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Giannakis University of Virginia 17.1 Introduction17.2 Definitions, Properties, Representations17.3 Estimation, Time-Frequency Links, Testing Estimating Cyclic Statistics •Links with Tim

Giannakis, G.B “Cyclostationary Signal Analysis”

Digital Signal Processing Handbook

Ed Vijay K Madisetti and Douglas B Williams Boca Raton: CRC Press LLC, 1999

17 Cyclostationary Signal Analysis

Georgios B Giannakis

University of Virginia

17.1 Introduction17.2 Definitions, Properties, Representations17.3 Estimation, Time-Frequency Links, Testing

Estimating Cyclic Statistics •Links with Time-Frequency

Rep-resentations•Testing for Cyclostationarity

17.4 CS Signals and CS-Inducing Operations

Amplitude Modulation •Time Index Modulation•Fractional

Sampling and Multivariate/Multirate Processing •Periodically

Varying Systems

17.5 Application Areas

CS Signal Extraction•Identification and Modeling

17.6 Concluding RemarksAcknowledgmentsReferences

17.1 Introduction

Processes encountered in statistical signal processing, communications, and time series analysisapplications are often assumed stationary The plethora of available algorithms testifies to the needfor processing and spectral analysis of stationary signals (see, e.g., [42]) Due to the varying nature

of physical phenomena and certain man-made operations, however, time-invariance and the relatednotion of stationarity are often violated in practice Hence, study of time-varying systems andnonstationary processes is well motivated

Research in nonstationary signals and time-varying systems has led both to the development ofadaptive algorithms and to several elegant tools, including short-time (or running) Fourier trans-forms, time-frequency representations such as the Wigner-Ville (a member of Cohen’s class of dis-tributions), Loeve’s and Karhunen’s expansions (leading to the notion of evolutionary spectra), andtime-scale representations based on wavelet expansions (see [37,45] and references therein) Adap-tive algorithms derived from stationary models assume slow variations in the underlying system

On the other hand, time-frequency and time-scale representations promise applicability to generalnonstationarities and provide useful visual cues for preprocessing When it comes to nonstationarysignal analysis and estimation in the presence of noise, however, they assume availability of multipleindependent realizations

In fact, it is impossible to perform spectral analysis, detection, and estimation tasks on signalsinvolving generally unknown nonstationarities, when only a single data record is available Forinstance, consider extracting a deterministic signals(n) observed in stationary noise v(n), using

regression techniques based on nonstationary datax(n) = s(n)+v(n), n = 0, 1, , N −1 Unless s(n) is finitely parameterized by a d θ s × 1 vector θ s(withd θ s < N), the problem is ill-posed because

adding a new datum, sayx(n0), adds a new unknown, s(n0), to be determined Thus, only structured

nonstationarities can be handled when rapid variations are present; and only for classes of finitelyparameterized nonstationary processes can reliable statistical descriptors be computed using a singletime series One such class is that of (wide-sense) cyclostationary processes which are characterized

by the periodicity they exhibit in their mean, correlation, or spectral descriptors

An overview of cyclostationary signal analysis and applications are the main goals of this tion Periodicity is omnipresent in physical as well as manmade processes, and cyclostationarysignals occur in various real life problems entailing phenomena and operations of repetitive nature:communications [15], geophysical and atmospheric sciences (hydrology [66], oceanography [14],meteorology [35], and climatology [4]), rotating machinery [43], econometrics [50], and biologicalsystems [48]

sec-In 1961 Gladysev [34] introduced key representations of cyclostationary time series, while in 1969Hurd’s thesis [38] offered an excellent introduction to continuous time cyclostationary processes.Since 1975 [22], Gardner and co-workers have contributed to the theory of continuous-time cyclo-stationary signals, and especially their applications to communications engineering Gardner [15]adopts a “non-probabilistic” viewpoint of cyclostationarity (see [19] for an overview and also [36]and [18] for comments on this approach) Responding to a recent interest in digital periodicallyvarying systems and cyclostationary time series, the exposition here is probabilistic and focuses ondiscrete-time signals and systems, with emphasis on their second-order statistical characterizationand their applications to signal processing and communications

