression problem of using coding to compress the amount of data used to represent information for the purpose of efficient storage or transmission. In many situations, the information contained in a complex temporal waveform (e.g., a speech segment) can be coded more efficiently in terms of the spectrum.

    There are two types of spectral representations. The more elementary of the two shall be referred to as simply the spectrum, and the other shall be referred to as the statistical spectrum. The term statistical indicates that averaging or smoothing is used to reduce random effects in the data that mask the spectral characteristics of the phenomenon under study. For time-functions, the spectrum is obtained from an invertible transformation from a time-domain description of a function, x(t), to a frequency-domain description, or more generally to a joint time- and frequency-domain description. The (complex) spectrum of a segment of data of length T centered at time t and evaluated at frequency f is


    for which . Because of the invertibility of this transformation, a function can be recovered from its spectrum,


    In contrast to this, a statistical spectrum involves an averaging or smoothing operation that is not invertible. For example, the statistical spectrum


    for which SxT(t,f) is the normalized squared magnitude spectrum


    is obtained from a temporal smoothing operation. Thus, a statistical spectrum is a summary description of a function from which the function x(t) cannot be recovered. Therefore, although the spectrum is useful for both signal extraction and data compression, the statistical spectrum is not directly useful for either. It is, however, quite useful indirectly for analysis, design, and adaptation of schemes for signal extraction and data compression. It is also useful for forecasting or prediction and more directly for other signal-processing tasks such as 1) the modeling and system-identification problems of determining the characteristics of a system from measurements on it, such as its response to excitation, and 2) decision problems, such as the signal-detection problem of detecting the presence of a signal buried in noise. As a matter of fact, the problem of detecting hidden periodicities in random data motivated the earliest work in the development of spectral analysis, as discussed in Section D.

    Statistical spectral analysis has diverse applications in areas such as mechanical vibrations, acoustics, speech, communications, radar, sonar, ultrasonics, optics, astronomy, meteorology, oceanography, geophysics, economics, biomedicine, and many other areas. To be more specific, let us briefly consider a few applications. Spectral analysis is used to characterize various signal sources. For example, the spectral purity of a sine wave source (oscillator) is determined by measuring the amounts of harmonics from distortion due, for example, to nonlinear effects in the oscillator and also by measuring the spectral content close in to the fundamental frequency of the oscillator, which is due to random phase noise. Also, the study of modulation and coding of sine wave carrier signals and pulse-train signals for communications, telemetry, radar, and sonar employs spectral analysis as a fundamental tool, as do surveillance systems that must detect and identify modulated and coded signals in a noisy environment. Spectral analysis of the response of electrical networks and components such as amplifiers to both sine wave and random-noise excitation is used to measure various properties such as nonlinear distortion, rejection of unwanted components, such as power-supply components and common-mode components at the inputs of differential amplifiers, and the characteristics of filters, such as center frequencies, bandwidths, pass-band ripple, and stop-band rejection. Similarly, spectral analysis is used to study the magnitude and phase characteristics of the transfer functions as well as nonlinear distortion of various electrical, mechanical, and other systems, including loudspeakers, communication channels and modems (modulator-demodulators), and magnetic tape recorders in which variations in tape motion introduce signal distortions. In the monitoring and diagnosis of rotating machinery, spectral analysis is used to characterize random vibration patterns that result from wear and damage that cause imbalances. Also, structural analysis of physical systems such as aircraft and other vehicles employs spectral analysis of vibrational response to random excitation to identify natural modes of vibration (resonances). In the study of natural phenomena such as weather and the behavior of wildlife and fisheries populations, the problem of identifying cause-effect relationships is attacked using techniques of spectral analysis. Various physical theories are developed with the assistance of spectral analysis, for example, in studies of atmospheric turbulence and undersea acoustical propagation. In various fields of endeavor involving large, complex systems such as economics, spectral analysis is used in fitting models to time-series for several purposes, such as simulation and forecasting. As might be surmised from this sampling of applications, the techniques of spectral analysis permeate nearly every field of science and engineering.

    Spectral analysis applies to both continuous-time functions, called waveforms,and discrete-time functions, called sampled data. Other terms are commonly used also; for example, the terms data and time-series are each used for both continuous-time and discrete-time functions. Since the great majority of data sources are continuous-time phenomena, continuous-time data are focused on in this book, because an important objective is to maintain a close tie between theory and empirical reality. Furthermore, since optical technology has emerged as a new frontier in signal processing and optical quantities vary continuously in time and space, this focus on continuous time data is well suited to upcoming technological developments. Nevertheless, since some of the most economical implementations of spectrum analyzers and many of the newly emerging parametric methods of spectral analysis operate with discrete time and discrete frequency and since some data are available only in discrete form, discrete-time and discrete-frequency methods also are described.


    1 The term spectrum, which derives from the Latin for image, was originally introduced by Sir Isaac Newton (see [Robinson 1982]).

