Cyclostationarity: Half a Century of Research

ARTICLE IN PRESS

Signal Processing 86 (2006) 639–697 www.elsevier.com/locate/sigpro Review Cyclostationarity: Half a century of research

William A. Gardnera, Antonio Napolitanob,Ã, Luigi Paurac

aStatistical Signal Processing Incorporated (SSPI), 1909 Jefferson Street, Napa, CA 94559, USA bUniversita` di Napoli ‘‘Parthenope’’, Dipartimento per le Tecnologie, via Acton 38, I-80133 Napoli, Italy cUniversita` di Napoli ‘‘Federico II’’, Dipartimento di Ingegneria Elettronica e delle Telecomunicazioni, via Claudio 21, I-80125 Napoli, Italy

Received 22 December 2004; accepted 29 June 2005 Available online 6 September 2005

Abstract

In this paper, a concise survey of the literature on cyclostationarity is presented and includes an extensive bibliography. The literature in all languages, in which a substantial amount of research has been published, is included. Seminal contributions are identified as such. Citations are classified into 22 categories and listed in chronological order. Both stochastic and nonstochastic approaches for signal analysis are treated. In the former, which is the classical one, signals are modelled as realizations of stochastic processes. In the latter, signals are modelled as single functions of time and statistical functions are defined through infinite-time averages instead of ensemble averages. Applications of cyclostationarity in communications, signal processing, and many other research areas are considered. r 2005 Elsevier B.V. All rights reserved.

Keywords: Cyclostationarity; Almost-cyclostationary time series; Almost-cyclostationary processes; Bibliography

Contents

1. Introduction ...... 641 2. General treatments...... 642 3. General properties and structure of stochastic processes and time-series ...... 642 3.1. Introduction ...... 642 3.2. Stochastic processes ...... 643 3.2.1. Continuous-time processes ...... 643 3.2.2. Discrete-time processes ...... 645

ÃCorresponding author. Tel.: +39 081 768 37 81; fax: +39 081 768 31 49. E-mail addresses: [email protected] (W.A. Gardner), [email protected] (A. Napolitano), [email protected] (L. Paura).

640 W.A. Gardner et al. / Signal Processing 86 (2006) 639–697

3.3. Time series ...... 646 3.3.1. Continuous-time time series ...... 646 3.3.2. Discrete-time time series ...... 648 3.4. Link between the stochastic and fraction-of-time approaches ...... 649 3.5. Complex processes and time series ...... 649 3.6. Linear filtering ...... 650 3.6.1. Structure of linear almost-periodically time-variant systems ...... 650 3.6.2. Input/output relations in terms of cyclic statistics ...... 650 3.7. Product modulation ...... 651 3.8. Supports of cyclic spectra of band limited signals ...... 652 3.9. Sampling and aliasing...... 653 3.10. Representations by stationary components ...... 654 3.10.1. Continuous-time processes and time series ...... 654 3.10.2. Discrete-time processes and time series...... 655 4. Ergodic properties and measurement of characteristics ...... 656 4.1. Estimation of the cyclic autocorrelation function and the cyclic spectrum ...... 656 4.2. Two alternative approaches to the analysis of measurements on time series ...... 657 5. Manufactured signals: modelling and analysis ...... 658 5.1. General aspects ...... 658 5.2. Examples of communication signals...... 659 5.2.1. Double side-band amplitude-modulated signal ...... 659 5.2.2. Pulse-amplitude-modulated signal ...... 659 5.2.3. Direct-sequence spread-spectrum signal ...... 660 6. Natural signals: modelling and analysis ...... 661 7. Communications systems: analysis and design ...... 661 7.1. General aspects ...... 661 7.2. Cyclic Wiener filtering ...... 661 8. Synchronization...... 663 8.1. Spectral line generation...... 663 9. Signal parameter and waveform estimation ...... 663 10. Channel identification and equalization ...... 664 10.1. General aspects ...... 664 10.2. LTI-system identification with noisy-measurements ...... 664 10.3. Blind LTI-system identification and equalization...... 665 10.4. Nonlinear-system identification ...... 665 11. Signal detection and classification, and source separation ...... 665 12. Periodic AR and ARMA modelling and prediction...... 666 13. Higher-order statistics ...... 666 13.1. Introduction ...... 666 13.2. Higher-order cyclic statistics ...... 667 13.2.1. Continuous-time time series ...... 667 13.2.2. Discrete-time time series ...... 669 14. Applications to circuits, systems, and control ...... 670 15. Applications to acoustics and mechanics ...... 670 16. Applications to econometrics ...... 671 17. Applications to biology ...... 671 18. Level crossings ...... 671 19. Queueing ...... 671 20. Cyclostationary random fields...... 671 21. Generalizations of cyclostationarity ...... 671 21.1. General aspects ...... 671 ARTICLE IN PRESS

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21.2. Generalized almost-cyclostationary signals ...... 672 21.3. Spectrally correlated signals ...... 672 References ...... 673 Further references ...... 697

1. Introduction autocorrelation functions that are independent of time, depending on only the lag para- Many processes encountered in nature arise meter, and all distinct spectral components are from periodic phenomena. These processes, uncorrelated. although not periodic functions of time, give rise As an alternative, the presence of periodicity in to random data whose statistical characteristics the underlying data-generating mechanism of a vary periodically with time and are called cyclosta- phenomenon can be described without modelling tionary processes [2.5]. For example, in telecom- the available data as a sample path of a stochastic munications, telemetry, radar, and sonar process but, rather, by modelling it as a single applications, periodicity is due to modulation, function of time [4.31]. Within this nonstochastic sampling, multiplexing, and coding operations. In framework, a time-series is said to exhibit second- mechanics it is due, for example, to gear rotation. order cyclostationarity (in the wide sense), as first In radio astronomy, periodicity results from defined in [2.8], if there exists a stable quadratic revolution and rotation of planets and on pulsa- time-invariant transformation of the time-series tion of stars. In econometrics, it is due to that gives rise to finite-strength additive sinewave seasonality; and in atmospheric science it is components. due to rotation and revolution of the earth. In this paper, a concise survey of the literature The relevance of the theory of cyclostationarity (in all languages in which a substantial amount of to all these fields of study and more was first research has been published) on cyclostationarity proposed in [2.5]. is presented and includes an extensive bibliography Wide-sense cyclostationary stochastic processes and list of issued patents. Citations are classified have autocorrelation functions that vary periodi- into 22 categories and listed, for each category, in cally with time. This function, under mild reg- chronological order. In Section 2, general treat- ularity conditions, can be expanded in a Fourier ments and tutorials on the theory of cyclostatio- series whose coefficients, referred to as cyclic narity are cited. General properties of processes autocorrelation functions, depend on the lag para- and time-series are presented in Section 3. In meter; the frequencies, called cycle frequencies, are Section 4, the problem of estimating statistical all multiples of the reciprocal of the period of functions is addressed. Models for manufactured cyclostationarity [2.5]. Cyclostationary processes and natural signals are considered in Sections 5 have also been referred to as periodically correlated and 6, respectively. Communications systems and processes [2.18,3.5]. More generally, if the frequen- related problems are treated in Sections 7–11. cies of the (generalized) Fourier series expansion of Specifically, the analysis and design of commu- the autocorrelation function are not commensu- nications systems is addressed in Section 7, the rate, that is, if the autocorrelation function is an problem of synchronization is addressed in Section almost-periodic function of time, then the process 8, the estimation of signal parameters and wave- is said to be almost-cyclostationary [3.26] or, forms is addressed in Section 9, the identification equivalently, almost-periodically correlated [3.5]. and equalization of channels is addressed in The almost-periodicity property of the autocorre- Section 10, and the signal detection and classifica- lation function is manifested in the frequency tion problems and the problem of source separa- domain as correlation among the spectral com- tion are addressed in Section 11. Periodic ponents of the process that are separated by autoregressive (AR) and autoregressive moving- amounts equal to the cycle frequencies. In contrast average (ARMA) modelling and prediction are to this, wide-sense stationary processes have treated in Section 12. In Section 13, theory and ARTICLE IN PRESS

642 W.A. Gardner et al. / Signal Processing 86 (2006) 639–697 applications of higher-order cyclostationarity are [13.4,13.5], and treated in depth in [13.9,13.13, presented. We address applications to circuits, 13.14,13.17]. An analogous treatment for stochas- systems, and control in Section 14, to acoustics tic processes is given in [13.10]. and mechanics in Section 15, to econometrics in Section 16, and to biology in Section 17. Applica- tions to the problems of level crossing and 3. General properties and structure of stochastic queueing are addressed in Sections 18 and 19, processes and time-series respectively. Cyclostationary random fields are treated in Section 20. In Section 21, some classes 3.1. Introduction of nonstationary signals that extend the class of almost-cyclostationary signals are considered. Fi- General properties of cyclostationary processes nally, some miscellaneous references are listed (see [3.1–3.91]) are derived in terms of the Fourier [22.1–22.8]. Further references only indirectly series expansion of the autocorrelation function. related to cyclostationarity are [23.1–23.15]. The frequencies, called cycle frequencies, are To assist readers in ‘‘going to the source’’, multiples of the reciprocal of the period of seminal contributions—if known— are identified cyclostationarity and the coefficients, referred to within the literature published in English. In some as cyclic autocorrelation functions, are continuous cases, identified sources may have been preceded in functions of the lag parameter. Cyclostationary the literature of another language, most likely processes are characterized in the frequency Russian. For the most part, the subject of this domain by the cyclic spectra, which are the Fourier survey developed independently in the literature transforms of the cyclic autocorrelation functions. published in English. The cyclic spectrum at a given cycle frequency represents the density of correlation between two spectral components of the process that are 2. General treatments separated by an amount equal to the cycle frequency. Almost-cyclostationary processes have General treatments on cyclostationarity are in autocorrelation functions that can be expressed in [2.1–2.18]. The first extensive treatments of the a (generalized) Fourier series whose frequencies theory of cyclostationary processes can be found are possibly incommensurate. in the pioneering works of Hurd [2.1] and Gardner The first contributions to the analysis of the [2.2].In[4.13,2.5,2.11], the theory of second-order general properties of cyclostationary stochastic cyclostationary processes is developed mainly with processes are in the Russian literature [3.1–3.3, reference to continuous-time stochastic processes, 3.5,3.7–3.9]. The problem of spectral analysis is but discrete-time is the focus in [4.13]. Discrete- mainly treated in [3.10,3.11,3.32,3.35,3.51,3.53, time processes are treated more generally in [2.17] 3.61,3.68,3.69,3.80,3.88]. The stationarizing effects in a manner largely analogous to that in [2.5]. The of random shifts are examined in [3.20,3.26,3.57] statistical characterization of cyclostationary time- and the important impact of random shifts on series in the nonstochastic framework is intro- cyclo-ergodic properties is exposed in [2.15]. The duced and treated in depth [2.6,2.8,2.12,2.14] with problem of filtering is considered in [3.9,3.13,3.67]. reference to continuous-time signals and in [2.15] The mathematical link between the stochastic for both continuous- and discrete-time signals. and nonstochastic approaches, which was first The case of complex signals is introduced and established in [2.5,2.12], is rigorously treated in treated in depth in [2.8,2.9]. Finally, a rigorous [3.87]. Wavelet analysis of cyclostationary pro- treatment of periodically correlated processes cesses is addressed in [3.70,3.81]. within the framework of harmonizable processes For further-related references, see the general is given in [2.18]. treatments in [2.1,2.2,2.4–2.8,2.11–2.13,2.15,2.18], The theory of higher-order cyclostationarity in and also [4.13,4.31,4.41,13.2,13.3,13.6,16.3,18.3, the nonstochastic framework is introduced in 21.9,21.14]. ARTICLE IN PRESS

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3.2. Stochastic processes frequencies. Wide-sense cyclostationary processes have also been called periodically correlated 3.2.1. Continuous-time processes processes (see e.g., [2.1,3.3,3.59]). As first shown Let us consider a continuous-time real-valued in [3.23], xðtÞ and its frequency-shifted version stochastic process fxðt; oÞ; t 2 R; o 2 Og, with ab- xðtÞej2pnt=T0 are correlated. The wide-sense station- breviated notation xðtÞ when this does not create ary processes are the special case of cyclostation- n=T0 c ambiguity, defined on a probability space ary processes for which Rx ðtÞ 0 only for ðO; F; PÞ, where O is the sample space, equipped n ¼ 0. It can be shown that if xðtÞ is cyclosta- with the s-field F,andP is a probability measure tionary with period T 0, then the stochastic process defined on the elements of F. xðt þ yÞ, where y is a random variable that is The process xðtÞ is said to be Nth-order uniformly distributed in ½0; T0Þ and is statistically cyclostationary in the strict sense [2.2,2.5] if its independent of xðtÞ, is wide sense stationary Nth-order distribution function [2.5,3.20,3.26,3.57]. A more general class of stochastic processes is F ðx ; ...; x ; x Þ xðtþt1ÞxðtþtN1ÞxðtÞ 1 N1 N obtained if the autocorrelation function Rxðt; tÞ is 9Pfxðt þ t1Þpx1; ...; xðt þ tN1Þ almost periodic in t for each t. A function zðtÞ is said to be almost periodic if it is pxN1; xðtÞpxN gð3:1Þ the limit of a uniformly convergent sequence of is periodic in t with some period, say T 0: trigonometric polynomials in t [23.2,23.3,23.4, Paragraphs 24–25,23.6,23.7, Part 5, 23.11]: F xðtþt1þT0ÞxðtþtN1þT0ÞxðtþT0Þðx1; ...; xN1; xN Þ X zðtÞ¼ z ej2pat, (3.8) ¼ F xðtþt1ÞxðtþtN1ÞxðtÞðx1; ...; xN1; xN Þ a N1 a2A 8t 2 R 8ðt1; ...; tN1Þ2R where A is a countable set, the frequencies a 2 A 8ð Þ2 N ð Þ x1; ...; xN R . 3:2 are possibly incommensurate, and the coefficients The process xðtÞ is said to be second-order za are given by cyclostationary in the wide sense [2.2,2.5] if its j2pat za9hzðtÞe it mean EfxðtÞg and autocorrelation function Z 1 T=2 ¼ lim zðtÞej2pat dt. ð3:9Þ Rxðt; tÞ9Efxðt þ tÞxðtÞg (3.3) T!1 T T=2 are periodic with some period, say T : 0 Such functions are called almost-periodic in the

Efxðt þ T 0Þg ¼ EfxðtÞg, (3.4) sense of Bohr [23.3, Paragraphs 84–92] or, equivalently, uniformly almost periodic in t in the sense of Besicovitch [23.2, Chapter 1]. Rxðt þ T0; tÞ¼Rxðt; tÞ (3.5) The process xðtÞ is said to be Nth-order almost- for all t and t. Therefore, by assuming that the cyclostationary in the strict sense [2.2,2.5] if its Nth- Fourier series expansion of Rxðt; tÞ is convergent order distribution function (3.1) is almost-periodic to Rxðt; tÞ, we can write in t. With reference to the autocorrelation proper- Xþ1 ties, a continuous-time real-valued stochastic pro- n=T0 j2pðn=T0Þt Rxðt; tÞ¼ Rx ðtÞe , (3.6) cess xðtÞ is said to be almost-cyclostationary (ACS) n¼1 in the wide sense [2.2,2.5,3.26] if its autocorrelation where the Fourier coefﬁcients function Rxðt; tÞ is an almost periodic function of t Z (with frequencies not depending on t). Thus, it is 1 T0=2 the limit of a uniformly convergent sequence of Rn=T0 ðtÞ9 R ðt; tÞej2pðn=T0Þt dt (3.7) x T x trigonometric polynomials in t: 0 T0=2 X a j2pat are referred to as cyclic autocorrelation functions Rxðt; tÞ¼ RxðtÞe ,(3.10) 2 and the frequencies fn=T0gn2Z are called cycle a A ARTICLE IN PRESS

644 W.A. Gardner et al. / Signal Processing 86 (2006) 639–697 where tive sinewave component: Z 1 T=2 j2pat a 9 j2pat hrxðt; tÞe it ¼ 0 8a 2 R. (3.16) RxðtÞ lim Rxðt; tÞe dt (3.11) T!1 T T=2 In the special case where limjtj!1 rxðt; tÞ¼0, xðtÞ is the cyclic autocorrelation function at cycle is said to be asymptotically almost cyclostationary frequency a. As ﬁrst shown in [3.26],ifxðtÞ is an [3.26]. ACS process, then the process xðtÞ and its Let us now consider the second-order (wide- frequency-shifted version xðtÞej2pat are correlated sense) characterization of ACS processes in the when a 2 A. Wide-sense almost-cyclostationary frequency domain. Let processes have also been called almost-periodically Z correlated processes (see e.g., [2.1,3.5,3.59]). The Xðf Þ9 xðtÞej2pft dt (3.17) wide-sense cyclostationary processes are obtained R as a special case of the ACS processes when A be the stochastic process obtained from Fourier fn=T 0gn2Z for some T 0. transformation of the ACS process xðtÞ, where the Let At be the set Fourier transform is assumed to exist, at least in 9 a a the sense of distributions (generalized functions) At fa 2 R : RxðtÞ 0g. (3.12) [23.10], with probability 1. By using (3.10), in the In [3.59], it is shown that the ACS processes are sense of distributions we obtain characterized by the following conditions: X a EfXðf 1ÞX ðf 2Þg ¼ Sxðf 1Þdðf 2 f 1 þ aÞ, (1) The set a2A [ (3.18) A9 At (3.13) t2R where dðÞ is the Dirac delta, superscript is is countable. complex conjugation, and Z (2) The autocorrelation function Rxðt; tÞ is uni- a 9 a j2pf t formly continuous in t and t. Sxðf Þ RxðtÞe dt (3.19) (3) The time-averaged autocorrelation function R 0 9 RxðtÞ hRxðt; tÞit is continuous for t ¼ 0 is referred to as the cyclic spectrum at cycle (and, hence, for every t). frequency a. Therefore, for an ACS process, (4) The process is mean-square continuous, that is correlation exists between spectral components sup Efjxðt þ tÞxðtÞj2g!0ast ! 0. that are separated by amounts equal to the cycle t2R frequencies. The support in theðf 1; f 2Þ plane of the (3.14) spectral correlation function E Xðf 1ÞX ðf 2Þ consists of parallel lines with unity slope. The density of spectral correlation on this support is described More generally, a stochastic process xðtÞ is said to by the cyclic spectra aðf Þ, a 2 A, which can be exhibit cyclostationarity at cycle frequency a if Sx expressed as [2.5] RaðtÞc0 [2.5]. In such a case, the autocorrelation x Z function can be expressed as T=2 X a 1 Sxðf Þ¼ lim lim EfDfX 1=Df ðt; f Þ a j2pat Df !0 T!1 T Rxðt; tÞ¼ RxðtÞe þ rxðt; tÞ, (3.15) T=2 a2A X 1=Df ðt; f aÞg dt, ð3:20Þ where the function X where the order of the two limits cannot be a j2pat RxðtÞe reversed, and a2A Z tþZ=2 is not necessarily continuous in t and the term j2pfs X Zðt; f Þ9 xðsÞe ds. (3.21) rxðt; tÞ does not contain any ﬁnite-strength addi- tZ=2 ARTICLE IN PRESS

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a Therefore, the cyclic spectrum Sxðf Þ is also called ACS, then it follows that [2.1,2.18,3.59] the spectral correlation density function. It repre- X ðf ; f Þ¼ aðf Þdðf f þ aÞ df df . sents the time-averaged statistical correlation (with dg 1 2 Sx 1 2 1 1 2 a2A zero lag) of two spectral components at frequencies f and f a, as the bandwidth approaches (3.26) zero. For a ¼ 0, the cyclic spectrum reduces to the Finally, note that symmetric definitions of cyclic 0 power spectrum or spectral density function Sxðf Þ autocorrelation function and cyclic spectrum have and (3.19) reduces to the Wiener–Khinchin Rela- been widely used (see, e.g., [2.5,2.8,2.11]). They are tion. Consequently, when (3.19) and (3.20) was linked to the asymmetric definitions (3.11) and discovered in [2.5] it was dubbed the Cyclic (3.20) by the relationships Wiener–Khinchin Relation. Z 1 T=2 In contrast, for wide-sense stationary processes lim Efxðt þ t=2Þxðt t=2Þg the autocorrelation function does not depend on t, T!1 T T=2 j2pat a jpat 0 e dt ¼ RxðtÞe ð3:27Þ Efxðt þ tÞxðtÞg ¼ RxðtÞ (3.22) and and, equivalently, no correlation exists between Z distinct spectral components, 1 T=2 lim lim Df EfX 1=Df ðt; f þ a=2Þ 0 Df !0 T!1 T T=2 EfXðf 1ÞX ðf 2Þg ¼ Sxðf 1Þdðf 2 f 1Þ. (3.23) X ðt f Þg t ¼ aðf þ Þ ð Þ A covariance (or autocorrelation) function 1=Df ; a=2 d Sx a=2 , 3:28 Efxðt1Þx ðt2Þg is said to be harmonizable if it can respectively. be expressed as Z 3.2.2. Discrete-time processes j2pðf 1t1f 2t2Þ Efxðt1Þx ðt2Þg ¼ e dgðf 1; f 2Þ, (3.24) R2 Let us consider a discrete-time real-valued stochastic process fxðn; oÞ; n 2 Z; o 2 Og, with where gðf ; f Þ is a covariance of bounded varia- 1 2 abbreviated notation xðnÞ when this does not tion on R R and the integral is a Fourier– create ambiguity. The stochastic process xðnÞ is Stieltjes transform [23.5]. Moreover, a second- said to be second-order almost-cyclostationary in order stochastic process xðtÞ is said to be the wide sense [2.2,2.5,4.13] if its autocorrelation harmonizable if there exists a second-order sto- function chastic process wðf Þ with covariance function e Efwðf 1Þw ðf 2Þg ¼ gðf 1; f 2Þ of bounded variation Rxðn; mÞ9Efxðn þ mÞxðnÞg (3.29) on R R such that Z is an almost-periodic function of the discrete-time parameter n. Thus, it can be expressed as xðtÞ¼ ej2pft dwðf Þ (3.25) X R ea e e ðn; mÞ¼ e ðmÞej2pan, (3.30) with probability one. In [23.5], it is shown that a Rx Rx e necessary condition for a stochastic process to be ea2A harmonizable is that it be second-order contin- where uous. Moreover, it is shown that a stochastic e XN process is harmonizable if and only if its covar- e a 9 1 e j2pean RxðmÞ lim Rxðn; mÞe N!1 þ iance is harmonizable. If a stochastic process 2N 1 n¼N is harmonizable, in the sense of distributions (3.31) [23.10], we have dwðf Þ¼Xðf Þ df and dgðf 1; f 2Þ¼ Efdwðf Þ dwðf Þg ¼ EfXðf ÞX ðf Þg df df ,and is the cyclic autocorrelation function at cycle 1 2 1 2 1 2 e wðf Þ is the indefinite integral of Xðf Þ or the frequency a and integrated Fourier transform of xðtÞ [23.4]. There- e e9 e 1 1 e a c fore, if the stochastic process is harmonizable and A fa 2½2; 2Þ : RxðmÞ 0g (3.32) ARTICLE IN PRESS

646 W.A. Gardner et al. / Signal Processing 86 (2006) 639–697 is a countable set. Note that the cyclic autocorre- proach turns out to be more appropriate when an ea ensemble of realizations does not exist and would lation function e ðmÞ is periodic in ea with period Rx have to be artificially introduced just to create a 1. Thus, the sum in (3.30) can be equivalently e mathematical model, that is, the stochastic pro- e 9 e e a c cess. Such a model can be unnecessarily abstract extended to the set A1 fa 2½0; 1Þ : RxðmÞ 0g. e f when there is only a single time series at hand. In general, the set A (or A1) contains possibly incommensurate cycle frequencies ea. In the special In the FOT probability approach, probabilistic parameters are defined through infinite-time Ae f N ðN Þ N g case where 1 0; 1= 0; ...; 0 1 = 0 for averages of a single time series (and functions of some integer N0, the autocorrelation function e this time series) rather than through expected Rxðn; mÞ is periodic in n with period N0 and the values or ensemble averages of a stochastic process xðnÞ is said to be cyclostationary in the wide process. Moreover, starting from this single time sense.IfN0 ¼ 1 then xðnÞ is wide-sense stationary. series, a (possibly time varying) probability dis- Let tribution function can be constructed and this X XeðnÞ9 xðnÞej2pnn (3.33) leads to an expectation operation and all the n2Z associated familiar probabilistic concepts and parameters, such as stationarity, cyclostationarity, be the stochastic process obtained from Fourier nonstationarity, independence, mean, variance, transformation (in a generalized sense) of the ACS moments, cumulants, etc. For comprehensive process xðnÞ. By using (3.30), in the sense of treatments of the FOT probability framework see distributions it can be shown that [3.59] [2.8,2.12,2.15] and, for more mathematical rigor e e on foundations and existence proofs, see [23.15]. EfXðn1ÞX ðn2Þg X e X The extension of the Wold isomorphism to e a e cyclostationary sequences was first introduced in ¼ Sxðn1Þ dðn2 n1 þ a ‘Þ, ð3:34Þ ‘2Z [2.5], treated more in depth in [2.12], and finally— ea2Ae with mathematical rigor—in [3.87]. where The time-variant FOT probability framework is e X e based on the decomposition of functions of a time e a 9 e a j2pnm SxðnÞ RxðmÞe (3.35) series into their (possibly zero) almost-periodic m2Z components and residual terms. Starting from is the cyclic spectrum at cycle frequency ea. The such a decomposition, the expectation operator is ea defined. Specifically, for any finite-average-power e ð Þ e cyclic spectrum Sx n is periodic in both n and a time series xðtÞ, let us consider the decomposition with period 1. In [3.3,4.41], it is shown that discrete-time xðtÞ9xapðtÞþxrðtÞ, (3.36) cyclostationary processes are harmonizable. where xapðtÞ is an almost-periodic function and xrðtÞ a residual term not containing finite-strength 3.3. Time series additive sinewave components; that is, hx ðtÞej2pati 0 8a 2 R. (3.37) 3.3.1. Continuous-time time series r t A statistical analysis framework for time series The almost-periodic component extraction operator that is an alternative to the classical stochastic- Efagfg is defined to be the operator that extracts all process framework is the fraction-of-time (FOT) the finite-strength additive sinewave components probability framework first introduced for ACS of its argument, that is, time series in [2.5,2.8,2.12]; see also [2.15]. In the EfagfxðtÞg9x ðtÞ. (3.38) FOT probability framework, signals are modelled ap as single functions of time (time series) rather than Let xðtÞ, t 2 R, be a real-valued continuous-time sample paths of stochastic processes. This ap- time series (a single function of time) and let us ARTICLE IN PRESS

W.A. Gardner et al. / Signal Processing 86 (2006) 639–697 647 assume that the set G1 of frequencies (for every x) let us assume that the set G2 of frequencies (for of the almost-periodic component of the function every x1; x2 and t) of the almost-periodic compo- of t 1fxðtÞpxg is countable, where nent of the function of t 1fxðtþtÞpx1g1fxðtÞpx2g is ( countable. Then, the function of x1 and x2 1; t : xðtÞpx; 9 1fxðtÞpxg (3.39) fag 0; t : xðtÞ4x F xðtþtÞxðtÞðx1; x2Þ fag 9E f1fxðtþtÞ x g1fxðtÞ x ggð3:43Þ is the indicator function of the set ft 2 R : xðtÞpxg. p 1 p 2 In [2.12] it is shown that the function of x is a valid second-order joint cumulative distribu- F fag ðxÞ9Efagf1 g (3.40) tion function for every fixed t and t, except for the xðtÞ fxðtÞpxg right-continuity property (in the discontinuity for any t is a valid cumulative distribution func- points) with respect to x1 and x2. Moreover, the second-order derivative, with respect to x1 and x2, tion except for the right-continuity property (in fag fag the discontinuity points). That is, it has values of F xðtþtÞxðtÞðx1; x2Þ, denoted by f xðtþtÞxðtÞðx1; x2Þ,is fag a valid second-order joint probability density in ½0; 1, is non decreasing, F xðtÞð1Þ ¼ 0, and fag function [2.12]. Furthermore, it can be shown that F xðtÞðþ1Þ ¼ 1. Moreover, its derivative with res- fag the function pect to x, denoted by f xðtÞðxÞ, is a valid probability density function and, for any well-behaved func- 9 fag tion gðÞ, it follows that Rxðt; tÞ E fxðt þ tÞxðtÞg (3.44) Z is a valid autocorrelation function and can be EfagfgðxðtÞÞg ¼ gðxÞf fag ðxÞ dx (3.41) xðtÞ characterized by R Z fag which reveals that E fg is the expecta- fag Rxðt; tÞ¼ x1x2f xðtþtÞxðtÞðx1; x2Þ dx1 dx2 tion operator with respect to the distribution R2 fag X function F xðtÞðxÞ for the time series xðtÞ. The a j2pat ¼ RxðtÞe , ð3:45Þ result (3.41), first introduced in [2.8], is referred a2A to as the fundamental theorem of temporal expectation, by analogy with the corresponding funda- where A is a countable set and mental theorem of expectation from probability a 9 j2pat theory. RxðtÞ hxðt þ tÞxðtÞe it (3.46) For an almost-periodic signal xðtÞ,wehave xðtÞxapðtÞ and, hence, is the (nonstochastic) cyclic autocorrelation func- fag tion at cycle frequency a ða 2 RÞ. E fxðtÞg ¼ xðtÞ. (3.42) The classification of the kind of nonstationarity That is, the almost-periodic functions are the of a time series is made on the basis of the elements deterministic signals in the FOT probability contained in the set A. In general, the set A can framework. All the other signals are the random contain incommensurate cycle frequencies a and, signals. Note that the term ‘‘random’’ here is not in such a case, the time series is said to be wide- intended to be synonymous with ‘‘stochastic’’. In sense almost cyclostationary. In the special case fact, the adjective stochastic is adopted, as usual, where A fk=T 0gk2Z the time series xðtÞ is said to when an ensemble of realizations or sample paths be wide-sense cyclostationary. If the set A contains exists, whereas the adjective random is used in only the element a ¼ 0, then the time series xðtÞ is reference to a single function of time. said to be wide-sense stationary. Analogously, a second-order characterization Finally, note that the periodic component with for the real-valued time series xðtÞ can be obtained period T 0 of an almost-periodic time series (or lag by using the almost-periodic component extraction product time series) zðtÞ can be extracted by operator as the expectation operator. Specifically, exploiting the synchronized averaging identity ARTICLE IN PRESS

648 W.A. Gardner et al. / Signal Processing 86 (2006) 639–697 introduced in [2.5,2.8]: the right-continuity property (in the discontinuity feag Xþ1 points) with respect to x1 and x2. In (3.50), E fg j2pðk=T0Þt denotes the discrete-time almost-periodic compo- zk=T0 e k¼1 nent extraction operator, which is defined analo- 1 XN gously to its continuous-time counterpart. The ¼ lim zðt nT 0Þ, ð3:47Þ second-order derivative with respect to x1 and x2 N!1 2N þ 1 n¼N feag feag of F xðnþmÞxðnÞðx1; x2Þ, denoted by f xðnþmÞxðnÞðx1; x2Þ, where the Fourier coefficients z are defined k=T0 is a valid second-order joint probability density according to (3.9). function [2.12]. Furthermore, it can be shown that ACS time series are characterized in the spectral the function domain by the (nonstochastic) cyclic spectrum or spectral correlation density function at cycle fre- e Re ðn; mÞ9Efagfxðn þ mÞxðnÞg (3.51) quency a: x Z T=2 is a valid autocorrelation function and can be a 9 1 Sxðf Þ lim lim Df characterized by Df !0 T!1 T T=2 Z X ðt; f ÞX ðt; f aÞ dt, ð3:48Þ 1=Df 1=Df e feag Rxðn; mÞ¼ x1x2f xðnþmÞxðnÞðx1; x2Þ dx1 dx2 where X ðt; f Þ is defined according to (3.21) and R2 1=Df X ea e the order of the two limits cannot be reversed. The ¼ Re ðmÞej2pan, ð3:52Þ Cyclic Wiener Relation introduced in [2.8] x Z ea2Ae a a j2pf t Sxðf Þ¼ RxðtÞe dt (3.49) R where links the cyclic autocorrelation function to the e e9 e 1 1 ea c cyclic spectrum. This relation generalizes that for A fa 2½2; 2Þ : RxðmÞ 0g (3.53) a ¼ 0, which was first dubbed the Wiener Relation in [2.5] to distinguish it from the Khinchin Relation is a countable set and (3.19), which is frequently called the Wiener– XN Khinchin Relation. ea e 9 1 j2pean RxðmÞ lim xðn þ mÞxðnÞe N!1 þ 2N 1 n¼N 3.3.2. Discrete-time time series (3.54) The characterization of discrete-time time series (sequences) is similar to that of continuous-time is the (nonstochastic) discrete-time cyclic autocor- time series. We consider here only the wide-sense relation function at cycle frequency ea. second-order characterization. Obviously, as in the stochastic framework, the Let xðnÞ, n 2 Z, be a real-valued discrete-time e e ea time series. Let us assume that the set G2 of cyclic autocorrelation function RxðmÞ is periodic in e frequencies (for every x1; x2 and m) of the almost- a with period 1. Thus, the sum in (3.52) can be e periodic component of the function of equivalently extended to the set A19fea 2½0; 1Þ : n 1 1 is countable. Then, the ea fxðnþmÞpx1g fxðnÞpx2g Re ðmÞc g function of x and x x 0 . Moreover, as in the continuous-time 1 2 case, the classification of the kind of nonstationarity e F fag ðx ; x Þ of discrete-time time series is made on the basis of xðnþmÞxðnÞ 1 2 e e the elements contained in set A. That is, in general, 9Efagf1 1 gð3:50Þ fxðnþmÞpx1g fxðnÞpx2g set Ae can contain incommensurate cycle frequencies is a valid second-order joint cumulative distribu- ea and, in such a case, the time series is said to be tion function for every fixed n and m, except for wide-sense almost cyclostationary. In the special case ARTICLE IN PRESS