The material in the remaining sections is organized as follows: Section17.2provides definitions,properties, and representations of cyclostationary processes, along with their relations with stationaryand general classes of nonstationary processes Testing a time series for cyclostationarity and retrieval

of possibly hidden cycles along with single record estimation of cyclic statistics are the subjects

of Section17.3 Typical signal classes and operations inducing cyclostationarity are delineated inSection17.4to motivate the key uses and selected applications described in Section17.5 Finally,Section17.6concludes and presents trade-offs, topics not covered, and future directions

17.2 Definitions, Properties, Representations

Letx(n) be a discrete-index random process (i.e., a time series) with mean µ x (n) := E{x(n)}, and

covariancec xx (n; τ) := E{[x(n) − µ x (n)][x(n + τ) − µ x (n + τ)]} For x(n) complex valued, let

also¯c xx (n; τ) := c xx∗ (n; τ), where ∗ denotes complex conjugation, and n, τ are in the set of integers Z.

DEFINITION 17.1 Processx(n) is (wide-sense) cyclostationary (CS) iff there exists an integer P

such thatµ x (n) = µ x (n + lP ), c xx (n; τ) = c xx (n + lP ; τ), or, ¯c xx (n; τ) = ¯c xx (n + lP ; τ),

∀n, l ∈ Z The smallest of all such P s is called the period Being periodic, they all accept Fourier

Series expansions over complex harmonic cycles with the set of cycles defined as: A c

the focus in engineering is on periodically and almost periodically correlated1time series, since realdata are often zero-mean, correlated, and with unknown distributions Almost periodicity is verycommon in discrete-time because sampling a continuous-time periodic process will rarely yield adiscrete-time periodic signal; e.g., sampling cos(ω c t + θ) every T sseconds results in cos(ω c nT s + θ) for which an integer period exists only if ω c T s = 2π/P Because 2π/(ω c T s ) is “almost an integer”

period, such signals accept generalized (or limiting) Fourier expansions (see also Eq (17.2) and [9]for rigorous definitions of almost periodic functions)

DEFINITION 17.2 Processx(n) is (wide-sense) almost cyclostationary (ACS) iff its mean and

correlation(s) are almost periodic sequences Forx(n) zero-mean and real, the time-varying and

cyclic correlations are defined as the generalized Fourier Series pair:

α k ∈A c xx

The set of cycles,A c

xx (τ) := {α k : C xx (α k ; τ) 6= 0 , −π < α k ≤ π}, must be countable and the

limit is assumed to exist at least in the mean-square sense [9, Thm 1.15]

Definition17.2and Eq (17.2) for ACS, subsume CS Definition17.1and Eq (17.1) Note thatthe latter require integer period and a finite set of cycles In theα-domain, ACS signals exhibit lines

but not necessarily at harmonically related cycles The following example will illustrate the cyclicquantities defined thus far:

EXAMPLE 17.1: Harmonic in multiplicative and additive noise

Let

wheres(n), v(n) are assumed real, stationary, and mutually independent Such signals appear when

communicating through flat-fading channels, and with weather radar or sonar returns when, inaddition to sensor noisev(n), backscattering, target scintillation, or fluctuating propagation media

give rise to random amplitude variations modeled bys(n) [33] We will consider two cases:Case 1:µ s 6= 0 The mean in (17.3) isµ x (n) = µ scos(ω0n) + µ v, and the cyclic mean:

Signal x(n) in (17.3) is thus (first-order) cyclostationary with set of cyclesA c

x = {±ω0, 0} If

X N (ω) :=PN−1 n=0 x(n) exp(−jωn), then from (17.4) we findC x (α) = lim N→∞ N−1E{X N (α)};

thus, the cyclic mean can be interpreted as an averaged DFT andω0can be retrieved by picking thepeak of|X N (ω)| for ω 6= 0.

