ReviewCyclostationarity: Half a century of research
Introduction
Many processes encountered in nature arise from periodic phenomena. These processes, although not periodic functions of time, give rise to random data whose statistical characteristics vary periodically with time and are called cyclostationary processes [2.5]. For example, in telecommunications, telemetry, radar, and sonar applications, periodicity is due to modulation, sampling, multiplexing, and coding operations. In mechanics it is due, for example, to gear rotation. In radio astronomy, periodicity results from revolution and rotation of planets and on pulsation of stars. In econometrics, it is due to seasonality; and in atmospheric science it is due to rotation and revolution of the earth. The relevance of the theory of cyclostationarity to all these fields of study and more was first proposed in [2.5].
Wide-sense cyclostationary stochastic processes have autocorrelation functions that vary periodically with time. This function, under mild regularity conditions, can be expanded in a Fourier series whose coefficients, referred to as cyclic autocorrelation functions, depend on the lag parameter; the frequencies, called cycle frequencies, are all multiples of the reciprocal of the period of cyclostationarity [2.5]. Cyclostationary processes have also been referred to as periodically correlated processes [2.18], [3.5]. More generally, if the frequencies of the (generalized) Fourier series expansion of the autocorrelation function are not commensurate, that is, if the autocorrelation function is an almost-periodic function of time, then the process is said to be almost-cyclostationary [3.26] or, equivalently, almost-periodically correlated [3.5]. The almost-periodicity property of the autocorrelation function is manifested in the frequency domain as correlation among the spectral components of the process that are separated by amounts equal to the cycle frequencies. In contrast to this, wide-sense stationary processes have autocorrelation functions that are independent of time, depending on only the lag parameter, and all distinct spectral components are uncorrelated.
As an alternative, the presence of periodicity in the underlying data-generating mechanism of a phenomenon can be described without modelling the available data as a sample path of a stochastic process but, rather, by modelling it as a single function of time [4.31]. Within this nonstochastic framework, a time-series is said to exhibit second-order cyclostationarity (in the wide sense), as first defined in [2.8], if there exists a stable quadratic time-invariant transformation of the time-series that gives rise to finite-strength additive sinewave components.
In this paper, a concise survey of the literature (in all languages in which a substantial amount of research has been published) on cyclostationarity is presented and includes an extensive bibliography and list of issued patents. Citations are classified into 22 categories and listed, for each category, in chronological order. In Section 2, general treatments and tutorials on the theory of cyclostationarity are cited. General properties of processes and time-series are presented in Section 3. In Section 4, the problem of estimating statistical functions is addressed. Models for manufactured and natural signals are considered in Sections 5 and 6, respectively. Communications systems and related problems are treated in Sections 7–11. Specifically, the analysis and design of communications systems is addressed in Section 7, the problem of synchronization is addressed in Section 8, the estimation of signal parameters and waveforms is addressed in Section 9, the identification and equalization of channels is addressed in Section 10, and the signal detection and classification problems and the problem of source separation are addressed in Section 11. Periodic autoregressive (AR) and autoregressive moving-average (ARMA) modelling and prediction are treated in Section 12. In Section 13, theory and applications of higher-order cyclostationarity are presented. We address applications to circuits, systems, and control in Section 14, to acoustics and mechanics in Section 15, to econometrics in Section 16, and to biology in Section 17. Applications to the problems of level crossing and queueing are addressed in Sections 18 and 19, respectively. Cyclostationary random fields are treated in Section 20. In Section 21, some classes of nonstationary signals that extend the class of almost-cyclostationary signals are considered. Finally, some miscellaneous references are listed [22.1], [22.2], [22.3], [22.4], [22.5], [22.6], [22.7], [22.8]. Further references only indirectly related to cyclostationarity are [23.1], [23.2], [23.3], [23.4], [23.5], [23.6], [23.7], [23.8], [23.9], [23.10], [23.11], [23.12], [23.13], [23.14], [23.15].
To assist readers in “going to the source”, seminal contributions—if known— are identified within the literature published in English. In some cases, identified sources may have been preceded in the literature of another language, most likely Russian. For the most part, the subject of this survey developed independently in the literature published in English.