W.A. Gardner et al. / Signal Processing 86 (2006) 639–697 649

e where A1 f0; 1=N0; ...; ðN0 1Þ=N0g for some ties (see Section 4), the stochastic and FOT integer N0, the time series xðnÞ is said to be wide- autocorrelation functions are identical for almost sense cyclostationary.IfthesetAe contains only the all sample paths xðtÞ, e element a ¼ 0, then the time series xðnÞ is said to be Rxðt; tÞ9Efxðt þ tÞxðtÞg wide-sense stationary. ¼ Efagfxðt þ tÞxðtÞg9R ðt; tÞð3:58Þ Discrete-time ACS time series are characterized x in the spectral domain by the (nonstochastic) and, hence, so too are their cyclic components and cyclic spectrum or spectral correlation density frequency-domain counterparts, function at cycle frequency ea: a a RxðtÞ¼RxðtÞ, (3.59) e XN ea 9 1 SxðnÞ lim lim Dn Dn!0 N!1 þ 2N 1 n¼N a a Sxðf Þ¼Sxðf Þ. (3.60) e X b1=Dncðn; nÞX b1=Dncðn; n aÞ, ð3:55Þ Analogous equivalences hold in the discrete-time where case. The link between the two approaches is treated nXþM in depth in [2.5,2.8,2.9,2.11,2.12,2.15,3.87,4.31, ð Þ9 ð Þ j2pnk X 2Mþ1 n; n x k e (3.56) 23.15]. In the following, most results are presented ¼ k n M in the FOT probability framework. and bc denotes the closest odd integer. The Cyclic Wiener relation [2.5] 3.5. Complex processes and time series X ea ea The case of complex-valued processes and time Se ðnÞ¼ Re ðmÞej2pnm (3.57) x x series, ﬁrst treated in depth in [2.9], is extensively m2Z treated in [2.6,2.8,3.43,13.13,13.14,13.20,13.25]. links the cyclic autocorrelation function to the Several results with reference to higher-order cyclic spectrum. statistics are reported in Section 13. Let xðtÞ be a zero-mean complex-valued continuous-time time series. Its wide-sense character- 3.4. Link between the stochastic and fraction-of- ization can be made in terms of the two second- time approaches order moments fag In the time-variant nonstochastic framework R ðt; tÞ9E fxðt þ tÞx ðtÞg xx X (for ACS time series), the almost-periodic func- a j2pat ¼ R ðtÞe , ð3:61Þ tions play the same role as that played by the xx a2A deterministic functions in the stochastic-process xx framework, and the expectation operator is the fag Rxxðt; tÞ9E fxðt þ tÞxðtÞg almost-periodic component extraction operator. X b j2pbt Therefore, in the case of processes exhibiting ¼ RxxðtÞe ð3:62Þ suitable ergodicity properties, results derived in b2Axx the FOT probability approach for time series can which are called the autocorrelation function and be interpreted in the classical stochastic process conjugate autocorrelation function, respectively. framework by substituting, in place of the almost- The Fourier coefﬁcients periodic component extraction operator Efagfg, a 9 j2pat the statistical expectation operator Efg. In fact, by Rxx ðtÞ hxðt þ tÞx ðtÞe it, (3.63) assuming that fxðt; oÞ; t 2 R; o 2 Og is a stochastic b 9 j2pbt process satisfying appropriate ergodicity proper- RxxðtÞ hxðt þ tÞxðtÞe it (3.64) ARTICLE IN PRESS

650 W.A. Gardner et al. / Signal Processing 86 (2006) 639–697 are referred to as the cyclic autocorrelation function More general time-variant linear filtering is ad- at cycle frequency a and the conjugate cyclic dressed in [21.10,21.11]. autocorrelation function at conjugate cycle frequency b, respectively. The Fourier transforms of 3.6.1. Structure of linear almost-periodically time- the cyclic autocorrelation function and conjugate variant systems cyclic autocorrelation function A linear time-variant system with input xðtÞ, Z output yðtÞ, impulse-response function hðt; uÞ, and a 9 a j2pf t Sxx ðf Þ Rxx ðtÞe dt, (3.65) input–output relation R Z Z yðtÞ¼ hðt; uÞxðuÞ du (3.69) b 9 b j2pf t R Sxxðf Þ Rxxðf Þe dt (3.66) R is said to be linear almost-periodically time-variant are called the cyclic spectrum and the conjugate (LAPTV) if the impulse-response function admits cyclic spectrum, respectively. the Fourier series expansion ðÞ X Let us denote by the superscript an optional j2psu complex conjugation. In the following, when it hðt; uÞ¼ hsðt uÞe , (3.70) does not create ambiguity, both functions s2G defined in (3.63) and (3.64) are simultaneously where G is a countable set. represented by By substituting (3.70) into (3.69) we see that the output yðtÞ can be expressed in the two equivalent a ð Þ9 ð þ Þ ðÞð Þ j2pat 2 RxxðÞ t x t t x t e t; a AxxðÞ . forms [2.8,9.51]: X (3.67) j2pst yðtÞ¼ hsðtÞ½xðtÞe (3.71a) Analogously, both functions defined in (3.65) and s2G (3.66) are simultaneously represented by X Z j2pst ¼ ½gsðtÞxðtÞe , (3.71b) a 9 a j2pf t s2G SxxðÞ ðf Þ RxxðÞ ðtÞe dt. (3.68) R where It should be noted that, in [2.6,2.8,2.9], for 9 j2pst example, a different conjugation notation in the gsðtÞ hsðtÞe . (3.72) subscripts is adopted to denote the autocorrelation From (3.71a) it follows that a LAPTV systems and the conjugate autocorrelation function. Spe- performs a linear time-invariant filtering of fag cifically, Rxxðt; tÞ¼E fxðt þ t=2Þx ðt t=2Þg and frequency-shifted version of the input signal. fag Rxx ðt; tÞ¼E fxðt þ t=2Þxðt t=2Þg. Moreover, For this reason LAPTV filtering is also referred an analogous notation is adopted in these refer- to as frequency-shift (FRESH) filtering [7.31]. ences and others for the (conjugate) cyclic auto- Equivalently, from (3.71b) it follows that a correlation function and the (conjugate) cyclic LAPTV systems performs a frequency shift spectrum. of linear time-invariant filtered versions of the input. 3.6. Linear filtering In the special case for which G fk=T 0gk2Z for some period T 0, the system is said to be linear The linear almost-periodically time-variant fil- periodically time-variant (LPTV). If G contains tering of ACS signals introduced in [2.2] and only the element s ¼ 0, then the system is linear followed by [2.5] for cyclostationary processes and time-invariant (LTI). generalized in [2.8,2.9] to ACS signals, is also considered in follow-on work in [7.60,9.51,13.9, 3.6.2. Input/output relations in terms of cyclic 13.14,13.20] as well as the early Russian work in statistics [3.9]. Properties of linear periodically time-varying Let xiðtÞ, i ¼ 1; 2, t 2 R, be two possibly- systems are analyzed in [23.8,23.12,23.13,23.14]. complex ACS continuous-time time series with ARTICLE IN PRESS

W.A. Gardner et al. / Signal Processing 86 (2006) 639–697 651 Z second-order (conjugate) cross-correlation func- Sb ðf Þ9 Rb ðtÞej2pf t dt tion y yðÞ y yðÞ 1 2 XR 1X2 fag ðÞ bs1ðÞs2 ð Þ9 f ð þ Þ ð Þg ¼ S ðÞ ðf s1Þ R ðÞ t; t E x1 t t x2 t x x x1x 1 2 2 X s12G1 s22G2 ¼ Ra ðtÞej2pat, ð3:73Þ ðÞ x xðÞ Hs ðf ÞH ððÞðb f ÞÞ, ð3:81Þ 1 2 1 s2 a2A12 where where Z a ðÞ j2pat j2pf t R ðÞ ðtÞ9hx1ðt þ tÞx ðtÞe i (3.74) H ðf Þ9 h ðtÞe dt (3.82) x x 2 t si si 1 2 R is the (conjugate) cyclic cross-correlation between and, in the sums in (3.80) and (3.81), only those x1 and x2 at cycle frequency a and s1 2 G1 and s2 2 G2 such that b s1 ðÞs2 2 9 a c A12 fa 2 R : R ðÞ ðtÞ 0g (3.75) A12 give nonzero contribution. x1x 2 Eqs. (3.80) and (3.81) can be specialized to is a countable set. If the set A12 contains at least one nonzero element, then the time series several cases of interest. For example, if x1 ¼ x2 ¼ x, h1 ¼ h2 ¼ h, y1 ¼ y2 ¼ y,andðÞ is x1ðtÞ and x2ðtÞ are said to be jointly ACS. Note conjugation, then we obtain the input–output that, in general, set A12 depends on whether ðÞ is conjugation or not and can be different from relations for LAPTV systems in terms of cyclic autocorrelation functions and cyclic spectra: the sets A11 and A22 (both deﬁned according to X X (3.75)). b bs1þs2 j2ps1t Ryy ðtÞ¼ ½Rxx ðtÞe Let us consider now two linear LAPTV systems s12G s22G whose impulse-response functions admit the Four- b r12ðtÞ, ð3:83Þ ier series expansionsX t j2psiu X X hiðt; uÞ¼ hs ðt uÞe ; i ¼ 1; 2. (3.76) i b bs1þs2 ð Þ¼ ð Þ si2Gi Syy f Sxx f s1 s12G s22G The (conjugate) cross-correlation of the outputs Z H ðf ÞH ðf bÞ, ð3:84Þ s1 s2 y ðtÞ¼ h ðt; uÞx ðuÞ du; i ¼ 1; 2 (3.77) i i i where R Z is given by b j2pbs r ðtÞ9 hs ðt þ sÞh ðsÞe ds. (3.85) fag ðÞ 12 1 s2 R ðÞ ðt; tÞ9E fy ðt þ tÞy ðtÞg R y y 1 2 1 2 X X X For further examples and applications, see Sec- ¼ ½Ra ðtÞej2ps1t x xðÞ 1 2 tions 7 and 10. a2A12 s12G1 s22G2

aþs1þðÞs2 j2pðaþs1þðÞs2Þt rs s ðÞ ðtÞe , ð3:78Þ t 1 2 3.7. Product modulation where denotes convolution with respect to t, ðÞ t Let xðtÞ and cðtÞ be two ACS signals with is an optional minus sign that is linked to ðÞ,and Z (conjugate) autocorrelation functions g 9 ðÞ j2pgs X rs s ðÞðtÞ hs1 ðt þ sÞhs ðsÞe ds. (3.79) ax j2paxt 1 2 2 R ðÞ ðt; tÞ¼ R ðtÞe , (3.86) R xx xxðÞ a 2A Thus, x xxðÞ b ðÞ X 9 j2pbt a j2pa t R ðÞ ðtÞ hy1ðt þ tÞy ðtÞe it c c y y 2 RccðÞ ðt; tÞ¼ R ðÞ ðtÞe . (3.87) 1 2 X X cc ðÞ ac2A ðÞ ¼ ½Rb s1 s2 ðtÞej2ps1t cc x xðÞ 1 2 s12G1 s22G2 If xðtÞ and cðtÞ are statistically independent in the b FOT probability sense, then their joint probability rs s ðÞðtÞ, ð3:80Þ t 1 2 density function factors into the product of the ARTICLE IN PRESS

652 W.A. Gardner et al. / Signal Processing 86 (2006) 639–697 marginal probability densities [2.8,2.9,2.12] and From (3.96) it follows that the support in the ða; f Þ the (conjugate) autocorrelation functions of the plane of the (conjugate) cyclic spectrum of yðtÞ is product waveform such that yðtÞ¼cðtÞxðtÞ (3.88) a 9 a a supp½SyyðÞ ðf Þ fða; f Þ2R R : SyyðÞ ðf Þ 0g also factor, fða; f Þ2R R : Hðf Þ ðÞ RyyðÞ ðt; tÞ¼RccðÞ ðt; tÞRxxðÞ ðt; tÞ. (3.89) H ððÞða f ÞÞa0g. ð3:97Þ Therefore, the (conjugate) cyclic autocorrelation Let xðtÞ be a strictly band-limited low-pass signal function and the (conjugate) cyclic spectrum of 0 with monolateral bandwidth B; that is, Sxx ðf Þ0 yðtÞ are [2.5,2.8]: for f eðB; BÞ. Then, X Ra ðtÞ¼ Rax ðtÞRaax ðtÞ, (3.90) yyðÞ xxðÞ ccðÞ xðtÞxðtÞhLPFðtÞ, (3.98) ax2A ðÞ xx where h ðtÞ is the impulse-response function of Z LPF X the ideal low-pass ﬁlter with harmonic-response Sa ðf Þ¼ Sax ðlÞSaax ðf lÞ dl, yyðÞ xxðÞ ccðÞ function: a 2A R x xxðÞ ( f 1; jf jpB; (3.91) H ðf Þ¼rect 9 (3.99) LPF 2B 0; jf j4B: where, in the sums, only those (conjugate) cycle frequencies ax such that a ax 2 AccðÞ give non- Accounting for (3.97), we have (see Fig. 1) zero contribution. a Observe that, if cðtÞ is an almost-periodic supp½SxxðÞ ðf Þ fða; f Þ2R R : HLPFðf Þ function ð Þa g ð Þ X HLPF f a 0 , 3:100 j2pgt cðtÞ¼ cge , (3.92) where the fact that rectðf Þ is real and even is g2G used. then ðÞ R ðÞ ðt; tÞ¼cðt þ tÞc ðtÞ cc X X ¼ c cðÞeðÞj2pg2t g1 g2 2B g12G g22G ej2pðg1þðÞg2Þt ð3:93Þ and X a ðÞ j2pðagÞt RccðÞ ðtÞ¼ cgcðÞðagÞe , (3.94) g2G X ðÞ a − SccðÞ ðf Þ¼ cgcðÞðagÞdðf þ g aÞ. (3.95) B B f g2G

3.8. Supports of cyclic spectra of band limited signals

By specializing (3.81) to the case x ¼ x ¼ x, 1 2 −2B h1 ¼ h2 ¼ h (LTI), and y1 ¼ y2 ¼ y, we obtain Sa ðf Þ¼Sa ðf ÞHðf ÞHðÞððÞða f ÞÞ. (3.96) yyðÞ xxðÞ Fig. 1. Cyclic-spectrum support of a low-pass signal. ARTICLE IN PRESS

W.A. Gardner et al. / Signal Processing 86 (2006) 639–697 653

Let xðtÞ be a strictly band-limited high-pass 0 signal; that is, Sxx ðf Þ0 for f 2ðb; bÞ. Then, 2B xðtÞxðtÞhHPFðtÞ, (3.101) where hHPFðtÞ is the impulse-response function of the ideal high-pass ﬁlter with harmonic-response function 2b f H ðf Þ¼1 rect . (3.102) HPF 2b −B −b b B f From (3.97), we have (see Fig. 2)

a − supp½SxxðÞ ðf Þ fða; f Þ2R R : HHPFðf Þ 2b

HHPFðf aÞa0g. ð3:103Þ Finally, let xðtÞ be a strictly band-limited band- 0 pass signal; that is, Sxx ðf Þ0 for f eðB; bÞ[ −2B ðb; BÞ, where 0oboB. Then, xðtÞxðtÞhBPFðtÞ, (3.104) Fig. 3. Cyclic-spectrum support of a band-pass signal. where hBPFðtÞ is the impulse-response function of the ideal band-pass ﬁlter with harmonic-response function for (3.97), we have (see Fig. 3) HBPFðf Þ¼HLPFðf ÞHHPFðf Þ, (3.105) a supp½SxxðÞ ðf Þ where H ðf Þ and H ðf Þ are given by (3.99) LPF HPF fða; f Þ2R R : H ðf ÞH ðf aÞa0g and (3.102), respectively. Therefore, accounting BPF BPF ¼fða; f Þ2R R : HLPFðf ÞHLPFðf aÞa0g

\fða; f Þ2R R : HHPFðf Þ HHPFðf aÞa0g. ð3:106Þ It is noted that supports for symmetric deﬁnitions of cyclic spectra (3.28) are reported in [2.6,2.8].

3.9. Sampling and aliasing 2b Let xðnÞ be the sequence obtained by uniformly sampling, with period T s ¼ 1=f s, the continuous- −b b f time signal xaðtÞ: 9 xðnÞ xaðtÞjt¼nTs . (3.107) −2b The (conjugate) autocorrelation function of the discrete-time signal xðnÞ can be shown to be the sampled version of the (conjugate) autocorrelation function of the continuous-time signal xaðtÞ: e Efagfxðn þ mÞxðÞðnÞg fag Fig. 2. Cyclic-spectrum support of a high-pass signal. ¼ E fxaðt þ tÞxaðtÞgjt¼nTs;t¼mTs . ð3:108Þ ARTICLE IN PRESS

654 W.A. Gardner et al. / Signal Processing 86 (2006) 639–697

However, the (conjugate) cyclic autocorrelation signal xðnÞ [13.20]: functions of xðnÞ are not sampled versions of the 8 > e (conjugate) cyclic autocorrelation functions of < ea f s RxxðÞ ðmÞje ; jajp ; ð Þ a a¼a=f s 2 xa t because of the presence of aliasing in the R ðÞ ðtÞjt¼mTs ¼ xaxa :> cycle-frequency domain. Consequently, for the 0 otherwise; (conjugate) cyclic spectra, aliasing in both the ð Þ spectral-frequency and cycle-frequency domains 3:113 occurs. Specifically, the (conjugate) cyclic autocorrelation functions and the (conjugate) cyclic a S ðÞ ðf Þ spectra of xðnÞ can be expressed in terms of the xaxa 8 e (conjugate) cyclic autocorrelation functions and < a f f e ð Þj j j s j j s T sSxxðÞ n n¼f =f ;ea¼a=f ; a p 2 ; f p 2 ; the (conjugate) cyclic spectra of xaðtÞ by the ¼ : s s relations [2.5,2.8,13.20,13.23]: 0 otherwise: X ea ð3:114Þ e ð Þ¼ apf s ð Þj RxxðÞ m R ðÞ t ¼mT ¼ef , (3.109) xax t s;a a s p2Z a The discrete-time signal obtained by sampling, with period T s, a cyclostationary continuous-time X X ea 1 signal with cyclostationarity period T 0 ¼ KT s, K e ð Þ¼ apf s ð Þj SxxðÞ n S ðÞ f qf s f ¼nf ;a¼eaf . integer, is a discrete-time cyclostationary signal T xax s s s p2Z q2Z a with cyclostationarity period K. If, however, (3.110) T 0 ¼ KT s þ , with 0ooT s and incommensurate with T s, then the discrete-time signal is Let xðtÞ be a strictly band-limited low-pass signal almost-cyclostationary [13.23].In[3.44], common with monolateral bandwidth B. From (3.100), it pitfalls arising in the application of the stationary follows that signal theory to time sampled cyclostationary a signals are examined. supp½SxxðÞ ðf Þ fða; f Þ2R R : jf jpB; ja f jpBg 3.10. Representations by stationary components fða; f Þ2R R : jf jpB; jajp2Bg. ð3:111Þ 3.10.1. Continuous-time processes and time series Thus, the support of each replica in (3.110) is A continuous-time wide-sense cyclostationary contained in the set signal (process or time-series) xðtÞ can be expressed in terms of singularly and jointly wide- fða; f Þ2R R : jf qf sjpB; ja pf sjp2Bg sense stationary signals with non-overlapping and, consequently, a sufficient condition for spectral bands [2.5,3.22,3.23,3.77,4.41]. That is, assuring that the replicas in (3.110) do not if T0 is the period of cyclostationarity, and we overlap is define

e j2pðk=T0Þt f sX4B. (3.112) xkðtÞ9½xðtÞe h0ðtÞ, (3.115) 9 In such a case, only the replica with p ¼ 0 gives where h0ðtÞ ð1=T0Þ sincðt=T 0Þ is the impulse- nonzero contribution in the base support region response function of an ideal low-pass ﬁlter in (3.109) and (3.110). Thus, the (conjugate) with monolateral bandwidth 1=ð2 T0Þ, then cyclic autocorrelation functions and the cyclic xðtÞ can be expressed by the harmonic series representation: spectra of the continuous-time signal xaðtÞ are amplitude- and/or time- or frequency-scaled ver- Xþ1 j2pðk=T0Þt sions of the (conjugate) cyclic autocorrelation xðtÞ¼ xekðtÞe , (3.116) functions and cyclic spectra of the discrete-time k¼1 ARTICLE IN PRESS

W.A. Gardner et al. / Signal Processing 86 (2006) 639–697 655 where with stationary components has been demon- strated to exist for ACS processes. fag E fxekðt þ tÞxe ðtÞg Z h 1=ð2 T0Þ k ðhkÞ=T0 j2pf t 3.10.2. Discrete-time processes and time series ¼ Sxx f þ e df ð3:117Þ 1=ð2 T0Þ T 0 A discrete-time wide-sense cyclostationary signal (process or time-series) xðnÞ can be expressed in which is independent of t. This result reflects the terms of a finite number of singularly and jointly fact that the Fourier transforms Xe ðf Þ of the k wide-sense stationary signals xe ðnÞ with non over- signals xe ðtÞ have non-overlapping support of k k lapping bands [2.1,2.5,2.17,4.41,12.3,12.30]. That width 1=T 0 is, if N is the period of cyclostationarity, and we ( 0 Xðf Þ; jf k=T j 1=ð2 T Þ; define e k 0 p 0 X k f ¼ j2pðk=N0Þn T 0 0 otherwise; xekðnÞ9½xðnÞe h0ðnÞ, (3.120)

ð3:118Þ where h0ðnÞ9ð1=N0Þ sincðn=N0Þ is the ideal low- pass ﬁlter with monolateral bandwidth 1=ð2N Þ, therefore, since only spectral components of xðtÞ 0 then xðnÞ can be expressed by the harmonic series with frequencies separated by an integer multiple representation: of 1=T0 can be correlated, then the only pairs of N spectral components in xekðtÞ and xehðtÞ that can be X0 1 j2pðk=N0Þn correlated are those with the same frequency f. xðnÞ¼ xekðnÞe . (3.121) Thus, there is no spectral cross-correlation at k¼0 e e distinct frequencies in xkðtÞ and xhðtÞ. Therefore, a discrete-time wide-sense cyclostation- It follows that a continuous-time wide-sense ary scalar signal xðnÞ is equivalent to the N0- cyclostationary scalar signal xðtÞ is equivalent to dimensional vector-valued wide-sense stationary the inﬁnite-dimensional vector-valued wide-sense signal stationary signal e e e ½x0ðnÞ; x1ðnÞ; ...; xN01ðnÞ. ½...; xe ðtÞ; ...; xe ðtÞ; xe ðtÞ; xe ðtÞ; ...; xe ðtÞ; .... k 1 0 1 k A further decomposition of a discrete-time cyclos- Another representation of a cyclostationary signal tationary signal can be obtained in terms of sub- by stationary components is the translation series sampled components. Let xðnÞ be a discrete-time representation in terms of any complete orthonor- real-valued wide-sense cyclostationary time-series mal set of basis functions [3.23]. One example uses with period N0. The sub-sampled (or decimated) the Karhunen–Loe` ve expansion of the cyclosta- time-series: tionary signal xðtÞ on the intervals t 2½nT 0; ðn þ xiðnÞ9xðnN0 þ iÞ; i ¼ 0; ...; N0 1 (3.122) 1ÞT0Þ for all integers n, where T0 is the period of cyclostationarity. constitute what is called the polyphase decomposi- A harmonic series representation can also be tion of xðnÞ. Given the set of time series xiðnÞ, obtained for an ACS process xðtÞ, provided that it i ¼ 0; ...; N0 1, the original signal xðnÞ can be belongs to the sub-class of the almost-periodically reconstructed by using the synthesis formula: unitary processes [3.65]. In this case, NX01 X Xþ1 xðnÞ¼ xið‘Þdni‘N0 , (3.123) j2plkt xðtÞ¼ xekðtÞe , (3.119) i¼0 ‘2Z k¼1 where dg is the Kronecker delta (dg ¼ 1 for g ¼ 0 where the frequencies lk are possibly incommen- and dg ¼ 0 for ga0). The signal xðnÞ is wide- surate and the processes xekðtÞ are singularly and sense cyclostationary with period N0 if and jointly wide-sense stationary but not necessarily only if the set of sub-sampled xiðnÞ are jointly band limited. No translation series representation wide-sense stationary [3.3]. In fact, due to the ARTICLE IN PRESS

656 W.A. Gardner et al. / Signal Processing 86 (2006) 639–697 cyclostationarity of xðnÞ: cyclic periodogram

fag a E fxiðn þ mÞxkðnÞg Sx ðt0; f ÞDf T Z ¼ Efagfxððn þ mÞN þ iÞxðnN þ kÞg f þDf =2 0 0 9 1 1 fag X T ðt0; lÞX T ðt0; l aÞ dl ð4:3Þ ¼ E fxðmN0 þ iÞxðkÞg, ð3:124Þ Df f Df =2 T which is independent of n. is a consistent estimator of the cyclic spectrum a Sxðf Þ. Moreover, pffiffiffiffiffiffiffiffiffiffiffi a a T Df ½Sx ðt; f ÞDf Sxðf Þ 4. Ergodic properties and measurement of T characteristics is an asymptotically (T !1, Df ! 0, with T Df !1) zero-mean complex normal random 4.1. Estimation of the cyclic autocorrelation variable for each f and t0. The first detailed study function and the cyclic spectrum of the variance of estimators of the cyclic spectrum is given in [2.8] and is based on the FOT Ergodic properties and measurements of char- framework. Consistency for estimators of the acteristics are treated in [4.1–4.61]. Consistent cyclic spectrum has been addressed in [2.1,4.14, estimates of second-order statistical functions of 4.20,4.30,4.35,4.43,4.44,4.55,4.59,13.12]. an ACS stochastic process can be obtained provided In [2.5,2.8,4.17], it is shown that the time- that the stochastic process has finite or ‘‘effectively smoothed cyclic periodogram: finite’’ memory. Such a property is generally Z 1 tþT=2 expressed in terms of mixing conditions or summa- Sa ðt; f Þ 9 Df x1=Df T bility of second- and fourth-order cumulants. Under T tT=2 such mixing conditions, the cyclic correlogram X 1=Df ðs; f ÞX ðs; f aÞ ds ð4:4Þ Z 1=Df t0þT=2 a 9 1 j2pat is asymptotically equivalent to the frequency Rxðt; t0; TÞ xðt þ tÞxðtÞe dt (4.1) T t0T=2 smoothed cyclic periodogram in the sense that is a consistent estimator of the cyclic autocorrelation a lim lim Sx ðt; f ÞDf a Df !0 T!1 T function RxðtÞ (see (3.11)). Moreover, ffiffiffiffi a p ¼ lim lim Sx ðt; f ÞT , ð4:5Þ a a Df !0 T!1 1=Df T½Rxðt; t0; TÞRxðtÞ is an asymptotically (T !1) zero-mean complex where the order of the two limits on each side normal random variable for each t and t0. cannot be reversed. Note that both the time- Consistency for estimators of the cyclic autocorrela- smoothed and frequency-smoothed cyclic period- tion function for cyclostationary and/or ACS ograms exhibit spectral frequency resolution on processes has been addressed in [2.1,3.1,4.3,4.13, the order of Df and a cycle frequency resolution 4.15,4.23,4.24,4.36,4.42,13.18]. The first treatment on the order of 1=T [4.17]. The problem of cyclic for ACS processes is in [4.13]. leakage in the estimate of a cyclic statistic at cycle In the frequency domain, the cyclic periodogram frequency a arising from cyclic statistics at cycle frequencies different from a, first observed in [2.8], 1 a 9 is addressed in [2.8,2.9,4.17]. In particular, it is I xðt; f Þ X T ðt; f ÞX T ðt; f aÞ, (4.2) T shown that a strong stationary spectrally over- where X T ðt; lÞ is defined according to (3.21), is lapping noise component added to a cyclostation- an asymptotically unbiased but not consistent ary signal degrades the performance (bias and a estimator of the cyclic spectrum Sxðf Þ (see (3.19) variance) of the estimators of cyclic statistics at and (3.20)). However, under the above-men- nonzero cycle frequencies because of the leakage tioned mixing conditions, the frequency-smoothed from the zero cycle frequency. ARTICLE IN PRESS

W.A. Gardner et al. / Signal Processing 86 (2006) 639–697 657

e−j2πft

× h∆f (t) x (t) × 〈⋅〉 ∆f S (t,f ) T x1/∆ f T × ⋅ h∆f (t) ( )*

e−j2π( f−)t

Fig. 4. Spectral correlation analyzer.

A spectral correlation analyzer can be realized 4.2. Two alternative approaches to the analysis of by frequency shifting the signal xðtÞ by two measurements on time series amounts differing by a, passing such frequency- shifted versions through two low-pass filters hDf ðtÞ In the FOT probability approach, probabilistic with bandwidth Df and unity pass-band height, parameters are defined through infinite-time and then correlating the output signals (see Fig. 4). averages of functions of a single time series (such a The spectral correlation density function Sxðf Þ is as products of time- and frequency-shifted ver- obtained by normalizing the output by Df ,and sions of the time series) rather than through taking the limit as the correlation time T !1 expected values or ensemble averages of a stochas- and the bandwidth Df ! 0, in this order [4.17]. tic process. Estimators of the FOT probabilistic Reliable spectral estimates with reduced com- parameters are obtained by considering finite-time putational requirements can be obtained by using averages of the same quantities involved in the nonlinear transformations of the data [4.39,4.51]. infinite-time averages. Therefore, assuming the Strict-sense ergodic properties of ACS processes, above-mentioned limits exist (that is, the infinite- referred to as cycloergodicity in the strict sense, time averages exist), their asymptotic estimators were first treated in depth in [4.13]; see also [2.15]. converge by definition to the true values, which are A survey of estimation problems is given in [4.41]. exactly the infinite-time averages, without the Cyclostationary feature measurements in the non- necessity of requiring ergodicity properties as in stochastic approach are treated in considerable the stochastic process framework. Thus, in the depth in [4.17] and also in [4.31]. Problems arising FOT probability framework, the kind of conver- from the presence of jitter in measurements are gence of the estimators to be considered as the addressed in [4.53,4.61]. Computationally efficient data-record length approaches infinity is the digital implementations of cyclic spectrum analy- convergence of the function sequence of the zers are developed and analyzed in [4.25,4.32, finite-time averages (indexed by the data-record 4.38,4.45,4.48]. length). Therefore, unlike the stochastic process For measurements of cyclic higher-order statis- framework where convergence must be defined, for tics see [4.50,4.56,4.57,13.7,13.9,13.12,13.14,13.18, example, in the ‘‘stochastic mean-square sense’’ 13.29]. [3.59,4.41,13.18,13.26] or ‘‘almost sure sense’’ For further references, see the general treat- [4.35,4.56] or ‘‘in distribution’’, the convergence ments [2.1,2.2,2.5,2.8,2.9,2.11,2.15,2.18] and also in the FOT probability framework must be see [3.22,3.35,3.39,3.59,3.72,11.10,11.13,11.20,12.49, considered ‘‘pointwise’’, in the ‘‘temporal mean- 12.51]. Measurements on cyclostationary random square sense’’ [2.8,2.9,2.31], or in the ‘‘sense of fields are treated in [20.3,20.5]. The problem of generalized functions (distributions)’’ [23.9]. measurement of statistical functions for more By following the guidelines in [2.8,2.9,23.1], let general classes of nonstationary signals is considered us consider the convergence of time series in the in [21.1,21.5,21.12,21.14,21.15]. temporal mean-square sense (t.m.s.s.). Given a ARTICLE IN PRESS

658 W.A. Gardner et al. / Signal Processing 86 (2006) 639–697 time series zðtÞ (such as a lag product of another where the approximation becomes exact equality time series), we define in the limit as T !1. Thus, unlike the stochastic Z process framework where the variance accounts tþT=2 9 1 j2pbu for fluctuations of the estimates over the ensemble zbðtÞT zðuÞe du, (4.6) T tT=2 of sample paths, in the FOT probability framework the variance accounts for the fluctuations of Z þT=2 the estimates in the time parameter t, viz., the 1 j2pbu zb9 lim zðuÞe du (4.7) central point of the finite-length time series T!1 T T=2 segment adopted for the estimation. Therefore, and we assume that the assumption that the estimator asymptotically approaches the true value (the infinite-time aver- lim zbðtÞT ¼ zb ðt:m:s:s:Þ8b 2 R, (4.8) age) in the mean-square sense is equivalent to the T!1 statement that the estimator is mean-square that is, consistent in the FOT probability sense. In fact, from (4.9), (4.11), and (4.12) it follows that 2 lim hjzbðtÞT zbj it ¼ 0 8b 2 R. (4.9) T!1 2 hjzbðtÞT zbj it ’ varfzbðtÞT g It can be shown that, if the time series zðtÞ has 2 þjbiasfzbðtÞT gj ð4:13Þ 2 finite-average-power (i.e., hjzðtÞj ito1), then the 9 a and this approximation become exact as T !1. setP B fb 2 R : zb 0g is countable, the series 2 In such a case, estimates obtained by using b2B jzbj is summable [2.9] and, accounting for (4.8), it follows that different time segments asymptotically do not X X depend on the central point of the segment. j2pbt j2pbt lim zbðtÞT e ¼ zbe ðt:m:s:s:Þ. T!1 b2B b2B (4.10) 5. Manufactured signals: modelling and analysis

The magnitude and phase of zb are the amplitude and phase of the finite-strength additive complex 5.1. General aspects sinewave with frequency b contained in the time series zðtÞ. Moreover, the right-hand side in (4.10) Cyclostationarity in manmade communications is just the almost-periodic component contained in signals is due to signal processing operations used the time series zðtÞ. in the construction and/or subsequent processing of the signal, such as modulation, sampling, The function zbðtÞT is an estimator of zb based on the observation fzðuÞ; u 2½t T=2; t þ T=2g.It scanning, multiplexing and coding operations is worthwhile to emphasize that, in the FOT [5.1–5.23]. probability framework, probabilistic functions are The analytical cyclic spectral analysis of math- defined in terms of the almost-periodic component ematical models of analog and digitally modulated extraction operation, which plays the same role as signals was first carried out in [2.5] for stochastic that played by the statistical expectation operation processes and in [5.9,5.10] for nonstochastic time in the stochastic process framework [2.8,2.15]. series. The effects of multiplexing are considered in Therefore, [5.13]. Continuous-phase frequency-modulated signals are treated in [5.12,5.18,5.19,5.23]. The 9 fag biasfzbðtÞT g E fzbðtÞT gzb effects of timing jitter on the cyclostationarity properties of communications signals are ad- ’hzbðtÞT it zb, ð4:11Þ dressed in [2.8,5.17,5.21]. fag fag 2 On this general subject, see the general treat- varfzbðtÞ g9E fjzbðtÞ E fzbðtÞ gj g T T T ments [2.5,2.8,2.9,2.11,2.13], and also see [2.15, 2 ’hjzbðtÞT hzbðtÞT itj it, ð4:12Þ 6.11,7.22]. ARTICLE IN PRESS

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5.2. Examples of communication signals

In this section, two fundamental examples of cyclostationary communication signals are considered and their wide-sense cyclic statistics are described. The derivations of the cyclic statistics can be accomplished by using the results of Sections 3.6 and 3.7. Then a third example of a more sophisticated communication signal is considered.