Case 2: µ s = 0 From (17.3) we find the correlationc xx (n; τ) = c ss (τ)[cos(2ω0n + ω0τ) +

cos(ω0τ)]/2 + c vv (τ) Because c xx (n; τ) is periodic in n, x(n) is (second-order) CS with cyclic

The set of cycles isA c

xx (τ) = {±2ω0, 0} provided that c ss (τ) 6= 0 and c vv (τ) 6= 0 The set A c

xx (τ)

is lag-dependent in the sense that some cycles may disappear while others may appear for different

τs To illustrate the τ -dependence, let s(n) be an MA process of order q Clearly, c ss (τ) = 0 for

|τ| > q, and thus A c

xx (τ) = {0} for |τ| > q.

The CS process in (17.3) is just one example of signals involving products and sums of stationaryprocesses such ass(n) with (almost) periodic deterministic sequences d(n), or, CS processes x(n).

For such signals, the following properties are useful:

Property 1 Finite sums and products of ACS signals are ACS If x i (n) is CS with period P i , then for λ i constants, y1(n) :=PI1

i=1 λ i x i (n) and y2(n) :=QI2

i=1 λ i x i (n) are also CS Unless cycle cancellations occur among x i (n) components, the period of y1(n) and y2(n) equals the least common multiple of the P i s Similarly, finite sums and products of stationary processes with deterministic (almost) periodic signals are also ACS processes.

As examples of random–deterministic mixtures, consider

wheres(n) is zero-mean, stationary, and d(n) is deterministic (almost) periodic with Fourier Series

coefficientsD(α) Time-varying correlations are, respectively,

c x1x1(n; τ) = c ss (τ) + d(n)d(n + τ) and c x2x2(n; τ) = c ss (τ)d(n)d(n + τ) (17.8)Both are (almost) periodic inn, with cyclic correlations

C x1x1(α; τ) = c ss (τ)δ(α) + D2(α; τ) and C x2x2(α; τ) = c ss (τ)D2(α; τ) , (17.9)whereD2(α; τ) = Pβ D(β)D(α − β) exp[j (α − β)τ], since the Fourier Series coefficients of

the productd(n)d(n + τ) are given by the convolution of each component’s coefficients in the α-domain To reiterate the dependence on τ, notice that if d(n) is a periodic ±1 sequence, then

c x2x2(n; 0) = c ss (0)d2(n) = c ss (0), and hence periodicity disappears at τ = 0.

ACS signals appear often in nature with the underlying periodicity hidden, unknown, or sible In contrast, CS signals are often man-made and arise as a result of, e.g., oversampling (by aknown integer factorP ) digital communication signals, or by sampling a spatial waveform with P

inacces-antennas (see also Section17.4)

Both CS and ACS definitions could also be given in terms of the Fourier Transforms (τ → ω)

ofc xx (n; τ) and C xx (α; τ), namely the time-varying and the cyclic spectra which we denote by

S xx (n; ω) and S xx (α; ω) Suppose c xx (n; τ) and C xx (α; τ) are absolutely summable w.r.t τ for all

Absolute summability w.r.t τ implies vanishing memory as the lag separation increases, and many

real life signals satisfy these so called mixing conditions [5, Ch 2] Power signals are not absolutelysummable, but it is possible to define cyclic spectra equivalently [for real-valuedx(n)] as

unless|ω1± ω2| = 0 (mod 2π) [5, Ch 4] Specifically, we have from (17.12) that:

Property 2 If x(n) is ACS or CS, the N-point Fourier transform X N (ω1) is correlated with X N (ω2) for

|ω1± ω2| = α k (mod 2π), and α k ∈ A s

xx .

Before dwelling further on spectral characterization of ACS processes, it is useful to note the sity of tools available for processing Stationary signals are analyzed with time-invariant correlations(lag-domain analysis), or with power spectral densities (frequency-domain analysis) However, CS,ACS, and generally nonstationary signals entail four variables:(n, τ, α, ω) :=(time, lag, cycle, fre-

diver-quency) Grouping two variables at a time, four domains of analysis become available and theirrelationship is summarized in Fig.17.1 Note that pairs(n; τ) ↔ (α; τ), or, (n; ω) ↔ (α; ω), have

τ or ω fixed and are Fourier Series pairs; whereas (n; τ) ↔ (n; ω), or, (α; τ) ↔ (α; ω), have n or

α fixed and are related by Fourier Transforms Further insight on the links between stationary and