Section snippets
General treatments
General treatments on cyclostationarity are in [2.1], [2.2], [2.3], [2.4], [2.5], [2.6], [2.7], [2.8], [2.9], [2.10], [2.11], [2.12], [2.13], [2.14], [2.15], [2.16], [2.17], [2.18]. The first extensive treatments of the theory of cyclostationary processes can be found in the pioneering works of Hurd [2.1] and Gardner [2.2]. In [4.13], [2.5], [2.11], the theory of second-order cyclostationary processes is developed mainly with reference to continuous-time stochastic processes, but discrete-time
Estimation of the cyclic autocorrelation function and the cyclic spectrum
Ergodic properties and measurements of characteristics are treated in [4.1], [4.2], [4.3], [4.4], [4.5], [4.6], [4.7], [4.8], [4.9], [4.10], [4.11], [4.12], [4.13], [4.14], [4.15], [4.16], [4.17], [4.18], [4.19], [4.20], [4.21], [4.22], [4.23], [4.24], [4.25], [4.26], [4.27], [4.28], [4.29], [4.30], [4.31], [4.32], [4.33], [4.34], [4.35], [4.36], [4.37], [4.38], [4.39], [4.40], [4.41], [4.42], [4.43], [4.44], [4.45], [4.46], [4.47], [4.48], [4.49], [4.50], [4.51], [4.52], [4.53], [4.54], [4.55]
General aspects
Cyclostationarity in manmade communications signals is due to signal processing operations used in the construction and/or subsequent processing of the signal, such as modulation, sampling, scanning, multiplexing and coding operations [5.1], [5.2], [5.3], [5.4], [5.5], [5.6], [5.7], [5.8], [5.9], [5.10], [5.11], [5.12], [5.13], [5.14], [5.15], [5.16], [5.17], [5.18], [5.19], [5.20], [5.21], [5.23].
The analytical cyclic spectral analysis of mathematical models of analog and digitally modulated
Natural signals: modelling and analysis
Cyclostationarity occurs in data arising from a variety of natural (not man-made) phenomena due to the presence of periodic mechanisms in the phenomena [6.1], [6.2], [6.3], [6.4], [6.5], [6.6], [6.7], [6.8], [6.9], [6.10], [6.11], [6.12], [6.13], [6.14], [6.15], [6.16], [6.17], [6.18], [6.19], [6.20], [6.21], [6.22], [6.23], [6.24], [6.25], [6.26], [6.27]. In climatology and atmospheric science, cyclostationarity is due to rotation and revolution of the earth [6.1], [6.2], [6.9], [6.12], [6.13]
General aspects
Cyclostationarity properties of modulated signals can be suitably exploited in the analysis and design of communications systems (see [7.1], [7.2], [7.3], [7.4], [7.5], [7.6], [7.7], [7.8], [7.9], [7.10], [7.11], [7.12], [7.13], [7.14], [7.15], [7.16], [7.17], [7.18], [7.19], [7.20], [7.21], [7.22], [7.23], [7.24], [7.25], [7.26], [7.27], [7.28], [7.29], [7.30], [7.31], [7.32], [7.33], [7.34], [7.35], [7.36], [7.37], [7.38], [7.39], [7.40], [7.41], [7.42], [7.43], [7.44], [7.45], [7.46], [7.47]
Spectral line generation
Let be a real-valued second-order wide-sense ACS time series. According to the results of Section 3.3, the second-order lag product can be decomposed into the sum of its almost-periodic component and a residual term not containing any finite-strength additive sinewave component (see (3.44) and (3.46)):where
For communications signals, the cycle frequencies are related to parameters
Signal parameter and waveform estimation
Cyclostationarity properties can be exploited to design signal selective algorithms for signal parameter and waveform estimation [9.1], [9.2], [9.3], [9.4], [9.5], [9.6], [9.7], [9.8], [9.9], [9.10], [9.11], [9.12], [9.13], [9.14], [9.15], [9.16], [9.17], [9.18], [9.19], [9.20], [9.21], [9.22], [9.23], [9.24], [9.25], [9.26], [9.27], [9.28], [9.29], [9.30], [9.31], [9.32], [9.33], [9.34], [9.35], [9.36], [9.37], [9.38], [9.39], [9.40], [9.41], [9.42], [9.43], [9.44], [9.45], [9.46], [9.47],
General aspects
Cyclostationarity-based techniques have been exploited for channel identification and equalization [10.1], [10.2], [10.3], [10.4], [10.5], [10.6], [10.7], [10.8], [10.9], [10.10], [10.11], [10.12], [10.13], [10.14], [10.15], [10.16], [10.17], [10.18], [10.19], [10.20], [10.21], [10.22], [10.23], [10.24], [10.25], [10.26], [10.27], [10.28], [10.29], [10.30], [10.31], [10.32], [10.33], [10.34], [10.35], [10.36], [10.37], [10.38], [10.39], [10.40], [10.41], [10.42], [10.43], [10.44], [10.45],
Signal detection and classification, and source separation
Signal detection techniques designed for cyclostationary signals take account of the periodicity or almost periodicity of the signal autocorrelation function [11.1], [11.2], [11.3], [11.4], [11.5], [11.6], [11.7], [11.8], [11.9], [11.10], [11.11], [11.12], [11.13], [11.14], [11.15], [11.16], [11.17], [11.18], [11.19], [11.20], [11.21], [11.22], [11.23], [11.24], [11.25], [11.26], [11.27], [11.28], [11.29], [11.30], [11.31], [11.32], [11.33], [11.34], [11.35], [11.36], [11.37], [11.38], [11.39].