5.2.1. Double side-band amplitude-modulated signal Let xðtÞ be the (real-valued) double side-band amplitude-modulated (DSB-AM) signal: 9 xðtÞ sðtÞ cosð2pf 0t þ f0Þ. (5.1) The cyclic autocorrelation function and cyclic spectrum of xðtÞ are [5.9]:

a 1 a RxðtÞ¼2 Rs ðtÞ cos ð2pf 0tÞ 1 aþ2f 0 j2pf 0t j2f0 þ 4 fRs ðtÞe e a2f 0 j2pf 0t j2f0 þ Rs ðtÞe e g, ð5:2Þ

Fig. 5. (a) Magnitude of the cyclic autocorrelation function a 1 a a a ð Þ Sxðf Þ¼4 fSs ðf f 0ÞþSs ðf þ f 0Þ Rx t , as a function of a and t, and (b) magnitude of the cyclic spectrum Sa ðf Þ, as a function of a and f, for the DSB-AM signal aþ2f 0 j2f0 x þ Ss ðf þ f 0Þe (5.1). a2f 0 j2f0 þ Ss ðf f 0Þe g, ð5:3Þ respectively. If sðtÞ is a wide-sense stationary signal and t, and in Fig. 5(b) the magnitude of the cyclic a 0 spectrum Saðf Þ, as a function of a and f, are then Rs ðtÞ¼Rs ðtÞda and x 8 reported for the DSB-AM signal (5.1) with 1 0 <> 2 Rs ðtÞ cosð2pf 0tÞ; a ¼ 0; stationary modulating signal sðtÞ having triangular RaðtÞ¼ 1 R0ðtÞej2pf 0tej2f0 ; a ¼2f ; autocorrelation function. x :> 4 s 0 0 otherwise; 5.2.2. Pulse-amplitude-modulated signal (5.4) Let xðtÞ be the complex-valued pulse-amplitude modulated (PAM) signal: 8 X > 1 fS0ðf f ÞþS0ðf þ f Þg; a ¼ 0; < 4 s 0 s 0 xðtÞ9 akqðt kT 0Þ, (5.6) a S ðf Þ¼ 1 S0ðf f Þej2f0 ; a ¼2f ; k2Z x :> 4 s 0 0 0 otherwise: where qðtÞ is a complex-valued square integrable (5.5) pulse and fakgk2Z, ak 2 C, is an ACS sequence whose cyclostationarity is possibly induced by Thus, xðtÞ is cyclostationary with period 1=ð2 f 0Þ. framing, multiplexing, or coding [13.21]. In Fig. 5(a) the magnitude of the cyclic The (conjugate) cyclic autocorrelation fun- a autocorrelation function RxðtÞ, as a function of a ction and (conjugate) cyclic spectrum of xðtÞ ARTICLE IN PRESS

660 W.A. Gardner et al. / Signal Processing 86 (2006) 639–697 are [5.9,5.10]: X e a 1 ea R ðÞ ðtÞ¼ ½RaaðÞ ðmÞe xx T a¼aT0 0 m2Z a rqqðÞ ðt mT 0Þ, ð5:7Þ

1 ea a ð Þ¼ ½e ð Þ SxxðÞ f SaaðÞ n n¼fT ;ea¼aT T 0 0 0 Qðf ÞQðÞððÞða f ÞÞ, ð5:8Þ respectively, where e XN ea 9 1 ðÞ j2peak RaaðÞ ðmÞ lim akþma e , N!1 2N þ 1 k k¼N (5.9) e X e ea 9 ea j2pnm SaaðÞ ðnÞ RaaðÞ ðmÞe (5.10) m2Z are the (conjugate) cyclic autocorrelation function and the (conjugate) cyclic spectrum, respectively, of the sequence fa g , Z k k2Z Fig. 6. (a) Magnitude of the cyclic autocorrelation function j2pft Qðf Þ9 qðtÞe dt (5.11) a RxðtÞ, as a function of a and t, and (b) magnitude of the cyclic R a spectrum Sxðf Þ, as a function of a and f, for the PAM signal and (5.6). ra ðtÞ9qðtÞ½qðÞðtÞej2pat qqðÞ Z ¼ qðt þ tÞqðÞðtÞej2pat dt. ð5:12Þ In Fig. 6(a) the magnitude of the cyclic R a autocorrelation function RxðtÞ, as a function of a If the sequence fakgk2Z is wide-sense stationary and t, and in Fig. 6(b) the magnitude of the cyclic and white, then a spectrum Sxðf Þ, as a function of a and f, are eea e0 reported for the PAM signal (5.6) with stationary R ðmÞ¼R ð0Þd d , (5.13) aa aa ðea mod 1Þ m modulating sequence fakgk2Z, and rectangular pulse qðtÞ9rectððt T =2Þ=T Þ. In this case, where mod denotes the modulo operation. In this 0 0 (5.12) reduces to case, the cyclic autocorrelation function and the cyclic spectrum of xðtÞ become t a jpaðT0tÞ rqqðtÞ¼e rect e0 2T0 a Raa ð0Þ a Rxx ðtÞ¼ dðaT0 mod 1Þrqq ðtÞ, (5.14) jtj T 0 1 T 0 sincðaðT 0 jtjÞÞ. ð5:16Þ T 0 e0 a Raa ð0Þ Sxx ðf Þ¼ dðaT0 mod 1ÞQðf ÞQ ðf aÞ, T 0 5.2.3. Direct-sequence spread-spectrum signal (5.15) Let xðtÞ be the direct-sequence spread-spectrum (DS-SS) baseband PAM signal respectively. Thus, xðtÞ exhibits cyclostationarity X with cycle frequencies a ¼ k=T0, k 2 Z; that is, xðtÞ xðtÞ9 akqðt kT 0Þ, (5.17) is cyclostationary with period T 0. k2Z ARTICLE IN PRESS

W.A. Gardner et al. / Signal Processing 86 (2006) 639–697 661 where fakgk2Z, ak 2 C, is an ACS sequence, T 0 is ered in [6.3,6.11,6.14,6.16,6.17,6.19,6.25,6.26]; for the symbol period, and periodic ARMA modelling and the prediction problem in hydrology, see [12.16,12.20,12.21, NXc1 12.23]; for the modelling of ocean waves as a qðtÞ9 cnpðt nT cÞ (5.18) n¼0 two-dimensional cyclostationary random field see [20.1]. is the spreading waveform. In (5.18), fc ; ...; 0 A patent on a speech recognition technique c g is the N -length spreading sequence (code) Nc1 c exploiting cyclostationarity is [6.27]. with cn 2 C, and T c is the chip period such that T 0 ¼ NcT c. The (conjugate) cyclic autocorrelation function 7. Communications systems: analysis and design and (conjugate) cyclic spectrum of xðtÞ are [2.8,8.24]: 7.1. General aspects X e a 1 ea R ðÞ ðtÞ¼ ½RaaðÞ ðmÞe xx T a¼aT0 0 m2Z Cyclostationarity properties of modulated signals can be suitably exploited in the analysis and ra ðt mT Þga ðtÞ, ð5:19Þ ppðÞ 0 ccðÞ design of communications systems (see [7.1–7.101]) since the signals involved are typically ACS (see 1 ea a ð Þ¼ ½e ð Þ Section 5). SxxðÞ f SaaðÞ n n¼fT ;ea¼aT T 0 0 0 The cyclostationary nature of interference in ðÞ Pðf ÞP ððÞða f ÞÞ communications systems is characterized in eea [7.5,7.6,7.8,7.9,7.18,7.25,7.47,7.98]. The problem G ðÞ ðnÞj e , ð5:20Þ cc n¼fTc;a¼aTc of optimum filtering (cyclic Wiener filtering) of where ACS signals is addressed in Section 7.2. The problems of synchronization, signal para- NXc1 NXc1 a ðÞ j2pan2Tc meter and waveform estimation, channel identifi- g ðÞ ðtÞ9 cn c e cc 1 n2 cation and equalization, and signal detection and n ¼0 n ¼0 1 2 classification are treated in Sections 8–11. dðt ðn1 n2ÞT cÞð5:21Þ On the analysis and design of communications and systems, see the general treatments [2.5,2.8, e 2.11,2.13], and also see [3.23,3.47,5.9,5.10,5.14, ea 9 ðÞ e 10.7,22.8]. GccðÞ ðnÞ CðnÞC ððÞða nÞÞ (5.22) with 7.2. Cyclic Wiener filtering NXc1 9 j2pnn CðnÞ cne . (5.23) The problem of optimum linear filtering consists n¼0 of designing the linear transformation of the data xðtÞ that minimizes the mean-squared error of the filter output relative to a desired signal, say dðtÞ.In 6. Natural signals: modelling and analysis the case of complex data, the optimum filter is obtained by processing both xðtÞ and xðtÞ, leading Cyclostationarity occurs in data arising from a to the linear-conjugate-linear (LCL) structure variety of natural (not man-made) phenomena due [2.8,2.9].IfdðtÞ and xðtÞ are jointly ACS signals, to the presence of periodic mechanisms in the the optimum filtering is referred to (as first phenomena [6.1–6.27]. In climatology and atmo- suggested in [7.31])ascyclic Wiener filtering (and spheric science, cyclostationarity is due to rotation also frequency-shift (FRESH) filtering). FRESH and revolution of the earth [6.1,6.2,6.9,6.12,6.13, filtering consists of periodically or almost-periodi- 6.21–6.24]. Applications in hydrology are consid- cally time-variant filtering of xðtÞ and xðtÞ and ARTICLE IN PRESS

662 W.A. Gardner et al. / Signal Processing 86 (2006) 639–697 adding the results, where the frequency shifts are filter design [7.31,9.51]: chosen in accordance with the cycle frequencies of X j2pst½ gsð Þ ð Þ xðtÞ and dðtÞ (e.g., cycle frequencies of the signal of e Rxx t hs t interest and interference, xðtÞ¼dðtÞþnðtÞ where s2G X j2pZt Zg c nðtÞ is the interference) [7.19,7.21,7.31]. The þ e ½Rxx ðtÞ hZðtÞ problem of optimum LPTV and LAPTV filtering Z2Gc g was first addressed in [2.2,3.23] and, in terms of ¼ Rdx ðtÞ8g 2 F dx , ð7:8aÞ cyclic autocorrelations and cyclic spectra in X [2.5,2.8]. j2pst gs e ½Rxx ðtÞhsðtÞ Let s2G X j2pZt Zg c b 9 þ e ½Rxx ðtÞ hZðtÞ dðtÞ yðtÞþycðtÞ Z Z Z2Gc c g ¼ hðt; uÞxðuÞ du þ h ðt; uÞx ðuÞ du ð7:1Þ ¼ RdxðtÞ8g 2 F dx, ð7:8bÞ R R b b where the fact that Rxx ðtÞ¼Rxx ðtÞ and be the LCL estimate of the desired signal dðtÞ a a R ðtÞ¼R ðtÞ is used and, in (7.8a), for each obtained from the data xðtÞ. To minimize the x x xx g 2 F , the sums are extended to all frequency mean-squared error dx shifts s 2 G and Z 2 Gc such that g s 2 Axx and EfagfjdbðtÞdðtÞj2g (7.2) Z g 2 Axx; and, in (7.8b), for each g 2 F dx, the sums are extended to all frequency shifts s 2 G a necessary and sufficient condition is that the and Z 2 Gc such that g s 2 Axx and Z g 2 Axx . error signal be orthogonal to the data (orthogon- Eqs. (7.8a) and (7.8b) can be re-expressed in the ality condition [2.5]); that is, frequency domain: X fag b gs E f½dðt þ tÞdðt þ tÞx ðtÞg ¼ 0 Sxx ðf sÞHsðf sÞ 8t 2 R 8t 2 R, ð7:3aÞ s2G X Zg c þ Sxx ðZ f Þ HZðf ZÞ 2G fag b Z c E f½dðt þ tÞdðt þ tÞxðtÞg ¼ 0 g ¼ S ðf Þ8g 2 F dx , ð7:9aÞ 8t 2 R 8t 2 R. ð7:3bÞ dx X gs If xðtÞ and dðtÞ are singularly and jointly ACS, Sxx ðf sÞHsðf sÞ X s2G fag ðÞ a j2pat X Zg E fxðt þ tÞx ðtÞg ¼ RxxðÞ ðtÞe , (7.4) c þ Sxx ðZ f Þ HZðf ZÞ a2A ðÞ xx Z2Gc g X ¼ S ðf Þ8g 2 F dx, ð7:9bÞ Efagfdðt þ tÞxðÞðtÞg ¼ Rg ðtÞej2pgt, (7.5) dx dxðÞ c g2F ðÞ where Hsðf Þ and HZðf Þ are the Fourier transforms dx c of hsðtÞ and hZðtÞ, respectively. then it follows that the optimum filters are Applications of FRESH filtering to interference LAPTV: suppression have been considered in [7.56,7.58, X 7.68,7.73,7.76,7.77,7.79–7.81,7.85,7.94,7.98]. hðt; uÞ¼ h ðt uÞej2psu, (7.6) s Adaptive FRESH filtering is addressed in [7.11, s2G 7.14,7.19,7.20,7.21,7.23,7.24,7.27,7.30,7.46,7.57,7.69, X 7.74,7.75,7.81]. hcðt; uÞ¼ hc ðt uÞej2pZu. (7.7) Z Cyclic Wiener filtering or FRESH filtering can Z2G c be recognized to be linked to FSE’s, RAKE filters, By substituting (7.1), (7.6), and (7.7) into (7.3a) adaptive demodulators, and LMMSE despreaders and (7.3b), we obtain the system of simultaneous for recovery of baseband symbol data from PAM, ARTICLE IN PRESS

W.A. Gardner et al. / Signal Processing 86 (2006) 639–697 663

GMSK/GFSK, and DSSS signals, and for separa- follows that tion of baseband symbol streams in overlapped X x2ðtÞ¼ q2ðt kT Þ signal environments, as well as multicarrier or 0 (8.3) k2 direct frequency diversity spread spectrum systems Z for adaptive transmission and combining. In and ‘xðt; 0Þ¼0. Therefore, the synchronization 2 particular, on blind and nonblind adaptive demo- signal x ðtÞ is periodic with period T0. dulation of PAM signals and LMMSE blind More generally, by definition, ACS signals despreading of short-code DSSS/CDMA signals enable spectral lines to be generated by passage using FSE’s and RAKE filtering structures, see through a stable nonlinear time-invariant trans- [7.12,7.15–7.17,7.28,7.29,7.36,7.38,7.39,7.44,7.48, formation (see Sections 10.4 and 13). That is, 7.50,7.51,7.53,7.59,7.61,7.65–7.67]. On direct fre- quadratic or higher-order nonlinear time-invariant quency-diverse multicarrier transceivers, see transformations of an ACS signal give rise to time [7.32–7.34,7.52]. series containing finite-strength additive sinewave Patents of inventions for the analysis and design components whose frequencies are the second or of communications systems exploiting cyclostatio- higher-order cycle frequencies of the original narity are [7.35,7.37,7.42,7.45,7.70,7.83,7.86,7.87, signal. All synchronization schemes can be recog- 7.92,7.95,7.96,7.99–7.101]. Patents of inventions nized to exploit the second- or higher-order exploiting the FRESH filtering are [7.55,7.93]. cyclostationarity features of signals [8.13]. Cyclo- stationarity properties are exploited for synchronization in [8.1–8.36]. 8. Synchronization The spectral analysis of timing waveforms with re-generated spectral lines is treated in 8.1. Spectral line generation [8.1,8.8,8.11,8.13–8.15]. Phase-lock loops are analyzed in [8.2,8.5–8.7]. Blind or non-data-aided Let xðtÞ be a real-valued second-order wide- synchronization algorithms are described in sense ACS time series. According to the results of [8.20–8.26,8.28–8.33,8.35,8.36]. See also [2.8,5.1]. Section 3.3, the second-order lag product xðt þ Patents on synchronization techniques exploit- tÞxðtÞ can be decomposed into the sum of its ing cyclostationarity are [8.17,8.27,8.34]. almost-periodic component and a residual term ‘xðt; tÞ not containing any finite-strength additive sinewave component (see (3.44) and (3.46)): 9. Signal parameter and waveform estimation xðt þ tÞxðtÞ¼Efagfxðt þ tÞxðtÞg þ ‘ ðt; tÞ X x Cyclostationarity properties can be exploited to a j2pat ¼ RxðtÞe þ ‘xðt; tÞ, ð8:1Þ design signal selective algorithms for signal para- a2A meter and waveform estimation [9.1–9.100].In where fact, if the desired and interfering signals have j2pat different cyclic parameters such as carrier fre- h‘xðt; tÞe it 0 8a 2 R. (8.2) quency or baud rate, then they exhibit cyclosta- For communications signals, the cycle frequen- tionarity at different cycle frequencies and, cies a 2 A are related to parameters such as consequently, parameters of the desired signal sinewave carrier frequency, pulse rate, symbol can be extracted by estimating cyclic statistics of rate, frame rate, sampling frequency, etc. (see the received data, consisting of desired signal plus Section 5). Therefore, the extraction of the almost- interfering signal, at a cycle frequency exhibited by periodic component in (8.1) leads to a signal the desired signal but not by the interference. suitable for synchronization purposes. For exam- This signal selectivity by exploitation of cyclosta- ple, if xðtÞ is the binary PAM signal defined in (5.6) tionarity was first suggested in [4.8]. Parameters with qðtÞ real and duration limited to an interval that can be estimated in the presence of inter- strictly less than T0,andak 2f1; 1g, then it ference and/or high noise include carrier frequency ARTICLE IN PRESS

664 W.A. Gardner et al. / Signal Processing 86 (2006) 639–697 and phase, pulse rate and phase, signal power Patents of inventions on blind system identifica- level, modulation indices, bandwidths, time- and tion exploiting cyclostationarity are [10.3,10.6, frequency-difference of arrival, direction of arri- 10.10,10.11,10.39,10.56]. val, and so on. Signal selective time-difference-of-arrival esti- 10.2. LTI-system identification with noisy- mation algorithms are considered in [9.11,9.12, measurements 9.19,9.31,9.33,9.36–9.38,9.54,9.56,9.72,9.75,9.83]. In [9.84,9.90], both time-difference-of-arrival and Cyclostationarity-based techniques can be frequency-difference-of arrival are estimated. Ar- exploited in channel identification and equaliza- ray processing problems, including spatial filtering tion problems in order to separate the desired and and direction finding, are treated in [7.15,9.7,9.9, disturbance contributions in noisy input/output 9.10,9.15,9.17,9.18,9.20–9.22,9.26–9.30,9.32,9.34, measurements, provided that there is at least one 9.39–9.44,9.47–9.50,9.53,9.57–9.65,9.68,9.73,9.74, cycle frequency of the desired signal that is not 9.79,9.80,9.82,9.86–9.88,9.91,9.94,9.95,9.97,9.99,9.100]. shared by the disturbance. The polynomial-phase signal parameter estima- Let us consider the problem of estimating the tion is addressed in [9.67,9.71,9.78,9.93]. Harmo- impulse-response function hðtÞ or, equivalently, the nics in additive and multiplicative noise are harmonic-response function: considered in [9.55,9.66,9.69,9.70,9.76,9.77]. But, Z there are difficulties in application of some of this Hðf Þ9 hðtÞej2pft dt (10.1) work—difficulties associated with lack of ergodic R properties of stochastic-process models adopted of an LTI system with input/output relation for polynomial-phase signals and signals with yðtÞ¼hðtÞxðtÞ (10.2) coupled harmonics. On this subject see the general treatments on the basis of the observed noisy signals vðtÞ and [2.2,2.5,2.8,2.9,2.11], and also see [3.23,3.60,5.12, zðtÞ 7.1,7.2,7.7,7.19,7.21,7.24,8.24,11.1,21.6]. vðtÞ¼xðtÞþnðtÞ, (10.3) Patents of inventions on signal parameter and waveform estimation exploiting cyclostationarity zðtÞ¼yðtÞþmðtÞ, (10.4) are [9.16,9.45,9.46,9.52,9.81,9.96]. where xðtÞ, nðtÞ, and mðtÞ are zero-mean time series. By assuming x1 ¼ x2 ¼ x, y1 ¼ y, y2 ¼ x, h1 ¼ h (LTI), and h2 ¼ d in (3.77), and choosing ðÞ to be 10. Channel identification and equalization conjugation, (3.81) specializes to a a Syx ðf Þ¼Sxx ðf ÞHðf Þ. (10.5) 10.1. General aspects Therefore, from the model (10.3) and (10.4), we Cyclostationarity-based techniques have been obtain exploited for channel identification and equaliza- a a a Svv ðf Þ¼Sxx ðf ÞþSnn ðf Þ, (10.6) tion [10.1–10.57]. Linear and nonlinear systems, a a a and time-invariant, periodically and almost-peri- Szv ðf Þ¼Syx ðf ÞþSmn ðf Þ odically time-variant systems, have been consid- a a ¼ S ðf ÞHðf ÞþS ðf Þð10:7Þ ered. Also, techniques for noisy input/output xx mn measurement, and blind adaptation algorithms, provided that xðtÞ is uncorrelated with both nðtÞ have been developed for LTI systems. and mðtÞ. On this subject see the general treatments Eqs. (10.6) and (10.7) reveal the ability of [2.2,2.5,2.8,2.9,2.11], and also see [7.25,10.36,8.24, cyclostationarity-based algorithms to be signal 9.59,13.20,13.28,14.11]. selective. In fact, under the assumption that nðtÞ ARTICLE IN PRESS

W.A. Gardner et al. / Signal Processing 86 (2006) 639–697 665 does not exhibit cyclostationarity at cycle fre- 10.4. Nonlinear-system identification a quency a (i.e., Snn ðf Þ0) and nðtÞ and mðtÞ do not exhibit joint cyclostationarity with cycle Let yðtÞ the output signal of a Volterra system a frequency a (i.e., Smn ðf Þ0), the harmonic- excited by the input signal xðtÞ: Z response function for the system is given by Xþ1 a a yðtÞ¼ knðt1; ...; tnÞxðt þ t1Þ Syx ðf Þ S ðf Þ zv Rn Hðf Þ¼ a ¼ a . (10.8) n¼1 Sxx ðf Þ Svv ðf Þ xðt þ tnÞ dt1 dtn. ð10:11Þ That is, Hðf Þ can be expressed in terms of the cyclic spectra of the noisy input and output Accounting for the results of Sections 3.3 and 13, signals. Therefore, (10.8) provides a system identi- the input lag-product waveform can be decom- fication formula that is intrinsically immune to the posed into the sum of an almost-periodic compo- effects of noise and interference as first observed in nent, referred to as the temporal moment function, [2.8,10.5]. Thus, this identification method is and a residual term not containing any finite- strength additive sinewave component (see (13.3)): highly tolerant to disturbances in practice, pro- X vided that a sufficiently long integration time is a j2pat xðt þ t1Þxðt þ tnÞ¼ RxðsÞe þ ‘xðt; sÞ. used for the cyclic spectral estimates. a2Ax The identification of LTI systems based on noisy (10.12) input/output measurements is considered in [2.5, 2.8,9.11,9.12,9.36,9.37,10.5,10.12,10.14,10.18,10.20, Thus, identification and equalization techniques 10.24,10.55]. for Volterra systems excited by ACS signals make use of higher-order cyclostationarity properties. In 10.3. Blind LTI-system identification and fact, there are potentially substantial advantages equalization to using cyclostationary input signals, relative to stationary input signals, for purposes of Volterra By specializing (3.84) to the case of the LTI system modelling and identification as first ob- system (10.2), we get served in [10.13]. Both time-invariant and almost-periodically a ð Þ¼ a ð Þ ð Þ ð Þ Syy f Sxx f H f H f a (10.9) time-variant nonlinear systems are treated in which, for a ¼ 0, reduces to the input/output [10.4,10.8,10.13,10.23,10.35,10.41,10.46,13.16]. relationship in terms of power spectra: 0 0 2 S ðf Þ¼S ðf ÞjHðf Þj . (10.10) yy xx 11. Signal detection and classification, and source From (10.9) and (10.10) it follows that input/output separation relationships for LTI systems in terms of cyclic statistics, unlike those in terms of autocorrelation Signal detection techniques designed for cyclos- functions and power spectra, preserve phase in- tationary signals take account of the periodicity or formation of the harmonic-response function Hðf Þ. almost periodicity of the signal autocorrelation Thus, cyclostationarity properties of the output function [11.1–11.39]. Single-cycle and multicycle signal are suitable to be exploited for recovering detectors exploit one or multiple cycle frequencies, both phase and magnitude of the system harmonic- respectively. The detection problem for additive response function as first observed in [10.7]. Gaussian noise is addressed in [11.2,11.5,11.7, Cyclostationarity properties are exploited for 11.9–11.11,11.15,11.19,11.21,11.26,11.27,11.32], blind identification (without measurements of the and for non-Gaussian noise, in [11.16,11.18, system input) of linear systems and for blind 11.24,11.25]. The problem of signal detection in equalization techniques in [9.59,10.1,10.7,10.15, cyclostationary noise is treated in [11.2,11.8, 10.16,10.19,10.21,10.22,10.25,10.26,10.28–10.33, 11.12,11.17]. Tests for the presence of cyclostatio- 10.36–10.38,10.42–10.45,10.47,10.48,10.50–10.53]. narity are proposed in [11.14,11.20,11.31,12.31] by ARTICLE IN PRESS

666 W.A. Gardner et al. / Signal Processing 86 (2006) 639–697 exploiting the asymptotic properties of the cyclic Periodic AR and ARMA systems are treated in correlogram and in [11.6,11.13] by exploiting the [12.1–12.53]. The modelling and prediction pro- properties of the support of the spectral correla- blem is treated in [12.1–12.8,12.12–12.16,12.19, tion function. Modulation classiﬁcation techni- 12.21,12.23–12.26,12.28–12.30,12.32,12.34–12.37, ques are proposed in [11.23,11.28–11.30]; see also 12.41,12.43,12.45,12.46,12.50,12.52]. The para- [13.22]. The problem of cyclostationary source meter estimation problem is addressed in [12.9, separation is considered in [11.34,11.35,11.37, 12.11,12.17,12.22,12.27,12.31,12.36,12.38–12.40, 11.39]. 12.47–12.49,12.51]. On this subject see the general treatments On the subject of periodic AR and ARMA [2.5,2.11,2.13], and also see [3.60,7.72,7.74,9.13, modelling and prediction, see the general treat- 9.14,9.19,11.11,13.30,13.31]. ments [2.5,2.8,2.11,2.13], and also see [3.30,3.69, Patents on detection and signal recognition 14.23,15.3] for the prediction problem; [4.3,4.5, exploiting cyclostationarity are [11.22,11.36,11.38]. 4.12,4.27,4.29] for the parameter estimation pro- See also the URL http://www.sspi-tech. blem; [6.2,6.9,6.13,6.16] for modelling of atmo- com for information on general purpose, auto- spheric and hydrologic signals; [16.1,16.2,16.5, matic, communication-signal classiﬁcation software 16.7] for applications to econometrics; and [21.4] systems. for application to modelling helicopter noise.

13. Higher-order statistics 12. Periodic AR and ARMA modelling and prediction 13.1. Introduction

Periodic autoregressive (AR) and autoregressive As first defined in [13.4], a signal xðtÞ is said to moving average (ARMA) (discrete-time) systems exhibit higher-order cyclostationarity (HOCS) if are characterized by input/output relationships there exists a homogeneous non-linear transfor- described by difference equations with periodically mation of xðtÞ of order greater than two such that time-varying coefficients and system orders the output of this transformation contains finite- [12.36,12.37]: strength additive sinewave components. Motiva- tions to study HOCS properties of signals include XPðnÞ XQðnÞ the following: akðnÞyðn kÞ¼ bmðnÞxðn mÞ, (12.1) k¼0 m¼0 (1) Signals not exhibiting second-order cyclosta- where xðnÞ and yðnÞ are the input and output tionarity can exhibit HOCS [13.13]. signals, respectively, and the coefficients akðnÞ and (2) Narrow-band filtering can destroy second- bmðnÞ and the orders PðnÞ and QðnÞ are periodic order (wide-sense) cyclostationarity. In fact, functions with the same period N0. Thus, periodic let us consider the input/output relationship ARMA systems are a special case of discrete-time for LTI systems in terms of cyclic spectra periodically time-varying systems. When they are (10.9). If the bandwidth of Hðf Þ is smaller than excited by a stationary or cyclostationary input the smallest nonzero second-order cycle fre- signal xðnÞ, they give rise to a cyclostationary quency of the input signal xðtÞ, then the output output signal yðnÞ. When they are excited by an signal yðtÞ does not exhibit second-order wide- almost-cyclostationary input signal, they give rise sense cyclostationarity. The signal yðtÞ, how- to an almost-cyclostationary output signal. The ever, can exhibit HOCS (see also [3.13]). problem of fitting an AR model to data yðnÞ is (3) The exploitation of HOCS can be useful for equivalent to the problem of solving for a linear signal classification [13.22]. Specifically, differ- predictor for yðnÞ, where xðnÞ is the model-fitting- ent communication signals, even if they error time series. exhibit the same second-order cyclostationarity ARTICLE IN PRESS

W.A. Gardner et al. / Signal Processing 86 (2006) 639–697 667

properties, can exhibit different cyclic features cross-moment functions (CTCMFs) of higher order. Moreover, different behaviors *+ YN can be obtained for different conjugation aðsÞ9 xðÞk ðt þ t Þej2pat , (13.1) configurations [13.9,13.14,13.17]. Rx k k k¼1 (4) Exploitation of HOCS can be useful for many t T estimation problems, as outlined below. where s9½t1; ...; tN is not identically zero. Thus, N time series exhibit wide-sense joint Nth-order On the subject of higher-order cyclostationarity cyclostationarity with cycle frequency aa0 if, for and its applications see [13.1–13.31]; in addition, some s, the lag product waveform see [3.81] for the wavelet decomposition; [4.56] for estimation issues; [5.23] for the cyclic higher-order YN ðt Þ9 xðÞk ðt þ Þ properties of CPM signals; [7.74] for blind Lx ; s k tk (13.2) adaptive detection; [8.8] for the spectral analysis k¼1 of PAM signals; [9.55,9.61,9.67,9.69–9.71,9.80, contains a finite-strength additive sinewave com- 9.85,9.93,9.99] for application to signal parameter ponent with frequency a, whose amplitude and and waveform estimation; [10.13,10.23,10.24, a phase are the magnitude and phase of RxðsÞ, 10.34,10.40,10.41,10.46] for applications to non- respectively. linear system identification and equalization; The Nth-oder temporal cross-moment function [11.20] for applications to signal detection; (TCMF) is defined by [11.28–11.30] for applications to signal classification; [11.34] for applications to source sepa- ðt; sÞ9EfagfL ðt; sÞg Rx X x ration; [15.18,15.21,15.23] for applications in a j2pat ¼ R ðsÞe ð13:3Þ acoustics and mechanics; [21.9–21.11,21.13] for x a2Ax the higher-order characterization of generalized almost-cyclostationary signals. where Ax is the countable set (not depending on s) of the Nth-order cycle frequencies of the time ðÞ1 ðÞN 13.2. Higher-order cyclic statistics series x1 ðt þ t1Þ; ...; xN ðt þ tN Þ (for the given conjugation configuration). In this section, cyclic higher-order (joint) statis- The N-dimensional Fourier transform of the tics for both continuous-time and discrete-time CTCMF time series are presented in the FOT probability Z framework as first introduced for continuous time a 9 a j2pf Ts Sxðf Þ RxðsÞe ds, (13.4) in [13.4,13.5] and developed in [13.9,13.13,13.14, RN 13.17], and, for discrete and continuous time, in where f 9½f ; ...; f T, is called the Nth-order [13.20]. For treatments within the stochastic 1 N cyclic spectral cross-moment function (CSCMF) framework, see [13.10,13.12,13.15,13.18,13.26] or and can be written as extend to higher-order statistics the link between the two frameworks discussed in Section 3.4. a a 0 T Sxðf Þ¼Sxðf Þdðf 1 aÞ, (13.5) 13.2.1. Continuous-time time series where 1 is the vector ½1; ...; 1T, and prime denotes Let us consider the column vector the operator that transforms a vector 9 ðÞ1 ðÞN T 9 T xðtÞ ½x1 ðtÞ; ...; xN ðtÞ whose components u ½u1; ...; uK into the reduced-dimension ver- 09 T a 0 are N not necessarily distinct complex-valued sion u ½u1; ...; uK1 . The function Sxðf Þ, re- continuous-time time-series and ðÞk represents ferred to as the reduced-dimension CSCMF (RD- optional complex conjugation of the kth signal CSCMF), can be expressed as xkðtÞ. The N time-series exhibit joint Nth-order Z 0 wide-sense cyclostationarity with cycle frequency a 0 a 0 j2pf Ts0 0 Sxðf Þ¼ Rxðs Þe ds , (13.6) aa0 if at least one of the Nth-order cyclic temporal RN1 ARTICLE IN PRESS