    2. Why Analyze Waveforms Into Sine Wave Components?2

    The primary reason why sine waves are especially appropriate components with which to analyze waveforms is our preoccupation with linear time-invariant (LTI) transformations of signals, which we often call signal filters. A secondary reason why (statistical or time-averaged) analysis into sine wave components is especially appropriate is our preoccupation with time-invariant phenomena (data sources). To be specific, a transformation of a waveform x(t) into another waveform, say y(t), is an LTI transformation if and only if there exists a weighting function h(t) (here assumed to be absolutely integrable in the generalized sense, which accommodates Dirac deltas) such that y(t) is the convolution (denoted by ) of x(t) with h(t) :



    The time-invariance property of a transformation is, more precisely, a translation-invariance property that guarantees that a translation, by  w, of  to x(t+w) has no effect on y(t) other than a corresponding translation to y(t+w). A phenomenon is said to be time-invariant only if it is persistent in the sense that it is appropriate to conceive of a mathematical model of x(t) for which the following limit time-average exists for each value of and is not identically zero 3



    This function is called the limit autocorrelation function4 for x(t). For , (6) is simply the time-averaged value of the instantaneous power.5

    Sine wave analysis is especially appropriate for studying a convolution because the principal components (eigenfunctions) of the convolution operator are the complex sine wave functions , for all real values of f. This follows from the facts that (1) the convolution operation produces a continuous linear combination of time-translates, that is, y(t) is a weighted sum (over v) of x(t-v), and (2) the complex sine wave is the only bounded function whose form is invariant (except for a scale factor) to time-translation; that is, a bounded function x(t) satisfies


    for all t if and only if


    for some complex X and real f. As a consequence, the form of a bounded function x(t) is invariant to convolution if and only if , in which case (5) yields


    for which


    This fact can be exploited in the study of convolution by decomposing a waveform x(t) into a continuous linear combination of sine waves, 6


    with weighting function



    because then substitution of (11) into (5) yields


    for which


    Thus, any particular sine wave component in y(t) , say



    can be determined solely from the corresponding sine wave component in x(t), since (14) and (15) yield


    The scale factor H(f) is the eigenvalue associated with the eigenfunction of the convolution operator. Transformations (11) and (12) are the Fourier transform and its inverse, abbreviated by

    Statistical (time-averaged) analysis of waveforms into sine wave components is especially appropriate for time-invariant phenomena because an ideal statistical spectrum, in which all random effects have been averaged out, exists if and only if the limit autocorrelation (6) exists. Specifically, it is shown in Chapter 3 that the ideal statistical spectrum obtained from (3) by smoothing over all time,

    exists if and only if the limit autocorrelation exists. Moreover, this ideal statistical spectrum can be characterized in terms of the Fourier transform of , denoted by




    for which Z1/Tfis the unit-area sinc-squared function with width parameter 1/T,


    As the time-interval of spectral analysis is made large, we obtain (in the limit)


    because the limit of Z1/Tf is the Dirac delta


    and convolution of a function with the Dirac delta as in(18) leaves the function unaltered. The ideal statistical spectrum defined by (20) is called the limit spectrum.

    Before leaving this topic of justifying the focus on sine wave components for time-series analysis, it is instructive (especially for the reader with a background in stochastic processes) to consider how the justification must be modified if we are interested in probabilistic (ensemble-averaged) statistical spectra rather than deterministic (time-averaged) statistical spectra. Let us therefore consider an ensemble of random samples of waveforms {x(t,s)}, indexed by s; for convenience in the ensuing heuristic argument, let us assume that the ensemble is a continuous ordered set for which the ensemble index, s, can be any real number. For each member x(t,s)of the ensemble, we can obtain an analysis into principal components (sine wave components). A characteristic property of a set of principal components is that they are mutually uncorrelated 7 in the sense that


    where * denotes complex conjugation. But in the probabilistic theory, it is required that the principal components be uncorrelated over the ensmble8


    as well as uncorrelated over time in order to obtain the desired simplicity in the study of time-series subjected to LTI transformations. If we proceed formally by substitution of the principal component,


    into (23), we obtain 9 (after reversing the order of the limit operation and the two integration operations)



    for which the function fx is the probabilistic autocorrelation defined by (for realx)


    It can be shown that (23) vanishes for all , as desired, if and only if


    for all translations w, in which as Rx depends on only the difference of its two arguments,


    (with some abuse of notation). Consequently principal-component methods of study of an LTI transformation of an ensemble of waveforms are applicable if and only if the correlation of the ensemble is translation invariant. Such an ensemble of random samples of waveforms is commonly said to have arisen from a wide-sense stationary stochastic process.10 But we must ask if ensembles with translation-invariant correlations are of interest in practice. As a matter of fact, they are for precisely the same reason that translation-invariant linear transformations are of practical interest. The reason is a preoccupation with time-invariance. That is, the ensemble of waveforms generated by some phenomenon will exhibit a translation-invariant correlation if and only if the data-generating mechanism of the phenomenon exhibits appropriate time-invariance. Such time-invariance typically results from a stable system being in a steady-state mode of operation—a statistical equilibrium. The ultimate in time-invariance of a data-generating mechanism is characterized by a translation-invariant ensemble, {x(t,s)} which is an ensemble for which the identity


    holds for all and all real ; that is, each translation by, for instance, of each ensemble member, such as x(t,s), yields another ensemble member, for example, x(t,s’). This time-invariance property (29) is more than sufficient for the desired time-invariance property (27). An ensemble that exhibits property (29) shall be said to have arisen from a strict-sense stationary stochastic process. For many applications, a natural way in which a translation-invariant ensemble would arise as a mathematical model is if the ensemble actually generated by the physical phenomenon is artificially supplemented with all translated versions of the members of the actual ensemble. In many situations, the most intuitively pleasing actual ensemble consists of one and only one waveform, x(t), which shall be called the ensemble generator. In this case, the supplemented ensemble is defined by


