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 treatises, 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.


This website is motivated by observations regarding the Ubiquity of Cyclicity in time-series data arising in science and engineering dating back to my doctoral dissertation from the University of Massachusetts, Amherst, reporting on research initiated in 1969—half a century before the construction of this website.

This Ubiquity of Cyclicity exists throughout what some refer to as God’s Creation: the World comprised of all natural phenomena on our planet Earth, our Solar System, our Galaxy, and the Universe; and it also exists throughout much of the machinery and process comprising mankind’s creation: technology in the form of electrical, mechanical, chemical, etc. machinery and processes. Because of this Ubiquity of Cyclicity, we find that a great deal of the observations, measurements, and other time-series data that we collect, analyze, and process in science and engineering exhibit a form of cyclicity. In the simplest cases, this cyclicity is simply periodicity—the more-or-less-exact repetition of data patterns; but it is far more common for the cyclicity to be statistical in its nature. By this, it is meant that appropriately (this is a critical modifier, to be explained in this website) calculated time averages of the data produce periodic patterns that are often not directly observable in the raw (non-averaged) data. In some cases, the averaging may be performed over the members of a preferably-large set of individual time series of data arising from some phenomenon such as might be obtained by repetition of some experiment, rather than over time (appropriately), but this is most often not the case for empirical data.

In many cases, it is found that the statistical cyclicity is regular (the statistics obtained by averaging (appropriately) long enough are essentially exactly periodic) and, in this case, the time-series is said to be cyclostationary. But in many more cases the cyclicity is irregular. Roughly speaking, this means the period of the cyclicity of the statistics, such as short-term empirical means and variances, and correlations, etc., of the time-series data changes over the long run in an irregular manner, which makes it quite difficult to perform averaging over the long term in the appropriate manner. The level of complication caused by irregular cyclicity was only recently (2015) reduced by the origination of theory and method for converting irregular cyclicity in time-series data to regular cyclicity [Statistically Inferred Time Warping]. This recent breakthrough opens the door, for many fields of science and engineering, to much broader application of the now-firmly-established theory and method for exploiting cyclostationarity.

That being said, what exactly is meant by “exploiting cyclostationarity”? As explained in considerable detail in this website, this means using knowledge of the cyclic statistical character of otherwise erratic or randomly fluctuating time-series data to achieve higher performance in various tasks of statistical inference than could otherwise be obtained; that is, making more precise and/or more reliable inferences about the physical source of time-series data on the basis of processing that data in various ways generally referred to as “signal processing”. Such inferences may consist of detection of the presence of signals in noise, estimation of parameters of such signals, filtering such signals out of noise, identifying signal types, locating the source of propagating signals, etc.

As an indication of how widespread exploitation of cyclostationarity in time-series data has become since its inception 50 years ago, a web search using https://scholar.google.com/ was performed in April 2018, [Statistically Inferred Time Warping] This search was based on just under 50 nearly-distinct applications areas in science and engineering, and the search terms were chosen to yield only results involving exploitation of cyclicity in time-series data. By “nearly distinct”, it is meant that the search terms were also selected to minimize redundancy (multiple search application areas producing the same “hits”). As shown in Table 1, the search found about 136,000 published research papers.

As another measure of the impact the cyclostationarity paradigm has had, Professor Antonio Napolitano, in Chapters 9 and 10 of his 2019 book, Cyclostationary Processes and Time Series: Theory, Applications, and Generalizations, surveys fields of application of the cyclostationarity paradigm, and identifies on the order of 100 distinct applications and cites about 500 specific published papers addressing these applications; his carefully selected bibliography on primarily cyclostationarity includes over 1500 published papers and books.

Table 1 Nearly Distinct Application Areasa

Serial NumberHeadingNumber
1"aeronautics OR astronautics OR navigation" AND "CS/CS"3,190
2"astronomy OR astrophysics" AND "CS/CS"864
3"atmosphere OR weather OR meteorology OR cyclone OR hurricane OR tornado" AND "CS/CS"2,230
4"cognitive radio" AND "CS/CS"8,540
5"comets OR asteroids" AND "CS/CS"155
6"cyclic MUSIC"512
7"direction finding" AND "CS/CS"1,170
8"electroencephalography OR cardiography" AND "CS/CS"742
9"global warming" AND "CS/CS"369
10"oceanography OR ocean OR maritime OR sea" AND "CS/CS"3,060
11"physiology" AND "CS/CS"673
12"planets OR moons" AND "CS/CS"274
13"pulsars" AND "CS/CS"115
14"radar OR sonar OR lidar" AND "CS/CS"5,440
15"rheology OR hydrology" AND "CS/CS"639
16"seismology OR earthquakes OR geophysics OR geology" AND "CS/CS"1.090
17"SETI OR extraterrestrial" AND "CS/CS"83
18autoregression AND "CS/CS"2,040
19bearings AND "CS/CS"3,980
20biology AND "CS/CS"2,030
21biometrics AND "CS/CS"309
22chemistry AND "CS/CS"2,020
23classification AND "CS/CS"10,900
24climatology AND "CS/CS"811
25communications AND "CS/CS"21,200
26cosmology AND "CS/CS"172
27ecology AND "CS/CS"356
28economics AND "CS/CS"2,050
29galaxies OR stars AND "CS/CS"313
30gears AND "CS/CS"2,000
31geolocation AND "CS/CS"676
32interception AND "CS/CS"2,270
33mechanical AND "CS/CS"4,770
34medical imaging OR scanning AND "CS/CS" 1,370
35medicine AND "CS/CS"2,990
36modulation AND "CS/CS"17,000
37physics AND "CS/CS"4,539
38plasma AND "CS/CS"542
39quasars AND "CS/CS"47
40Sun AND "CS/CS"4,320
41UAVs AND "CS/CS"238
42universe AND "CS/CS"209
43vibration OR rotating machines AND "CS/CS"3,240
44walking AND "CS/CS"990
45wireless AND "CS/CS"15,100

a “CS/CS” is an abbreviation for “cyclostationary OR cyclostationarity”

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.