ReviewCyclostationarity: New trends and applications
Introduction
The mixture of a periodic phenomenon and a random one or the modulation of a periodic process by a random one generates a process that is not periodic. The underlying periodicity, even if not present in the process itself, is present in its statistical characteristics. Such a process is called cyclostationary. If more periodic phenomena with incommensurate periods are present in the underlying generation mechanism of the process, then the resulting statistical characteristics of the process are almost-periodic functions of time [26], [30], [58], [254, Section 1.2], and the process is referred to as almost-cyclostationary (ACS). In contrast, the classical model of stationary process considers statistical functions not depending on time.
ACS processes have almost-periodic autocorrelation function whose (generalized) Fourier series expansion has coefficients, called cyclic autocorrelation functions, depending on the lag parameter and frequencies, called cycle frequencies, not depending on the lag parameter. Equivalently, ACS processes have spectral components that are correlated when the frequency separation equals one of the cycle frequencies. The density of spectral correlation is described by the cyclic spectra which are the Fourier transforms of the cyclic autocorrelation functions [110], [113], [120], [131], [159].
ACS processes are appropriate models for signals in several fields such as telecommunications, radar, sonar, telemetry, mechanics, acoustics, climatology, biology, econometrics, and astronomy [110], [113], [120], [131], [159]. In communications and radar/sonar, almost-cyclostationarity properties have been exploited for a variety of applications. They include weak-signal detection [122], [123], synchronization [111], [55], [54], minimum mean-squared error (MMSE) filtering [115], source location [2], [49], [119], [123], [127], [154], [332], [342], source separation [169], modulation format classification [81], [327], and cognitive radio [23], [144], [233], [329].
In severe noise and interference environments, cyclostationarity-based signal processing techniques have been shown to significantly outperform classical techniques that model signals as stationary. This is due to the inherent signal selectivity property of cyclostationarity-based algorithms and the existence of consistent estimators of statistical functions when the data-record length approaches infinity [62], [64], [71], [159].
A survey of the literature on cyclostationarity up to year 2005 is presented in the review article [120]. Moreover, in [120], general properties of ACS processes at second and higher orders, the problems of linear almost-periodically time-variant (LAPTV) filtering, uniform sampling, and measurements of characteristics are addressed in a tutorial way. Examples of man-made signals and applications in several fields are also discussed.
Since 2005, cyclostationarity properties of signals have been extensively exploited in several classical problems and applications in new fields have been considered, as proved by the large number of published works. In this paper, a survey of the literature on cyclostationarity since year 2005 is presented and recent applications of the theory of ACS signals are reviewed. The problems of signal detection and cycle frequency estimation, not covered in [120], are reviewed and their relevant application to spectrum sensing for cognitive radio [23], [144], [233], [329] is discussed since in this field the theory of ACS signals has been widely and successfully exploited. The recently introduced subsampling and block-bootstrap methods [69], [92], [96], [202] for constructing confidence intervals of estimators of cyclic statistical functions are described. In addition, since ACS signals are sparse in the cycle frequency domain [18], [57], [210], [222], [345], [382], the issue of compressive sensing is addressed.
The recent literature on cyclostationarity in several other fields is also reviewed. Applications are considered in the following fields: acoustics and mechanics [10], [14], radio astronomy and astrophysics [78], [100], [145], optics and spectroscopy [105], [310], [309], analysis of genome and biological signals [19], [97], [128], [132], [235], finance and econometrics [27], [43], [160], [204], and climatology [84], [125], [151], [190], [308].
Limits to the applicability of the ACS model in high mobility scenarios in communications and radar are discussed in the companion paper [247]. In order to overcome these limits, the new classes of the generalized almost-cyclostationary (GACS) processes [163], [164], [165], [166], [250], [251], [254], [259], and the spectrally correlated (SC) processes [249], [252], [253], [254, Chapter 4] have been introduced. These classes of processes generalize the class of the ACS ones and are reviewed in the companion paper [247] where they are shown to be appropriate models for the output signals of several Doppler channels of interest.
The paper is organized as follows. The second-order characterization of ACS processes is reviewed in Section 2. Cyclic statistic estimators, higher-order statistics, and discrete-time ACS signals are reviewed in 3 Cyclic statistic estimators, 4 Higher-order statistics, 5 Discrete-time, respectively. The problems of signal detection and cycle frequency estimation are addressed in 6 Signal detection and significance tests, 7 Cycle frequency estimation. Spectrum sensing and signal classification for cognitive radio are considered in Section 8. Further applications in communications and in other fields are discussed in 9 Further applications in communications, 10 Other applications. Conclusions are drawn in Section 11 and an extensive list of references ends the paper.
In this section, the abbreviations adopted in the paper are listed. Acronyms of specific modulation formats of communications signals are not included in this list.