668 W.A. Gardner et al. / Signal Processing 86 (2006) 639–697 where [13.4], the Nth-order generalization of the Cyclic Wiener Relation. að 0Þ9 að Þj Rx s Rx s tN ¼0 (13.7) a 0 Let us note that in general the function Rxðs Þ is is the reduced-dimension CTCMF (RD-CTCMF). not absolutely integrable because it does not in The RD-CSCMF can also be expressed as general decay as ks0k!1, but rather it oscillates. a 0 Z Thus, Sxðf Þ can contain impulses and, conse- tþZ=2 a a 0 1 1 ð Þ S ðf Þ¼ lim lim quently, Sx f can contain products of impulses. x T!1 Z!1 b 0 Z tZ=2 T In [13.13], it is shown that the RD-CSCMF Sxðf Þ 0 ðÞN T can containP impulsive terms if the vector f with X N;T ðu; ðÞN ða f 1ÞÞ N1 f N ¼ b k¼1 f k lies on the b-submanifold; i.e., NY1 ðÞk if there exists at least one partition fm1; ...; mpg of X k;T ðu; ðÞkf kÞ du, ð13:8Þ f1; ...; Ng with p41 such that each sum am ¼ k¼1 P i k2m f k is a jmijth-order cycle frequency of xðtÞ, where i where jmij is the number of elements in mi. Z tþT=2 In the spectral-frequency domain, a well-be- 9 j2pf ks X k;T ðt; f kÞ xkðsÞe ds (13.9) haved function can be introduced starting from the tT=2 Nth-order temporal cross-cumulant function (TCCF): and ðÞk denotes an optional minus sign that is linked to the optional conjugation ðÞk. Eq. (13.8) ðÞk C ðt; sÞ9cumfx ðt þ tkÞ; k ¼ 1; ...; Ng reveals that N time-series exhibit joint Nth-order x k wide-sense cyclostationarity with cycle frequency a @N ¼ðÞN a c a 0 c j loge (i.e., RxðsÞ 0 or, equivalently, Sxðf Þ 0) if and ()@o1"#@oN only if the Nth-order temporal cross-moment of XN fag ðÞk ð þ Þ their spectral components at frequencies f k, whose E exp j okxk t tk sum is equal to a, is nonzero. In fact, by using "#k¼1 x¼0 X Yp 9 p1 hBðtÞ B rectðBtÞ (13.10) ¼ ð1Þ ðp 1Þ! ðt; s Þ , Rxmi mi 9 1 P i¼1 with B 1=T and rectðtÞ¼1ifjtjp2 and rectðtÞ¼ 1 ð13:12Þ 0ifjtj42, Eq. (13.8) can be re-written as Z T tþZ=2 where x9½o1; ...; oN , P is the set of distinct a 0 1 1 ðÞN Sxðf Þ¼ lim lim N1 ½ðxN ðuÞ partitions of f1; ...; Ng, each constituted by the B!0 Z!1 Z tZ=2 B 0 subsets fmi; i ¼ 1; ...; pg, xmi is the jmij-dimen- j2pðaf T1Þu e ÞhBðuÞ sional vector whose components are those of x NY1 having indices in mi. In (13.12), the almost-periodic ðÞk j2pf ku fag ½ðxk ðuÞe ÞhBðuÞ du ð13:11Þ component extraction operator E fg extracts the k¼1 frequencies of the 2N-variate fraction-of-time joint which is the Nth-order temporal cross-moment of probability density function of the real and low-pass-ﬁltered versions of the frequency-shifted imaginary parts of the time-series xkðtÞðk ¼ ðÞ k j2pf kt 1; ...; NÞ according to signals xk ðtÞe when the sum of the frequency shifts is equal to a and the bandwidth B fag ðÞ1 ðÞN E fgðx ðt þ t1Þ; ...; x ðt þ tN ÞÞg approaches zero. Such a property is the general- Z 1 N ization to the order N42 of the spectral correla- ðÞ1 ðÞN ¼ gðx1 ; ...; xN Þ tion property of signals that exhibit second-order R2N cyclostationarity. Moreover, relation (13.4) be- f fag x1rðtþt1Þx1iðtþt1ÞxNrðtþtN ÞxNiðtþtN Þ tween the Nth-order cyclic spectral moment ðx ; x ; ...; x ; x Þ function (13.5), (13.8) and Nth-order cyclic tem- 1r 1i Nr Ni poral moment (13.1) is, at ﬁrst pointed out in dx1r dx1i dxNr dxNi, ð13:13Þ ARTICLE IN PRESS

W.A. Gardner et al. / Signal Processing 86 (2006) 639–697 669 where xkrðtÞ9Re½xkðtÞ, xkiðtÞ9Im½xkðtÞ, xkr9 can be re-written as Re½xk, xki9Im½xk, and 0T "# b 0 1 ðÞN j2pðaf 1Þt Pxðf Þ¼ lim cumf½ðxN ðtÞe Þ XN B!0 BN1 ðÞ1 ðÞN ðÞk gðx ; ...; x Þ9 exp j okx . (13.14) ðÞ 1 N k h ðtÞ; ½ðx k ðtÞej2pf ktÞh ðtÞ, k¼1 B k B ¼ gðÞ In fact, by taking the N-dimensional Fourier k 1; ...; N 1 13:20 transform of the coefficient of the Fourier series which is the Nth-order temporal cross-cumulant of expansion of the almost-periodic function (13.12) low-pass filtered versions of the frequency-shifted ðÞk j2pf t b j2pbt signals x ðtÞe k when the sum of the fre- C ðsÞ9hC ðt; sÞe i (13.15) k x x t quency shifts is equal to b and the bandwidth B which is referred to as the Nth-order cyclic approaches zero. temporal cross-cumulant function (CTCCF), one As first shown in [13.5] and then, in more detail, obtains the Nth-order cyclic spectral cross-cumu- in [13.9,13.13,13.15,13.17], the Nth-order temporal b lant function (CSCCF) Pxðf Þ. It can be written as cumulant function of a time series provides a b b 0 T mathematical characterization of the notion of a P ðf Þ¼P ðf Þdðf 1 bÞ, (13.16) x x pure Nth-order sinewave. It is that part of the where sinewave present in the Nth-order lag product Z waveform that remains after removal of all parts b 0 9 b 0 j2pf 0Ts0 0 Pxðf Þ Cxðs Þe ds (13.17) that result from products of sinewaves in lower RN1 order lag products obtained by factoring the Nth- is the Nth-order cyclic cross-polyspectrum (CCP), order product. and bð 0Þ9 bð Þj Cx s Cx s tN ¼0 (13.18) 13.2.2. Discrete-time time series is called the reduced-dimension CTCCF (RD- In accordance with the definition for the CTCCF). The cyclic cross-polyspectrum is a well- continuous-time case, N not necessarily distinct behaved function under the mild conditions that discrete-time complex-valued time series xkðnÞ, n 2 Z, exhibit joint Nth-order wide-sense cyclos- the time-series xN ðtÞ and xkðt þ tkÞðk ¼ tationarity with cycle frequency eaa0 ðea 2½1; 1½Þ if 1; ...; N 1Þ are asymptotically (jtkj!1) inde- 2 2 pendent (in the FOT probability sense) so that at least one of the Nth-order CTCMF’s b 0 0 C ðs Þ!0asks k!1 and, moreover, there XN YN x ea b 0 0 Nþ1 e 1 ðÞk j2pean exists an 40 such that jC ðs Þj ¼ oðks k Þ as R ðmÞ9 lim x ðn þ mkÞe x x N!1 2N þ 1 k ks0k!1. Furthermore, except on a b-submani- n¼N k¼1 b 0 fold, the CCP Pxðf Þ is coincident with the RD- (13.21) CSCMF Sbðf 0Þ. x is not identically zero. In (13.21), xðnÞ9 The CCP can also be expressed as ðÞ1 ðÞN T 9 T ½x1 ðnÞ; ...; xN ðnÞ and m ½m1; ...; mN . 0 b 0 1 ðÞN T The magnitude and phase of the CTCMF P ðf Þ¼ lim cumfX N T ðt; ðÞN ða f 1ÞÞ, x T!1 T ; (13.21) are the amplitude and phase of the ðÞk sinewave component with frequency ea contained X ðt; ðÞkf kÞ, k;T in the discrete-time lag product whose temporal k ¼ 1; ...; N 1g, ð13:19Þ expected value is the discrete-time Nth-order where, in the computation of the cumulant in TCMF. (13.19), the stationary FOT expectation operation The N-fold discrete Fourier transform of the can be adopted as T !1. Eq. (13.19) reveals that CTCMF the CCP of N time series is the Nth-order cross- e X e e a 9 e a j2pmTm cumulant of their spectral components at frequen- SxðmÞ RxðmÞe , (13.22) 2 N cies f k whose sum is equal to b. In fact, Eq. (13.19) m Z ARTICLE IN PRESS

670 W.A. Gardner et al. / Signal Processing 86 (2006) 639–697

T where m9½n1; ...; nN , is called the Nth-order 14. Applications to circuits, systems, and control CSCMF and can be written as e e Xþ1 Applications of cyclostationarity to circuits, e a ea 0 e T systems, and control are in [14.1–14.31]. In circuit SxðmÞ¼Sxðm Þ dða m 1 rÞ, (13.23) r¼1 theory, cyclostationarity has been exploited in e ea 0 modelling noise [14.1,14.2,14.12,14.17,14.19,14.20, where the function Sxðm Þ is the Nth-order RD- CSCMF, which can be expressed as the ðN 1Þ- 14.24,14.26,14.28]. In system theory, cyclostation- fold discrete Fourier transform ary or almost cyclostationary signals arise in dealing with periodically or almost-periodically e X e 0 ea 0 ea 0 j2pm Tm0 time variant systems (see Section 3.6) [14.4–14.8, Sxðm Þ¼ Rxðm Þe (13.24) m02ZN1 14.13,14.14,14.23,14.25]. In control theory, cyclostationarity has been exploited in [14.3,14.10,14.16, of the Nth-order RD-CTCMF 14.18,14.21,14.22]. e e ea 0 e a On this subject, also see [9.4,9.59,10.5,10.24, R ðm Þ9R ðmÞj . (13.25) x x mN ¼0 13.15,13.28,15.1,16.3]. e Once the Nth-order discrete-time TCCF Cxðk; mÞ Patents on applications of cyclostationarity to is deﬁned analogously to the continuous-time case, system and circuit analysis and design are the Nth-order CTCCF is given by [14.27,14.29,14.31]. e XN e eb 1 e j2pbn C ðmÞ9 lim Cxðn; mÞe . x N!1 þ 2N 1 n¼N (13.26) 15. Applications to acoustics and mechanics Its discrete Fourier transform, referred to as the Applications of cyclostationarity to acoustics Nth-order CSCCF, is given by and mechanics are in [15.1–15.24]. Cyclostationar- e e Xþ1 b b ity has been exploited in acoustics and mechanics e ð Þ¼Pe ð 0Þ ðe T rÞ Px m x m d b m 1 , (13.27) for modelling road trafﬁc noise [15.2,15.3], for r¼1 e analyzing music signals [15.6], and for describing eb 0 where the function Pxðm Þ is referred to as the Nth- the vibration signals in mechanical systems. In order CCP, which can be expressed as the ðN 1Þ- mechanical systems with moving parts, such as fold discrete Fourier transform of the Nth-order engines, if some parameter such as speed, tem- RD-CTCCF perature, and load torque can be assumed to be

eb eb constant, then the dynamic physical processes e ð 0Þ9e ð Þj Cx m Cx m mN ¼0. (13.28) generate vibrations that can be modelled as Finally, it can be easily shown that the following originating from periodic mechanisms such as periodicity properties hold: rotation and reciprocation of gears, belts, chains, shafts, propellers, pistons, and so on [2.5]. Conse- e e e a e aþp quently, vibration signals exhibit periodic behavior RxðmÞ¼Rx ðmÞ; p 2 Z, (13.29) of one type or another with periods related to the ea eaþp engine cycle. Often, the observed vibration signals e ðmÞ¼ e ðm þ Þ; p 2 ; 2 N , (13.30) Sx Sx q Z q Z contain both an almost-periodic and an almost- e e cyclostationary component [15.1,15.5,15.8,15.10, eb ebþp CxðmÞ¼Cx ðmÞ; p 2 Z, (13.31) 15.14,15.18,15.23]. Even if the almost-cyclostationary component has a power smaller than the e e b bþp power of the almost-periodic component, it can be e ðmÞ¼ e ðm þ qÞ; p 2 ; q 2 N . (13.32) Px Px Z Z useful; for example it can be successfully exploited Analogous relations can be stated for the reduced- in early diagnosis of gear faults [15.4,15.8,15.9, dimension statistics. 15.11,15.13,15.14,15.16,15.19–15.21,15.24]. ARTICLE IN PRESS

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Additional applications in the ﬁeld of acoustics 19. Queueing and mechanics can be found in [4.33,4.43,4.49, 14.21,21.2]. Exploitation of cyclostationarity in queueing Patents on applications of cyclostationarity to theory in computer networks is treated in engine diagnosis are [15.12,15.17]. [19.1–19.4,19.6]. Queueing theory in car trafﬁc is treated in [19.5]. On this subject, also see [14.8]. 16. Applications to econometrics

In high-frequency financial time series, such as 20. Cyclostationary random fields asset return, the repetitive patterns of openings and closures of markets, the number of active A periodically correlated or cyclostationary markets throughout the day, seasonally varying random field is a second-order random field whose preferences, and so forth, are sources of periodic mean and correlation have periodic structure variations in financial-market volatility and other [20.6,20.7]. Specifically, a random field xðn1; n2Þ statistical parameters. Autoregressive models with indexed on Z2 is called strongly periodically periodically varying parameters provide appropri- correlated with period ðN01; N02Þ if and only if ate descriptions of seasonally varying economic there exists no smaller N0140andN0240 for time series [16.1–16.13]. Furthermore, neglecting which the mean and correlation satisfy the periodic behavior gives rise to a loss in forecast Efxðn þ N ; n þ N Þg ¼ Efxðn ; n Þg, (20.1) efficiency [2.15,16.8,16.10]. 1 01 2 02 1 2 On this subject see also [12.40]. 0 0 Efxðn1 þ N01; n2 þ N02Þx ðn1 þ N01; n2 þ N02Þg 0 0 ¼ Efxðn1; n2Þx ðn1; n2Þg ð20:2Þ 0 0 17. Applications to biology for all n1, n2, n1, n2 2 Z. Cyclostationary random fields are treated in [20.1–20.7]. Applications of cyclostationarity to biology are in [17.1–17.18]. Applications of the concept of spectral redundancy (spectral correlation 21. Generalizations of cyclostationarity or cyclostationarity) have been proposed in medical image signal processing [17.6,17.8],and 21.1. General aspects nondestructive evaluation [17.2]. Methods of averaging were developed for estimating the Generalizations of cyclostationary processes generalized spectrum that allow for the meaningful and time series are treated in [21.1–21.17]. The characterization of phase information when problem of statistical function estimation for classic assumptions of stationarity do not hold. general nonstationary persistent signals is ad- This has led to many significant performance dressed in [21.1,21.5] and limitations of previously improvements in ultrasonic tissue characteriza- proposed approaches are exposed. In tion, particularly for liver and breast tissues [21.2–21.4,21.8], the class of the correlation auto- [17.5,17.10–17.18]. regressive processes is studied. Nonstationary A patent on the application of cyclostationarity signals that are not ACS can arise from linear to cholesterol detection is [17.4]. time-variant, but not almost-periodically time- variant, transformations of ACS signals. Such transformations occur, for example, in mobile 18. Level crossings communications when the product of transmitted- signal bandwidth and observation interval is not Level crossings of cyclostationary signals have much smaller than the ratio between the pro- been characterized in [18.1–18.10]. pagation speed in the medium and the relative ARTICLE IN PRESS

672 W.A. Gardner et al. / Signal Processing 86 (2006) 639–697 radial speed between transmitter and receiver signals, the set [8.35,21.11]. Two models for the output sig- [ A9 A (21.5) nals of such transformations are the generalized t t2R almost-cyclostationary signals [21.7,21.9–21.11, 21.13,21.15–21.17], and the spectrally correlated is not necessarily countable. signals [21.6,21.12,21.14]. Moreover, communica- The ACS signals are obtained as the special case tions signals with parameters, such as the carrier of GACS signals for which the functions anðtÞ are frequency and the baud rate, that vary slowly constant with respect to t and are equal to the with time cannot be modelled as ACS but, cycle frequencies. In such a case the support of the rather, can be modelled as generalized almost- cyclic autocorrelation function in the ða; tÞ plane cyclostationary if the observation interval is large consists of lines parallel to the t axis and the enough [21.9]. generalized cyclic autocorrelation functions are coincident with the cyclic autocorrelation functions. Moreover, the set A turns out to be 21.2. Generalized almost-cyclostationary signals countable in this case (see (3.13)). The higher-order characterization of the GACS A continuous-time complex-valued time series signals in the FOT probability framework is xðtÞ is said to be wide-sense generalized almost- provided in [21.9]. Linear filtering is addressed in cyclostationary (GACS) if the almost-periodic [21.10,21.11] where the concept of expectation of component of its second-order lag product admits the impulse-response function in the FOT prob- a (generalized) Fourier series expansion with both ability framework is also introduced. The problem coefficients and frequencies depending on the lag of sampling a GACS signal is considered in [21.13] parameter t [21.9,21.10]: where it is shown that the discrete-time signal X fag ðnÞ j2panðtÞt obtained by uniformly sampling a continuous-time E fxðt þ tÞx ðtÞg ¼ Rxx ðtÞe . (21.1) n2I GACS signal is an ACS signal. The estimation of the cyclic autocorrelation function for GACS In (21.1), I is a countable set, the frequencies anðtÞ processes is addressed in [21.15,21.17] in the are referred to as lag-dependent cycle frequencies, stochastic process framework. In [21.16], a survey and the coefficients, referred to as generalized of GACS signals is provided. cyclic autocorrelation functions, are given by

ðnÞ 9 j2panðtÞt 21.3. Spectrally correlated signals Rxx ðtÞ hxðt þ tÞx ðtÞe it. (21.2) For GACS signals, the cyclic autocorrelation A continuous-time complex-valued second-or- function can be expressed in terms of the general- der harmonizable stochastic process xðtÞ is said to ized cyclic autocorrelation functions by the follow- be spectrally correlated (SC) if its Loe`ve bifre- ing relationship: quency spectrum can be expressed as [21.14] X a ðnÞ R ðtÞ¼ R ðtÞd . (21.3) S ðf ; f Þ9EfXðf ÞX ðf Þg xx xx aanðtÞ xx 1 2 X 1 2 n2I ðnÞ ¼ Sxx ðf 1Þdðf 2 Cnðf 1ÞÞ, ð21:6Þ Moreover, it results that n2I a where I is a countable set, the curves A 9fa 2 R : R ðtÞa0g t [ xx f 2 ¼ Cnðf 1Þ; n 2 I, describe the support of ¼ fa 2 R : a ¼ anðtÞg. ð21:4Þ ðnÞ Sxx ðf 1; f 2Þ, and the functions Sxx ðf 1Þ, referred n2I to as the spectral correlation density functions, That is, for GACS signals, the support in the ða; tÞ describe the density of the Loe` ve bifrequency a plane of the cyclic autocorrelation function Rxx ðtÞ spectrum on its support curves. The case of linear consists of a countable set of curves described by support curves is considered in [21.6,21.12]. The the equations a ¼ anðtÞ, n 2 I. For the GACS ACS processes are obtained as the special case of ARTICLE IN PRESS

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SC processes for which the support curves are lines [2.10] W.A. Gardner, Exploitation of spectral correlation in with unity slope. cyclostationary signals, in: Fourth Annual ASSP The bifrequency spectral correlation density Workshop on Spectrum Estimation and Modeling, Minneapolis, MN, 1988, pp. 1–6. function [2.11] W.A. Gardner, Introduction to Random Processes with X ðnÞ Applications to Signals and Systems, second ed., ¯ 9 Sxx ðf 1; f 2Þ Sxx ðf 1Þdf 2Cnðf 1Þ (21.7) McGraw-Hill, New York, 1990 (Chapter 12). n2I [2.12] W.A. Gardner, W.A. Brown, Fraction-of-time probability for time-series that exhibit cyclostationarity, is the density of the Loe` ve bifrequency spectrum Signal Processing 23 (1991) 273–292. on its support curves f 2 ¼ Cnðf 1Þ.In[21.14] it is [2.13] W.A. Gardner, C.-K. Chen, The Random Processes shown that, if the location of the support curves is Tutor, McGraw-Hill, New York, 1991. unknown, the bifrequency spectral correlation [2.14] W.A. Gardner, Exploitation of spectral redundancy in cyclostationary signals, IEEE Signal Process. Mag. 8 density function S¯ xx ðf ; f Þ can be reliably 1 2 (1991) 14–36. estimated by the time-smoothed cross-periodo- [2.15] W.A. Gardner, An introduction to cyclostationary gram only if the slope of the support curves is not signals, in: W.A. Gardner (Ed.), Cyclostationarity in too far from unity. Communications and Signal Processing, IEEE Press, New York, 1994, pp. 1–90. [2.16] W.A. Gardner, C.M. Spooner, Cyclostationary signal processing, in: C.T. Leondes (Ed.), Digital Signal References Processing Techniques and Applications, Academic Press, New York, 1994. [2.1] H.L. Hurd, An investigation of periodically correlated [2.17] G.B. Giannakis, Cyclostationary signal analysis, stochastic processes, Ph.D. Dissertation, Duke Uni- in: V.K. Madisetti, D.B. Williams (Eds.), The Digi- versity, Durham, NC, 1969. tal Signal Processing Handbook, CRC Press and [2.2] W.A. Gardner, Representation and estimation of IEEE Press, Boca Raton, FL and New York, 1998 cyclostationary processes, Ph.D. Dissertation, Depart- (Chapter 17). ment of Electrical and Computer Engineering, Uni- [2.18] H.L. Hurd, A.G. Miamee, Periodically Correlated versity of Massachusetts, reprinted as Signal and Image Random Sequences: Spectral Theory and Practice, Processing Lab Technical Report No. SIPL-82-1, Wiley, New York, 2006. Department of Electrical and Computer Engineering, [3.1] L.I. Gudzenko, On periodic nonstationary processes, University of California, Davis, CA, 95616, 1982, 1972. Radio Eng. Electron. Phys. 4 (1959) 220–224 (in [2.3] K.S. Voychishin, Y.P. Dragan, Example of formation Russian). of periodically correlated random processes, Radio [3.2] V.L. Lebedev, On random processes having nonstatio- Eng. Electron. Phys. 18 (1973) 1426–1429 (English narity of periodic character, Nauchn. Dokl. Vysshch. translation of Radiotekh. Elektron. 18, 1957,1960). Shchk. Ser. Radiotekh. Elektron. 2 (1959) 32–34 (in [2.4] Y.P. Dragan, Structure and Representations of Models Russian). of Stochastic Signals, Part 4, Naukova Dumka, Kiev, [3.3] E.G. Gladyshev, Periodically correlated random se- 1980. quences, Sov. Math. Dokl. 2 (1961) 385–388. [2.5] W.A. Gardner, Introduction to Random Processes with [3.4] J. Kampe´de Fe´riet, Correlations and spectra for Applications to Signals and Systems, Macmillan, New nonstationary random functions, Math. Comput. 16 York, 1985 (Chapter 12). (77) (1962) 1–21. [2.6] W.A. Gardner, The spectral correlation theory of [3.5] E.G. Gladyshev, Periodically and almost periodically cyclostationary time series, Signal Processing 11 correlated random processes with continuous time para- (1986) 13–36 (Erratum: Signal Processing 11 405.) meter, Theory Probab. Appl. (1963) 137–177 (in Russian). Received EURASIP’s Best Paper of the Year award. [3.6] D.M. Willis, The statistics of a particular non-homo- [2.7] A.M. Yaglom, Correlation Theory of Stationary and geneous Poisson process, Biometrika 51 (1964) 399–404. Related Random Functions, vols. I and II, Springer, [3.7] V.A. Markelov, Extrusions and phases of the periodi- New York, 1986. cally nonstationary random process, Izv. VUZ Radio- [2.8] W.A. Gardner, Statistical Spectral Analysis: A Non- tekhn. 9 (5) (1966). probabilistic Theory, Prentice-Hall, Englewood Cliffs, [3.8] A.A. Kayatskas, Periodically correlated random processes, NJ, 1987. Telecommun. Radio Eng. 23 (Part 2) (1968) 136–141. [2.9] W.A. Brown, On the theory of cyclostationary signals, [3.9] Y.P. Dragan, Periodically correlated random processes Ph.D. Dissertation, Department of Electrical Engineer- and transformations with periodically varying para- ing and Computer Science, University of California, meters, Otbor Peredacha Inform. 22 (1969) 27–33 (in Davis, CA, 1987. Russian). ARTICLE IN PRESS

674 W.A. Gardner et al. / Signal Processing 86 (2006) 639–697

[3.10] Y.P. Dragan, The spectral properties of periodically [3.28] M.G. Bilyik, Some transformations of periodic inho- correlated stochastic processes, Otbor Peredacha In- mogeneous random fields, Otbor Peredacha Inform. 57 form. 30 (1971) 16–24 (in Russian). (1979) 21–25 (in Russian). [3.11] H. Ogura, Spectral representation of a periodic [3.29] I. Honda, Sample periodicity of periodically correlated nonstationary random process, IEEE Trans. Inform. processes, Keio Math. Rep. 5 (1980) 13–18. Theory IT-17 (1971) 143–149. [3.30] A.G. Miamee, H. Salehi, On the prediction of [3.12] Y.P. Dragan, On foundations of the stochastic model of periodically correlated stochastic processes, in: P.R. rhythmic phenomena, Otbor Peredacha Inform. 31 Krishnaiah (Ed.), Multivariate Analysis—V, North- (1972) 21–27 (in Russian). Holland Publishing Company, Amsterdam, 1980, [3.13] K.N. Gupta, Stationariness provided by filtration to a pp. 167–179. periodic non-stationary random process, J. Sound Vib. [3.31] P. Kochel, Periodically stationary Markovian decision 23 (1972) 319–329. models, Elektron. Informationsverarb. Kybernet. 16 [3.14] K.S. Voychishin, Y.P. Dragan, Some properties of the (1980) 553–567 (in German). stochastic model of systems with periodically varying [3.32] I. Honda, Spectral representation of periodically parameters, Algorithms Prod. Process. (Kiev) (1972) correlated stochastic processes and approximate Four- 11–20 (in Russian). ier series, Keio Math. Sem. Rep. 6 (1981) 11–16. [3.15] K.S. Voychishin, Y.P. Dragan, The elimination [3.33] A.I. Kapustinskas, Properties of periodically nonsta- of rhythm from periodically correlated random pro- tionary processes, Trudy Akad. Nauk Litov. SSR Ser. B cesses, Otbor Peredacha Inform. 33 (1972) 12–16 1 (122) (1981) 87–95 (in Russian). (in Russian). [3.34] B.S. Rybakov, Correlation properties of the envelope and [3.16] R. Lugannani, Sample stability of periodically phase of a periodically nonstationary Normal process, correlated pulse trains, J. Franklin Inst. 296 (1973) Telecommun. Radio Eng. 35–36 (1981) 96–97 (English 179–190. translation of Elektrosvyaz and Radiotekhnika). [3.17] V.A. Melititskiy, V.O. Radzievskii, Statistical charac- [3.35] I. Honda, On the spectral representation and related teristics of a nonstationary Normal process and ways of properties of a periodically correlated stochastic pro- its modeling, Radio Electron. Commun. Syst. 16 (1973) cess, Trans. IECE of Japan E 65 (1982) 723–729. 82–89. [3.36] H. Ogura, Y. Yoshida, Time series analysis of a [3.18] Y.P. Dragan, I.N. Yavorskii, A paradox of the periodic stationary random process, Trans. Inst. Electr. rhythmic model, Otbor Peredacha Inform. 41 (1974) Comm. Eng. Japan 365-A (1982) 22–29 (in Japanese). 11–15 (in Russian). [3.37] M. Pourahmadi, H. Salehi, On subordination and linear [3.19] H.L. Hurd, Periodically correlated processes with transformation of harmonizable and periodically corre- discontinuous correlation functions, Theory Probab. lated processes, in: Probability Theory on Vector Appl. 9 (1974) 804–807. Spaces, III, Springer (Lublin), Berlin, New York, [3.20] H.L. Hurd, Stationarizing properties of random shifts, 1983, pp. 195–213. SIAM J. Appl. Math. 26 (1) (1974) 203–212. [3.38] A.I. Kamolor, Kolmogorov diameters of a class of [3.21] Y.P. Dragan, General properties of the stochastic random processes, Dokl. Akad. Nauk USSR 8 (1984) model of rhythmic phenomena, Otbor Peredacha 8–10 (in Russian). Inform. 44 (1975) 3–14 (in Russian). [3.39] Y.P. Dragan, I.N. Yavorskii, Statistical analysis of [3.22] Y.P. Dragan, The representation of a periodically periodic random processes, Otbor Peredacha Inform. 71 correlated random process by stationary components, (1985) 20–29 (in Russian). Otbor Peredacha Inform. 45 (1975) 7–20 (in Russian). [3.40] Y.P. Dragan, Periodic and periodically nonstationary [3.23] W.A. Gardner, L.E. Franks, Characterization of random processes, Otbor Peredacha Inform. 72 (1985) cyclostationary random signal processes, IEEE Trans. 3–17 (in Russian). Inform. Theory IT-21 (1975) 4–14. [3.41] I.N. Yavorskii, Statistical analysis of periodically [3.24] Z.A. Piranashvili, L.O. Kavtaradze, Certain properties correlated random processes, Radiotekh. Elektron. 30 of periodic nonstationary random processes, V.I. Lenin. (1985) 1096–1104 (in Russian). Sakharth. Politekh. Inst. Samecn. Shrom 3 (176), Math. [3.42] B.S. Rybakov, Correlation coefficients of the envelope Mehk. (1975) 93–105 (in Russian). and phase of a signal for periodically nonstationary [3.25] Y.P. Dragan, Harmonizability and spectral distri- Gaussian noise, Radiotekhnika 41 (1986) 3–7 (in bution of random processes with finite mean power, Russian). Dokl. Arad. Nauk Ukrain 8 (1978) 679–684 (in [3.43] W.A. Gardner, Rice’s representation for cyclostation- Russian). ary processes, IEEE Trans. Commun. COM-35 (1987) [3.26] W.A. Gardner, Stationarizable random processes, 74–78. IEEE Trans. Inform. Theory IT-24 (1978) 8–22. [3.44] W.A. Gardner, Common pitfalls in the application of [3.27] J. Rootenberg, S.A. Ghozati, Generation of a class of stationary process theory to time-sampled and modu- nonstationary random processes, Internat. J. Systems lated signals, IEEE Trans. Commun. COM-35 (1987) Sci. 9 (1978) 935–947. 529–534. ARTICLE IN PRESS

W.A. Gardner et al. / Signal Processing 86 (2006) 639–697 675

[3.45] V.A. Melititskiy, A.I. Mosionzhik, A probabilistic [3.61] A. Makagon, A.G. Miamee, H. Salehi, Periodically model of non Gaussian periodically nonstationary correlated processes and their spectrum, in: A.G. radio signals, Sov. J. Commun. Technol. Electron. 32 Miamee (Ed.), Proceedings of the Workshop on (1987) 100–106. Nonstationary Stochastic Processes and their Applica- [3.46] I.N. Yavorskii, Statistical analysis of periodically tions, World Scientific, Virginia, VA, 1991, pp. 147–164. correlated vector random processes, Otbor Peredacha [3.62] H.L. Hurd, G. Kallianpur, Periodically correlated Inform. 76 (1987) 3–12. processes and their relationship to L2½0; T-valued [3.47] N.S. Akinshin, V.L. Rumyantsev, A.V. Mikhailov, stationary sequences, in: A.G. Miamee (Ed.), Proceed- V.V. Melititskaya, Statistical characteristics of the ings of the Workshop on Nonstationary Stochastic envelope of an additive mixture of a non-Gaussian Processes and their Applications, World Scientific, periodically nonstationary radio signal and non-Gaus- Virginia, VA, 1991, pp. 256–284. sian interference, Radioelectron. Commun. Systems 31 [3.63] G.D. Z˘ivanovic´, On the instantaneous frequency of (1988) 89–92. cyclostationary random signals, IEEE Trans. Signal [3.48] A.Y. Dorogovtsev, L. Tkhuan, The existence of Process. 39 (7) (1991) 1604–1610. periodic and stationary regimes of discrete dynamical [3.64] W.A. Gardner, A unifying view of coherence in signal systems in a Banach space, Kibernetika (Kiev) (6) processing, Signal Processing 29 (1992) 113–140. (1988) 121–123, 136. [3.65] H.L. Hurd, Almost periodically unitary stochastic [3.49] S.M. Rytov, Y.A. Kratsov, V.I. Tatarskii, Principles of processes, Stochastic Process. Appl. 43 (1992) 99–113. Statistical Radiophysics. Correlation Theory of Ran- [3.66] D.G. Konstantinides, V.I. Pieterbarg, Extreme values dom Processes, vol. 2, Springer, Berlin, 1988. of the cyclostationary Gaussian random process, J. [3.50] R. Ballerini, W.P. McCormick, Extreme value theory Appl. Probab. 30 (1993) 82–97. for processes with periodic variances, Stochastic Models [3.67] V.P. Sathe, P.P. Vaidyanathan, Effects of multirate 5 (1989) 45–61. systems on statistical properties of random signals, [3.51] H.L. Hurd, Representation of strongly harmonizable IEEE Trans. Signal Process. 41 (1) (1993) 131–146. periodically correlated processes and their covariances, [3.68] D. Dehay, Spectral analysis of the covariance of the J. Multivariate Anal. 29 (1989) 53–67. almost periodically correlated processes, Stochastic [3.52] I.N. Yavorskii, Statistical analysis of poly- and nearly Process. Appl. 50 (1994) 315–330. periodically-correlated random processes, Otbor Per- [3.69] A. Makagon, A.G. Miamee, H. Salehi, Continuous edacha Inform. 79 (1989) 1–10 (in Russian). time periodically correlated processes: spectrum and [3.53] V.G. Alekseev, On the construction of spectral densities prediction, Stochastic Process. Appl. 49 (1994) 277–295. of a periodically correlated random process, Problemy [3.70] S. Cambanis, C. Houdre´, On the continuous wavelet Peredachi Inform. 26 (1990) 106–108. transform of second-order random processes, IEEE [3.54] A.Y. Dorogovtsev, Stationary and periodic solutions of Trans. Inform. Theory 41 (3) (1995) 628–642. a stochastic difference equation in a Banach space, [3.71] S. Lambert, Extension of autocovariance coefficients Teor. Veroyatnost. Mat. Statist. ( 42) (1990) 35–42. sequence for periodically correlated random processes [3.55] A.Y. Dorogovtsev, Necessary and sufficient conditions by using the partial autocorrelation function, in: for existence of stationary and periodic solutions of a Proceedings of VIII European Signal Processing Con- stochastic difference equation in Hilbert space, Com- ference (EUSIPCO’96), Trieste, Italy, 1996. put. Math. Appl. 19 (1) (1990) 31–37. [3.72] R.J. Swift, Almost periodic harmonizable processes, [3.56] A.Y. Dorogovtsev, Stationary and periodic solutions of Georgian Math. J. 3 (3) (1996) 275–292. stochastic difference and differential equations in [3.73] L.E. Franks, Random processes, in: J. Gibson (Ed.), Banach space, in: New Trends in Probability and The Communications Handbook, CRC Press and IEEE Statistics, vol. 1, Bakuriani, 1990, VSP, Utrecht, 1991, Press, New York, 1997 (Chapter 5). pp. 375–390. [3.74] H. Zhang, Maximum entropy modeling of periodically [3.57] A.G. Miamee, Periodically correlated processes and correlated processes, IEEE Trans. Inform. Theory 43 their stationary dilations, SIAM J. Appl. Math. 50 (4) (6) (1997) 2033–2035. (1990) 1194–1199. [3.75] G. De Nicolao, G. Ferrari-Trecate, On the Wold [3.58] D. Dehay, Contributions to the spectral analysis of decomposition of discrete-time cyclostationary pro- nonstationary processes, Dissertation for the Habilita- cesses, IEEE Trans. Signal Process. 47 (7) (1999) tion to Direct Research, Universite des Sciences et 2041–2043. Techniques de Lille Flandres Artois, 1991. [3.76] B. Lall, S.D. Joshi, R.K.P. Bhatt, Second-order [3.59] H.L. Hurd, Correlation theory of almost periodically statistical characterization of the filter bank and its correlated processes, J. Multivariate Anal. 37 (1991) elements, IEEE Trans. Signal Process. 47 (6) (1999) 24–45. 1745–1749. [3.60] G.D. Z˘ivanovic´, W.A. Gardner, Degrees of cyclosta- [3.77] A. Makagon, Induced stationary process and structure tionarity and their application to signal detection and of locally square integrable periodically correlated estimation, Signal Processing 22 (1991) 287–297. processes, Stud. Math. 136 (1) (1999) 71–86. ARTICLE IN PRESS