FIGURE 17.1: Four domains for analyzing cyclostationary signals

cyclostationary processes is gained through the uniform shift (or phase) randomization concept Let

x(n) be CS with period P , and define y(n) := x(n + θ), where θ is uniformly distributed in [0, P )

and independent ofx(n) With c yy (n; τ) := E θ {E x [x(n + θ)x(n + τ + θ)]}, we find:

Such a mapping is often used with harmonic signals; e.g.,x(n) = A exp[j (2πn/P + θ)] + v(n) is

according to Property 2 a CS signal, but can be stationarized by uniform phase randomization Analternative trick for stationarizing signals which involve complex harmonics is conjugation Indeed,

c xx∗ (n; τ) = A2exp(−j2πτ/P ) + c vv (τ) is not a function of n — but why deal with CS or ACS

processes if conjugation or phase randomization can render them stationary?

Revisiting Case 2 of Example17.1offers a partial answer when the goal is to estimate the frequency

ω0 Phase randomization of x(n) in (17.3) leads to a stationary y(n) with correlation found by

substitutingα = 0 in (17.6) This leads toc yy (τ) = (1/2)c ss (τ) cos(ω0τ) + c vv (τ), and shows that

ifs(n) has multiple spectral peaks, or if s(n) is broadband, then multiple peaks or smearing of the

spectral peak hamper estimation ofω0(in fact, it is impossible to estimateω0from the spectrum of

y(n) if s(n) is white) In contrast, picking the peak of C xx (α; τ) in (17.6) yieldsω0, provided that

ω0 ∈ (0, π) so that spectral folding is prevented [33] Equation (17.13) provides a more generalanswer Phase randomization restricts a CS process only to one cycle, namelyα = 0 In other words,

the cyclic correlationC xx (α; τ) contains the “stationarized correlation” C xx (0; τ) and additional

axes at equidistant points 2π/P far apart from each other More specifically, we have [34]:

2 Nonstationary processes with Fourier transformable 2-D correlations are called harmonizable processes.

FIGURE 17.2: Support of 2-D spectrumS xx (ω1, ω2) for CS processes.

Property 5 A CS process with period P is a special case of a nonstationary (harmonizable) process with 2-D spectral density given by

For stationary processes, only thek = 0 term survives in (17.15) and we obtainS xx (ω1, ω2) =

S xx (0; ω1)δ D (ω2 −ω1); i.e., the spectral mass is concentrated on the diagonal of Fig.17.2 Thewell-structured spectral support for CS processes will be used to test for presence of cyclostationarityand estimate the periodP Furthermore, the superposition of lines parallel to the diagonal hints

towards representing CS processes as a superposition of stationary processes Next we will examinetwo such representations introduced by Gladysev [34] (see also [22,38,49], and [56])

We can uniquely writen0= nP + i and express x(n0) = x(nP + i), where the remainder i takes

values 0, 1, , P −1 For each i, define the subprocess x i (n) := x(nP +i) In multirate processing,

theP × 1 vector x(n) := [x0(n) x P −1 (n)]0constitutes the so-called polyphase decomposition of

x(n) [51, Ch 12] As shown in Fig.17.3, eachx i (n) is formed by downsampling an advanced copy

We maintain that subprocesses{x i (n)} P −1 i=0 are (jointly) stationary, and thus x(n) is vector stationary.

Suppose for simplicity thatE{x(n)} = 0, and start with E{x i1(n)x i2(n+τ)} = E{x(nP +i1)x(nP +

τP + i2)} := c xx (i1+ nP ; i2− i1+ τP ) Because x(n) is CS, we can drop nP and c xx becomesindependent ofn establishing that x i1(n), x i2(n) are (jointly) stationary with correlation:

c x i1 x i2 (τ) = c xx (i1; i2− i1+ τP ) , i1, i2 ∈ [0, P − 1] (17.17)

FIGURE 17.3: Representation 1: (a) analysis, (b) synthesis.

