Invited Review
Cyclostationarity by examples

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Abstract

This paper is a tutorial on cyclostationarity oriented towards mechanical applications. The approach is voluntarily intuitive and accessible to neophytes. It thrives on 20 examples devoted to illustrating key concepts on actual mechanical signals and demonstrating how cyclostationarity can be taken advantage of in machine diagnostics, identification of mechanical systems and separation of mechanical sources.

Introduction

Cyclostationarity pertains to the new wave of signal processing techniques that is currently revolutionising the field of mechanical signature analysis. Briefly stated, a cyclostationary signal is one that exhibits some hidden periodicity of its energy flow. Such a definition includes a broad class of mechanical signals such as vibrations or acoustic measurements. This is so because many mechanical systems—e.g., reciprocating mechanisms, gears, fans, electrical motors—sustain periodic motion of their components which in turn periodically modulate the vibration or noise they radiate. The cyclostationary property covers a rich statistical typology of signals (including periodic signals and stationary random signals as particular cases) which is particularly appealing since the same “cyclostationary” formalism will apply generally and independently of the object of study.

However, the formalism of cyclostationary signals may seem difficult to grasp at first because it is quite different from the stationary logic which most of us have been trained to work with. It is the aim of this paper to introduce cyclostationarity from an intuitive approach that proceeds essentially from generalising our common experience gained from stationary signals. In particular, because it is intuitive and physically meaningful, the approach relies as much as possible on the key concept of the energy conveyed by a signal: this will make easier the introduction of the various types of densities that characterise cyclostationary signals in the time, the frequency and the cyclic domains. Moreover, it was decided not to make any use of the mathematical expectation operator nor of the Dirac distribution, which although useful for simplifying the mathematics, both require an additional abstraction effort that hinders intuition. Finally, a special effort has been made to illustrate every concept on real-world signals, mainly coming from acoustic and vibration measurements: this will provide the reader with another route to comprehend the subject. These precautions should make the paper accessible to most readers, and in particular to engineers or young researchers who are looking for a first introduction to cyclostationarity.

Why should one invest so much effort in going beyond the traditional and well-established “stationary” approach, and consider an alternative such as cyclostationarity? It all relies on the fact that stationarity is more a matter of convenience than of actuality.

By definition, stationary signals are representative of physical phenomena that maintain a constant statistical behaviour in time. Yet, this property is hardly met by mechanical systems consisting of some machinery or rotating parts which, by nature, undergo a nonstationary operation. Even under constant operating conditions (speed, torque, temperature), a succession of phenomena usually takes place within the machine cycle so as to release energy on a rhythmic basis: meshing of teeth in gears, combustion of gas in internal combustion (IC) engines, inversion of forces in reciprocating or cam mechanisms, admission and exhaust of fluids in pumps, turbulence around fan blades, and so on. Such phenomena typically produce transient signatures in mechanical signals, which in turn are likely to carry critical information on the operating condition of the machine. Let us insist on the assertion that nonstationarity—as evidenced by the presence of transients—is intimately related to the concept of information. This is completely analogous to speech or music signals that can carry a message or a melody onlt because they consist of a succession of nonstationarities. Similarly, the identification of noise sources in an IC engine requires tracking their chronology and localising their temporal occurrences in the engine cycle.

However, when brief in time, transients are extremely difficult to track inside the machine cycle. For many decades the traditional approach has been to regard mechanical signals as if they were stationary—i.e., exempt from local information—because signal processing was not developed enough to proceed otherwise. Although time–frequency techniques have now become quite popular in industry, they are mainly “analysis” tools—as opposed to “processing” tools—and in any case they are unable to propose a versatile methodology that applies to all mechanical signals in all circumstances. This is because nonstationarity is intrinsically a non-property, defined by opposition to stationarity. For all these reasons, it is often reassuring to rely on the stationary assumption and to benefit from the many “on-the-shelf” tools offered by that framework. Because it is a choice by default, it is also a poor choice that deprives the user of the information conveyed by nonstationarities. This is what is referred here to as the “antinomy of nonstationarity”.

Cyclostationarity comes to the fore at this juncture. Because it defines a certain type of nonstationarity, it is a property endowed with a rigorous framework of signal processing tools liable to apply to a broad class of mechanical signals.

The construction of the theory of cyclostationarity is closely linked to that of modern signal processing. Nowadays the subject has probably reached its state of maturity, as evidenced by the two excellent bibliographies in Refs. [24], [25], and the very complete states of the art in Refs. [17], [23].

