Generalized Almost-Cyclostationary Signals*

ADVANCES IN IMAGING AND ELECTRON PHYSICS, VOL. 135

Generalized Almost-Cyclostationary Signals* LUCIANO IZZO AND ANTONIO NAPOLITANO Dipartimento di Ingegneria Elettronica e delle Telecomunicazioni Universita` di Napoli Federico II, 80125 Napoli, Italy

I. Introduction . . . . . . . . . . . . . . . . . . . . A. Generalized Almost-Cyclostationary Signals . . . . . . . . B. Nonstochastic Approach for Signal Analysis . . . . . . . . C. Review on Higher-Order Cyclostationarity . . . . . . . . D. Outline . . . . . . . . . . . . . . . . . . . . . II. Higher-Order Characterization . . . . . . . . . . . . . . A. Introduction . . . . . . . . . . . . . . . . . . . B. Strict Sense Characterization . . . . . . . . . . . . . C. Generalized Cyclic Moments . . . . . . . . . . . . . 1. Temporal Parameters . . . . . . . . . . . . . . . 2. Spectral Parameters . . . . . . . . . . . . . . . D. Generalized Cyclic Cumulants . . . . . . . . . . . . . 1. Temporal Parameters . . . . . . . . . . . . . . . 2. Spectral Parameters . . . . . . . . . . . . . . . E. Estimation of the Generalized Cyclic Statistics . . . . . . . F. Examples of GACS Signals . . . . . . . . . . . . . . 1. Chirp Signal . . . . . . . . . . . . . . . . . . 2. Nonuniformly Sampled Signal . . . . . . . . . . . . G. Summary . . . . . . . . . . . . . . . . . . . . III. Linear Time-Variant Transformations of GACS Signals . . . . . A. Introduction . . . . . . . . . . . . . . . . . . . B. FOT Deterministic and Random Linear Systems . . . . . . 1. FOT Deterministic and Random Systems . . . . . . . . 2. FOT Deterministic Linear Systems . . . . . . . . . . 3. Impulse-Response Function Decomposition for FOT Random LTV Systems . . . . . . . . . . . . . . . . . . C. Higher-Order System Characterization in the Time Domain . . D. Higher-Order System Characterization in the Frequency Domain E. Ergodicity of the Output Signal of a LTV System . . . . . .

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* Reprinted with permission from: L. Izzo and A. Napolitano, ‘‘The higher-order theory of generalized almost-cyclostationary time-series,’’ IEEE Trans. Signal Processing, Vol. 46, pp. 2975–2989, Nov. 1998. L. Izzo and A. Napolitano, ‘‘Linear time-variant transformations of generalized almost-cyclostationary signals, Part I: Theory and method,’’ IEEE Trans. signal Processing, Vol. 50, pp. 2947–2961, Dec. 2002. L. Izzo and A. Napolitano, ‘‘Linear time-variant transformations of generalized almost-cyclostationary signals, Part II: Developments and applications,’’ IEEE Trans. Signal Processing, Vol. 50, pp. 2962–2975, Dec. 2002. L. Izzo and A. Napolitano, ‘‘Sampling of generalized almost-cyclostationary signals,’’ IEEE Trans. Signal Processing, Vol. 51, pp. 1546–1556, June 2003. 103 ISSN 1076-5670/05 DOI: 10.1016/S1076-5670(04)35003-2

Copyright 2004, IEEE All rights reserved.