Using (17.17), it can be shown that auto- and cross-spectra ofx i1(n), x i2(n) can be expressed in terms

of the cyclic spectra ofx(n) as [56],

S x i1 x i2 (ω)e jω(i2−i1) e −j2P π ki2 . (17.19)

Based on (17.16) through (17.19), we infer that cyclostationary signals with periodP can be analyzed

as stationaryP × 1 multichannel processes and vice versa In summary, we have:

Representation 1 (Decimated Components) CS process x(n) can be represented as a P -variate

sta-tionary multichannel process x (n) with components x i (n) = x(nP + i), i = 0, 1, , P − 1 Cyclic spectra and stationary auto- and cross-spectra are related as in ( 17.18 ) and ( 17.19 ).

An alternative means of decomposing a CS process into stationary components is by splitting the

(−π, π] spectral support of X N (ω) into bands each of width 2π/P [22] As shown in Fig.17.4, thiscan be accomplished by passing modulated copies ofx(n) through an ideal low-pass filter H0(ω) with

spectral support(−π/P, π/P ] The resulting subprocesses ¯x m (n) can be shifted up in frequency

and recombined to synthesize the CS process as: x(n) =PP −1 m=0 ¯x m (n) exp(−j2πmn/P ) Within

each band, frequencies are separated by less than 2π/P and according to Property 2, there is no

correlation between spectral components ¯X m,N (ω1) and ¯X m,N (ω2); hence, ¯x m (n) components are

stationary with auto- and cross-spectra having nonzero support over−π/P < ω < π/P They are

related with the cyclic spectra as follows:

FIGURE 17.4: Representation 2: (a) analysis, (b) synthesis.

cross-spectra of ¯x m (n) can be found from the cyclic spectra of x(n) as in ( 17.20 ).

Because ideal low-pass filters cannot be designed, the subband decomposition seems less practical.However, using Representation 1 and exploiting results from uniform DFT filter banks, it is possibleusing FIR low-pass filters to obtain stationary subband components (see e.g., [51, Ch 12]) We willnot pursue this approach further, but Representation 1 will be used next for estimating time-varyingcorrelations of CS processes based on a single data record

17.3 Estimation, Time-Frequency Links, Testing

The time-varying and cyclic quantities introduced in (17.1), (17.2), and (17.10) through (17.12),entail ideal expectations (i.e., ensemble averages) and unless reliable estimators can be devised fromfinite (and often noisy) data records, their usefulness in practice is questionable For stationaryprocesses with (at least asymptotically) vanishing memory,3sample correlations and spectral densityestimators converge to their ensembles as the record lengthN → ∞ Constructing reliable (i.e.,

consistent) estimators for nonstationary processes, however, is challenging and generally impossible.Indeed, capturing time-variations calls for short observation windows, whereas variance reductiondemands long records for sample averages to converge to their ensembles

Fortunately, ACS and CS signals belong to the class of processes with “well-structured” variations that under suitable mixing conditions allow consistent single record estimators The key

time-is to note that althoughc xx (n; τ) and S xx (n; ω) are time-varying, they are expressed in terms of

cyclic quantities,C xx (α k ; τ) and S xx (α k ; ω), which are time-invariant Indeed, in (17.2) and (17.10)time-variation is assigned to the Fourier basis

3 Well-separated samples of such processes are asymptotically independent Sufficient (so-called mixing) conditions include absolute summability of cumulants and are satisfied by many real life signals (see [ 5 , 12 , Ch 2]).

17.3.1 Estimating Cyclic Statistics

First we will consider ACS processes with known cyclesα k Simpler estimators for CS processes and cle estimation methods will be discussed later in the section Ifx(n) has nonzero mean, we estimate the

cy-cyclic mean as in Example17.1using the normalized DFT: ˆC xx (α k ) = N−1PN−1

If the set of cycles is finite, we estimate the time-varying mean as: ˆc xx (n) =Pα k ˆC xx (α k ) exp(jα k n).

Similarly, for zero-mean ACS processes we estimate first cyclic and then time-varying correlationsusing:

Note that ˆC xxcan be computed efficiently using the FFT of the productx(n)x(n + τ).