Pioneering works in cyclostationarity date back to the early sixties, but it is truly since the eighties that cyclostationarity has become a subject of active research. Most of the precursory works pertain to the field of communications, where it was recognised that the process of modulating a signal for Hertzian transmission naturally led to a cyclostationary behaviour. A great tribute is due to William A. Gardner who first established many of the theoretical foundations, laid down the currently used terminology, and also foresaw many applications. As a matter of fact, W. A. Gardner was probably the first to recognise that the cyclostationary framework is appropriate for any physical phenomenon that gives rise to data with periodic statistical characteristics: “in mechanical-vibration monitoring and diagnosis for machinery, periodicity arises from rotation, revolution, and reciprocating of gears, belts, chains, shafts, propellers, bearings, pistons, and so on” [10].

Despite this encouraging perspective, actual contributions of cyclostationarity to mechanics have remained very limited. The bibliographical compilation [24] listed 52 references related to mechanics out of a total of 1559 until 2005, and [25] listed 29 out of 786 until the same year. In both cases, this is no more that three per thousand. Most of these references—plus some others—are listed in our reference section. They will be referred to throughout the text and in particular in the last part of the paper concerned with cyclostationary applications.

Writing a tutorial paper on cyclostationarity is a difficult task that requires finding the right balance between the temptation to provide a maximum of knowledgeable material on the subject and the necessity to follow a pedagogical—and hence simplistic—presentation. Therefore, the mathematical level was purposely kept as low as possible for the benefit of a more intuitive presentation of the key concepts that are underlying the theory of cyclostationary processes. It is hoped that this approach will facilitate the interested reader to persevere with more theoretical presentations of the subject such as [8], [13], [23], as well as providing him/her with the prerequisites to foresee potential applications in his/her field of interest. The key concepts that have been deemed essential towards that purpose are:

  • the concept of hidden periodicities in a random signal,

  • the equivalence between nonstationarity in the time domain and correlation in the frequency domain,

  • the interpretation of cyclic and spectral frequencies as modulation and carrier frequencies, respectively,

  • the interrelations between temporal, spectral and cyclic decomposition of the energy displayed by a signal,

  • the distinction between first, second, and higher-order cyclostationarity,

  • the implication of the uncertainty principle when choosing between various cyclostationary tools.

The paper is constructed around two questions and an assertion, each of which deserve a different part. The first part addresses the question “What is cyclostationarity?” and consists of the aforementioned key concepts. Whenever possible, these are introduced from first principles and illustrated on real-world examples. The second part of the paper addresses the question “How to implement cyclostationary tools?” and introduces some general guidelines as to how to construct a cyclostationary-based estimator and make it reliable. Finally the last part of the paper illustrates the assertion “Don’t ignore cyclostationarity: use it to advantage” by means of several case studies concerned with the diagnostics and identification of mechanical systems, and the separation of mechanical sources.

Section snippets

What is cyclostationarity?

Two examples of real-world signals are introduced from the onset which will provide a common basis to illustrate the various concepts to be discussed later in this section.

From theory to practice

Thus far, several cyclostationary have been introduced tools by means of the Pα and P-operators which involve limiting values as the measurement time T becomes infinitely large and, in some circumstances, the frequency bandwidth Δf infinitely small. This was the case in formulae (5), (7), (8), (11), (26). Obviously, in the real world, all measured signals are of finite length and, consequently, infinite(simal) limits do not apply. This means that (5), (7), (8), (11), (26) have to be

Do not ignore cyclostationarity: use it to advantage

The object of this last part is to portray a rapid overview of some of the recent breakthroughs that cyclostationarity has permitted in mechanical applications. Examples are mainly drawn from the fields of acoustics and vibrations, but are obviously not restricted to these. Even then, the spectrum of potential applications is so large that an effort to organise them is necessary. Therefore it was decided to single out four major topics concerned respectively with (i) the cyclostationary

Conclusion

One recognised difficulty with the analysis of mechanical signals is the characterisation of nonstationary behaviours. Cyclostationarity offers an elegant and powerful solution when nonstationarity originates from periodical mechanisms, such as in rotating and reciprocating machines. This paper aimed at providing an introduction to cyclostationarity from first principles, with a special emphasis on intuition rather than on mathematics. It is subdivided in three main parts which are briefly

Acknowledgments

Grateful acknowledgments are due to Professor R. B. Randall from the University of New South Wales (Sydney, Australia), S. Sieg-Zieba from the French Industrial and Mechanical Technical Centre (Senlis, France), F. Gautier and N. Ducleaux from Renault S.A. (Rueil-Malmaison, France) for providing some of the data used in this paper.

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