- ACS
almost cyclostationary
- AWGN
additive white Gaussian noise
- CDP
cycle frequency domain profile
- DOA
direction of arrival
- ECG
electrocardiogram
- FRESH
frequency shift
- GACS
generalized almost-cyclostationary
- (G)(L)LR(T)
(generalized) (log) likelihood ratio (test)
- GNSS
global navigation satellite system
- LAPTV
linear almost-periodically time-variant
- LTI
linear time-invariant
- MIMO
multi-input multi-output
- (M)MSE
(minimum) mean-squared error
- MUSIC
multiple signal classification
- (N)LMS
(normalized) least mean square
- PAM
pulse-amplitude modulated
- PAR(MA)
periodic autoregressive (moving average)
- SC
spectrally correlated
- SIR
signal-to-interference ratio
- SNR
signal-to-noise ratio
- SOI
signal of interest
- STFT
short-time Fourier transform
- TDOA
time difference of arrival
- WSS
wide-sense stationary
Section snippets
Second-order characterization of ACS processes
In this section, the second-order characterization of continuous-time ACS processes in the time and frequency domains is briefly reviewed to introduce definitions and notation. See [247, Section 3] for the definitions of second-order probabilistic functions of general nonstationary stochastic processes and [120, Section 3] for a comprehensive review of ACS processes.
The complex-valued (finite-power) continuous-time process x(t) is said to be second-order cyclostationary in the wide-sense or
Cyclic statistic estimators
Let be a data-tapering window with a(t) with unit-area and nonzero in . The (conjugate) cyclic correlogramis the natural estimator of the (conjugate) cyclic autocorrelation function on the basis of the observation of x(t) for . In the left-hand side of (3.1) and in the following, when it does not create ambiguity, the dependence on t0 is omitted in the notation.
Let x(t) be a zero-mean ACS process with
Higher-order statistics
A more complete characterization of stochastic processes can be obtained considering not only second-order statistics, but also higher-order statistics. Processes that have almost-periodically time-variant moment and cumulant functions are called higher-order almost-cyclostationary. They have been characterized in the time and frequency domains in [62], [64], [124], [248], [328]. The issues of sampling and Rice׳s representation are addressed in [161], [162].
The Fourier coefficients of the
Discrete-time
ACS processes, such as WSS processes, can also be defined in discrete time. The definitions are similar to those in continuous time with the obvious modifications [62], [116], [120], [131], [248].
Let xd(n), , be obtained by uniformly sampling the continuous-time signal x(t), , with sampling period , that is, . The (conjugate) cyclic autocorrelation function of xd(n) at cycle frequency is given by
Signal detection and significance tests
The problem of cyclostationary signal detection is addressed in [110], [113]. The test for the presence of a signal-of-interest (SOI) x(t) in the corrupted observation (hypothesis H1) versus its absence (hypothesis H0) is a formidable problem if the disturbance n(t) is a non-Gaussian and possibly nonstationary signal with unknown or complicate probability density function.
If the likelihood ratio test (LRT) cannot be built, alternative (sub-optimum) detectors can be
Cycle frequency estimation
(Conjugate) cyclic autocorrelation, (conjugate) cyclic spectrum, and (conjugate) spectral coherence functions are discrete in the cycle frequency domain. That is, they are nonzero only in correspondence of a finite or at most countable set of values of α. However, in cycle frequency estimation problems, since the true values of the cycle frequencies are unknown, it is convenient to consider α varying with continuity.
Let denote the (conjugate) cyclic autocorrelation function defined
Spectrum sensing and signal classification for cognitive radio
The signal selectivity properties of cyclostationarity-based signal processing techniques and the consistency and asymptotic complex normality of estimates of second- and higher-order statistics of ACS signals have been suitably exploited for the purpose of signal classification in cognitive radio [23], [81], [144], [283], [327], [329]. In fact, even if some modulation formats exhibit identical or very similar power spectral density or even classical time-invariant higher-order statistics, they
Further applications in communications
Cyclostationarity-based detection and estimation algorithms are signal selective. Specifically, in the case of additive noise uncorrelated with the SOI, if the SOI does not share at least one cycle frequency, say α0, with the disturbance, then the cyclic autocorrelation function of the SOI plus noise at α0 is coincident with the cyclic autocorrelation function of the SOI alone. Therefore, cyclostationarity-based detection or estimation algorithms operating at α0 are potentially immune to the
Mechanics and acoustics
Mechanical systems with moving parts generate vibro-acoustic signals that contain both an additive almost-periodic component and a zero-mean ACS component (see (2.3)) [10], [14], [120], [284]. The almost-periodic component in the signal generates an almost-periodic term in the (conjugate) cyclic autocorrelation function (see (2.5)) and, hence, Dirac impulses in the (conjugate) cyclic spectrum (see (2.9)). In general, the zero-mean ACS component has a power smaller than that of the
Conclusion
A survey of the recent literature on cyclostationarity is presented and an extensive bibliography included. For the ACS signals, the problems of statistical function estimation, signal detection, and cycle frequency estimation are reviewed.
Detectors based on cyclostationary features of the signal-of-interest turn out to be useful when the distribution of noise and/or of the useful signal is unknown or difficult to be treated. In addition, estimates of second- and higher-order cycle frequencies
Acknowledgments
The author is grateful to the Reviewers for their insightful comments on the original manuscript.
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