For cyclic spectral estimation, two options are available: (1) smoothed cyclic periodograms and(2) smoothed cyclic correlograms The first is motivated by (17.12) and smooths the cyclic peri-odogram,I xx (α; ω) := N−1X N (ω)X N (α − ω), using a frequency-domain window W(ω) The

second follows (17.2) and Fourier transforms ˆC xx (α; τ) after smoothing it by a lag-window w(τ)

with supportτ ∈ [−M, M] Either one of the resulting estimates:

12,24,39] and references therein

Whenx(n) is CS with known integer period P , estimation of time-varying correlations and spectra

becomes easier Recall that thanks to Representations 1 and 2, not onlyc xx (n; τ) and S xx (n; ω), but

the processx(n) itself can be analyzed into P stationary components Starting with (17.16), it can

be shown thatc xx (i; τ) = c x i x i+τ (0), where i = 0, 1, , P − 1 and subscript i + τ is understood

mod(P ) Because the subprocesses x i (n) and x i+τ (n) are stationary, their cross-covariances can be

estimated consistently using sample averaging; hence, the time-varying correlation can be estimatedas:

where the integer part[N/P ] denotes the number of samples per subprocess x i (n), and the last

equal-ity follows from the definition ofx i (n) in Representation 1 Similarly, the time-varying periodogram

can be estimated using: I xx (n; ω) = P−1PP −1

then smoothed to obtain a consistent estimate ofS xx (n; ω).

17.3.2 Links with Time-Frequency Representations

Consistency (and hence reliability) of single record estimates is a notable difference between stationary and time-frequency signal analyses Short-time Fourier transforms, the Wigner-Ville,and derivative representations are valuable exploratory (and especially graphical) tools for analyz-ing nonstationary signals They promise applicability on general nonstationarities, but unless slowvariations are present and multiple independent data records are available, their usefulness in es-timation tasks is rather limited In contrast, ACS analysis deals with a specific type of structuredvariation, namely (almost) periodicity, but allows for rapid variations and consistent single recordsample estimates Intuitively speaking, cyclostationarity provides within a single record, multipleperiods that can be viewed as “multiple realizations.” Interestingly, for ACS processes there is a closerelationship between the normalized asymmetric ambiguity functionA(α; τ) [37], and the samplecyclic correlation in (17.21):

τ=−(N−1) x(n) x(n+τ) exp(−jωτ) In fact, the aforementioned equivalences and the consistency

results of [12] establish that ambiguity and Wigner-Ville processing of ACS signals is reliable evenwhen only a single data record is available The following example uses a chirp signal to stress thispoint and shows how some of our sample estimates can be extended to complex processes

EXAMPLE 17.2: Chirp in multiplicative and additive noise

Consider x(n) = s(n) exp(jω0n2) + v(n), where s(n), v(n), are zero mean, stationary, and

mutually independent;c xx (n; τ) is nonperiodic for almost every ω0, and hencex(n) is not

(second-order) ACS Even whenE{s(n)} 6= 0, E{x(n)}isalsononperiodic, implyingthatx(n)isnotfirst-order

ACS either However,

˜c xx∗ (n; τ) := c xx∗ (n + τ; −2τ) := E{x(n + τ)x∗(n − τ)}

exhibits (almost) periodicity and its cyclic correlation is given by: ˜C xx∗ (α; τ) = c ss (τ)δ(α−4ω0τ)+

c vv∗ (2τ)δ(α) Assuming c ss (τ) 6= 0, the latter allows evaluation of ω0by picking the peak of thesample cyclic correlation magnitude evaluated at, e.g.,τ = 1, as follows:

The ˆ˜C xx∗ (α; τ) estimate in (17.27) is nothing but the symmetric ambiguity function Becausex(n)

is ACS, ˆ˜C xx∗can be shown to be consistent This provides yet one more reason for the success of

time-frequency representations with chirp signals Interestingly, (17.27) shows that exploitation ofcyclostationarity allows not only for additive noise tolerance [by avoiding theα = 0 cycle in (17.27)],but also permits parameter estimation of chirps modulated by stationary multiplicative noises(n).

