This website has been established for the sole purpose of supporting students and users of the statistical theory and methodology of cyclostationarity, including researchers and practitioners in academia and industry--engineers, scientists, and other researchers working with time-series data representing cyclic phenomena. The primary objective is to assist those trying to learn the fundamentals of the existing body of knowledge on this topic, but a selection of new research results also is expected to be included, although the scope of this secondary part of the website remains to be determined. More specifically, this website is a study guide and overview of cyclostationarity, a subfield of statistical signal processing theory and methodology, which provides recommended study materials including narratives, expository commentary, essays and linked references to monographs and expository treatise, all intended to elucidate, clarify, illuminate, explicate, and critique the subject of cyclostationarity to the best of my (William A. Gardner, website content manager) ability.

At Issue

Considering that tutorials on this topic have been appearing in published form (journals, magazines, text books, reference books, etc.) and on websites, like Wikipedia more recently, for over thirty years now—since publication of the first comprehensive text-book treatment [Statistical Spectral Analysis: A Nonprobabilistic Theory, Part II, Periodic Phenomena, Prentice-Hall, 1987 {link}]—the issue being addressed with this website is not a lack of sources, but rather a perceived need to recapture the original perspective from which this subject was first developed in earnest: a perspective that avoids unnecessary abstraction and introduces concepts in a carefully chosen manner that follows a step-by-step method that avoids conceptual leaps that too often span gaps that are larger than students can comfortably jump across. This original perspective and the conceptual clarity it provided has come to be masked by the unnecessary abstraction of the stochastic process promulgated by authors who’s own training was unfortunately based on this mathematical construct, which was invented by mathematicians for mathematicians to facilitate developing/proving theorems at the often hidden or at least glossed over expense of not being directly related to empirical time-series data. This unfortunate development began in the 1940s and rather quickly led to its wholesale promotion by mathematicians and its resultant adoption in the 1950s and 1960s by engineers and scientists who were not forewarned of the absence of any practical necessity for this particularly abstract mathematization of the theretofore empirical subject of time-series analysis as initially developed by empirically minded scientists and engineers prior to this transition, cf. [Statistical Spectral Analysis: A Nonprobabilistic Theory, Part I, Constant Phenomena ].


This website presents an introduction to the fundamental concepts, history, basic theory, and applications of cyclostationarity and its exploitation for purposes of statistical inference or information extraction from time-series data; and it includes a bibliography that directs users and provides links to carefully selected reference sources for expansions on all the theoretical and methodological topics addressed here, as well as on the practice of exploiting cyclostationarity—a practice that is essentially defined by the signal processing algorithms used. (The terms data and signal are typically used interchangeably when the data is an information-bearing time-series.)


The Content Manager of this website is the author of the great majority of website content and linked material and has published a considerable amount of explanatory material on the topic of cyclostationarity since 1971, including a number of seminal contributions in research journals, graduate-level textbooks, and professional reference books that introduce and develop a comprehensive statistical theory and methodology for understanding and utilizing this special property of time-series data from cyclic phenomena. This explanatory material includes original published research results spanning nearly half a century that establish the conceptual and mathematical foundations of the subject, tutorial treatments, and the philosophical considerations that motivated the author of these publications to discover and teach the duality of two distinct models for conceptualizing and mathematizing the statistical nature of cyclostationarity: stochastic and non-stochastic models—the latter of which is also called fraction-of-time (FOT) probabilistic models and sometimes function or functional models. This published material also includes comprehensive histories, research reviews, and bibliographies on the topic of cyclostationarity.


Because the publication-industry's commercial interests often create economic impedance to potential users’ access to previously published material link, it is hoped that this website will circumvent this impedance by bringing much of the author's seminal work and subsequent complementary work on cyclostationarity by other experts on this subject together in one place for educational purposes, and by providing users with perspective and careful guidance for gaining a command of this body of knowledge or at least those parts of this body that may serve individual users’ more specific purposes.


In order to maintain a close link between physical reality and mathematical models concerning cyclostationarity, the classes of continuous-time, non-stochastic, scalar-valued time-series exhibiting regular cyclostationarity or regular polycyclostationarity are preferred by the Content Manager as vehicles for tutorial purposes, although his work and this website address to varying degrees other classes listed here. However, because digital computers require that time be quantized, the algorithms produced by the methodology of cyclostationarity can be implemented on digital computers only in terms of discrete-time models and processing, as seen in the algorithm-oriented material addressed herein.