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F. Countability of the Set of the Output Cycle Frequencies. . . . . . 1. Analysis of LTV Systems. . . . . . . . . . . . . . . . 2. The Special Case of FOT Deterministic LTV Systems . . . . . G. LAPTV Filtering . . . . . . . . . . . . . . . . . . . H. Product Modulation . . . . . . . . . . . . . . . . . . I. Multipath Doppler Channels . . . . . . . . . . . . . . . J. Summary . . . . . . . . . . . . . . . . . . . . . . IV. Sampling of GACS Signals . . . . . . . . . . . . . . . . . A. Introduction . . . . . . . . . . . . . . . . . . . . . B. Discrete-Time ACS Signals . . . . . . . . . . . . . . . . C. Sampling of GACS Signals . . . . . . . . . . . . . . . . D. Conjecturing the Nonstationarity Type of the Continuous-Time Signal E. Summary . . . . . . . . . . . . . . . . . . . . . . V. Time-Frequency Representations of GACS Signals. . . . . . . . . A. Introduction . . . . . . . . . . . . . . . . . . . . . B. Second-Order GACS Signals . . . . . . . . . . . . . . . C. Time-Frequency Representations of GACS Signals . . . . . . . D. Signal Feature Extraction . . . . . . . . . . . . . . . . Appendices . . . . . . . . . . . . . . . . . . . . . . Appendix A . . . . . . . . . . . . . . . . . . . . . . Appendix B . . . . . . . . . . . . . . . . . . . . . . Appendix C . . . . . . . . . . . . . . . . . . . . . . Appendix D . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . .

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165 166 171 176 181 183 189 190 190 191 194 199 201 201 201 204 206 208 211 211 212 213 216 220

I. INTRODUCTION This section introduces the class of second-order generalized almost-cyclostationary signals in the classical stochastic approach. Then, the alternative characterization in the nonstochastic (or fraction-of-time probability) framework is considered. Moreover, within such a framework, a review on the higher-order cyclostationarity is provided. Finally, the outline of the topics treated in Section II through V is presented. A. Generalized Almost-Cyclostationary Signals The theory of second- and higher-order almost-cyclostationary (ACS) signals has been developed and applied to several signal-processing and communication problems, such as weak-signal detection, parameter estimation, system identification, blind-adaptive spatial filtering, and so faith (see Dandawate´ and Giannakis, 1994, 1995; Dehay and Hurd, 1994; Gardner, 1988a,b, 1993, 1994; Gardner and Spooner, 1994; Gladyshev, 1963; Hurd, 1991; Izzo and Napolitano, 1996c, 1997a; Napolitano, 1995; Spooner and Gardner, 1994, and references therein).

GENERALIZED ALMOST-CYCLOSTATIONARY SIGNALS

105

A finite-power continuous-time complex-valued stochastic process x(t) is said second-order almost-cyclostationary in the wide sense or, equivalently, almost-periodically correlated, if its autocorrelation function E{x(t þ t)x*(t)}, with E{
} denoting statistical expectation, is an almost-periodic function (Besicovitch, 1932; Bohr, 1933; Corduneanu, 1989) of t with frequencies not depending on t. That is, the autocorrelation function is the limit of an uniformly convergent sequence of trigonometric polynomials in t: X an
ðtÞe j2pan t ; Efxðt þ tÞx  ðtÞg ¼ ð1:1Þ R xx n2I

where I is a countable set, the frequencies an (not depending on t) are referred
an ðtÞ, called cyclic autocorrelato as cycle frequencies, and the coeYcients R xx tion functions, are given by Z T=2
an ðtÞ ≜ lim 1 Efxðt þ tÞx ðtÞge j2pan t dt R xx ð1:2Þ T!1 T T=2

hEfxðt þ tÞx  ðtÞge j2pan t it : Let us define the function of the two variables (a, t)
xx ða; tÞ ≜ hEfxðt þ tÞx ðtÞge j2pat i ; R t

ð1:3Þ

which is called two-variable cyclic autocorrelation function or, if it does not generate ambiguity, cyclic autocorrelation function. Its magnitude and phase are the amplitude and phase, respectively, of the finite-strength additive complex sinewave component at frequency a contained in the autocorrelation function with lag t. According to Eqs. (1.1) and (1.2), the two-variable cyclic autocorrelation function Eq. (1.3) of ACS processes is nonzero only in correspondence of a countable set of values of a. That is, 8 a
n ðtÞ; a ¼ an n 2 I
ð1:5Þ

Hurd (1991) shows that the ACS processes are characterized by the following conditions: 1. The set A ≜ [ At t2R

is countable.