676 W.A. Gardner et al. / Signal Processing 86 (2006) 639–697

[3.78] S. Akkarakaran, P.P. Vaidyanathan, Bifrequency and [4.4] A. Renger, Spectral analysis of periodically nonsta- bispectrum maps: a new look at multirate systems with tionary stochastic impulse processes, Z. Angew. Math. stochastic inputs, IEEE Trans. Signal Process. 48 (3) Mech. 57 (1977) 681–692 (in German). (2000) 723–736. [4.5] A.I. Kapustinskas, Covariational estimation of the [3.79] D. Alpay, A. Chevreuil, P. Loubaton, An extension parameters of a periodically nonstationary autoregres- problem for discrete-time periodically correlated sto- sive process, Liet. Tsr Mokslu Akad. Darbai B Ser. 104 chastic processes, J. Time Series Anal. 22 (1) (2001) (1978) 113–119 (in Russian). 1–12. [4.6] Y.P. Dragan, V.P. Mezentsev, I.N. Yavorskii, Algo- [3.80] A. Makagon, Characterization of the spectra of rithm and program for calculating estimates of the periodically correlated processes, J. Multivariate Anal. characteristics of periodically correlated random pro- 78 (1) (2001) 1–10. cesses, Otbor Peredacha Inform. 58 (1979) 10–15 (in [3.81] S. Touati, J.-C. Pesquet, Statistical properties of the Russian). wavelet decomposition of cyclostationary processes, in: [4.7] S.M. Porotskii, M.Y. Tsymbalyuk, Extension of Proceedings of the IEEE-EURASIP Workshop on stochastic approximation procedures to periodically Nonlinear Signal and Image Processing, Baltimore, nonstationary random processes, Automat. Remote MD, 2001. Control 40 (Part 2) (1979) 606–609. [3.82] P. Borgnat, P. Flandrin, P.-O. Amblard, Stochastic [4.8] W.A. Gardner, Cyclo-ergodicity, Part I: Application to discrete scale invariance, IEEE Signal Process. Lett. 9 spectral extraction, in: D.G. Lainiotis, N.S. Tzannes (6) (2002) 181–184. (Eds.), Advances in Communications, D. Reidel, Hol- [3.83] A.Y. Dorogovtsev, .I.Discretesystems Periodicity in land, 1980, pp. 203–209. distribution, Internat. J. Math. Math. Sci. 30 (2) [4.9] S.I. Yuran, Methods for the instrumental analysis of (2002) 65–127. periodically nonstationary random processes, Meas. [3.84] A.G. Miamee, G.H. Shahkar, Shift operator for Tech. 23 (1980) 1069–1071. periodically correlated processes, Indian J. Pure Appl. [4.10] D. Osteyee, Testing Markov properties of time-series, Math. 33 (5) (2002) 705–712. in: Proceedings of Time-Series Analysis, Houston, TX, [3.85] H.J. Woerdeman, The Carathe´odory–Toeplitz problem North-Holland, Amsterdam, 1981, pp. 385–400. with partial data, Linear Algebra Appl. 342 (2002) [4.11] Y.P. Dragan, V.P. Mezentsev, I.N. Yavorskii, Symme- 149–161. try of the covariance matrix of measurements of a [3.86] R. Guidorzi, R. Diversi, Minimal representations periodically correlated random process, Otbor Pereda- of MIMO time-varying systems and realization of cha Inform. 66 (1982) 3–6 (in Russian). cyclostationary models, Automatica 39 (2003) [4.12] H.J. Newton, Using periodic autoregressions for multi- 1903–1914. ple spectral estimation, Technometrics 24 (1982) [3.87] H.L. Hurd, T. Koski, The Wold isomorphism for 109–116. cyclostationary sequences, Signal Processing 84 (2004) [4.13] R.A. Boyles, W.A. Gardner, Cycloergodic properties 813–824. of discrete parameter non-stationary stochastic pro- [3.88] H.L. Hurd, T. Koski, Spectral theory of cyclostationary cesses, IEEE Trans. Inform. Theory IT-29 (1983) arrays, in: A.C. Krinik, R.J. Swift (Eds.), Stochastic 105–114. Processes and Functional Analysis: Recent Advances. A [4.14] I. Honda, A note on periodogram analysis of periodi- Volume in Honor of Professor M.M. Rao, Marcel cally correlated stochastic processes, Keio Math. Sem. Dekker, New York, 2004. Rep. 8 (1983) 1–7. [3.89] D.G. Konstantinides, V.I. Pieterbarg, S. Stamotovic, [4.15] I.N. Yavorskii, Properties of estimators of mathema- Gnedenko-type limit theorems for cyclostationary w2- tical expectation and correlation function of periodi- processes, Lithuanian Math. J. 44 (2) (2004) 157–167. cally-correlated random processes, Otbor Peredacha [3.90] P. Borgnat, P. Amblard, P. Flandrin, Scale invariances Inform. 67 (1983) 22–28 (in Russian). and Lamperti transformations for stochastic processes, [4.16] Y.P. Dragan, V.P. Mezentsev, I.N. Yavorskii, Compu- J. Phys. A: Math. Gen. 38 (2005) 2081–2101. tation of estimates for spectral characteristics of [3.91] S. Lambert-Lacroix, Extension of autocovariance coef- periodically correlated random processes, Otbor Per- ﬁcients sequence for periodically correlated processes, J. edacha Inform 69 (1984) 29–35 (in Russian). Time Ser. Anal. 26 (3) (2005) 423–435. [4.17] W.A. Gardner, Measurement of spectral correlation, [4.1] L.J. Herbst, Almost periodic variances, Ann. Math. IEEE Trans. Acoust. Speech Signal Process. ASSP-34 Stat. 34 (1963) 1549–1557. (1986) 1111–1123. [4.2] L.J. Herbst, The statistical Fourier analysis of var- [4.18] I.N. Yavorskii, Component estimates of the probability iances, J. Roy. Statist. Soc. B 27 (1965) 159–165. characteristics of periodically correlated random pro- [4.3] A.I. Kapustinskas, Estimation of the parameters of a cesses, Avtomatika 19 (1986) 44–48 (in Russian). periodically nonstationary autoregressive process, Tru- [4.19] I.N. Yavorskii, Interpolation of estimates of probability dy Akad. Nauk Litov. SSR Ser. B 4 (101) (1977) characteristics of periodically correlated random pro- 115–121 (in Russian). cesses, Avtometrika 1 (1987) 36–41 (in Russian). ARTICLE IN PRESS

W.A. Gardner et al. / Signal Processing 86 (2006) 639–697 677

[4.20] V.G. Alekseev, Estimating the spectral densities of a [4.37] J. Les´kow, A. Weron, Ergodic behavior and estimation Gaussian periodically correlated stochastic process, for periodically correlated processes, Statist. Probab. Problems Inform. Transmission 24 (1988) 109–115. Lett. 15 (1992) 299–304. [4.21] W.A. Brown, H.H. Loomis Jr., Digital implementations [4.38] W.A. Brown, H.H. Loomis Jr., Digital implementation of spectral correlation analyzers, in: Proceedings of the of spectral correlation analyzers, IEEE Trans. Signal Fourth Annual ASSP Workshop on Spectrum Estima- Process. 41 (2) (1993) 703–720. tion and Modeling, Minneapolis, MN, 1988, pp. 264–270 [4.39] W.A. Gardner, R.S. Roberts, One-bit spectral correla- [4.22] V.P. Mezentsev, I.N. Yavorskii, Estimation of prob- tion algorithms, IEEE Trans. Signal Process. 41 (1) ability characteristics of rhythmic signals as a problem (1993) 423–427. of linear filtration, Radioelectron. Commun. Systems 31 [4.40] S. Cambanis, C.H. Houdre´, H.L. Hurd, J. Les´kow, (1988) 70–72. Laws of large numbers for periodically and almost [4.23] C.J. Tian, A limiting property of sample autocovar- periodically correlated processes, Stochastic Process. iances of periodically correlated processes with applica- Appl. 53 (1994) 37–54. tion to period determination, J. Time Ser. Anal. 9 [4.41] D. Dehay, H.L. Hurd, Representation and estimation (1988) 411–417. for periodically and almost periodically correlated [4.24] H.L. Hurd, Nonparametric time series analysis for random processes, in: W.A. Gardner (Ed.), Cyclosta- periodically correlated processes, IEEE Trans. Inform. tionarity in Communications and Signal Processing, Theory 35 (1989) 350–359. IEEE Press, New York, 1994, pp. 295–326 (Chapter 6). [4.25] R.S. Roberts, Architectures for digital cyclic spectral [4.42] M.J. Genossar, H. Lev-Ari, T. Kailath, Consistent analysis, Ph.D. Dissertation, Department of Electrical estimation of the cyclic autocorrelation, IEEE Trans. Engineering and Computer Science, University of Signal Process. 42 (3) (1994) 595–603. California, Davis, CA, 1989. [4.43] N.L. Gerr, C. Allen, The generalized spectrum and [4.26] I. Honda, On the ergodicity of Gaussian periodically spectral coherence of harmonizable time series, Digital correlated stochastic processes, Trans. Inst. Electron. Signal Process. 4 (1994) 222–238. Inform. Commun. Eng. E 73 (10) (1990) 1729–1737. [4.44] J. Les´kow, Asymptotic normality of the spectral density [4.27] C. Jacob, Ergodicity in periodic autoregressive models, estimator for almost periodically correlated stochastic C. R. Acad. Sci. Ser. I 310 (1990) 431–434 (in French). processes, Stochastic Process. Appl. 52 (1994) 351–360. [4.28] D.E. Martin, Estimation of the minimal period of [4.45] R.S. Roberts, W.A. Brown, H.H. Loomis Jr., A review periodically correlated processes, Ph.D. Dissertation, of digital spectral correlation analysis: theory and University of Maryland, 1990. implementation, in: W.A. Gardner (Ed.), Cyclostatio- [4.29] H. Sakai, Spectral analysis and lattice filter for periodic narity in Communications and Signal Processing, IEEE autoregressive processes, Electron. Commun. Japan Press, New York, 1994, pp. 455–479. (Part 3) 73 (1990) 9–15. [4.46] T. Biedka, L. Mili, J.H. Reed, Robust estimation [4.30] V.G. Alekseev, Spectral density estimators of a of cyclic correlation in contaminated Gaussian noise, periodically correlated stochastic process, Problems in: Proceedings of 29th Asilomar Conference on Inform. Transmission 26 (1991) 286–288. Signals, Systems and Computers, Pacific Grove, CA, [4.31] W.A. Gardner, Two alternative philosophies for 1995 estimation of the parameters of time-series, IEEE [4.47] D. Dehay, Asymptotic behavior of estimators of cyclic Trans. Inform. Theory 37 (1991) 216–218. functional parameters for some nonstationary pro- [4.32] R.S. Roberts, W.A. Brown, H.H. Loomis Jr., Compu- cesses, Statist. Decisions 13 (1995) 273–286. tationally efficient algorithms for cyclic spectral analy- [4.48] R.S. Roberts, J.H.H. Loomis, Parallel computation sis, IEEE Signal Process. Mag. 8 (1991) 38–49. structures for a class of cyclic spectral analysis [4.33] S. Yamaguchi, Y. Kato, A statistical study for algorithms, J. VLSI Signal Process. 10 (1995) 25–40. determining the minimum sample size for Leq estima- [4.49] B.M. Sadler, Acousto-optic cyclostationary signal tion of periodic nonstationary random noise, Appl. processing, Appl. Opt. 34 (23) (1995) 5091–5099. Acoust. 32 (1991) 35–48. [4.50] S.V. Schell, Asymptotic moments of estimated cyclic [4.34] J. Wilbur, J. Bono, Wigner/cycle spectrum analysis of correlation matrices, IEEE Trans. Signal Process. 43 (1) spread spectrum and diversity transmissions, IEEE J. (1995) 173–180. Ocean. Eng. 16 (1) (1991) 98–106. [4.51] M.C. Sullivan, E.J. Wegman, Estimating spectral [4.35] H.L. Hurd, J. Les´kow, Estimation of the Fourier correlations with simple nonlinear transformations, coefficient functions and their spectral densities for f- IEEE Trans. Signal Process. 43 (6) (1995) 1525–1526. mixing almost periodically correlated processes, Statist. [4.52] D. Dehay, J. Les´kow, Functional limit theory for the Probab. Lett. 14 (1992) 299–306. spectral covariance estimator, J. Appl. Probab. 33 [4.36] H.L. Hurd, J. Les´kow, Strongly consistent and asymp- (1996) 1077–1092. totically normal estimation of the covariance for almost [4.53] D. Dehay, V. Monsan, Random sampling estimation periodically correlated processes, Statist. Decisions 10 for almost-periodically correlated processes, J. Time (1992) 201–225. Ser. Anal. 17 (5) (1996) 425–445. ARTICLE IN PRESS

678 W.A. Gardner et al. / Signal Processing 86 (2006) 639–697

[4.54] A. Makagon, A.G. Miamee, Weak law of large [5.11] V.E. Gurevich, S.Y. Korshunov, Statistical character- numbers for almost periodically correlated processes, istics of a narrow-band cyclo-stationary process, Tele- Proc. Amer. Math. Soc. 124 (6) (1996) 1899–1902. commun. Radio Eng. (Part 2) 42 (1987) 87–91. [4.55] F. Dominique, J.H. Reed, Estimating spectral correla- [5.12] C.-K. Chen, Spectral correlation characterization of tions using the least mean square algorithm, Electron. modulated signals with application to signal detection Lett. 33 (3) (1997) 182–184. and source location, Ph.D. Dissertation, Department of [4.56] H. Li, Q. Cheng, Almost sure convergence analysis Electrical Engineering and Computer Science, Univer- of mixed time averages and kth-order cyclic sta- sity of California, Davis, CA, 1989. tistics, IEEE Trans. Inform. Theory 43 (4) (1997) [5.13] D.C. Bukofzer, Characterization and applications of 1265–1268. multiplexed PAM/PPM processes, in: Proceedings of [4.57] P.W. Wu, H. Lev-Ari, Optimized estimation of the International Symposium on Information Theory moments for nonstationary signals, IEEE Trans. on and Applications (ISITA’90), Hawaii, USA, 1990, Signal Process. 45 (5) (1997) 1210–1221. pp. 915–918. [4.58] L.G. Hanin, B.M. Schreiber, Discrete spectrum of [5.14] A.I. Kozlov, A.I. Mosionzhik, V.R. Rusinov, Statistical nonstationary stochastic processes on groups, J. Theo- characteristics of polarization parameters of non- ret. Probab. 11 (4) (1998) 1111–1133. Gaussian periodically non stationary radio signals, [4.59] B.M. Sadler, A.V. Dandawate´, Nonparametric estima- Sov. J. Commun. Technol. Electron. 35 (1990) 130–134. tion of the cyclic cross spectrum, IEEE Trans. Inform. [5.15] J. Wilbur, R.J. McDonald, Nonlinear analysis of Theory 44 (1) (1998) 351–358. cyclically correlated spectral spreading in modulated [4.60] Z. Huang, W. Wang, Y. Zhou, W. Jiang, Asymptotic signals, J. Acoust. Soc. Amer. 92 (1992) 219–230. analysis of estimated cyclic cross-correlation function [5.16] M.R. Bell, R.A. Grubbs, JEM modeling and measure- between stationary and cyclostationary processes, ments for radar target identification, IEEE Trans. System Eng. Electron. 14 (3) (2003) 87–91. Aerospace 29 (1993) 73–87. [4.61] D. Dehay, V. Monsan, Discrete periodic sampling with [5.17] J.M.H. Elmirghani, R.A. Cryan, F.M. Clayton, Spec- jitter and almost periodically correlated processes, tral analysis of timing jitter effects on the cyclosta- Statist. Inference Stochastic Process. (2005). tionary PPM format, Signal Processing 43 (3) (1995) [5.1] W.R. Bennett, Statistics of regenerative digital trans- 269–277. mission, Bell System Tech. J. 37 (1958) 1501–1542. [5.18] P. Gournay, P. Viravau, Theoretical spectral correla- [5.2] L.E. Franks, Signal Theory, Prentice-Hall, Englewood tion of CPM modulations—Part I: Analytic results for Cliffs, NJ, 1969 (Chapter 8). two-states CPFSK modulation (1REC), Ann. Te´lećom- [5.3] A.A. Kayatskas, Quasi-periodically correlated random mun. 53 (7–8) (1998) 267–278 (in French). processes, Telecommun. Radio Eng. (Part 2) 25 (1970) [5.19] P. Viravau, P. Gournay, Theoretical spectral correla- 145–146 (English translation of Radiotekhnika). tion of CPM modulations—Part II: General calculation [5.4] P. van Der Wurf, On the spectral density of a method and analysis, Ann. Te´lećommun. 53 (7–8) cyclostationary process, IEEE Trans. Commun. 22 (1998) 279–288 (in French). (10) (1974) 1727–1730. [5.20] M.Z. Win, On the power spectral density of digital [5.5] Y.V. Berezin, I.V. Krasheninnikov, Distribution func- pulse streams generated by M-ary cyclostationary tions of the envelope and phase of a periodically non sequences in the presence of stationary timing jitter, stationary process frequently encountered in radio IEEE Trans. Commun. 46 (9) (1998) 1135–1145. physics, Geomagn. Aeron. 18 (1978) 309–312. [5.21] D. Vuc˘ic´, M. Obradovic´, Spectral correlation evalua- [5.6] O.P. Dragan, Y.P. Dragan, Y.V. Karavan, A pro- tion of MSK and offset QPSK modulation, Signal babilistic model of rhythmicity in the radiolysis of Processing 78 (1999) 363–367. solids, Otbor Peredacha Inform. 62 (1980) 26–31 (in [5.23] A. Napolitano, C.M. Spooner, Cyclic spectral analysis Russian). of continuous-phase modulated signals, IEEE Trans. [5.7] A. Papoulis, Random modulation: a review, IEEE Signal Process. 49 (1) (2001) 30–44. Trans. Acoust. Speech Signal Process. ASSP-31 (1983) [6.1] A.S. Monin, Stationary and periodic time series in the 96–105. general circulation of the atmosphere, in: E.M. Rosen- [5.8] A. Papoulis, Probability, Random Variables, and blatt (Ed.), Proceedings of the Symposium on Time Stochastic Processes, second ed., McGraw-Hill, New Series Analysis, Wiley, New York, 1963, pp. 144–151. York, 1984. [6.2] R.H. Jones, W.M. Brelsford, Time series with periodic [5.9] W.A. Gardner, Spectral correlation of modulated structure, Biometrika 54 (1967) 403–408. signals, Part I—Analog modulation, IEEE Trans. [6.3] R.K. Bhuiya, Stochastic analysis of periodic hydrologic Commun. COM-35 (6) (1987) 584–594. process, J. Hydraulics Division, American Society of [5.10] W.A. Gardner, W.A. Brown, C.-K. Chen, Spectral Civil Engineers 97 (1971) 949–963. correlation of modulated signals, Part II—Digital [6.4] A. Renger, Behaviour of linear oscillatory systems modulation, IEEE Trans. Commun. COM-35 (6) which are energized by stochastic impulse trains, (1987) 595–601. Maschinenbau Tech. 20 (1971) 596–600 (in German). ARTICLE IN PRESS

W.A. Gardner et al. / Signal Processing 86 (2006) 639–697 679

[6.5] K.S. Voychishin, Y.P. Dragan, A simple stochastic [6.21] K.-Y. Kim, G.R. North, Seasonal cycle and second- model of the natural rhythmic processes, Otbor moment statistics of a simple coupled climate system, Peredacha Inform. 2 (1971) 7–15 (in Russian). J. Geophys. Res. 97 (20) (1992) 437–448. [6.6] D.F. Dubois, B. Besserides, Nonlinear theory of [6.22] P. Bloomfield, H.L. Hurd, R. Lund, Periodic correla- parametric instabilities in plasmas, Phys. Rev. A 14 tion in stratospheric ozone time series, J. Time Ser. (1976) 1869–1882. Anal. 15 (2) (1994) 127–150. [6.7] B. Arciplani, F. Carloni, M. Marseguerra, Neutron [6.23] R.B. Lund, H.L. Hurd, P. Bloomfield, R.L. Smith, counting statistics in a subcritical cyclostationary Climatological time series with periodic correlation, multiplying system, Nucl. Instrum. Methods (Part I J. Climate 11 (1995) 2787–2809. (1979) 465–473. [6.24] K.-Y. Kim, G.R. North, J. Huang, EOFs of one- [6.8] B. Arciplani, F. Carloni, M. Marseguerra, Neutron dimensional cyclostationary time series: computations, counting statistics in a subcritical cyclostationary examples and stochastic modeling, J. Atmos. Sci. 53 multiplying system, Nucl. Instrum. Methods (Part II) (1996) 1007–1017. 172 (1980) 531. [6.25] A.R. Kacimov, Y.V. Obnosov, N.D. Yakimov, [6.9] K. Hasselmann, T.P. Barnett, Techniques of linear Groundwater flow in a medium with parquet-type prediction for systems with periodic statistics, J. Atmos. conductivity distribution, J. Hydrol. 226 (1999) Sci. 38 (1981) 2275–2283. 242–249. [6.10] W.K. Johnson, The dynamic pneumocardiogram: an [6.26] F. Gini, M. Greco, Texture modelling, estimation and application of coherent signal processing to cardiovas- validation using measured sea clutter, IEE Proc.-Radar, cular measurement, IEEE Trans. Biomed. Eng. BME- Sonar Navig. 149 (3) (2002) 115–124. 28 (1981) 471–475. [6.27] X. Menendez-Pidal, G.H. Abrego, System and method [6.11] Y.P. Dragan, I.N. Yavorskii, Rhythmics of sea waves for performing speech recognition in cyclostationary and underwater acoustic signals, Naukova Dumka, noise environments, US Patent No. 6,785,648, August Kiev, 1982 (in Russian). 31, 2004. [6.12] T.P. Barnett, Interaction of the monsoon and pacific [7.1] W.A. Gardner, The structure of least-mean-square trade wind system at interannual time scales. I. The linear estimators for synchronous M-ary signals, IEEE equatorial zone, Mon. Weather Rev. III (1983) Trans. Inform. Theory IT-19 (1973) 240–243. 756–773. [7.2] W.A. Gardner, An equivalent linear model for marked [6.13] T.P. Barnett, H.D. Heinz, K. Hasselmann, Statistical and filtered doubly stochastic Poisson processes with prediction of seasonal air temperature over Eurasia, application to MMSE estimation for synchronous M- Tellus Ser. A (Sweden) 36A (1984) 132–146. ary optical data signals, IEEE Trans. Commun. [6.14] Y.P. Dragan, V.A. Rozhkov, I.N. Yavorskii, Applica- Technol. COM-24 (1976) 917–921. tions of the theory of periodically correlated random [7.3] M.F. Mesiya, P.J. McLane, L.L. Campbell, Optimal processes to the probabilistic analysis of oceanological receiver filters for BPSK transmission over a bandlim- time series, in: V.A. Rozhkov (Ed.), In Probabilistic ited nonlinear channel, IEEE Trans. Commun. COM- Analysis and Modeling of Oceanological Processes, 26 (1978) 12–22. Gidrometeoizdat, Leningrad, USSR, 1984, pp. 4–23 (in [7.4] T.H.E. Ericson, Modulation by means of linear periodic Russian). filtering, IEEE Trans. Inform. Theory IT-27 (1981) [6.15] W. Dunsmuir, Time series regression with periodically 322–327. correlated errors and missing data, in: Time Series [7.5] J.C. Campbell, A.J. Gibbs, B.M. Smith, The cy- Analysis of Irregularly Observed Data, Springer, New clostationary nature of crosstalk interference from York, Berlin, 1984, pp. 78–107. digital signals in multipair cable—Part I: Fundamen- [6.16] C.M. Johnson, P. Lemke, T.P. Barnett, Linear predic- tals, IEEE Trans. Commun. COM-31 (5) (1983) tion of sea ice anomalies, J. Geophys. Res. 90 (1985) 629–637. 5665–5675. [7.6] J.C. Campbell, A.J. Gibbs, B.M. Smith, The cyclosta- [6.17] M.J. Ortiz, A. Ruiz de Elvira, A cyclo-stationary model tionary nature of crosstalk interference from digital of sea surface temperature in the pacific ocean, Tellus signals in multipair cable—Part II: Applications and Ser. A 37A (1985) 14–23. further results, IEEE Trans. Commun. COM-31 (5) [6.18] V.O. Alekseev, Extraction of the trend in a periodically (1983) 638–649. correlated time series, Atmos. Ocean. Phys. 23 (1987) [7.7] F.K. Graef, Joint optimization of transmitter and 187–191. receiver for cyclostationary random signal processes, [6.19] Y.P. Dragan, V.A. Rozhkov, I.N. Yavorskii, Methods in: Proceedings of the NATO Advanced Study Institute of Probabilistic Analysis of Oceanological Rhythmics, on Nonlinear Stochastic Problems, Reidel, Algarve, Gidrometeoizdat, Leningrad, USSR, 1987 (in Russian). Portugal, 1983, pp. 581–592. [6.20] Y.P. Dragan, Principles of a linear theory of stochastic [7.8] J.P.A. Albuquerque, O. Shimbo, L.N. Ngugen, Mod- test signals and their statistical analysis, Otbor Per- ulation transfer noise effects from a continuous digital edacha Inform. 77 (1988) 2–7 (in Russian). carrier to FDM/FM carriers in memoryless nonlinear ARTICLE IN PRESS

680 W.A. Gardner et al. / Signal Processing 86 (2006) 639–697

devices, IEEE Trans. Commun. COM-32 (1984) [7.22] J.-P. Leduc, Image modelling for digital TV codecs, in: 337–353. Proceedings of the SPIE, Lausanne, Switzerland, 1990, [7.9] B.M. Smith, P.G. Potter, Design criteria for crosstalk pp. 1160–1170. interference between digital signals in multipair cable, [7.23] J.H. Reed, T.C. Hsia, The performance of time- IEEE Trans. Commun. COM-34 (1986) 593–599. dependent adaptive fitters for interference rejection, [7.10] J.W.M. Bergmans, A.J.E.M. Janssen, Robust data IEEE Trans. Acoust. Speech Signal Process. 38 (8) equalization, fractional tap spacing and the Zak trans- (1990) 1373–1385. form, Philips J. Res. 23 (1987) 351–398. [7.24] R. Mendoza, J.H. Reed, T.C. Hsia, B.G. Agee, [7.11] J.H. Reed, Time-dependent adaptive filters for inter- Interference rejection using time-dependent constant ference rejection, Ph.D.Dissertation, Department of modulus algorithms (CMA) and the hybrid CMA/ Electrical Engineering and Computer Science, Univer- spectral correlation discriminator, IEEE Trans. Signal sity of California, Davis, CA, 1987. Process. 39 (9) (1991) 2108–2111. [7.12] B.G. Agee, The baseband modulus restoral approach to [7.25] B.R. Petersen, D.D. Falconer, Minimum mean square blind adaptive signal demodulation, in: Proceedings of equalization in cyclostationary and stationary inter- Digital Signal Processing Workshop, 1988. ference—analysis and subscriber line calculations, IEEE [7.13] V. Joshi, D.D. Falconer, Reduced state sequence J. Sel. Areas Commun. 9 (6) (1991) 931–940. estimation techniques for digital subscriber loop appli- [7.26] M. Abdulrahman, D.D. Falconer, Cyclostationary cation, in: Proceedings of GLOBECOM ’88, vol. 2, cross-talk suppression by decision feedback equaliza- Hollywood, CA, 1988, pp. 799–803. tion on digital subscriber loops, IEEE J. Sel. Areas [7.14] J.H. Reed, A.A. Quilici, T.C. Hsia, A frequency domain Commun. 12 (4) (1992) 640–649. time dependent adaptive fliter for interference rejection, [7.27] H. Ding, M.M. Fahmy, Synchronised linear almost in: Proceedings of MILCOM 88, vol. 2, San Diego, CA, periodically time-varying adaptive filters, IEE Proc.-I 1988, pp. 391–397. 139 (24) (1992) 429–436. [7.15] B.G. Agee, The property restoral approach to blind [7.28] C. Pateros, G. Saulnier, Interference suppression and adaptive signal extraction, Ph.D. Dissertation, Depart- multipath mitigation using an adaptive correlator direct ment of Electrical Engineering and Computer Science, sequence spread spectrum receiver, in: Proceedings of University of California, Davis, CA, 1989. IEEE ICC, 1992, pp. 662–666. [7.16] B.G. Agee, The property-restoral approach to blind [7.29] B.G. Agee, Solving the near-far problem: exploitation adaptive signal extraction, in: Proceedings of CSI-ARO of spectral and spatial coherence in wireless personal Workshop on Advanced Topics in Communications, communication networks, in: Proceedings of Virginia 1989. Tech. Third Symposium on Wireless Personal Commu- [7.17] B.G. Agee, S. Venkataraman, Adaptive demodu- nications, vol. 15, 1993, pp. 1–12. lation of PCM signals in the frequency domain, in: [7.30] H. Ding, M.M. Fahmy, Performance of output-mod- Proceedings of the 23rd Asilomar Conference on ulator-structured linear almost periodically time-varying Signals, Systems and Computers, Pacific Grove, CA, adaptive filters, IEE Proc.-I 140 (2) (1993) 114–120. 1989. [7.31] W.A. Gardner, Cyclic Wiener filtering: theory and [7.18] A. Fung, L.S. Lee, D.D. Falconer, A facility for near method, IEEE Trans. Commun. 41 (1) (1993) 151–163. end crosstalk measurements on ISDN subscriber loops, [7.32] J. Proakis, Reduced-complexity simultaneous beam- in: Proceedings of GLOBECOM ’89, IEEE Global forming and equalization for underwater acoustic Telecommunications Conference and Exhibition, vol. 3, communications, in: Proceedings of Oceans’93, 1993. Dallas, TX, 1989, pp. 1592–1596. [7.33] M. Stojanovic, J. Catipovic, J. Proakis, Adaptive [7.19] W.A. Gardner, W.A. Brown, Frequency-shift filtering receivers for underwater acoustic communications: their theory for adaptive co-channel interference removal, in: relation to beamforming and diversity combining, in: Conference Record, 23rd Asilomar Conference on Proceedings of COMCON, vol. 4, 1993. Signals, Systems and Computers, vol. 2, Pacific Grove, [7.34] M. Stojanovic, J. Catipovic, J. Proakis, Adaptive CA, 1989, pp. 562–567. multichannel combining and equalization for under- [7.20] J.H. Reed, C. Greene, T.C. Hsia, Demodulation of a water acoustic communications, J. Acoust. Soc. Amer. direct sequence spread-spectrum signal using an optimal 94 (1993) 1621–1631. time-dependent filter, in: Proceedings of IEEE Militar [7.35] J.W.M. Bergmans, Data receiver comprising a control Communications Conference (MILCOM’89), Boston, loop with reduced sampling frequency, US Patent No. MA, 1989, pp. 657–662. 5,311,558, May 10, 1994. [7.21] W.A. Gardner, S. Venkataraman, Performance of [7.36] I. Fijalkow, F. Lopez de Victoria, C. Johnson, Adaptive optimum and adaptive frequency-shift filters for fractionally spaced blind equalization, in: Proceedings co-channel interference and fading, in: Proceedings of of Digital Signal Processing Workshop, 1994. the 24th Annual Asilomar Conference on Signals, [7.37] G.D. Golden, Extended bandwidth transmitter for Systems, and Computers, Pacific Grove, CA, 1990, crosstalk channels, US Patent No. 5,377,230, December pp. 242–247. 27, 1994. ARTICLE IN PRESS