    The way in which a probabilistic model can, in principle, be derived from this ensemble is explained in Chapter 5, Section B. This most intuitively pleasing translation-invariant ensemble shall be said to have arisen from an ergodic11 stationary stochastic process. Ergodicity is the property that guarantees equality between time-averages, such as (22), and ensemble-averages, such as (23). The ergodic relation (30) is known as Herman O. A. Wold’s isomorphism between an individual time-series and a stationary stochastic process [Wold 1948].

    In summary, statistical sine wave analysis—spectral analysis as we shall call it—is especially appropriate in principle if we are interested in studying linear time-invariant transformations of data and data from time-invariant phenomena. Nevertheless, in practice, statistical spectral analysis can be used to advantage for slowly time-variant linear transformations and for data from slowly time-variant phenomena (as explained in Chapter 8) and in other special cases, such as periodic time-variation (as explained in Part II) and the study of the departure of transformations from linearity (as explained in Chapter 7).


    2 Readers in need of a brief remedial review of the prerequisite topic of linear time-invariant transformations and the Fourier transform should consultAPPENDIX1-1.

    3 In Part II, it is explained that periodic and almost periodic phenomena as well as constant (time-invariant) phenomena satisfy (6). For to be from a constant phenomenon, it must satisfy not only (6) but also for all .

    4 In some treatments of time-series analysis (see [Jenkins and Watts 1968]), the function (6) modified by subtraction of the mean

    from x(t), is called the autocovariance function, and when normalized by it is called the autocorrelation function.

    5 If x(t) is the voltage (in volts) across a one-ohm resistance, then x2(t)is the power dissipated in the resistance (in watts).

    6 If x(t) is absolutely integrable, then (11) and (12) are the usual Fourier transform pair, but if x(t) is a persistent waveform (which does not die out as ) from a time-invariant phenomenon, then (11) and (12) must be replaced with the generalized (integrated) Fourier transform [Wiener 1930], in which case (14) becomes the Stieltjes integral [Gardner 1985].

    7 For a persistent waveform (which does not die out as ) from a time-invariant phenomenon, the property of sine wave components being mutually uncorrelated is deeper than suggested by (22). In particular, the envelopes (from (1)),Xt(t,f) and Xt(t,v), of the local sine wave components (cf. Chapter 4, Section E) become uncorrelated in the limit for all as explained in Chapter 7, Section C.

    8 The limit averaging operation in (23) can be interpreted (via the law of large numbers) as the probabilistic expectation operation.

    9 To make the formal manipulation used to obtain (25) rigorous,X(f,s) must be replaced with the envelope of the local sine wave component, which is obtained from (1) with x(u) replaced by x(u,s); then the limit, , must be taken. An in-depth treatment of this topic of spectral correlation is introduced in Chapter 7, Section C, and is the major focus of Part II.

    10 The term stochastic comes from the Greek to aim (guess) at.

    11 The term ergodic comes from the Greek for work path, which—in the originating field of statistical mechanics—relates to the path, in one dimension, described by x(-,s), of an energetic particle in a gas.


    2. Fundamental vs Superficial Understanding of PSD for Stationary Ergodic Processes12


    Essentially all treatments of the concepts of what are here called the limit spectrum, stationarity, and ergodicity throughout the vast signal processing, engineering, statistics, and mathematics literature, as well as a great deal of the physics literature, simply *define* the limit spectrum—typically called the power spectral density(PSD)—using (17) and (6), or much more frequently the probabilistic counterpart of (6), which is (26)—typically represented by the abstract expectation operation, E{x(t)x(v)}.

    In contrast, the treatment here provides the fundamental definition (20) in terms of quantities with concrete empirical meanings given in the text preceding (20). This treatment also explains that the PSD is an abbreviation for the explicit terminology spectral density of time-averaged instantaneous power or, for the probabilistic counterpart, spectral density of expected instantaneous power). Similarly, typical treatments provide no fundamental conceptual origin of the stationarity and ergodicity properties such as those given here; rather, these properties are usually just posited as assumptions such as (28) and equality between (6) and (28) with .

    12 This section was added here in May 2020 to explicitly elucidate the difference in character between the treatment from [Bk2] given here (fundamental and intuitive) and that (superficial or abstract mathematical) given in the great majority of all other textbooks on this subject.

  • 3.4 The Theory of Cyclic Wiener Filtering

    Content coming soon…

  • 3.5 Algorithms for RF-Emitter Location

    Content coming soon…

  • 3.6 Signal Classification Based on Cyclostationarity

    Content coming soon…