3. Learning About Cyclostationarity

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. 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, which can be found here, for the first recommended reading.

Fifteen years later, in 2006, the most comprehensive survey of cyclostationarity at that time and still, as of this writing 12 years later, 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 has been cited in 740 research papers. 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, which can be found here, for the econd 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” is recommended and can be found here. 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 has been cited in only 351 research papers. 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) 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). Read book reviews; Read book.
      • Statistical Spectral Analysis: A Nonprobabilistic Theory, Prentice-Hall, Englewood Cliffs, NJ, 565 pages, 1987. AUTHOR’S COMMENTS. SIAM Review, Vol. 36, No. 3, pp. 528-530, 1994. Read book reviews; Read book.
      • Cyclostationarity in Communications and Signal Processing, IEEE Press, Piscataway, NJ, sponsored by IEEE Communications Society, 1994, 504 pages. Read book reviews; Read book.

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 the page Learning About Generalizations of Cyclostationarity, which can be found here.

An even more recent development of the cyclostationarity paradigm is its extension to signals that exhibit irregular cyclicity, rather than the regular cyclicity, which 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 in EURASIP Journal on Advances in Signal Processing, Vol. XX, pp. xxx-xxx, which can be found here.

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 the page Ensemble Statistics, Probability, Stochastic Processes, and their Temporal Counterparts, which can be found here, 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.