17.3.3 Testing for Cyclostationarity

In certain applications involving man-made (e.g., communication) signals, presence of arity and knowledge of the cycles is assured by design (e.g., baud rates or oversampling factors) Inother cases, however, only a time series{x(n)} N−1 n=0 is given and two questions arise: How does onedetect cyclostationarity, and ifx(n) is confirmed to be CS of a certain order, how does one estimate

cyclostation-the cycles present? The former is addressed by testing hypocyclostation-theses of nonzero ˆC x (α k ), ˆC xx (α k ; τ) or

ˆS xx (α k ; ω) over a fine cycle-frequency grid obtained by sufficient zero-padding prior to taking the

FFT

Specifically, to test whetherx(n) exhibits cyclostationarity in { ˆC xx (α; τ l )} L

l=1for at least one lag,

we form the(2L + 1) × 1 vector ˆc xx (α) := [ ˆC R

xx (α; τ1) ˆC R

xx (α; τ L ); ˆC I

xx (α; τ1) ˆC I

xx (α; τ L )]0

where superscriptR(I) denotes real (imaginary) part Similarly, we define the ensemble vector

cxx (α) and the error e xx (α) := ˆc xx (α) − c xx (α) For N large, it is known that√N e xx (α) is

Gaussian with pdfN (0, 6 c ) An estimate ˆ6 cof the asymptotic covariance can be computed fromthe data [12] Ifα is not a cycle for all {τ l}L

l=1, then cxx (α) ≡ 0, e xx (α) = ˆc xx (α) will have zero

mean, and ˆD2c (α) := ˆc0

xx (α) ˆ6†

c (α)ˆc xx (α) will be central chi-square For a given false-alarm rate,

we find fromχ2tables a threshold0 and test [10]

EXAMPLE 17.3: Cyclostationarity test

Considerx(n) = s1(n) cos(πn/8) + s2(n) cos(πn/4) with s1(n), s2(n), and v(n) zero-mean,

Gaussian, and mutually independent To test for cyclostationarity and retrieve the possible periodspresent,N = 2, 048 samples were generated; s1(n) and s2(n) were simulated as AR(1) with variances

σ2

s1 = σ2

s2 = 2, while v(n) was white with variance σ2

v = 0.1 Figure17.5a shows| ˆC xx (α; 0)|

peaking atα = ±2(π/8), ±2(π/4), 0 as expected, while Fig.17.5b depictsρ xx (ω1, ω2) computed as

in (17.29) withM = 64 The parallel lines in Fig.17.5b are seen at|ω1−ω2| = 0, π/8, π/4 revealing

the periods present One can easily verify from (17.11) thatC xx (α; 0) = (2π)−1Rπ

using the FFT ofx2(n), ρ xx (ω1, ω2) is generally more informative.

Because cyclostationarity is lag-dependent, as an alternative toρ xx (ω1, ω2) one can also plot

| ˆC xx (α; τ)| or | ˆS xx (α; ω)| for all τ or ω Figures17.6and17.7show perspective and contour plots

FIGURE 17.5: (a) Cyclic cross-correlationC xx (α; 0), and (b) coherence ρ xx (ω1, ω2) (Example17.3).

of| ˆC xx (α; τ)| for τ ∈ [−31, 31] and | ˆS xx (α; ω)| for ω ∈ (−π, π], respectively Both sets exhibit

planes (lines) parallel to theτ-axis and ω-axis, respectively, at cycles α = ±2(π/8), ±2(π/4), 0, as

expected

FIGURE 17.6: Cycle detection and estimation (Example17.3): 3D and contour plots of ˆC xx (α; τ).

17.4 CS Signals and CS-Inducing Operations

We have already seen in Examples17.1and17.2that amplitude or index transformations of repetitivenature give rise to one class of CS signals A second category consists of outputs of repetitive (e.g.,periodically varying) systems excited by CS or even stationary inputs Finally, it is possible to have

FIGURE 17.7: Cycle detection and estimation (Example17.3): 3D and contour plots of ˆS xx (α; ω).

cyclostationarity emerging in the output due to the data acquisition process (e.g., multiple sensors

or fractional sampling)