W.A. Gardner et al. / Signal Processing 86 (2006) 639–697 681

[7.38] U. Madhow, M. Honig, MMSE interference suppres- access communication systems, IEEE Trans. Commun. sion for direct-sequence spread-spectrum CDMA, IEEE 46 (3) (1998) 392–399. Trans. Commun. 42 (12) (1994) 3178–3188. [7.54] J. Cui, D.D. Falconer, A.U.H. Sheikh, Blind adapta- [7.39] P. Rapajic, B. Vucetic, Adaptive receiver structures for tion of antenna arrays using a simple algorithm based asynchronous CDMA systems, IEEE J. Sel. Areas on small frequency offset, IEEE Trans. Commun. 46 (1) Commun. 12 (4) (1994) 685–697. (1998) 61–70. [7.40] S. Roy, J. Yang, P.S. Kumar, Joint transmitter/receiver [7.55] W.A. Gardner, S.V. Schell, GMSK signal processors optimization for multiuser communications, in: W.A. for improved communications capacity and quality, US Gardner (Ed.), Cyclostationarity in Communications Patent No. 5,848,105, December 8, 1998. and Signal Processing, IEEE Press, New York, 1994, [7.56] G. Gelli, L. Paura, A.M. Tulino, Cyclostationarity- pp. 329–361. based filtering for narrowband interference suppression [7.41] G.K. Yeung, W.A. Brown, W.A. Gardner, A new in direct-sequence spread-spectrum systems, IEEE J. algorithm for blind-adaptive frequency-shift filtering, Sel. Areas Commun. 16 (1998) 1747–1755. in: 28th Annual Conference on Information Science and [7.57] H. Lev-Ari, Adaptive RLS filtering under the cyclo- Systems, Princeton, NJ, 1994. stationary regime, in: Proceedings of International [7.42] J.W.M. Bergmans, Transmission system with increased Conference on Acoustics, Speech, and Signal Processing sampling rate detection, US Patent No. 5,463,654, (ICASSP’98), 1998, pp. 2185–2188. October 31, 1995. [7.58] M. Lops, G. Ricci, A.M. Tulino, Narrow-band-inter- [7.43] W.A. Gardner, G.K. Yeung, W.A. Brown, Signal ference suppression in multiuser CDMA systems, IEEE reconstruction after spectral excision, in: Proceedings Trans. Commun. 46 (9) (1998) 1163–1175. of the 29th Annual Asilomar Conference on Signals, [7.59] U. Madhow, Blind adaptive interference suppression Systems, and Computers, Pacific Grove, CA, 1995. for direct-sequence CDMA, Proc. IEEE (1998) [7.44] S. Miller, An adaptive direct sequence code division 2049–2069. multiple access receiver for multiuser interference [7.60] D. McLernon, Inter-relationships between different rejection, IEEE Trans. Commun. 43 (2) (1995) structures for periodic systems, in: Proceedings of IX 1746–1755. European Signal Processing Conference (EUSIP- [7.45] J.J. Nicolas, J.S. Lim, Method and apparatus for CO’98), Rhodes, Greece, 1998, pp. 1885–1888. decoding broadcast digital HDTV in the presence of [7.61] X. Wang, H.V. Poor, Blind multiuser detection: a quasi-cyclostationary interference, US Patent No. subspace approach, IEEE Trans. Inform. Theory 44 (2) 5,453,797, September 26, 1995. (1998) 30–44. [7.46] J.H. Reed, N.M. Yuen, T.C. Hsia, An optimal receiver [7.62] A. Duverdier, B. Lacaze, D. Roviras, Introduction of using a time-dependent adaptive filter, IEEE Trans. linear cyclostationary filters to model time-variant Commun. 43 (2/3/4) (1995) 187–190. channels, in: Proceedings of Global Telecom- [7.47] N. Uzun, R.A. Haddad, Cyclostationary modeling, and munications Conference (GLOBECOM’99), 1999, optimization compensation of quantization errors in pp. 325–329. subband codecs, IEEE Trans. Signal Process. 43 (9) [7.63] B. Francis, S. Dasgupta, Signal compression by (1995) 2109–2119. subband coding, Automatica 35 (1999) 1895–1908. [7.48] S. Bensley, B. Aazhang, Subspace-based channel [7.64] A. Pandharipande, S. Dasgupta, Subband coding of estimation for code division multiple access systems, cyclostationary signals with static bit allocation, IEEE IEEE Trans. Commun. 44 (8) (1996) 1009–1020. Signal Process. Lett. 6 (11) (1999) 284–286. [7.49] S. Ohno, H. Sakai, Optimization of filter banks using [7.65] U. Madhow, M. Honig, On the average near-far cyclostationary spectral analysis, IEEE Trans. Signal resistance for MMSE detection of direct-sequence Process. 44 (11) (1996) 2718–2725. CDMA signals with random spreading, IEEE Trans. [7.50] C. Pateros, G. Saulnier, An adaptive correlator receiver Inform. Theory 45 (9) (1999). for direct-sequence spread-spectrum communication, [7.66] N. Mangalvedhe, Development and analysis of adap- IEEE Trans. Commun. 44 (1996) 1543–1552. tive interference rejection techniques for direct sequence [7.51] E. Strom, S. Parkvall, S. Miller, B. Ottersten, Propaga- code division multiple access systems, Ph.D. Disserta- tion delay estimation in asynchronous direct-sequence tion, Bradley Department of Electrical Engineering, code-division multiple access systems, IEEE Trans. Virginia Polytechnic Institute and State University, Commun. 44 (1) (1996) 84–93. 1999. [7.52] B.G. Agee, P. Kelly, D. Gerlach, The backtalk airlink [7.67] E. Strom, S. Miller, Properties of the single-bit single- for full exploitation of spectral and spatial diversity in user MMSE receiver for DS-CDMA systems, IEEE wireless communication systems, in: Proceedings of Trans. Commun. 47 (3) (1999) 416–425. Fourth Workshop on Smart Antennas in Wireless [7.68] A.M. Tulino, Interference suppression in CDMA Mobile Communications, 1997. systems, Ph.D. Dissertation, Dipartimento di Ingegner- [7.53] S. Bensley, B. Aazhang, Maximum-likelihood synchro- ia dell’Informazione, Seconda Universita` di Napoli, nization of a single user for code-division multiple- Aversa, Italy, January 1999 (in Italian). ARTICLE IN PRESS

682 W.A. Gardner et al. / Signal Processing 86 (2006) 639–697

[7.69] J. Zhang, K.M. Wong, Z.Q. Luo, P.C. Ching, Blind communications, US Patent No. 6,240,098, May 29, adaptive FRESH filtering for signal extraction, IEEE 2001. Trans. Signal Process. 47 (5) (1999) 1397–1402. [7.84] A.M. Tulino, S. Verdu`, Asymptotic analysis of im- [7.70] B.G. Agee, Stacked-carrier discrete multiple tone proved linear receivers for BPSK-CDMA subject to communication technology and combinations with code fading, IEEE J. Sel. Areas Commun. 19 (2001) nulling, interference cancellation, retrodirective com- 1544–1555. munication adaptive antenna arrays, US Patent No. [7.85] F. Verde, Equalization and interference suppression 6,128,276, October 3, 2000. techniques for high-capacity digital systems, Ph.D. [7.71] A. Duverdier, B. Lacaze, On the use of periodic clock Dissertation, Dipartimento di Ingegneria Elettronica e changes to implement linear periodic time-varying delle Telecomunicazioni, Universita` di Napoli Federico filters, IEEE Trans. Circuits Systems II: Analog Digital II, Napoli, Italy, November 2001 (in Italian). Signal Process. 47 (11) (2000) 1152–1159. [7.86] B.G. Agee, M. Bromberg, D. Gerlach, D. Gibbons, J.T. [7.72] G. Gelli, L. Paura, A.R.P. Ragozzini, Blind widely Golden, M. Ho, E. Hole, M. Jesse, R.L. Maxwell, R.G. linear multiuser detection, IEEE Commun. Lett. 4 (6) Mechaley Jr., R.R. Naish, D.J. Nix, D.J. Ryan, D. (2000) 187–189. Stephenson, High bandwidth efficient communications, [7.73] M. Lops, A.M. Tulino, Simultaneous suppression of US Patent No. 6,359,923, March 19, 2002. multiaccess and narrowband interference in asynchro- [7.87] Z. Ding, P.H. Thomas, R. Wang, W. Tong, Co-channel nous CDMA networks, IEEE Trans. Veh. Technol. 49 interference reduction in wireless communications (2000) 1705–1718. systems, US Patent No. 6,396,885, May 28, 2002. [7.74] M. Martone, Blind adaptive detection of DS/CDMA [7.88] A. Gameiro, Simple receiver for systems with spectrally signals on time-varying multipath channels with anten- overlapping narrowband and broadband signals, Wire- na array using higher-order statistics, IEEE Trans. less Personal Commun. 23 (2002) 311–320. Commun. 48 (9) (2000) 1590–1600. [7.89] Z. Huang, Y. Zhou, W. Jiang, On cyclic correlation [7.75] S.-J. Yu, F.-B. Ueng, Implementation of cyclostation- matched filtering, Acta Electron. Sinica 30 (12) (2002) ary signal-based adaptive arrays, Signal Processing 80 122–126 (in Chinese). (2000) 2249–2254. [7.90] B. Lacaze, D. Roviras, Effect of random permutations [7.76] S. Buzzi, M. Lops, A.M. Tulino, Partially blind applied to random sequences and related applications, adaptive MMSE interference rejection in asynchronous Signal Processing 82 (6) (2002) 821–831. DS/CDMA networks over frequency-selective fading [7.91] G.D. Mandyam, Digital-to-analog conversion of pulse channels, IEEE Trans. Commun. 49 (1) (2001) 94–108. amplitude modulated systems using adaptive quantiza- [7.77] S. Buzzi, M. Lops, A.M. Tulino, A new family of tion, Wireless Personal Commun. 23 (2002) 253–281. MMSE multiuser receiver for interference sup- [7.92] B.G. Agee, Stacked carrier discrete multiple tone pression in DS/CDMA systems employing BPSK communication systems, US Patent No. 6,512,737, modulation, IEEE Trans. Commun. 49 (1) (2001) January 28, 2003. 154–167. [7.93] H. Arslan, K.J. Molnar, Methods of receiving co- [7.78] S. Buzzi, M. Lops, A.M. Tulino, Blind adaptive channel signals by channel separation and successive multiuser detection for asynchronous dual rate DS/ cancellation and related receivers, US Patent No. CDMA systems, IEEE J. Sel. Areas Commun. 19 (2001) 6,574,235, June 3, 2003. 233–244. [7.94] G. Gelli, F. Verde, Blind FSR-LPTV equalization and [7.79] W.A. Gardner, Suppression of cochannel interference interference rejection, IEEE Trans. Commun. 51 (2) in GSM by pre-demodulation signal processing, in: (2003) 145–150. Proceedings of the 11th Virginia Tech. Symposium on [7.95] G.D. Mandyam, Adaptive quantization in a pulse- Wireless Personal Communications, 2001, pp. 217–228. amplitude modulated system, US Patent No. 6,535,564, [7.80] W.A. Gardner, C.W. Reed, Making the most out of March 18, 2003. spectral redundancy in GSM: Cheap CCI suppression, [7.96] F. Trans, Means and method for increasing perfor- in: Proceedings of the 36th Annual Asilomar Con- mances of interference-suppression based receivers, US ference on Signals, Systems and Computers, Pacific Patent No. 6,553,085, April 22, 2003. Grove, CA, 2001. [7.97] J.H. Cho, Joint transmitter and receiver optimization in [7.81] G. Gelli, F. Verde, Adaptive minimum variance additive cyclostationary noise, IEEE Trans. Inform. equalization with interference suppression capabilities, Theory 50 (12) (2004) 3396–3405. IEEE Commun. Lett. 5 (12) (2001) 491–493. [7.98] G.-H. Im, H.-C. Won, Performance analysis of [7.82] A. Sabharwal, U. Mitra, R. Moses, MMSE receivers cyclostationary interference suppression for multiuser for multirate DS-CDMA systems, IEEE Trans. Com- wired communication systems, J. Commun. Networks 6 mun. 49 (12) (2001) 2184–2197. (2) (2004) 93–105. [7.83] J. Thibault, P. Chevalier, F. Pipon, J.-J. Monot, G. [7.99] T.L. Carroll, Low-interference communications device Multedo, Method and device for space division multi- using chaotic signals, US Patent No. 6,854,058, plexing of radio signals transmitted in cellular radio February 8, 2005. ARTICLE IN PRESS

W.A. Gardner et al. / Signal Processing 86 (2006) 639–697 683

[7.100] F. Gourge, F. Roosen, Device enabling different Cyclostationarity in Communications and Signal Pro- spreading factors while preserving a common rambling cessing, IEEE Press, New York, 1994, pp. 362–390. code, in particular for transmission in a code division [8.16] J. Riba, G. Vazquez, Bayesian recursive estimation of multiple access cellular mobile radio system, US Patent frequency and timing exploiting the cyclostationarity No. 6,868,077, March 15, 2005. property, Signal Processing 40 (1994) 21–37. [7.101] O. Vourinen, T. Seppanen, J. Roning, J. Lilleberg, T. [8.17] K.E. Scott, M. Kaube, K. Anvari, Timing and Kolehmainen, Method for distinguishing signals from automatic frequency control of digital receiver using one another, and filter, US Patent No. 6,847,689, the cyclic properties of a non-linear operation, US January 25, 2005. Patent No. 5,282,228, January 25, 1994. [8.1] L.E. Franks, J. Bubrouski, Statistical properties of [8.18] K.E. Scott, E.B. Olasz, Simultaneous clock phase and timing jitter in a PAM timing recovery scheme, IEEE frequency offset estimation, IEEE Trans. Commun. 43 Trans. Commun., COM-22 (1974) 913–930. (7) (1995) 2263–2270. [8.2] U. Mengali, E. Pezzani, Tracking properties of phase- [8.19] B.G. Agee, R.J. Kleinman, J.H. Reed, Soft synchroni- locked loops in optical communication systems, IEEE zation of direct sequence spread-spectrum signals, IEEE Trans. Commun. COM-26 (1978) 1811–1818. Trans. Commun. 44 (11) (1996) 1527–1536. [8.3] L.E. Franks, Carrier and bit synchronization in data [8.20] F. Gini, G.B. Giannakis, Frequency offset and symbol communication—A tutorial review, IEEE Trans. Com- timing recover in flat-fading channels: a cyclostationary mun. COM-28 (1980) 1107–1121. approach, IEEE Trans. Commun. 46 (1998) 400–411. [8.4] M. Moeneclaey, Comment on ‘Tracking performance [8.21] P. Ciblat, Some estimation problems relative to non co- of the filter and square bit synchronizer’, IEEE Trans. operative communications, Ph.D. Dissertation, Univ. Commun. COM-30 (1982) 407–410. Marne-la-Valleé, Marne-la-Valleé, France, 2000 (in [8.5] M. Moeneclaey, Linear phase-locked loop theory for French). cyclostationary input disturbances, IEEE Trans. Com- [8.20] E. Serpedin, A. Chevreuil, G.B. Giannakis, P. Louba- mun. COM-30 (10) (1982) 2253–2259. ton, Blind channel and carrier frequency offset estima- [8.6] C.Y. Yoon, W.C. Lindsay, Phase-locked loop perfor- tion using periodic modulation precoders, IEEE Trans. mance in the presence of CW interference and additive Signal Process. 8 (8) (2000) 2389–2405. noise, IEEE Trans. Commun. COM-30 (1982) [8.23] H. Bo¨lcskei, Blind estimation of symbol timing and 2305–2311. carrier frequency offset in wireless OFDM systems, [8.7] M. Moeneclaey, The optimum closed-loop transfer IEEE Trans. Commun. 49 (6) (2001) 988–999. function of a phase-locked loop used for synchroniza- [8.24] A. Napolitano, M. Tanda, Blind parameter estimation tion purposes, IEEE Trans. Commun. COM-31 (1983) in multiple-access systems, IEEE Trans. Commun. 49 549–553. (4) (2001) 688–698. [8.8] C. Bilardi, S. Pupolin, Spectral analysis of the powers of [8.25] P. Ciblat, P. Loubaton, E. Serpedin, G.B. Giannakis, a PAM digital signal, Alta Frequenza 53 (2) (1984) Performance analysis of blind carrier frequency offset 70–76. estimators for noncircular transmissions through fre- [8.9] M. Moeneclaey, A fundamental lower bound on the quency-selective channels, IEEE Trans. Signal Process. performance of practical joint carrier and bit synchro- 50 (1) (2002) 130–140. nizers, IEEE Trans. Commun. COM-32 (1984) [8.26] P. Ciblat, P. Loubaton, E. Serpedin, G.B. Giannakis, 1007–1012. Asymptotic analysis of blind cyclic correlation-based [8.10] J.J. O’Reilly, Timing extraction for baseband digital symbol-rate estimators, IEEE Trans. Inform. Theory 48 transmission, in: K.W. Cattermole, J.J. O’Reilly (Eds.), (7) (2002) 1922–1934. Problems of Randomness in Communication Engineer- [8.27] I. Lakkis, D. O’Shea, M.K. Tayebi, B. Hatim, ing, Plymouth, London, 1984. Performance analysis of a class of nondata-aided [8.11] S. Pupolin, C. Tomasi, Spectral analysis of line frequency offset and symbol timing estimators for flat- regenerator time jitter, IEEE Trans. Commun. COM- fading channels, IEEE Trans. Signal Process. 50 (9) 32 (1984) 561–566. (2002) 2295–2305. [8.12] C.M. Monti, G.L. Pierobon, Block codes for linear [8.28] Y. Wang, P. Ciblat, E. Serpedin, P. Loubaton, timing recovery in data transmission systems, IEEE Performance analysis of a class of nondata-aided Trans. Commun. COM-33 (1985) 527–534. frequency offset and symbol timing estimators for flat- [8.13] W.A. Gardner, The role of spectral correlation in fading channels, IEEE Trans. Signal Process. 50 (9) design and performance analysis of synchronizers, (2002) 2295–2305. IEEE Trans. Commun. COM-34 (11) (1986) 1089–1095. [8.29] P. Ciblat, E. Serpedin, On a blind fractionally sampling- [8.14] Z.H. Qiu, F.A. Cassara, Steady-state analysis of the based carrier frequency offset estimator for noncircular clock synchronizer for a narrowband communication transmissions, IEEE Signal Process. Lett. 10 (4) (2003) system, Int. J. Electron. 66 (1989) 551–570. 89–92. [8.15] N.M. Blachman, Beneficial effects of spectral correla- [8.30] P. Ciblat, L. Vandendorpe, Blind carrier frequency tion on synchronization, in: W.A. Gardner (Ed.), offset estimation for noncircular constellation-based ARTICLE IN PRESS

684 W.A. Gardner et al. / Signal Processing 86 (2006) 639–697

transmissions, IEEE Trans. Signal Process. 51 (5) Spectral Estimation and Modeling, Minneapolis, MN, (2003) 1378–1389. 1988, pp. 277–282. [8.31] Y. Wang, E. Serpedin, P. Ciblat, An alternative blind [9.10] W.A. Gardner, Simplification of MUSIC and ESPRIT feedforward symbol timing estimator using two samples by exploitation of cyclostationarity, Proc. IEEE 76 (7) per symbol, IEEE Trans. Commun. 51 (9) (2003) (1988) 845–847. 1451–1455. [9.11] W.A. Gardner, C.-K. Chen, Selective source location by [8.32] Y. Wang, E. Serpedin, P. Ciblat, Optimal blind carrier exploitation of spectral coherence, in: Proceedings of recovery for MPSK burst transmissions, IEEE Trans. the Fourth Annual ASSP Workshop on Spectrum Commun. 51 (9) (2003) 1571–1581. Estimation and Modeling, Minneapolis, MN, 1988, [8.33] P. Ciblat, E. Serpedin, A fine blind frequency offset pp. 271–276. estimator for OFDM/OQAM systems, IEEE Trans. [9.12] W.A. Gardner, C.-K. Chen, Interference-tolerant time- Signal Process. 52 (1) (2004) 291–296. difference-of-arrival estimation for modulated signals, [8.34] I. Lakkis, D. O’Shea, M.K. Tayebi, Non-data aided IEEE Trans. Acoustics Speech Signal Process. ASSP-36 maximum likelihood based feedforward timing syn- (1988) 1385–1395. chronization method, US Patent No. 6,768,780, July 27, [9.13] V.A. Omel’chenko, A.V. Omel’chenko, Y.P. Dragan, 2004. O.A. Kolesnikov, .I Recognising Gaussian periodically- [8.35] A. Napolitano, M. Tanda, Doppler-channel blind correlated random signals, Radiotekhnika 85 (1988) identification for non-circular transmissions in multi- 75–79 (in Russian). ple-access systems, IEEE Trans. Commun. 52 (12) [9.14] B. Sankar, N. Tugbay, The use of the Wigner-Ville (2004) 2172–2191. distribution (WVD) for the classification and the [8.36] A. Napolitano, S. Ricciardelli, M. Tanda, Blind analysis of modulation techniques, in: Proceedings of estimation of amplitudes, time delays and frequency ISCIS III, Cesme, Turkey, 1988, pp. 75–82. offsets in multiple access systems with circular transmis- [9.15] S.V. Schell, B.G. Agee, Application of the SCORE sions, Signal Processing 85 (8) (2005) 1588–1601. algorithm and SCORE extensions to sorting in the [9.1] K.L. Jordan, Discrete representations of random rank-L spectral self-coherence environment, in: Con- signals, Ph.D. Dissertation, Department of Electrical ference Record, 22nd Asilomar Conference on Signals, Engineering, Massachusetts Institute of Technology, Systems and Computers, Pacific Grove, CA, 1988, pp. Cambridge, MA, July 1961. 274–278. [9.2] Y.P. Dragan, Expansion of random processes and their [9.16] J.S. Friedman, J.P. King, J.P. Pride III, Time difference noncommutative transformations, Otbor Peredacha of arrival geolocation method, etc., US Patent No. Inform. 22 (1969) 22–27 (in Russian). 4,888,593, December 19, 1989. [9.3] F.L. Rocha, Adaptive deconvolution of cyclostationary [9.17] S.V. Schell, R.A. Calabretta, W.A. Gardner, B.G. signals. I, Rev. Telegr. Electron. 68 (1980) 1026–1030 Agee, Cyclic MUSIC algorithms for signal-selective (in Spanish). direction estimation, in: Proceedings of the Interna- [9.4] E.R. Ferrara Jr., B. Widrow, The time-sequenced tional Conference on Acoustics, Speech, and Signal adaptive filter, IEEE Trans. Acoust. Speech Signal Processing (ICASSP’89), Glasgow, UK, 1989, pp. Process. ASSP-29 (1981) 679–683. 2278–2281. [9.5] M. Hajivandi, Tracking performance of stochastic [9.18] S.V. Schell, W.A. Gardner, Signal-selective direction gradient algorithms for nonstationary processes, Ph.D. finding for fully correlated signals, in: Abstracts of Sixth Dissertation, Department of Electrical and Com- Multidimensional Signal Processing Workshop of the puter Engineering, University of California, Davis, IEEE/ASSP Society, Pacific Grove, CA, 1989, pp. CA, 1982. 139–140. [9.6] C.A. French, W.A. Gardner, Despreading spread- [9.19] C.M. Spooner, C.-K. Chen, W.A. Gardner, Maximum spectrum signals without the code, IEEE Trans. likelihood two-sensor detection and TDOA estimation Commun. COM (1986) 404–407. for cyclostationary signals, in: Abstracts of Sixth [9.7] B.G. Agee, S.V. Schell, W.A. Gardner, Self-coherence Multidimensional Signal Processing Workshop of the restoral: a new approach to blind adaptation of antenna IEEE/ASSP Society, Pacific Grove, CA, 1989, pp. arrays, in: Proceedings of the 21st Annual Asilomar 119–120. Conference on Signals, Systems, and Computers, Pacific [9.20] G. Xu, T. Kailath, Array signal processing via Grove, CA, 1987, pp. 589–593. exploitation of spectral correlation—a combination of [9.8] W.A. Gardner, Nonstationary learning characteristics temporal and spatial processing, in: Conference Record, of the LMS algorithm, IEEE Trans. Circuits Systems 23rd Asilomar Conference on Signals, Systems and CAS-34 (1987) 1199–1207. Computers, Pacific Grove, CA, 1989, pp. 945–949. [9.9] B.G. Agee, S.V. Schell, W.A. Gardner, The SCORE [9.21] B.C. Agee, S.V. Schell, W.A. Gardner, Spectral self- approach to blind adaptive signal extraction: an coherence restoral: a new approach to blind adaptive application of the theory of spectral correlation, in: signal extraction using antenna arrays, Proc. IEEE 78 Proceedings of the IEEE/ASSP Fourth Workshop on (1990) 753–767. ARTICLE IN PRESS

W.A. Gardner et al. / Signal Processing 86 (2006) 639–697 685

[9.22] B.G. Agee, D. Young, Blind capture and geolocation of Treizie` me Colloque sur le Traitement du Signal et des stationary waveforms using multiplatform temporal Images (GRETSI’91), Juan- Les-Pins, France, 1991. and spectral self-coherence restoral, in: Proceedings of [9.35] D.C. McLernon, Analysis of LMS algorithm with 24th Asilomar Conference on Signals, Systems and inputs from cyclostationary random processes, Elec- Computers, Pacific Grove, CA, 1990. tron. Lett. 27 (1991) 136–138. [9.23] H. Ding, M.M. Fahmy, Inverse of linear periodically [9.36] W.A. Gardner, C.-K. Chen, Signal-selective time- time-varying filtering, in: Proceedings of International difference-of-arrival estimation for passive location of Conference on Acoustic, Speech, and Signal Processing manmade signal sources in highly corruptive environ- (ICASSP’90), vol. 3, Albuquerque, NM, 1990, ments. Part I: Theory and method IEEE Trans. Signal pp. 1217–1220. Process. 40 (1992) 1168–1184. [9.24] M. Hajivandi, W.A. Gardner, Measures of tracking [9.37] C.-K. Chen, W.A. Gardner, Signal-selective time- performance for the LMS algorithm, IEEE Trans. difference-of-arrival estimation for passive location of Acoust. Speech Signal Process. 38 (1990) 1953–1958. manmade signal sources in highly corruptive environ- [9.25] V. Joshi, D.D. Falconer, Sequence estimation techni- ments. Part II: Algorithms and performance, IEEE ques for digital subscriber loop transmission with Trans. Signal Process. 40 (1992) 1185–1197. crosstalk interference, IEEE Trans. Commun. 38 [9.38] W.A. Gardner, C.M. Spooner, Comparison of auto- (1990) 1367–1374. correlation and cross-correlation methods for signal- [9.26] S.V. Schell, Exploitation of spectral correlation for selective TDOA estimation, IEEE Trans. Signal Pro- signal-selective direction finding, Ph.D. Dissertation, cess. 40 (10) (1992) 2606–2608. Department of Electrical Engineering and Computer [9.39] L. Izzo, L. Paura, G. Poggi, An interference-tolerant Science, University of California, Davis, CA, 1990. algorithm for localization of cyclostationary-signal [9.27] S.V. Schell, W.A. Gardner, Signal-selective high-resolu- sources, IEEE Trans. Signal Process. 40 (1992) 1682–1686. tion direction finding in multipath, in: Proceedings of [9.40] S.V. Schell, W.A. Gardner, The Crame´r–Rao lower International Conference on Acoustic, Speech, and bound for parameters of Gaussian cyclostationary Signal Processing (ICASSP’90), Albuquerque, NM, signals, IEEE Trans. Inform. Theory 38 (4) (1992) 1990, pp. 2667–2670. 1418–1422. [9.28] S.V. Schell, W.A. Gardner, Progress on signal-selective [9.41] G. Xu, T. Kailath, Direction-of-arrival estimation via direction finding, in: Proceedings of the Fifth IEEE/ exploitation of cyclostationarity—a combination of ASSP Workshop on Spectrum Estimation and Model- temporal and spatial filtering, IEEE Trans. Signal ing, Rochester, NY, 1990, pp. 144–148. Process. 40 (7) (1992) 1775–1785. [9.29] S.V. Schell, W.A. Gardner, Detection of the number of [9.42] T. Biedka, Subspace-constrained SCORE algorithms, cyclostationary signals in unknown interference and in: Proceedings of 27th Asilomar Conference on noise, in: Proceedings of the 24th Annual Asilomar Signals, Systems and Computers, Pacific Grove, CA, Conference on Signals, Systems, and Computers, Pacific 1993, pp. 716–720. Grove, CA, 1990, pp. 473–477. [9.43] L. Castedo, C. Tseng, L. Griffiths, A new cost function [9.30] S.V. Schell, W.A. Gardner, The Cramer–Rao lower for adaptive beamforming using cyclostationary signal bound for parameters of Gaussian cyclostationary properties, in: Proceedings of IEEE International signals, in: Proceedings of International Symposium Conference on Acoustics, Speech, and Signal Processing on Information Theory and its Applications (ISI- (ICASSP’93), vol. 4, 1993, pp. 284–287. TA’90), Honolulu, HI, 1990, pp. 255–258. [9.44] L. Castedo, C. Tseng, Behavior of adaptive beamfor- [9.31] G. Xu, T. Kailath, A simple and effective algorithm for mers based on cyclostationary signal properties in estimating time delay of communication signals, in: multipath environments, in: Proceedings of 27th Asilo- Proceedings of International Symposium on Informa- mar Conference on Signals, Systems and Computers, tion Theory and its Applications (ISITA’90), 1990, vol. 1, Pacific Grove, CA, 1993, pp. 653–657. pp. 267–270. [9.45] W.A. Gardner, B.G. Agee, Self-coherence restoring [9.32] T. Biedka, B.G. Agee, Subinterval cyclic MUSIC— signal extraction apparatus and method, US Patent No. robust DF with inaccurate knowledge of cycle fre- 5,255,210, October 19, 1993. quency, in: Proceedings of 25th Asilomar Conference [9.46] W.A. Gardner, S.V. Schell, Method and apparatus for on Signals, Systems and Computers, Pacific Grove, CA, multiplexing communications signals through blind 1991. adaptive spatial filtering, US Patent No. 5,260,968, [9.33] G. Gelli, A. Napolitano, L. Paura, Spectral-correlation November 9, 1993. based estimation of channel parameters by noncoherent [9.47] S.V. Schell, W.A. Gardner, High resolution direction processing, in: Proceedings of China 1991 International finding, in: N.K. Bose, C.R. Rao (Eds.), Handbook of Conference on Circuits and Systems, Sheuzhen, China, Statistics, Elsevier, Amsterdam, NL, 1993. 1991, pp. 354–357. [9.48] S.V. Schell, W.A. Gardner, Blind adaptive spatio- [9.34] G. Gelli, L. Izzo, L. Paura, G. Poggi, A cyclic SVD- temporal filtering for wideband cyclostationary signals, based algorithm for multiple source localization, in: IEEE Trans. Signal Process. 41 (1993) 1961–1964. ARTICLE IN PRESS

686 W.A. Gardner et al. / Signal Processing 86 (2006) 639–697

[9.49] T. Biedka, M. Kahn, Methods for constraining a CMA Speech, and Signal Processing (ICASSP’95), 1995, beamformer to extract a cyclostationary signal, in: pp. 1828–1831. Proceedings of Second Workshop on Cyclostationary [9.64] L. Castedo, A.R. Figueiras-Vidal, An adaptive beam- Signals, Yountville, CA, 1994. forming technique based on cyclostationary signal [9.50] L. Castedo, A. Fueiras-Vidal, C. Tseng, L. Griffiths, properties, IEEE Trans. Signal Process. 43 (7) (1995) Linearly constrained adaptive beamforming using 1637–1650. cyclostationary signal properties, in: Proceedings of [9.65] G. Gelli, Power and timing parameter estimation of IEEE International Conference on Acoustics, Speech, multiple cyclostationary signals from sensor array data, and Signal Processing (ICASSP’94), vol. 4, 1994, Signal Processing 42 (1) (1995) 97–102. pp. 249–252. [9.66] G.B. Giannakis, G. Zhou, Harmonics in multiplicative [9.51] L.E. Franks, Polyperiodic linear filtering, in: W.A. and additive noise: parameter estimation using cyclic Gardner (Ed.), Cyclostationarity in Communications statistics, IEEE Trans. Signal Process. 43 (9) (1995) and Signal Processing, IEEE Press, New York, 1994, 2217–2221. pp. 240–266 (Chapter 4). [9.67] S. Shamsunder, G.B. Giannakis, B. Friedlander, [9.52] W.A. Gardner, S.V. Schell, B.G. Agee, Self-coherence Estimating random amplitude polynomial phase sig- restoring signal extraction and estimation of signal nals: a cyclostationary approach, IEEE Trans. Signal direction of arrival, US Patent No. 5,299,148, March Process. 43 (2) (1995) 492–505. 29, 1994. [9.68] S.V. Schell, W.A. Gardner, Programmable canonical [9.53] G. Gelli, Sensor array-based parameter estimation of correlation analysis: a flexible framework for blind cyclostationary signals, Ph.D. Dissertation, Diparti- adaptive spatial filtering, IEEE Trans. Signal Process. mento di Ingegneria Elettronica, Universita` di Napoli 43 (12) (1995) 2898–2908. Federico II, Napoli, Italy, February 1994 (in Italian). [9.69] G. Zhou, G.B. Giannakis, Retrieval of self-coupled [9.54] N.L. Gerr, C. Allen, Time-delay estimation for harmo- harmonics, IEEE Trans. Signal Process. 43 (5) (1995) nizable signals, Digital Signal Process. 4 (1994) 49–62. 1173–1185. [9.55] G.B. Giannakis, G. Zhou, Parameter estimation of [9.70] G. Zhou, G.B. Giannakis, Harmonics in multiplicative cyclostationary AM time series with application to and additive noise: performance analysis of cyclic missing observations, IEEE Trans. Signal Process. 42 estimators, IEEE Trans. Signal Process. 43 (6) (1995) (9) (1994) 2408–2419. 1445–1460. [9.56] L. Izzo, A. Napolitano, L. Paura, Modified cyclic [9.71] G. Zhou, Random amplitude and polynomial phase methods for signal selective TDOA estimation, IEEE modeling of nonstationary processes using higher-order Trans. Signal Process. 42 (11) (1994) 3294–3298. and cyclic statistics, Ph.D. Dissertation, University of [9.57] M. Kahn, M. Mow, W.A. Gardner, T. Biedka, A Virginia, Charlottesville, VA, January 1995. recursive programmable canonical correlation analyzer, [9.72] F. Mazzenga, F. Vatalaro, Parameter estimation in in: Proceedings of Second Workshop on Cyclostation- CDMA multiuser detection using cyclostationary sta- ary Signals, Yountville, CA, 1994. tistics, Electron. Lett. 32 (3) (1996). [9.58] S.V. Schell, An overview of sensor array processing for [9.73] Q. Wu, M. Wong, Blind adaptive beamforming for cyclostationary signals, in: W.A. Gardner (Ed.), Cy- cyclostationary signals, IEEE Trans. Signal Process. 44 clostationarity in Communications and Signal Proces- (11) (1996) 2757–2767. sing, IEEE Press, New York, 1994, pp. 168–239 [9.74] G. Gelli, L. Izzo, Minimum-redundancy linear arrays (Chapter 3). for cyclostationarity-based source location, IEEE [9.59] S.V. Schell, W.A. Gardner, Spatio-temporal filtering Trans. Signal Process. 45 (10) (1997) 2605–2608. and equalization for cyclostationary signals, in: C.T. [9.75] Y. Hongyi, B. Zhang, Fully blind estimation of time Leondes (Ed.), Control and Dynamic Systems, vol. 64, delays and spatial signatures for cyclostationary signals, Academic Press, New York, 1994 (Chapter 1). Electron. Lett. 34 (25) (1998) 2378–2380. [9.60] S.V. Schell, Performance analysis of the cyclic MUSIC [9.76] M. Ghogo, A. Swami, A.K. Nandi, Nonlinear least- method of direction estimation for cyclostationary squares estimation for harmonics in multiplicative and signals, IEEE Trans. Signal Process. 42 (11) (1994) additive noise, Signal Processing 79 (2) (1999) 43–60. 3043–3050. [9.77] M. Ghogo, A. Swami, B. Garel, Performance analysis [9.61] S. Shamsunder, G.B. Giannakis, Signal selective loca- of cyclic statistics for the estimation of harmonics in lization of nonGaussian cyclostationary sources, IEEE multiplicative and additive noise, IEEE Trans. Signal Trans. Signal Process. 42 (10) (1994) 2860–2864. Process. 47 (1999) 3235–3249. [9.62] Q. Wu, K.M. Wong, R. Ho, Fast algorithm for [9.78] F. Gini, G.B. Giannakis, Hybrid FM-polynomial phase adaptive beamforming of cyclic signals, IEE Proc.- signal modeling: parameter estimation and Crame´r–- Radar Sonar Navig. 141 (6) (1994) 312–318. Rao bounds, IEEE Trans. Signal Process. 47 (2) (1999) [9.63] T. Biedka, A method for reducing computations in 363–377. cyclostationarity-exploiting beamforming, in: Proceed- [9.79] J.-H. Lee, Y.-T. Lee, Robust adaptive array beamform- ings of IEEE International Conference on Acoustics, ing for cyclostationary signals under cycle frequency ARTICLE IN PRESS