Periodic AR and ARMA modelling and prediction
Periodic autoregressive (AR) and autoregressive moving average (ARMA) (discrete-time) systems are characterized by input/output relationships described by difference equations with periodically time-varying coefficients and system orders [12.36], [12.37]:where and are the input and output signals, respectively, and the coefficients and and the orders and are periodic functions with the same period . Thus, periodic ARMA
Applications to circuits, systems, and control
Applications of cyclostationarity to circuits, systems, and control are in [14.1], [14.2], [14.3], [14.4], [14.5], [14.6], [14.7], [14.8], [14.9], [14.10], [14.11], [14.12], [14.13], [14.14], [14.15], [14.16], [14.17], [14.18], [14.19], [14.20], [14.21], [14.22], [14.23], [14.24], [14.25], [14.26], [14.27], [14.28], [14.29], [14.30], [14.31]. In circuit theory, cyclostationarity has been exploited in modelling noise [14.1], [14.2], [14.12], [14.17], [14.19], [14.20], [14.24], [14.26], [14.28].
Applications to acoustics and mechanics
Applications of cyclostationarity to acoustics and mechanics are in [15.1], [15.2], [15.3], [15.4], [15.5], [15.6], [15.7], [15.8], [15.9], [15.10], [15.11], [15.12], [15.13], [15.14], [15.15], [15.16], [15.17], [15.18], [15.19], [15.20], [15.21], [15.22], [15.23], [15.24]. Cyclostationarity has been exploited in acoustics and mechanics for modelling road traffic noise [15.2], [15.3], for analyzing music signals [15.6], and for describing the vibration signals in mechanical systems. In
Applications to econometrics
In high-frequency financial time series, such as asset return, the repetitive patterns of openings and closures of markets, the number of active markets throughout the day, seasonally varying preferences, and so forth, are sources of periodic variations in financial-market volatility and other statistical parameters. Autoregressive models with periodically varying parameters provide appropriate descriptions of seasonally varying economic time series [16.1], [16.2], [16.3], [16.4], [16.5], [16.6]
Applications to biology
Applications of cyclostationarity to biology are in [17.1], [17.2], [17.3], [17.4], [17.5], [17.6], [17.7], [17.8], [17.9], [17.10], [17.11], [17.12], [17.13], [17.14], [17.15], [17.16], [17.17], [17.18]. Applications of the concept of spectral redundancy (spectral correlation or cyclostationarity) have been proposed in medical image signal processing [17.6], [17.8], and nondestructive evaluation [17.2]. Methods of averaging were developed for estimating the generalized spectrum that allow for
Level crossings
Level crossings of cyclostationary signals have been characterized in [18.1], [18.2], [18.3], [18.4], [18.5], [18.6], [18.7], [18.8], [18.9], [18.10].
Queueing
Exploitation of cyclostationarity in queueing theory in computer networks is treated in [19.1], [19.2], [19.3], [19.4], [19.6]. Queueing theory in car traffic is treated in [19.5].
On this subject, also see [14.8].
Cyclostationary random fields
A periodically correlated or cyclostationary random field is a second-order random field whose mean and correlation have periodic structure [20.6], [20.7]. Specifically, a random field indexed on is called strongly periodically correlated with period if and only if there exists no smaller and for which the mean and correlation satisfyfor all , , , .
General aspects
Generalizations of cyclostationary processes and time series are treated in [21.1], [21.2], [21.3], [21.4], [21.5], [21.6], [21.7]. The problem of statistical function estimation for general nonstationary persistent signals is addressed in [21.1], [21.5] and limitations of previously proposed approaches are exposed. In [21.2], [21.3], [21.4], [21.8], the class of the correlation autoregressive processes is studied. Nonstationary signals that are not ACS can arise from linear time-variant, but
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