Cyclostationarity: New trends and applications

doi:10.1016/j.sigpro.2015.09.011

Signal Processing

Volume 120, March 2016, Pages 385-408

https://doi.org/10.1016/j.sigpro.2015.09.011 Get rights and content

Highlights

•
Update of Cyclostationarity: Half a century of research, Sig. Proc., April 2006.
•
Review of cyclostationarity-based detectors and cycle frequency estimators.
•
Emphasis on communications and cognitive radio applications.

Abstract

A concise survey of the literature on cyclostationarity of the last 10 years is presented and an extensive bibliography included. The problems of statistical function estimation, signal detection, and cycle frequency estimation are reviewed. Applications in communications are addressed. In particular, spectrum sensing and signal classification for cognitive radio, source location, MMSE filtering, and compressive sensing are discussed. Limits to the applicability of the cyclostationary signal processing and generalizations of cyclostationarity to overcome these limits are addressed in the companion paper “Cyclostationarity: Limits and generalizations”.

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)

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 $w_{T} (t) ≜ a (t / T) / T$ be a data-tapering window with a(t) with unit-area and nonzero in $(- 1 / 2, 1 / 2)$ . The (conjugate) cyclic correlogram $R_{x}^{(T)} (α, τ) ≜ \int_{R} w_{T} (t - t_{0}) x (t + τ) x^{(⁎)} (t) e^{- j 2 π α t} d t$ is the natural estimator of the (conjugate) cyclic autocorrelation function $R_{x}^{α} (τ)$ on the basis of the observation of x(t) for $t \in [t_{0} - T / 2, t_{0} + T / 2]$ . 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 x_d(n), $n \in Z$ , be obtained by uniformly sampling the continuous-time signal x(t), $t \in R$ , with sampling period $T_{s} = 1 / f_{s}$ , that is, $x_{d} (n) = x (t) |_{t = {nT}_{s}}$ . The (conjugate) cyclic autocorrelation function of x_d(n) at cycle frequency $\tilde{α} \in [- 1 / 2, 1 / 2)$ is given by ${\tilde{R}}_{x_{d}}^{\tilde{α}} (m) ≜ \lim_{N \to \infty} \frac{1}{2 N + 1} \sum_{n = - N}^{N} E {x_{d} (n + m) x_{d}^{(⁎)} (n)} e^{- j 2 π \tilde{α} n}$ $= \sum_{p \in Z}$

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 $r (t) = x (t) + n (t)$ (hypothesis H₁) versus its absence $r (t) = n (t)$ (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 $R_{{rr}^{(⁎)}} (α, τ)$ 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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ReviewCyclostationarity: New trends and applications

Highlights

Abstract

Introduction

Section snippets

Second-order characterization of ACS processes

Cyclic statistic estimators

Higher-order statistics

Discrete-time

Signal detection and significance tests

Cycle frequency estimation

Spectrum sensing and signal classification for cognitive radio

Further applications in communications

Mechanics and acoustics

Conclusion

Acknowledgments

J. Stat. Plann. Inference

Mech. Syst. Signal Process.

J. Sound Vib.

Mech. Syst. Signal Process.

Mech. Syst. Signal Process.

Mech. Syst. Signal Process.

Signal Process.

Curr. Opin. Neurobiol.

Stat. Probab. Lett.

Mech. Syst. Signal Process.

Mech. Syst. Signal Process.

Mech. Syst. Signal Process.

Comput. Stat. Data Anal.

Mech. Syst. Signal Process.

Mech. Syst. Signal Process.

Mech. Syst. Signal Process.

Mech. Syst. Signal Process.

Mech. Syst. Signal Process.

Signal Process.

Physica A: Stat. Mech. Appl.

Signal Process.

Appl. Acoust.

J. Sound Vib.

Mech. Syst. Signal Process.

J. Stat. Plann. Inference

Stat. Probab. Lett.

Mech. Syst. Signal Process.

Mech. Syst. Signal Process.

Signal Process.

Sea clutter texture estimationexploiting decorrelation and cyclostationarity

IEEE Trans. Aerosp. Electron. Syst.

Spectral self-coherence restorala new approach to blind adaptive signal extraction using antenna arrays

Proc. IEEE

An algorithm for the estimation of the signal-to-noise ratio in surface myoelectric signals generated during cyclic movements

IEEE Trans. Biomed. Eng.

A SARS method for reliable spectrum sensing in multiband communication systems

IEEE Trans. Signal Process.

Second-order cyclostationarity of mobile WiMAX and LTE OFDM signals and application to spectrum awareness in cognitive radio systems

IEEE J. Sel. Top. Signal Process.

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IET Commun.

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Review
Cyclostationarity: New trends and applications