W.A. Gardner et al. / Signal Processing 86 (2006) 639–697 687

error, IEEE Trans. Antennas Propagation 47 (2) (1999) [9.95] G. Gelli, L. Izzo, Cyclostationarity-based coherent 233–241. methods for wideband-signal source location, IEEE [9.80] M. Martone, Adaptive multistage beamforming using Trans. Signal Process. 51 (10) (2003) 2471–2482. cyclic higher order statistics, IEEE Trans. Signal [9.96] K.V. Krikorian, R.A. Rosen, Technique for robust Process. 47 (10) (1999) 2867–2873. characterization of weak RF emitters and accurate time [9.81] M.C. Sullivan, Wireless geolocation system, US Patent difference of arrival estimation for passive ranging of No. 5, 999, 131, December 7, 1999. RF emitters, US Patent No. 6, 646, 602, November 11, [9.82] J.-H. Lee, Y.-T. Lee, W.-H. Shih, Efficient robust 2003. adaptive beamforming for cyclostationary signals, [9.97] J.-H. Lee, Y.-T. Lee, Robust technique for estimating IEEE Trans. Signal Process. 48 (7) (2000) 1893–1901. the bearings of cyclostationary signals, Signal Proces- [9.83] F. Mazzenga, Blind adaptive parameter estimation for sing 83 (5) (2003) 1035–1046. CDMA systems using cyclostationary statistics, Eur. [9.98] P. Bianchi, P. Loubaton, F. Sirven, Non data-aided Trans. Telecommun. Relat. Technol. 11 (5) (2000) estimation of the modulation index of continuous phase 495–500. modulations, IEEE Trans. Signal Process. 52 (10) [9.84] D.A. Streight, G.K. Lott, W.A. Brown, Maximum (2004) 2847–2861. likelihood estimates of the time and frequency differ- [9.99] G. Gelli, D. Mattera, L. Paura, Blind wide-band spatio- ences of arrival of weak cyclostationary digital com- temporal filtering based on higher-order cyclostationar- munication signals, in: Proceedings of the 21st Century ity, IEEE Trans. Signal Process. 53 (4) (2005) Military Communications Conference (MILCOM 1282–1290. 2000), 2000, 957–961. [9.100] H. Yan, H.H. Fan, Wideband cyclic MUSIC algorithm, [9.85] J. Xin, H. Tsuji, A. Sano, Higher-order cyclostationar- Signal Processing 85 (2005) 643–649. ity based direction estimation of coherent narrow-band [10.1] R. Cerrato, B.A. Eisenstein, Deconvolution of cyclosta- signals, IEICE Trans. Fundam. E 83-A (8) (2000) tionary signals, IEEE Trans. Acoust. Speech Signal 1624–1633. Process., ASSP-25 (1977) 466–476. [9.86] T. Biedka, Analysis and development of blind ad- [10.2] R.D. Gitlin, S.B. Weinstein, Fractionally-spaced equal- aptive beamforming algorithms, Ph.D. Dissertation, ization: an improved digital transversal equalizer, Bell. Bradley Department of Electrical Engineering, Virginia Syst. Tech. J. 60 (1981) 275–296. Polytechnic Institute and State University, Virginia, [10.3] J.W.M. Bergmans, P.J. van Gerwen, K.J. Wouda, Data 2001. signal transmission system using decision feedback [9.87] J.-H. Lee, Y.-T. Lee, A novel direction finding method equalization, US Patent No. 4,870,657, September 26, for cyclostationary signals, Signal Processing 81 (6) 1989. (2001) 1317–1323. [10.4] T.L. Archer, W.A. Gardner, New methods for identify- [9.88] Y.-T. Lee, J.-H. Lee, Robust adaptive array beamforming the Volterra kernels of a nonlinear system, in: ing with random error in cycle frequency, IEE Proc.- Proceedings of the 24th Annual Asilomar Conference Radar, Sonar Navig. 148 (4) (2001) 193–199. on Signals, Systems, and Computers, Pacific Grove, [9.89] J. Xin, A. Sano, Linear prediction approach to CA, 1990, pp. 592–597. direction estimation of cyclostationary signals in multi- [10.5] W.A. Gardner, Identification of systems with cyclosta- path environments, IEEE Trans. Signal Process. 49 (4) tionary input and correlated input/output measurement (2001) 710–720. noise, IEEE Trans. Automat. Control 35 (4) (1990) [9.90] Z.-T. Huang, Y.-Y. Zhou, W.-L. Jiang, Q.-Z. Lu, Joint 449–452. estimation of Doppler and time-difference-of-arrival [10.6] R.D. McCallister, D. Ronald, D.D. Shearer III, exploiting cyclostationarity property, IEE Proc. (4) Compensating for distortion in a communication (2002) 161–165. channel, US Patent No. 4,922,506, May 1, 1990. [9.91] Z. Huang, Y. Zhou, W. Jiang, Direction-of-arrival of [10.7] W.A. Gardner, A new method of channel identification, signal sources exploiting cyclostationarity property, IEEE Trans. Commun. 39 (1991) 813–817. Acta Electron. Sinica 30 (3) (2002) 372–375 (in [10.8] W.A. Gardner, T.L. Archer, Simplified methods for Chinese). identifying the Volterra kernels of nonlinear systems, in: [9.92] J.-H. Lee, C.-H. Tung, Estimating the bearings of near- Proceedings of the 34th Midwest Symposium on field cyclostationary signals, IEEE Trans. Signal Pro- Circuits and Systems, Monterey, CA, 1991. cess. 50 (1) (2002) 110–118. [10.9] A. Duel-Hallen, Equalizers for multiple input/multiple [9.93] M.R. Morelande, A.M. Zoubir, On the performance of output channels and PAM systems with cyclostationary cyclic moment-based parameter estimators of amplitude input sequence, IEEE J. Sel. Areas Commun. 10 (3) modulated polynomial phase signals, IEEE Trans. (1992) 630–639. Signal Process. 50 (3) (2002) 590–605. [10.10] G.D. Golden, Use of cyclostationary signal to con- [9.94] P. Charge´, Y. Wang, J. Sillard, An extended cyclic strain the frequency response of a fractionally spa- MUSIC algorithm, IEEE Trans. Signal Process. 51 (7) ced equalizer, US Patent No. 5,095,495, March 10, (2003) 1695–1701. 1992. ARTICLE IN PRESS

688 W.A. Gardner et al. / Signal Processing 86 (2006) 639–697

[10.11] G.D. Golden, Use of a fractionally spaced equalizer to frequency domain approach, IEEE Trans. Inform. perform echo cancellation in a full-duplex modem, US Theory 41 (1) (1995) 329–334. Patent No. 5,163,044, November 10, 1992. [10.26] J.K. Tugnait, On blind identifiability of multipath [10.12] L. Izzo, A. Napolitano, M. Tanda, Spectral-correlation channels using fractional sampling and second-order based methods for multipath channel identification, cyclostationary statistics, IEEE Trans. Inform. Theory Eur. Trans. Telecommun. Relat. Technol. 3 (1) (1992) 41 (1) (1995) 308–311. 341–348. [10.27] C. Andrieu, P. Duvat, A. Doucet, Bayesian deconvolu- [10.13] W.A. Gardner, T.L. Archer, Exploitation of cyclostation of cyclostationary processes based on point tionarity for identifying the Volterra kernels of non- processes, in: Proceedings of VIII European Signal linear systems, IEEE Trans. Inform. Theory 39 (3) Processing Conference (EUSIPCO’96), Trieste, Italy, (1993) 535–542. 1996. [10.14] G. Gelli, L. Izzo, A. Napolitano, L. Paura, Multipath [10.28] H. Liu, G. Xu, L. Tong, T. Kailath, Recent develop- channel identification by an improved Prony algorithm ments in blind channel equalization: from cyclostatio- based on spectral correlation measurements, Signal narity to subspaces, Signal Processing 50 (1996) 83–99. Processing 31 (1) (1993) 17–29. [10.29] J.K. Tugnait, Blind equalization and estimation of FIR [10.15] Y. Chen, C.L. Nikias, Identifiability of a band limited communication channels using fractional sampling, system from its cyclostationary output autocorrelation, IEEE Trans. Commun. 44 (3) (1996) 324–336. IEEE Trans. Signal Process. 42 (2) (1994) 483–485. [10.30] H.E. Wong, J.A. Chambers, Two-stage interference [10.16] Z. Ding, Blind channel identification and equalization immune blind equaliser which exploits cyclostationary using spectral correlation measurements, Part I: Fre- statistics, Electron. Lett. 32 (19) (1996) 1763–1764. quency-domain analysis, in: W.A. Gardner (Ed.), [10.31] A. Chevreuil, P. Loubaton, Blind second-order identi- Cyclostationarity in Communications and Signal Pro- fication of FIR channels: forced cyclostationarity and cessing, IEEE Press, New York, 1994, pp. 417–436. structured subspace methods, IEEE Signal Process. [10.17] D. Hatzinakos, Nonminimum phase channel deconvo- Lett. 4 (1997) 204–206. lution using the complex cepstrum of the cyclic [10.32] A. Chevreuil, Blind equalization in a cyclostationary autocorrelation, IEEE Trans. Signal Process. 42 (11) context, Ph.D. Dissertation, ENST-Telecom Paris, (1994) 3026–3042. Department of Signal Processing, Paris, France, 1997 [10.18] L. Izzo, A. Napolitano, L. Paura, Cyclostationarity- (in French). exploiting methods for multipath-channel identifica- [10.33] G.B. Giannakis, S.D. Halford, Asymptotically optimal tion, in: W.A. Gardner (Ed.), Cyclostationarity in blind fractionally spaced channel estimation and Communications and Signal Processing, IEEE Press, performance analysis, IEEE Trans. Signal Process. 45 New York, 1994, pp. 91–416. (7) (1997) 1815–1830. [10.19] Y. Li, Z. Ding, ARMA system identification based on [10.34] Y.-C. Liang, A.R. Leyman, B.-H. Soong, (Almost) second-order cyclostationarity, IEEE Trans. Signal periodic FIR system identification using third-order Process. 42 (12) (1994) 3483–3494. cyclic-statistics, Electron. Lett. 33 (5) (1997) 356–357. [10.20] A. Napolitano, Spectral-correlation based methods for [10.35] S. Prakriya, D. Hatzinakos, Blind identification of multipath channel identification, Ph.D. Dissertation, linear subsystems of LTI-ZMNL-LTI models with Universita` di Napoli Federico II, Napoli, Italy, cyclostationary inputs, IEEE Trans. Signal Process. 45 Dipartimento di Ingegneria Elettronica, February (8) (1997) 2023–2036. 1994 (in Italian). [10.36] M. Tsatsanis, G.B. Giannakis, Transmitter induced [10.21] L. Tong, G. Xu, T. Kailath, Blind identification and cyclostationarity for blind channel equalization, IEEE equalization based on second order statistics: a time Trans. Signal Process. 45 (7) (1997) 1785–1794. domain approach, IEEE Trans. Inform. Theory 40 (2) [10.37] H.H. Zeng, L. Tong, Blind channel estimation using the (1994) 340–349. second-order statistics: algorithms, IEEE Trans. Signal [10.22] L. Tong, G. Xu, T. Kailath, Blind channel identification Process. 45 (8) (1997) 1919–1930. and equalization using spectral correlation measure- [10.38] G.B. Giannakis, E. Serpedin, Blind identification of ments, Part II: A time-domain approach, in: W.A. ARMA channels with periodically modulated inputs, Gardner (Ed.), Cyclostationarity in Communications IEEE Trans. Signal Process. 46 (11) (1998) 3099–3104. and Signal Processing, IEEE Press, New York, 1994, [10.39] F. Guglielmi, C. Luschi, A. Spalvieri, Fractionality pp. 437–454. spaced equalizing circuits and method, US Patent No. [10.23] W.A. Gardner, L. Paura, Identification of polyperiodic 5,751,768, May 12, 1998. nonlinear systems, Signal Processing 46 (1995) 75–83 [10.40] Y.-C. Liang, A.R. Leyman, (Almost) periodic moving [10.24] G.B. Giannakis, Polyspectral and cyclostationary ap- average system identification using higher order cylic- proaches for identification of closed-loop systems, statistics, IEEE Trans. Signal Process. 46 (3) (1998) IEEE Trans. Automat. Control 40 (5) (1995) 882–885 779–783. [10.25] L. Tong, G. Xu, B. Hassibi, T. Kailath, Blind channel [10.41] D. Mattera, L. Paura, Higher-order cyclostationarity- identification based on second order statistics: a based methods for identifying Volterra systems by ARTICLE IN PRESS

W.A. Gardner et al. / Signal Processing 86 (2006) 639–697 689

input–output noisy measurements, Signal Processing 67 [10.56] S. Celebi, Methods and devices for simplifying blind (1) (1998) 77–98. channel estimation of the channel impulse response for a [10.42] F. Mazzenga, Blind multipath channel identification for DMT signal, US Patent No. 6,782,042, August 24, 2004. wideband communication systems based on cyclosta- [10.57] S.V. Narasimhan, M. Hazarataiah, P.V.S. Giridhar, tionary statistics, Eur. Trans. Telecommun. Relat. Channel blind identification based on cyclostationarity Technol. 9 (1) (1998) 27–31. and group delay, Signal Processing 85 (7) (2005) [10.43] E. Serpedin, G.B. Giannakis, Blind channel identifica- 1275–1286. tion and equalization with modulation induced cyclos- [11.1] M.G. Bilyik, Periodic nonstationary processes, Otbor tationarity, IEEE Trans. Signal Process. 46 (7) (1998) Peredacha Inform. 50 (1977) 28–34 (in Russian). 1930–1944. [11.2] D. Bukofzer, Coherent and noncoherent detection of [10.44] A. Chevreuil, P. Loubaton, MIMO blind second- signals in cyclostationary noise and of cyclostationary order equalization method and conjugate cyclostatio- signals in stationary noise, Ph.D. Dissertation, Depart- narity, IEEE Trans. Signal Process. 47 (2) (1999) ment of Electrical and Computer Engineering, Uni- 572–578. versity of California, Davis, 1979. [10.45] R.W. Heat, G.B. Giannakis, Exploiting input cyclosta- [11.3] Y.P. Dragan, V.P. Mezentsev, I.N. Yavorskii, The tionarity for blind channel identification in OFDM problem of verification of a stochastic rhythmicity systems, IEEE Trans. Signal Process. 47 (3) (1999) model, Otbor Peredacha Inform. 59 (1980) 3–12 (in 848–856. Russian). [10.46] D. Mattera, Identification of polyperiodic Volterra [11.4] B.S. Rybakov, Optimum detection of a class of random systems by means of input–output noisy measurements, Gaussian pulsed signals in the presence of noise with a Signal Processing 75 (1) (1999) 41–50. priori parametric uncertainty, Radio Eng. Electron. [10.47] A. Chevreuil, E. Serpedin, P. Loubaton, G.B. Gianna- Phys. 25 (1980) 62–66 (English translation of Radio- kis, Blind channel identification and equalization using tekhnika Electron. 25, 1198–1202). nonredundant periodic modulation precoders: perfor- [11.5] W.A. Gardner, Structural characterization of locally mance analysis, IEEE Trans. Signal Process. 48 (6) optimum detectors in terms of locally optimum (2000) 1570–1586. estimators and correlators, IEEE Trans. Inform. [10.48] F. Mazzenga, On the identifiability of a channel transfer Theory IT-28 (1982) 924–932. function from cyclic and conjugate cyclic statistics, [11.6] H.L. Hurd, Clipping degradation in a system used to European Trans. Telecommun. Relat. Technol. 11 (3) detect time-periodic variance fluctuations, J. Acoust. (2000) 293–296. Soc. Amer. 72 (1982) 1827–1830. [10.49] H. Bo¨lcskei, P. Duhamel, R. Hleiss, A subspace-based [11.7] W.A. Gardner, Non-parametric signal detection and approach to blind channel identification in pulse identification of modulation type for signals hidden in shaping OFDM/OQAM systems, IEEE Trans. Signal noise: an application of the new theory of cyclic spectral Process. 49 (7) (2001) 1594–1598. analysis, in: Proceedings of Ninth Symposium on Signal [10.50] H. Bo¨lcskei, R.W. Health, A.J. Paulraj, Blind channel Processing and its Applications, 1983, pp. 119–124. identification and equalization in OFDM-based multi- [11.8] D. Bukofzer, Optimum and suboptimum detector antenna systems, IEEE Trans. Signal Process. 50 (1) performance for signals in cyclostationary noise, IEEE (2002) 96–109. J. Ocean. Eng. OE-12 (1987) 97–115. [10.51] G. Gelli, F. Verde, Two-stage interference-resistant [11.9] W.A. Gardner, Signal interception: a unifying theore- adaptive periodically time-varying CMA blind equal- tical framework for feature detection, in: Proceedings of ization, IEEE Trans. Signal Process. 50 (3) (2002) the 11th GRETSI Symposium on Signal and Image 662–672. Processing, Nice, France, 1987, pp. 69–72. [10.52] J.K. Tugnait, W. Luo, Linear prediction error method [11.10] W.A. Gardner, C.M. Spooner, Cyclic spectral analysis for blind identification of periodically time-varying for signal detection and modulation recognition, in: channels, IEEE Trans. Signal Process. 50 (12) (2002) Proceedings of MILCOM 88, San Diego, CA, 1988, pp. 3070–3082. 419–424. [10.53] I. Bradaric, A.P. Petropulu, K.I. Diamantaras, Blind [11.11] W.A. Gardner, Signal interception: a unifying theore- MIMO FIR channel identification based on second- tical framework for feature detection, IEEE Trans. order spectra correlations, IEEE Trans. Signal Process. Commun. COM-36 (1988) 897–906, awarded S.O. Rice 51 (6) (2003) 1668–1674. Prize Paper from the IEEE Communications Society. [10.54] S. Houcke, A. Chevreuil, P. Loubaton, Blind equaliza- [11.12] L. Izzo, L. Paura, M. Tanda, Optimum and subopti- tion-case of unknown symbol period, IEEE Trans. mum detection of weak signals in cyclostationary non- Signal Process. 51 (3) (2003) 781–793. Gaussian noise, Eur. Trans. Telecommun. Relat. [10.55] J. Antoni, P. Wagstaff, J.-C. Henrio, Ha—A consistent Technol. 1 (3) (1990) 233–238. estimator for frequency response function with input [11.13] N.L. Gerr, H.L. Hurd, Graphical methods for deter- and output noise, IEEE Trans. Instrumentat. Meas. 53 mining the presence of periodic correlation, J. Time Ser. (2) (2004) 457–465. Anal. 12 (1991) 337–350. ARTICLE IN PRESS

690 W.A. Gardner et al. / Signal Processing 86 (2006) 639–697

[11.14] A.V. Vecchia, R. Ballerini, Testing for periodic auto- Dissertation, INPG-Nat. Poly. Inst. Grenoble, Greno- correlations in seasonal time series data, Biometrika 78 ble, France, 1998 (in French). (1) (1991) 53–63. [11.30] P. Marchand, J.L. Lacoume, C. Le Martret, Multiple [11.15] W.A. Gardner, C.M. Spooner, Signal interception: hypothesis modulation classification based on cyclic performance advantages of cyclic feature detectors, cumulants of different orders, in: Proceedings of Inter- IEEE Trans. Commun. 40 (1) (1992) 149–159. national Conference on Acoustics, Speech, and Signal [11.16] L. Izzo, L. Paura, M. Tanda, Signal interception in non- Processing (ICASSP’98), vol. 4, 1998, pp. 2157–2160. Gaussian noise, IEEE Trans. Commun. 40 (1992) [11.31] O. Besson, P. Stoica, Simple test for distinguishing 1030–1037. constant from time varying amplitude in harmonic [11.17] A. Napolitano, L. Paura, M. Tanda, Signal detection in retrieval problem, IEEE Trans. Signal Process. 47 (4) cyclostationary generalized Gaussian noise with un- (1999) 1137–1141. known parameters, Eur. Trans. Telecommun. Relat. [11.32] P. Rostaing, E. Thierry, T. Pitarque, Asymptotic Technol. 3 (1) (1992) 39–43. performance analysis of cyclic detectors, IEEE Trans. [11.18] M. Tanda, Cyclostationarity-based algorithms for Commun. 47 (1) (1999) 10–13. signal interception in non-Gaussian noise, Ph.D. Dis- [11.33] N.S. Subotic, B.J. Thelen, D.A. Carrara, Cyclostation- sertation, Dipartimento di Ingegneria Elettronica, Uni- ary signal models for the detection and characterization versita` di Napoli Federico II, Napoli, Italy, February of vibrating objects in SAR data, in: Proceedings of the 1992 (in Italian). 32nd Asilomar Conference on Signals, Systems, and [11.19] W.A. Gardner, C.M. Spooner, Detection and source Computers, Pacific Grove, CA, 1998, pp. 1304–1308. location of weak cyclostationary signals: simplifications [11.34] A. Ferreól, P. Chevalier, On the behavior of current of the maximum-likelihood receiver, IEEE Trans. second and higher order blind source separation Commun. 41 (6) (1993) 905–916. methods for cyclostationary sources, IEEE Trans. [11.20] A.V. Dandawate´, G.B. Giannakis, Statistical test for Signal Process. 48 (6) (2000) 1712–1725. presence of cyclostationarity, IEEE Trans. Signal [11.35] K. Abed-Meraim, Y. Xiang, J.H. Manton, Y. Hua, Process. 42 (9) (1994) 2355–2369. Blind source separation using second-order cyclosta- [11.21] C.M. Spooner, W.A. Gardner, Robust feature detection tionary statistics, IEEE Trans. Signal Process. 49 (4) for cyclostationary signals, IEEE Trans. Commun. 42 (2001) 694–701. (5) (1994) 2165–2173. [11.36] A. Ferreol, Process of cyclic detection in diversity of [11.22] R.W. Gardner, Detection of a digitally modulated polarization of digital cyclostationary radioelectric signal using binary line enhancement, US Patent No. signals, US Patent No. 6,430,239, August 6, 2002. 5,455,846, October 3, 1995. [11.37] A. Ferreól, P. Chevalier, L. Albera, Second-order blind [11.23] P. Gournay, P. Nicolas, Cyclic spectral analysis and separation of first- and second-order cyclostationary time-frequency analysis for automatic transmission sources—application to AM, and deterministic sources, classification, in: Proceedings of Quinzieme Colloque IEEE Trans. Signal Process. 52 (4) (2004) 845–861. GRETSI, Juan-les-Pins, F, 1995 (in French). [11.38] J.A. Sills, Q.R. Black, Signal recognizer for commu- [11.24] L. Izzo, M. Tanda, Delay-and-multiply detectors for nications signals, US Patent No. 6,690,746, February signal interception in non-Gaussian noise, in: Proceed- 10, 2004. ings of 15th GRTESI Symposium on Signal and Image [11.39] J. Antoni, F. Guillet, M. El Badaoui, F. Bonnardot, Processing, Juan-Les-Pins, France, 1995. Blind separation of convolved cyclostationary pro- [11.25] G. Gelli, L. Izzo, L. Paura, Cyclostationarity-based cesses, Signal Processing 85 (2005) 51–66. signal detection and source location in non-Gaussian [12.1] W.M. Brelsford, Probability predictions and time series noise, IEEE Trans. Commun. 44 (3) (1996) 368–376. with periodic structure, Ph.D. Dissertation, Johns [11.26] P. Rostaing, Detection of modulated signals by Hopkins University, Baltimore, MD, 1967. exploiting their cyclostationarity properties: appli- [12.2] C.W.J. Granger, Some new time series models: non- cation to sonar signals, Ph.D. Dissertation, University linear, bilinear and non-stationary, Statistician 27 (3–4) of Nice-Sophia Antipolis, Nice, France, 1996 (in (1978) 237–254. French). [12.3] M. Pagano, On periodic and multiple autoregressions, [11.27] G.K. Yeung, W.A. Gardner, Search-efficient methods Ann. Statist. 6 (1978) 1310–1317. of detection of cyclostationary signals, IEEE Trans. [12.4] B.M. Troutman, Some results in periodic autoregres- Signal Process. 44 (5) (1996) 1214–1223. sion, Biometrika 66 (1979) 219–228. [11.28] D. Boiteau, C. Le Martret, A general maximum [12.5] J.D. Salas, R.A. Smith, Correlation properties of likelihood framework for modulation classification, in: periodic ARðpÞ models, in: Proceedings of the Third Proceedings of International Conference on Acoustics, International Symposium on Stochastic Hydraulics, Speech, and Signal Processing (ICASSP’98), vol. 4, Tokyo, 1980,107–115 1998, pp. 2165–2168. [12.6] G.C. Tiao, M.R. Grupe, Hidden periodic autoregres- [11.29] P. Marchand, Detection and classification of digital sive-moving average models in time series data, modulations by higher order cyclic statistics, Ph.D. Biometrika 67 (1980) 365–373. ARTICLE IN PRESS

W.A. Gardner et al. / Signal Processing 86 (2006) 639–697 691

[12.7] S. Shell, K.K. Biowas, A.K. Sihka, Modelling and [12.25] W.K. Li, Y.V. Hui, An algorithm for the exact predictions of stochastic processes involving periodi- likelihood of periodic autoregressive moving average cities, Appl. Math. Modeling 5 (1981) 241–245. models, Commun. Statist. Simulat. Comput. 17 (1988) [12.8] H. Sakai, Circular lattice filtering using Pagano’s 1483–1494. method, IEEE Trans. Acoust. Speech Signal Process. [12.26] J. Andel, Periodic autoregression with exogenous ASSP (1982) 279–287. variables and periodic variances, Cesk. Akod. Ved. [12.9] J.D. Salas, D.C. Boes, R.A. Smith, Estimation of Aplikace Motemotiky 34 (1989) 387–395. ARMA models with seasonal parameters, Water [12.27] H. Sakai, F. Sakaguchi, Simultaneous confidence bands Resour. Res. 18 (1982) 1006–1010. for the spectral estimate of two-channel autoregressive [12.10] H. Sakai, Covariance matrices characterization by a set processes, J. Time Ser. Anal. 11 (1990) 49–56. of scalar partial autocorrelation coefficients, Ann. [12.28] T.A. Ula, Periodic covariance stationarity of multi- Statist. 11 (1983) 337–340. variate periodic autoregressive moving average pro- [12.11] A.V. Vecchia, Aggregation and estimation for periodic cesses, Water Resour. Res. 26 (1990) 855–861. autoregressive moving average models, Ph.D. Disserta- [12.29] S. Bittanti, P. Bolzern, G. De Nicolao, L. Piroddi, D. tion, Colorado State University, Fort Collins, CO, Purassanta, A minimum prediction error algorithm for 1983. estimation of periodic ARMA models, in: Proceedings [12.12] J. Andel, Periodic autoregression with exogenous of the European Control Conference, Grenoble, variables and equal variances, in: W. Grossman, et al. France, 1991 (Eds.), Probability Theory and Mathematical Statistics [12.30] H. Sakai, On the spectral density matrix of a periodic with Applications, Reidel, Dordrecht, NL, Boston, ARMA process, J. Time Ser. Anal. 12 (1991) 73–82. MA, 1985, pp. 237–245. [12.31] P.L. Anderson, A.V. Vecchia, Asymptotic results for [12.13] J. Andel, On periodic autoregression with unknown periodic autoregressive moving-average processes, J. mean, Appl. Math. 30 (1985) 126–139. Time Ser. Anal. 14 (1993) 1–18. [12.14] T. Cipra, Periodic moving average process, Appl. Math. [12.32] M. Bentarzi, M. Hallin, On the invertibility of periodic 30 (1985) 218–229. moving-average models, J. Time Ser. Anal. 15 (1993) [12.15] R.M. Thompstone, K.W. Hipel, A.I. McLeod, Group- 263–268. ing of periodic autoregressive models, in: O.D. Ander- [12.33] A.I. McLeod, Parsimony, model adequacy and periodic son, J.K. Ord, E.A. Robinson (Eds.), Time Series correlation in time series forecasting, Int. Statist. Rev. Analysis, vol. 6, Elsevier, Amsterdam, Holland, 1985, 61 (1993) 387–393. pp. 35–49. [12.34] A.G. Miamee, Explicit formulas for the best linear [12.16] A.V. Vecchia, Periodic autoregressive-moving average predictor and predictor error matrix of a periodically (PARMA) modeling with applications to water re- correlated process, SIAM J. Math. Anal. 24 (1993) sources, Water Resources Bull. 21 (1985) 721–730. 703–711. [12.17] A.V. Vecchia, Maximum likelihood estimation for [12.35] T.A. Ula, Forecasting of multivariate periodic auto- periodic autoregressive moving average models, Tech- regressive moving-average processes, J. Time Ser. Anal. nometrics 27 (1985) 375–384. 14 (6) (1993). [12.18] J. Andel, A. Rubio, On interpolation on periodic [12.36] S. Bittanti, P. Bolzern, G. De Nicolao, L. Piroddi, autoregressive processes, Appl. Math. 31 (1986) Representation, in: W.A. Gardner (Ed.), predic- 480–485. tion, IEEE Press, New York, 1994, pp. 267–294 [12.19] S. Bittanti, The periodic prediction problem for (Chapter 5). cyclostationary processes—an introduction, in: Pro- [12.37] A.G. Miamee, On recent developments in prediction ceedings of the NATO Advanced Research Workshop, theory for cyclostationary processes, in: W.A. Gardner Groningen, 1986, 239–249. (Ed.), Cyclostationarity in Communications and Signal [12.20] B. Fernandez, J.D. Salas, Periodic gamma autoregres- Processing, IEEE Press, New York, 1994, pp. 480–493. sive processes for operational hydrology, Water Re- [12.38] G.J. Adams, G.C. Goodwin, Parameter estimation for sour. Res. 22 (1986) 1385–1396. periodic ARMA models, J. Time Ser. Anal. 16 (1995) [12.21] J.T.B. Obeysekera, J.D. Salas, Modeling of aggregated 127–146. hydrologic time series, J. Hydrol. 86 (1986) 197–219. [12.39] G.N. Boshnakov, Recursive computation of the para- [12.22] T. Cipra, P. Tusty, Estimation in multiple autoregres- meters of periodic autoregressive moving average sive-moving average models using periodicity, J. Time processes, J. Time Ser. Anal. 17 (1995) 333–349. Ser. Anal. 8 (1987) 293–300. [12.40] M. Bentarzi, M. Hallin, Locally optimal tests against [12.23] A.I. McLeod, D.J. Noakes, K.W. Hipel, R.M. Thomp- periodical autocorrelation: parametric and nonpara- stone, Combining hydrologic forecasts, J. Water metric approaches, Econometric Theory 12 (1996) Resour. Plan. Manage. 113 (1987) 29–41. 88–112. [12.24] P. Bartolini, J.D. Salas, J.T.B. Obeysekera, Multi- [12.41] A. Dandawate´, G.B. Giannakis, Modeling periodic variate periodic ARMA(1,1) processes, Water Resour. moving average processes using cyclic statistics, IEEE Res. 24 (1988) 1237–1246. Trans. Signal Process. 44 (3) (1996) 673–684. ARTICLE IN PRESS