17.4.1 Amplitude Modulation

General examples in this class include signalsx1(n) and x2(n) of (17.7) or their combinations asdescribed by Property 1 More specifically, we will focus on communication signals where random(often i.i.d.) information dataw(n) are D/A converted with symbol period T0, to obtain the process:

w c (t) = Pl w(l)δ D (t − lT0), which is CS in the continuous variable t The continuous-time

signalw c (t) is subsequently pulse shaped by the transmit filter h (tr) c (t), modulated with the carrier

exp(jω c t), and transmitted over the linear time-invariant (LTI) channel h (ch) c (t) On reception, the

carrier is removed and the data are passed through the receive filterh (rec) c (t) to suppress stationary

additive noise Defining the composite channelh c (t) := h (tr) c ? h (ch) c ? h (rec) c (t), the continuous time

received signal at the baseband is:

Ifω e = 0, x(n) (and thus v(n)) is stationary, whereas ω e 6= 0 renders r(n) similar to the ACS

signal in Example17.1 Whenw(n) is zero-mean, i.i.d., complex symmetric, we have: E{w(n)} ≡ 0,

However, peak-picking the cyclic fourth-order correlation [Fourier coefficients ofr4(n)] yields 4ω e

uniquely, providedω e < π/4 If E{w4(n)} ≡ 0, higher powers can be used to estimate and recover

ω e

Having estimatedω e, we form exp(−jω e n) r(n) in order to demodulate the signal in (17.31).Traditionally, cyclostationarity is removed from the discrete-time information signal, although itmay be useful for other purposes (e.g., blind channel estimation) to retain cyclostationarity at thebaseband signalx(n) This can be accomplished by multiplying w(n) with a P -periodic sequence p(n) prior to pulse shaping The noise-free signal in this case is x(n) =Pl p(l)w(l)h(n − l), and

has correlation,¯c xx (n; τ) = σ2

wP

l |p(n−l)|2h(l)h∗(l +τ), which is periodic with period P Cyclic

correlations and spectra are given by [28]

m=0 |p(m)|2exp(−jαm) and H (ω) := PL l=0 h(l) exp(−jωl) As we will

see later in this section, cyclostationarity can also be introduced at the transmitter using multirateoperations, or at the receiver by fractional sampling With a CS input, the channelh(n) can be

identified using noisy output samples only [28,64,65] — an important step towards blind equalization

of (e.g., multipath) communication channels

p(n)s(n)+v(n) can be used to model systematically missing observations Periodically, the stationary

signals(n) is observed in noise v(n) for P1samples and disappears for the nextP − P1data Using

C xx (α; τ) = P2(α; τ)c ss (τ), the period P [and thus P2(α; τ)] can be determined Subsequently,

c ss (τ) can be retrieved and used for parametric or nonparametric spectral analysis of s(n); see [32]and references therein

17.4.2 Time Index Modulation

Suppose that a random CS signals(n) is delayed by D samples and received in zero-mean stationary

noisev(n) as: x(n) = s(n − D) + v(n) With s(n) independent of v(n), the cyclic correlation is

C xx (α; τ) = C ss (α; τ) exp(jαD)+δ(α)c vv (τ) and the delay manifests itself as a phase of a complex

exponential But even whens(n) models a narrowband deterministic signal, the delay appears in the

exponent sinces(n − D(n)) ≈ s(n) exp(jD(n)) [53] Time-delay estimation of CS signals appearsfrequently in sonar and radar for range estimation whereD(n) = νn and ν denotes velocity of

propagation.D(n) is also used to model Doppler effects that appear when relative motion is present.

Note that with time-varying (e.g., accelerating) motion we haveD(n) = γ n2and cyclostationarityappears in the complex correlation as explained in Example17.2

Polynomial delays are one form of time scale transformations Another one isd(n) = λn + p(n),

whereλ is a constant and p(n) is periodic with period P (e.g., [38]) For stationarys(n), signal x(n) = s[d(n)] is CS because c xx (n + lP ; τ) = c ss [d(n + lP + τ) − d(n + lP )] = c ss [λτ + p(n) −

p(n + τ)] = c xx (n; τ) A special case is the familiar FM model with d(n) = ω c n + h sin(ω0n) where

h here denotes the modulation index The signal and its periodically varying correlation are given

by:

In addition to communications, frequency modulated signals appear in sonar and radar when rotatingand vibrating objects (e.g., propellers or helicopter blades) induce periodic variations in the phase ofincident narrowband waveforms [2,67]