ð1:6Þ

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2. The autocorrelation function E{x(t þ t)x*(t)} is uniformly continuous in t and t.
xx ð0; tÞ ≜ hEfxðt þ tÞ 3. The time-averaged autocorrelation function R x  ðtÞgit is continuous for t ¼ 0 (and, hence, for every t). 4. The process is mean-square continuous, that is sup Efjxðt þ tÞ xðtÞj2 g ! 0

as t ! 0

ð1:7Þ

t2R

In Figure 1, the support in the (a, t) plane of the two-variable cyclic
xx ða; tÞ of an ACS signal is reported. According autocorrelation function R to Condition 1, such a support is constituted by the lines a ¼ an, n 2 I, that is, lines parallel to the t axis in correspondence of the cycle frequencies. If the autocorrelation function is an almost-periodic function of t whose (generalized) Fourier series expansion has both coeYcients and frequencies depending on the lag parameter t, then the process is said to be generalized almost-cyclostationary (GACS): X ðnÞ
ðtÞe j2pan ðtÞt : Efxðt þ tÞx ðtÞg ¼ ð1:8Þ R xx n2I

In Eq. (1.8), the frequencies an(t) are referred to as lag-dependent cycle frequencies and the coeYcients, referred to as generalized cyclic autocorrelation functions, are given by

FIGURE 1. Support in the (a, t) plane of the cyclic autocorrelation function of an ACS signal.

GENERALIZED ALMOST-CYCLOSTATIONARY SIGNALS



ðnÞ ðtÞ ≜ Efxðt þ tÞx ðtÞge j2pan ðtÞt : R xx t

107 ð1:9Þ

It can be shown that for GACS signals the two-variable cyclic autocorrelation function can be expressed in terms of the generalized cyclic autocorrelation functions by the following relationship X ðnÞ
xx ða; tÞ ¼
ðtÞda a ðtÞ R ð1:10Þ R xx n n2I

where da is the Kronecker delta, that is, da ¼ 1 for a ¼ 0 and da ¼ 0 for a 6¼ 0. Moreover, it results that At ¼ [ fa 2 R : a ¼ an ðtÞg: n2I

ð1:11Þ


xx ða; tÞ is Thus, for GACS signals the support in the (a, t) plane of R constituted by a countable set of curves described by the equations a ¼ an ðtÞ; n 2 I. In Figure 2, the support in the (a, t) plane of the cyclic autocorrelation
xx ða; tÞ of a GACS signal is reported. function R From the previous considerations, it follows that the ACS signals are the sub-class of GACS signals such that the lag-dependent cycle frequencies an(t) are constant with respect to t and are equal to the cycle frequencies. Moreover, for GACS signals that are not ACS the set A defined in Eq. (1.6) is uncountable.

FIGURE 2. Support in the (a, t) plane of the cyclic autocorrelation function of a GACS signal.

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B. Nonstochastic Approach for Signal Analysis The signal analysis approach used in the following sections is nonstochastic, and is also referred to as the fraction-of-time (FOT) probability framework (Gardner, 1988a, 1994). It is an alternative approach where signals are modeled as single functions of time (time series) rather than sample paths of stochastic processes. Such an approach is more appropriate when the ensemble of realizations does not exist and is introduced just to create an (abstract) mathematical model, the stochastic process, for the only time series at hand. Thus, the FOT probability framework turns out to be particularly useful when the stochastic process is not ergodic or it is subject to a processing destroying ergodicity properties (see Section III). In the FOT probability approach, statistical parameters are defined through infinite-time averages of a single time series (and of products of time- and frequency-shifted versions of this time series) rather than ensemble averages of a stochastic process. Moreover, starting from this single time series, a (possibly time-varying) probability distribution function can be constructed, which leads to the expectation operation and all the familiar probabilistic concepts and parameters, such as stationarity, cyclostationarity, nonstationarity, independence, mean, variance, moments, cumulants, and so on. Estimators of the FOT probabilistic parameters are obtained by considering finite-time averages of the same quantities involved in the infinitetime averages. Therefore, assuming the previously-mentioned parameters exist, that is, the infinite-time averages exist, their asymptotic estimators converge by definition to the true values, which are exactly the infinite-time averages without the necessity of requiring ergodicity properties as in the stochastic process framework. Thus, in the FOT probability framework, the kind of convergence of the estimators to be considered as the data-record length approaches infinity is the convergence of the function sequence of the finite-time averages (indexed by the data-record length). Therefore, unlike the stochastic process framework where convergence must be intended, for example, in the ‘‘stochastic meansquare sense’’ (Dandawate´ and Gianakis, 1995; Dehay and Hurd, 1994; Hurd, 1991; Lacoume et al., 1997) or ‘‘almost sure sense’’ (Hurd and Les´kow, 1992; Li and Cheng, 1997) or ‘‘in distribution,’’ the convergence in the FOT probability framework must be considered ‘‘pointwise,’’ in the ‘‘temporal mean-square sense’’ (Weiner, 1930) (see also Appendix A), or in the ‘‘sense of generalized functions (distributions)’’ (PfaVelhuber, 1975). The functional analysis approach for signal processing based on infinitetime averages was first introduced by Norbert Wiener (1930) in the classical paper with reference to time-invariant statistics of ordinary functions of time; subsequently, it was developed in Bass (1971), Brown (1958),