692 W.A. Gardner et al. / Signal Processing 86 (2006) 639–697

[12.42] P.L. Anderson, M.M. Meerschaert, Periodic moving Information Theory and its Applications (ISITA’90), averages of random variables with regularly varying Honolulu, HI, 1990, pp. 355–358 tails, Ann. Statist. 25 (1997) 771–785. [13.6] C.M. Spooner, W.A. Gardner, An overview of higher- [12.43] R. Lund, I. Basawa, Modeling and inference for order cyclostationarity, in: A.G. Miamee (Ed.), Pro- periodically correlated time series, in: S. Gosh (Ed.), ceeding of the Workshop on Nonstationary Stochastic Asymptotics, Nonparametrics and Time Series, 1997 Processes and their Applications, World Scientific, [12.44] M. Bentarzi, M. Hallin, J. Appl. Probab. 35 (1) (1998) Virginia, VA, 1991, pp. 110–125. 46–54. [13.7] C.M. Spooner, W.A. Gardner, Estimation of cyclic [12.45] P.L. Anderson, M.M. Meerschaert, A. Vecchia, In- polyspectra, in: Proceedings of 25th Annual Asilomar novations algorithm for periodically stationary time Conference on Signals, Systems, and Computers, Pacific series, Stochastic Process. Appl. 83 (1999) 149–169. Grove, CA, 1991 [12.46] A. Makagon, Theoretical prediction of periodically [13.8] G.D. Z˘ivanovic´, Some aspects of the higher-order correlated sequences, Probab. Math. Statist. 19 (2) cyclostationary theory, in: Proceedings of the Interna- (1999) 287–322. tional Conference on Acoustics, Speech, and Signal [12.47] S. Lambert-Lacroix, On periodic autoregressive pro- Processing (ICASSP’91), 1991 cesses estimation, IEEE Trans. Signal Process. 48 (6) [13.9] C.M. Spooner, Theory and application of higher-order (2000) 1800–1803. cyclostationarity, Ph.D. Dissertation, Department of [12.48] R. Lund, I.V. Basawa, Recursive prediction and like- Electrical and Computer Engineering, University of lihood evaluation for periodic ARMA models, J. Time California, Davis, CA, 1992. Ser. Anal. 21 (1) (2000) 75–93. [13.10] A.V. Dandawate´, Exploiting cyclostationarity and [12.49] I.V. Basawa, R. Lund, Large sample properties of higher-order statistics in signal processing, Ph.D. parameter estimates for periodic ARMA models, J. Dissertation, University of Virginia, Charlottesville, Time Ser. Anal. 22 (6) (2001) 651–664. VA, January 1993 [12.50] H.L. Hurd, A. Makagon, A.G. Miamee, On AR(1) [13.11] A. Dandawate´, G.B. Giannakis, Nonparametric cyclic- models with periodic and almost periodic coefficients, polyspectral analysis of AM signals and processes with Stochastic Process. Appl. 100 (2002) 167–185. missing observations, IEEE Trans. Inform. Theory 39 [12.51] A. Bibi, C. Franq, Consistent and asymptotically (6) (1993) 1864–1876. Normal estimators for cyclically time-dependent linear [13.12] A.V. Dandawate´, G.B. Giannakis, Nonparametric models, Ann. Inst. Statist. Math. 55 (1) (2003) 41–68. polyspectral estimators for kth-order (almost) cyclosta- [12.52] Q. Shao, R. Lund, Computation and characterization tionary processes, IEEE Trans. Inform. Theory 40 (1) of autocorrelations and partial autocorrelations in (1994) 67–84. periodic ARMA models, J. Time Ser. Anal. 25 (3) [13.13] W.A. Gardner, C.M. Spooner, The cumulant theory of (2004) 359–372. cyclostationary time-series, Part I: Foundation, IEEE [12.53] P.L. Anderson, M.M. Meerschaert, Parameter estima- Trans. Signal Process. 42 (12) (1994) 3387–3408. tion for periodically stationary time series, J. Time Ser. [13.14] C.M. Spooner, W.A. Gardner, The cumulant theory of Anal. 26 (4) (2005) 489–518. cyclostationary time-series, Part II: Development and [13.1] V.O. Alekseev, Symmetry properties of high-order applications, IEEE Trans. Signal Process. 42 (12) (1994) spectral densities of stationary and periodic-nonsta- 3409–3429. tionary stochastic processes, Problems Inform. Trans- [13.15] W.A. Gardner, C.M. Spooner, Cyclostationary signal mission 23 (1987) 210–215. processing, in: C.T. Leondes (Ed.), Control and [13.2] D.E. Reed, M.A. Wickert, Nonstationary moments of a Dynamic Systems, vol. 65, Academic Press, New York, random binary pulse train, IEEE Trans. Inform. 1994 (Chapter 1). Theory 35 (1989) 700–703. [13.16] L. Izzo, A. Napolitano, L. Paura, MIMO Volterra system [13.3] A.V.T. Cartaxo, A.A.D. Albuquerque, A general input/output relations for cyclic higher-order statistics, in: property of the n-order moment generating function Proceedings of VII European Signal Processing Confer- of strict-sense cyclostationary processes, in: E. Arikan ence (EUSIPCO’94), Edinburgh, UK, 1994 (Ed.), Proceedings of the 1990 Bilkent International [13.17] C.M. Spooner, Higher-order statistics for nonlinear Conference on New Trends in Communication, Control processing of cyclostationary signals, in: W.A. Gardner and Signal Processing, vol. 2, Ankara, Turkey, Elsevier (Ed.), Cyclostationarity in Communications and Signal Science, Amsterdam, Netherlands, 1990, pp. 1548–1552 Processing, IEEE Press, New York, 1994, pp. 91–167 [13.4] W.A. Gardner, Spectral characterization of n-th order (Chapter 2). cyclostationarity, in: Proceedings of the Fifth IEEE/ [13.18] A.V. Dandawate´, G.B. Giannakis, Asymptotic theory ASSP Workshop on Spectrum Estimation and Model- of mixed time averages and Kth-order cyclic moment ing, Rochester, NY, 1990, pp. 251–255 and cumulant statistics, IEEE Trans. Inform. Theory 41 [13.5] W.A. Gardner, C.M. Spooner, Higher order cyclosta- (1) (1995) 216–232. tionarity, cyclic cumulants, and cyclic polyspectra, in: [13.19] P. Marchand, P.-O. Amblard, J.-L. Lacoume, Proceedings of the 1990 International Symposium on Higher than second-order statistics for complex-valued ARTICLE IN PRESS

W.A. Gardner et al. / Signal Processing 86 (2006) 639–697 693

cyclostationary signals, in: Proceedings of 15th GRET- terms of linear time-varying filters, Otbor Peredacha SI Symposium, Juan-Les-Pins, France, 1995, pp. 69–72 Inform. 39 (1974) 23–31 (in Russian). [13.20] A. Napolitano, Cyclic higher-order statistics: input/ [14.5] J. Rootenberg, S.A. Ghozati, Stability properties of output relations for discrete- and continuous-time periodic filters, Int. J. Systems Sci. 8 (1977) 953–959. MIMO linear almost-periodically time-variant systems, [14.6] T. Strom, S. Signell, Analysis of periodically switched Signal Processing 42 (2) (1995) 147–166, received linear circuits, IEEE Trans. Circuits Systems CAS EURASIP’s Best Paper of the Year award (1977) 531–541. [13.21] S.V. Schell, Higher-order cyclostationarity properties of [14.7] A.M. Silaiev, A.V. Yakimov, Periodic nonstationarity coded communication signals, in: Proceedings of the of noise in harmonic systems, Radiotekhnika Elektron. 29th Annual Asilomar Conference on Signals, Systems, 24 (1979) 1806–1811 (in Russian). and Computers, Pacific Grove, CA, 1995 [14.8] V.A. Zaytsev, A.V. Pechinkin, Optimal control of [13.22] C.M. Spooner, Classification of cochannel com- servicing in a multichannel system, Eng. Cybernet. 17 munication signals using cyclic cumulants, in: Proceed- (1979) 36–42 ings of the 29th Annual Asilomar Conference on [14.9] L. Pelkowitz, Frequency domain analysis of wrap- Signals, Systems, and Computers, Pacific Grove, CA, around error in fast convolution algorithms, IEEE 1995 Trans. Acoust. Speech Signal Process. ASSP (1981) [13.23] L. Izzo, A. Napolitano, Higher-order cyclostationarity 413–422. properties of sampled time-series, Signal Processing 54 [14.10] S. Bittanti, G. Guardabassi, Optimal cyclostationary (1996) 303–307. control: a parameter-optimization frequency-domain [13.24] G. Zhou, G.B. Giannakis, Polyspectral analysis of approach, in: H. Akashi (Ed.), Proceedings of the mixed processes and coupled harmonics, IEEE Trans. Eighth Triennial World Congress of the International Inform. Theory 42 (3) (1996) 943–958. Federation of Automatic Control, vol. 2, Kyoto, Japan, [13.25] L. Izzo, A. Napolitano, Higher-order statistics for 1982, pp. 857–862 Rice’s representation of cyclostationary signals, Signal [14.11] Y. Isokawa, An identification problem in almost and Processing 56 (1997) 279–292. asymptotically almost periodically correlated processes, [13.26] J.L. Lacoume, P.O. Amblard, P. Common, Higher J. Appl. Probab. 19 (1982) 456–462. Order Statistics for Signal Processing, Masson, Paris, [14.12] K. Vokurka, Application of the group pulse processes 1997 (in French). in the theory of Barkhausen noise, Czech. J. Phys. B 32 [13.27] L. Izzo, A. Napolitano, Multirate processing of time- (1982) 1384–1398. series exhibiting higher-order cyclostationarity, IEEE [14.13] V.A. Semenov, A.I. Smirnov, On the stability of linear Trans. Signal Process. 46 (2) (1998) 429–439. stochastic systems with periodically nonstationary [13.28] F. Flagiello, L. Izzo, A. Napolitano, A computationally parametric excitation, Mech. Solids 18 (1983) 14–20. efficient and interference tolerant nonparametric algo- [14.14] E.R. Ferrara Jr., Frequency-domain implementations rithm for LTI system identification based on higher- of periodically time-varying filters, IEEE Trans. order cyclic statistics, IEEE Trans. Signal Process. 48 Acoust. Speech Signal Process. ASSP (1985) 883–892. (4) (2000) 1040–1052. [14.15] G.V. Obrezkov, Y.A. Sizyakov, Approximate analysis [13.29] A. Napolitano, C.M. Spooner, Median-based cyclic of pulsed tracking systems when there is fluctuating polyspectrum estimation, IEEE Trans. Signal Process. interference, Radioelectron. Commun. Systems 28 48 (2000) 1462–1466. (1985) 63–65 (English translation of Izv. VUZ Radio- [13.30] C.M. Spooner, W.A. Brown, G.K. Yeung, Automatic electron.). radio-frequency environment analysis, in: Proceedings [14.16] D. Williamson, K. Kadiman, Cyclostationarity in the of the 34th Annual Asilomar Conference on Signals, digital regulation of continuous time systems, in: C.I. Systems, and Computers, Pacific Grove, CA, 2000 Byrnes, A. Lindquist (Eds.), Frequency Domain and [13.31] C.M. Spooner, On the utility of sixth-order cyclic State Space Methods for Linear Systems, North-Hol- cumulants for RF signal classification, in: Proceedings land, Amsterdam, 1985, pp. 297–309. of the 35th Annual Asilomar Conference on Signals, [14.17] Y.P. Dragan, B.I. Yavorskii, A model of the test signal Systems, and Computers, Pacific Grove, CA, 2001 noise in channels with digital signal processing, Radio- [14.1] A.N. Bruyevich, Fluctuations in auto-oscillators for electron. Commun. Systems 29 (1986) 17–21. periodically nonstationary shot noise, Radio Eng. 23 [14.18] K. Kadiman, D. Williamson, Discrete minimax linear (1968) 91–96. quadratic regulation of continuous-time systems, Auto- [14.2] V.I. Voloshin, Frequency fluctuations in isochronous matica 23 (1987) 741–747. self-oscillators at periodically nonstationary random [14.19] V.G. Leonov, L.Y. Mogilevskaya, Y.L. Khotuntsev, actions, Radiophys. Quantum Electron. A.V. Frolov, Correlation relations between the com- [14.3] E.N. Rozenvasser, Periodically Nonstationary Control plex amplitudes of the noise components in bipolar Systems, Izdat. Nauka, Moskow, 1973 (in Russian). and field-effect transistors in the presence of strong [14.4] Y.P. Dragan, I.N. Yavorskii, Hydroacoustic commu- signals, Sov. J. Commun. Technol. Electron. 32 (1987) nication channels with surface scattering represented in 126–130. ARTICLE IN PRESS

694 W.A. Gardner et al. / Signal Processing 86 (2006) 639–697

[14.20] J. Goette, C. Gobet, Exact noise analysis of SC circuits [15.5] A.C. McCormick, A.K. Nandi, Cyclostationarity in and an approximate computer implementation, IEEE rotating machinery vibrations, Mech. Systems Signal Trans. Circuits Systems 36 (1989) 508–521. Process. 12 (2) (1998) 225–242. [14.21] S. Bittanti, D.B. Hernandez, G. Zerbi, The simple [15.6] T.R. Black, K.D. Donohue, Pitch determination of pendulum and the periodic LQG control problem, J. music signals using the generalized spectrum, in: Franklin Inst. 328 (1991) 299–315. Proceedings of IEEE South-East Conference ’00, [14.22] D. Williamson, Digital Control and Implementation, Nashville, TN, 2000, pp. 104–109. Prentice-Hall, Inc., Englewood Cliffs, NJ, 1991. [15.7] T.R. Black, K.D. Donohue, Frequency correlation [14.23] S. Bittanti, G. De Nicolao, Spectral factorization of analysis for periodic echoes, in: Proceedings of IEEE linear periodic systems with application to the optimal South-East Conference ’00, Nashville, TN, 2000, prediction of periodic ARMA models, Automatica pp. 131–138. (1992). [15.8] C. Capdessus, M. Sidahmed, J.L. Lacoume, Cyclosta- [14.24] M. Okumura, H. Tanimoto, T. Itakura, T. Sugawara, tionary processes: application in gear faults early Numerical noise analysis for nonlinear circuits with a diagnosis, Mech. Systems Signal Process. 14 (3) (2000) periodic large signal excitation including cyclostation- 371–385. ary noise sources, IEEE Trans. Circuits Systems (1993) [15.9] G. Dalpiaz, A. Rivola, R. Rubini, Effectiveness and 581–590. sensitivity of vibration processing techniques for local [14.25] H.L. Hurd, C.H. Jones, Dynamical systems with fault detection in gears, Mech. Systems Signal Process. cyclostationary orbits, in: R. Katz (Ed.), The Chaos 14 (3) (2000) 387–412. Paradigm, AIP Press, New York, 1994. [15.10] I. Antoniadis, G. Glossiotis, Cyclostationary analysis of [14.26] J. Roychowdhury, D. Long, P. Feldmann, Cyclosta- rolling-element bearing vibration signals, J. Sound Vib. tionary noise analysis of large RF circuits with multi- 248 (5) (2001) 829–845. tone excitations, IEEE J. Solid (3) (1998) 324–336. [15.11] L. Bouillaut, M. Sidahmed, Cyclostationary approach [14.27] J. Roychowdhury, P. Feldmann, D.E. Long, Method and bilinear approach: comparison, applications to of making an integrated circuit including noise model- early diagnosis for helicopter gearbox and classification ing and prediction, US Patent No. 6, 072, 947, June 6, method based on HOCS, Mech. Systems Signal Process. 2000. 15 (5) (2001) 923–943. [14.28] F. Bonani, S. Donati, G. Ghione, Noise source [15.12] G.F.P. Dusserre-Telmon, D. Flores, F. Prieux, Damage modeling for cyclostationary noise analysis in large- detection of motor pieces, European Patent No. signal device operation, IEEE Trans. Electron. Devices 1111364, June 27, 2001. 49 (9) (2002) 1640–1647. [15.13] R.B. Randall, J. Antoni, S. Chobsaard, The relation- [14.29] K.K. Gullapalli, Method and apparatus for analyzing ship between spectral correlation and envelope analysis small signal response and noise in nonlinear circuits, US in the diagnostics of bearing faults and other cyclosta- Patent No. 6,536,026, March 18, 2003. tionary machine signals, Mech. Systems Signal Process. [14.30] P. Shiktorov, E. Starikov, V. Gruz˘inskis, L. Reggiani, 15 (5) (2001) 945–962. L. Varani, J.C. Vaissie` re, S.P.T. Gonza´les, Monte Carlo [15.14] J. Antoni, J. Daniere, F. Guillet, Effective vibration simulation in electronic noise in semiconductor materi- analysis of IC engines using cyclostationarity, Part I—A als and devices operating under cyclostationary condi- methodology for condition monitoring, J. Sound Vib. tions, J. Comput. Electron. 2 (2003) 455–458. 257 (5) (2002) 815–837. [14.31] J.S. Roychowdhury, Apparatus and method for re- [15.15] J. Antoni, J. Daniere, F. Guillet, R.B. Randall, duced-order modeling of time-varying systems and Effective vibration analysis of IC engines using cyclos- computer storage medium containing the same, US tationarity, Part II —New results on the reconstruction Patent No. 6,687,658, February 3, 2004. of the cylinder pressure, J. Sound Vib. 257 (5) (2002) [15.1] S. Braun, B. Seth, Analysis of repetitive mechanism 839–856. signature, J. Sound Vib. 70 (1980) 513–526. [15.16] J. Antoni, R.B. Randall, Differential diagnosis of gear [15.2] S. Yamaguchi, Y. Kato, A practical method of and bearing faults, ASME J. Vib. Acoust. 124 (2) (2002) predicting noise produced by road traffic controlled 165–171. by traffic signals, J. Acoust. Soc. Amer. 86 (1989) [15.17] G.F.P. Dusserre-Telmon, D. Flores, F. Prieux, Process 2206–2214. for the detection of damage to components of an [15.3] Y. Kato, S. Yamaguchi, A prediction method for engine, US Patent No. 6, 389, 887, May 21, 2002. probability distribution of road traffic noise at an [15.18] A. Raad, J. Antoni, M. Sidahmed, Third-order cyclic intersection, Acoust. Australia 18 (1990) 46–50. characterization of vibration signals in rotating [15.4] D. Koenig, J. Boehme, Application of cyclostationarity machinery, in: Proceedings of XI European Signal and time-frequency analysis to engine car diagnostics, Processing Conference (EUSIPCO’02), Tolouse, in: Proceedings of International Conference on Acous- France, 2002. tics, Speech, and Signal Processing (ICASSP), Adelaide, [15.19] J. Antoni, R.B. Randall, A stochastic model for Australia, 1994, 149–152. simulation and diagnostics of rolling element bearings ARTICLE IN PRESS

W.A. Gardner et al. / Signal Processing 86 (2006) 639–697 695

with localized faults, ASME J. Vib. Acoust. 125 (3) correlation in financial data, Physica A Statist. Mech. (2003) 282–289. Appl. 336 (1–2) (2004) 196–205. [15.20] L. Li, L. Qu, Cyclic statistics in rolling bearing [17.1] C.J. Finelli, J.M. Jenkins, A cyclostationary least mean diagnosis, J. Sound Vib. 267 (2003) 253–265. squares algorithm for discrimination of ventricular [15.21] A. Raad, Contributions to the higher-order cyclic tachycardia from sinus rhythm, in: Proceedings of statistics: applications to gear faults, Ph.D. Disserta- the Annual International Conference of the IEEE tion, Universite´ de Technologie Compie` gne, Com- Engineering in Medicine and Biology Society, vol. 13, pie` gne, France, November 2003 (in French). 1991. [15.22] S. Tardu, Characterization of unsteady time periodical [17.2] K.D. Donohue, J. Bressler, T. Varghese, N.M. Bilgu- turbulent flows, Comptes Rendus Mecanique 331 tay, Spectral correlation in ultrasonic pulse-echo signal (2003) 767–774 (in French). processing, IEEE Trans. Ultrason. Ferroelectrics Fre- [15.23] J. Antoni, F. Bonnardot, A. Raad, M. El Badaoui, quency Control 40 (3) (1993) 330–337. Cyclostationary modelling of rotating machine vibra- [17.3] K.D. Donohue, T. Varghese, N.M. Bilgutay, Spectral tion signal, Mech. Systems Signal Process. 18 (6) (2004) redundancy in characterizing scatterer structures from 1285–1314. ultrasonic echoes, in: Review of Progress in Quantita- [15.24] Z.K. Zhu, Z.H. Feng, F.R. Kong, Cyclostationarity tive Non-Destructive Evaluation, vol. 13, Brunswick, analysis for gearbox condition monitoring: approaches Maine, 1993, pp. 951–958. and effectiveness, Mech. Systems Signal Process. 19 [17.4] M. Savic, Detection of cholestrol deposits in arteries, (2005) 467–482. US Patent No. 5,327,893, July 12, 1994. [16.1] E. Parzen, M. Pagano, An approach to modeling [17.5] T. Varghese, K.D. Donohue, Characterization of tissue seasonally stationary time-series, J. Econometrics 9 microstructure with spectral crosscorrelation, Ultra- (1979) 137–153. sound Imaging 15 (1994) 238–254. [16.2] D.R. Osborn, J.P. Smith, The performance of periodic [17.6] T. Varghese, K.D. Donohue, Mean-scatter spacing autoregressive models in forecasting seasonal UK estimates with spectral correlation, J. Acoust. Soc. consumption, J. Bus. Econom. Statist. 7 (1989) Amer. 96 (6) (1994) 3504–3515. 117–127 (see also J. Appl. Econometrics 3 (1989) 255- [17.7] K.D. Donohue, H.Y. Cheah, Spectral correlation filters 266). for flaw detection, in: Proceedings of 1995 Ultrasonics [16.3] R.M. Todd, Periodic linear-quadratic models of sea- Symposium, Seattle, WA, 1995, pp. 725–728. sonality, J. Econom. Dyn. Control 14 (1990) 763–796. [17.8] T. Varghese, K.D. Donohue, Estimating mean scatterer [16.4] D.R. Osborn, The implications of periodically varying spacing with frequency-smoothed spectral autocorrela- coefficients for seasonal time-series processes, J. Econo- tion function, IEEE Trans. Ultrason. Ferroelectrics metrics 48 (1991) 373–384. Frequency Control 42 (3) (1995) 451–463. [16.5] P.H. Franses, R. Paap, Model selection in periodic [17.9] T. Varghese, K.D. Donohue, J.P. Chatterjee, Specular autoregressions, Oxford Bull. Econom. Statist. 56 echo imaging with spectral correlation, in: Proceedings (1994) 421–439. of 1995 Ultrasonics Symposium, Seattle, WA, 1995, pp. [16.6] H.P. Boswijk, P.H. Franses, Testing for periodic 1315–1318. integration, Econom. Lett. 48 (1995) 241–248. [17.10] T. Varghese, K.D. Donohue, Spectral redundancy in [16.7] P.H. Franses, The effects of seasonally adjusting a tissue characterization, in: Twentieth International periodic autoregressive process, Comput. Statist. Data Symposium on Ultrasonic Imaging and Tissue Char- Anal. 19 (1995) 683–704. acterization, 1995. [16.8] T. Bollerslev, E. Ghysels, Periodic autoregressive [17.11] K.D. Donohue, T. Varghese, F. Forsberg, E.J. Halpern, conditional heteroscedasticity, Amer. Statist. Assoc. J. The analysis and classification of small-scale tissue Bus. Econom. Statist. 14 (2) (1996) 139–151. structures using the generalized spectrum, in: Proceed- [16.9] D. Dehay, J. Les´kow, Testing stationarity for stock ings of the International Congress on Acoustics and market data, Econom. Lett. 50 (1996) 205–212. Acoustical Society of America, Seattle WA, 1998, [16.10] E. Ghysels, A. Hall, H.S. Lee, On periodic structures pp. 2685–2686. and testing for seasonal unit roots, J. Amer. Statist. [17.12] K.D. Donohue, F. Forsberg, C.W. Piccoli, B.B. Gold- Assoc. 91 (1996) 1551–1559. berg, Analysis and classification of tissue with scatterer [16.11] P.H. Franses, M. McAleer, Cointegration analysis of structure templates, IEEE Trans. Ultrason. Ferro- seasonal time series, J. Econom. Survey 12 (1998) electrics Frequency Control 46 (2) (1999) 300–310. 651–678. [17.13] K.D. Donohue, L. Haung, V. Genis, F. Foresberg, [16.12] E. Ghysels, D.R. Osborne, The Econometric Analysis Duct size estimation in breast tissue, in: Proceedings of Seasonal Time Series, Themes in Modern Econo- of 1999 Ultrasonics Symposium, Reno, NV, 1999, metrics, Cambridge University Press, Cambridge, UK, pp. 1353–1356. 2001. [17.14] L. Huang, K.D. Donohue, V. Genis, F. Forsberg, Duct [16.13] E. Broszkiewicz-Suwaj, A. Makagon, R. Weron, A. detection and wall spacing estimation in breast tissue, Wy"oman´ska, On detecting and modeling periodic Ultrason. Imaging 22 (3) (2000) 137–152. ARTICLE IN PRESS

696 W.A. Gardner et al. / Signal Processing 86 (2006) 639–697

[17.15] K.D. Donohue, L. Huang, T. Burks, F. Forsberg, C.W. IEEE INFOCOM’83, San Diego, CA, 1983, Piccoli, Tissue classification with generalized spectrum pp. 229–238. parameters, Ultrasound Med. Biol. 27 (11) (2001) [19.3] M. Kaplan, A single-server queue with cyclostationary 1505–1514. arrivals and arithmetic service, Oper. Res. 31 (1983) [17.16] K.D. Donohue, L. Huang, Ultrasonic scatterer struc- 184–205. ture classification with the generalized spectrum, in: [19.4] M.H. Ackroyd, Stationary and cyclostationary finite Proceedings of the IEEE International Conference on buffer behaviour computation via Levinson’s method, Acoustics, Speech and Signal Processing, Salt Lake AT (1984) 2159–2170. City, UT, 2001. [19.5] T. Nakatsuka, Periodic property of streetcar congestion [17.17] K.D. Donohue, L. Huang, G. Georgiou, F.S. Cohen, at the first station, J. Oper. Res. Soc. Japan 29 (1986) C.W. Piccoli, F. Forsberg, Malignant and benign breast 1–20. tissue classification performance using a scatterer [19.6] D. Habibi, D.J.H. Lewis, Fourier analysis for modelling structure preclassifier, IEEE Trans. Ultrason. Ferro- some cyclic behaviour of networks, Comput. Commun. electrics Frequency Control 50 (6) (2003) 724–729. 19 (1996) 426–434. [17.18] S. Gefen, O.J. Tretiak, C. Piccoli, K.D. Donohue, A.P. [20.1] Y.P. Dragan, I.N. Yavorskii, The periodic correla- Petropulu, P.M. Shankar, V.A. Dumane, L. Huang, tion-random field as a model for bidimensional ocean M.A. Kutay, V. Genis, F. Forsberg, J.M. Reid, B.B. waves, Otbor Peredacha Inform. 51 (1982) 15–21 (in Goldberg, ROC analysis of classifiers based on ultra- Russian). sonic tissue characterization features, IEEE Trans. [20.2] V.G. Alekseev, On spectral density estimates of Med. Imaging 22 (2) (2003) 170–177. Gaussian periodically correlated random fields, Probab. [18.1] V.A. Markelov, On extrusions and relative time of Math. Statist. 11 (2) (1991) 157–167. staying of the periodically nonstationary random [20.3] H.L. Hurd, Spectral correlation of randomly jittered process, Izv. VUZ Radiotekhnika 9 (4) (1996) (in periodic functions of two variables, in: Proceedings of Russian). the 29th Annual Asilomar Conference on Signals, [18.2] V.A. Markelov, Axis crossings and relative time of Systems, and Computers, Pacific Grove, CA, 1995. existence of a periodically nonstationary random [20.4] W. Chen, G.B. Giannakis, N. Nandhakumar, Spatio- process, Sov. Radiophys. 9 (1966) 440–443 (in Russian). temporal approach for time-varying image motion [18.3] Y.P. Dragan, Properties of counts of periodically estimation, IEEE Trans. Image Process. 10 (1996) correlated random processes, Otbor Peredacha Inform. 1448–1461. 33 (1972) 9–12. [20.5] H. Li, Q. Cheng, B. Yuan, Strong laws of large numbers [18.4] M.G. Bilyik, Up-crossings of periodically nonstationary for two dimensional processes, in: Proceedings of processes, Otbor Peredacha Inform. 36 (1973) 28–31 (in Fourth International Conference on Signal Processing Russian). (ICSP’98), 1998. [18.5] M.G. Bilyik, On the theory of up-crossings of [20.6] D. Dehay, H. Hurd, Spectral estimation for strongly periodically nonstationary processes, Otbor Peredacha periodically correlated random fields defined on R2, Inform. 38 (1974) 15–21 (in Russian). Math. Methods Statist. 11 (2) (2002) 135–151. [18.6] M.G. Bilyik, Models and level crossings of periodically [20.7] H. Hurd, G. Kallianpur, J. Farshidi, Correlation and nonstationary processes, Otbor Peredacha Inform. 39 spectral theory for periodically correlated random fields (1974) 3–7 (in Russian). indexed on Z2, J. Multivariate Anal. 90 (2) (2004) [18.7] M.G. Bilyik, Some properties of periodically nonsta- 359–383. tionary processes and inhomogeneous fields, Otbor [21.1] W.A. Gardner, Correlation estimation and time-series Peredacha Inform. 48 (1976) 27–33 (in Russian). modeling for nonstationary processes, Signal Processing [18.8] M.G. Bilyik, Statistical properties of level crossings of a 15 (1988) 31–41. periodic nonstationary process and a periodic inhomo- [21.2] J.C. Hardin, A.G. Miamee, Correlation autoregressive geneous field, Otbor Peredacha Inform. 52 (1977) 3–8 processes with application to helicopter noise, J. Sound (in Russian). Vib. 142 (1990) 191–202. [18.9] M.G. Bilyik, High crossings of periodically nonsta- [21.3] A.G. Miamee, J.C. Hardin, On a class of nonstationary tionary Gaussian process, Otbor Peredacha Inform. 59 stochastic processes, Sankhya: Indian J. Statist. Ser. A (1980) 24–31 (in Russian). (Part 2) 52 (1990) 145–156. [18.10] D.E.K. Martin, Detection of periodic autocorrelation [21.4] J.C. Hardin, A. Makagon, A.G. Miamee, Correlation in time series data via zero crossing, J. Time Ser. Anal. autoregressive sequences: a summary, in: A.G. Miamee 20 (4) (1999) 435–452. (Ed.), Proceeding of the Workshop on Nonstationary [19.1] R.R. Anderson, G.T. Foschini, B. Gopinath, A queuing Stochastic Processes and their Applications, World model for a hybrid data multiplexer, Bell System Tech. Scientific, Virginia, VA, 1991, pp. 165–175. J. 58 (1979) 279–300. [21.5] W.A. Gardner, On the spectral coherence of nonsta- [19.2] L. Clare, I. Rubin, Queueing analysis of TDMA with tionary processes, IEEE Trans. Signal Process. 39 limited and unlimited buffer capacity, in: Proceedings of (1991) 424–430. ARTICLE IN PRESS

W.A. Gardner et al. / Signal Processing 86 (2006) 639–697 697

[21.6] J. Allen, S. Hobbs, Detecting target motion by [22.5] B.N. Gartshtein, V.V. Smelyakov, Assessing character- frequency-plane smoothing, in: Proceedings of the istics of periodic random signals, Radiotekhnika 60 26th Asilomar Conference on Systems and Computers, (1982) 116–123 (in Russian). Pacific Grove, CA, 1992, pp. 1042–1047. [22.6] P. Kochel, A dynamic multilocation supply model with [21.7] L. Izzo, A. Napolitano, Time-frequency representations redistribution between the stores, Math. Oper. Stat. Ser. of generalized almost-cyclostationary processes, in: Optim. 13 (1982) 267–286 (in German). Proceedings of 16th GRETSI Symposium on Signal [22.7] G.G. Kuznetsov, P.A. Chernenko, Mathematical model and Image Processing, 1997. and device for diagnostics of periodically nonstationary [21.8] A. Makagon, A.G. Miamee, On the spectrum of random processes, Elektron. Model. 6 (1984) 45–48 (in correlation autoregressive sequences, Stochastic Pro- Russian). cess. Appl. 69 (1997) 179–193. [22.8] A.J.E.M. Janssen, The Zak transform: a signal trans- [21.9] L. Izzo, A. Napolitano, The higher-order theory of form for sampled time-continuous signals, Philips J. generalized almost-cyclostationary time-series, IEEE Res. 43 (1988) 23–69. Trans. Signal Process. 46 (11) (1998) 2975–2989. [21.10] L. Izzo, A. Napolitano, Linear time-variant transformations of generalized almost-cyclostationary signals, Further references Part I: Theory and method, IEEE Trans. Signal Process. 50 (12) (2002) 2947–2961. [23.1] N. Wiener, Generalized harmonic analysis, Acta Math. [21.11] L. Izzo, A. Napolitano, Linear time-variant transfor- 55 (1930) 117–258. mations of generalized almost-cyclostationary signals, [23.2] A.S. Besicovitch, Almost Periodic Functions, Cam- Part II: Development and applications, IEEE Trans. bridge University Press, London, 1932 (also New York: Signal Process. 50 (12) (2002) 2962–2975. Dover, 1954). [21.12] K.-S. Lii, M. Rosenblatt, Spectral analysis for [23.3] H. Bohr, Almost Periodic Functions, Springer, Berlin, harmonizable processes, Ann. Statist. 30 (1) (2002) 1933 (also Chelsea, New York, 1974). 258–297. [23.4] N. Wiener, The Fourier Integral and Certain of its [21.13] L. Izzo, A. Napolitano, Sampling of generalized almost- Applications, Cambridge University Press, London, cyclostationary signals, IEEE Trans. Signal Process. 51 1933 (also New York: Dover, 1958). (6) (2003) 1546–1556. [23.5] M. Loe` ve, Probability Theory, Van Nostrand, Prince- [21.14] A. Napolitano, Uncertainty in measurements on ton, NJ, 1963. spectrally correlated stochastic processes, IEEE Trans. [23.6] L. Amerio, Prouse, Almost-Periodic Functions and Inform. Theory 49 (9) (2003) 2172–2191. Functional Equations, Van Nostrand, New York, 1971. [21.15] A. Napolitano, Mean-square consistency of statistical- [23.7] J. Bass, Cours de Mathe´matiques, vol. III, Masson & function estimators for generalized almost-cyclostation- Cie, Paris, 1971. ary processes, in: Proceedings of XII European Signal [23.8] R.A. Meyer, C.S. Burrus, A unified analysis of multi- Processing Conference (EUSIPCO 2004), Vienna, rate and periodically time-varying digital filters, IEEE Austria, 2004. Trans. Circuits Systems CAS (1975) 162–168. [21.16] L. Izzo, A. Napolitano, Generalized almost-cyclosta- [23.9] E. Pfaffelhuber, Generalized harmonic analysis for tionary signals, in: P.W. Hawkes (Ed.), Advances in distributions, IEEE Trans. Inform. Theory IT-21 Imaging and Electron Physics, vol. 135, Elsevier, (1975) 605–611. Amsterdam, 2005, pp. 103–223 (Chapter 3). [23.10] A.H. Zemanian, Distribution Theory and Transform [21.17] A. Napolitano, Asymptotic Normality of statistical- Analysis, Dover, New York, 1987. function estimators for generalized almost-cyclostation- [23.11] C. Corduneanu, Almost Periodic Functions, Chelsea, ary processes, in: Proceedings of XIII European Signal New York, 1989. Processing Conference (EUSIPCO 2005), Antalya, [23.12] T. Chen, L. Qiu, Linear periodically time-varying Turkey, 2005. discrete-time systems: aliasing and LTI approximations, [22.1] O.I. Koronkevich, Sci. Ann. L’vovskii Inst. 44 (5) (1957). Systems Control Lett. 30 (1997) 225–235. [22.2] R.L. Stratonovich, Topics in the Theory of Random [23.13] A.S. Mehr, T. Chen, On alias-component matrices of Noise, vols. I and II, Gordon and Breach, New York, discrete-time periodically time-varying systems, IEEE 1967 (Revised English edition translated by R.A. Signal Process. Lett. 8 (2001) 114–116. Silverman). [23.14] A.S. Mehr, T. Chen, Representation of linear periodi- [22.3] K.S. Voychishin, Y.P. Dragan, et al., Coll. Algorithmi- cally time-varying multirate systems, IEEE Trans. zation of Industrial Processes, Cybernetics Institute, Signal Process. 50 (9) (2002) 2221–2229. AN USSR, 1972 (in Russian). [23.15] J. Leskow, A. Napolitano, Foundations of the non- [22.4] V. Zalud, V.N. Kulesov, Low-Noise Semiconductor stochastic approach for time series analysis, Signal Circuits, SNTL, Prague, 1980, Section 1.18 (in Czech). Process., submitted.