109

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Furstenberg (1960), Hofstetter (1964), Kac and Steinhaus (1938), Kac (1959), Les´kow and Napolitano (2002), and Urbanik (1958) and then extended to the case of distributions (generalized functions) in PfaVelhuber (1975). The term fraction-of-time probability was first introduced in Hofstetter (1964). Moreover, in Wold (1948) an isometric isomorphism (Wold isomorphism) between a stationary stochastic process and the Hilbert space generated by a single sample path was singled out and a rigorous link between the FOT probability and the stochastic process frameworks in the stationary case was established. The case of time-variant FOT statistics of ACS time series was widely treated by William A. Gardner in (1988a), and then in Gardner (1991, 1994) with reference to the second-order statistics and in Gardner and Brown (1991), Gardner and Spooner (1994) and Spooner and Gardner (1994) for the higher-order statistics. Furthermore, the Wold isomorphism was extended to cyclostationary sequences in Hurd and Koski (2004). Moreover, in Izzo and Napolitano (1996a, 1998b, 2002b,c) the class of the GACS time series was introduced and characterized in the FOT probability framework. In the time-variant nonstochastic framework for ACS and GACS time series, the almost-periodic functions (Besicovitch, 1932; Bohr, 1933; Corduneanu, 1989) play the same role played by the deterministic signals in the stochastic-process framework and the expectation operator is the almostperiodic component extraction operator, that is, the operator that extracts all the finite-strength additive sine wave components of its argument. Therefore, in the case of processes exhibiting suitable ergodicity properties, all the results presented here can be interpreted in the stochastic process framework by substituting the almost-periodic component extraction operator with the statistical expectation operator. Specifically, for any finite-power time series x(t), let us consider the decomposition xðtÞ ≜ xp ðtÞ þ xr ðtÞ where xp ðtÞ ≜

X


x
e j2p
t

ð1:12Þ ð1:13Þ

is an almost-periodic function and xr(t) a residual term not containing any finite-strength additive sinewave component, that is, hxr ðtÞe j2pat it 0; 8a 2 R:

ð1:14Þ

The almost-periodic component extraction operator E{a}{
} is defined by Efag fxðtÞg ≜ xp ðtÞ:

ð1:15Þ

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Let x(t) be real valued and let us denote by G the set of frequencies (for any x) of the almost-periodic component of the function of t U(x x(t)), where ( 1; x  0 UðxÞ ≜ ð1:16Þ 0; x < 0 is the unit step function. It can be shown that the function of x fGg

FxðtÞ ðxÞ ≜ Efag fUðx xðtÞÞg

ð1:17Þ

for any t is a valid cumulative distribution function except for the rightcontinuity property. That is, it has values in [0, 1], is nondecreasing, fGg fGg FxðtÞ ð 1Þ ¼ 0, and FxðtÞ ðþ1Þ ¼ 1. Moreover, its derivative with respect fGg to x, say fxðtÞ ðxÞ, is a valid probability density function, and it results that Z fGg Efag fxðtÞg ¼ x fxðtÞ ðxÞ dx ð1:18Þ R

which is a result formally analogous to that in the classical stochastic process framework. For an almost-periodic signal x(t), it results x(t) xp(t) and, hence, Efag fxðtÞg ¼ xðtÞ:

ð1:19Þ

That is, the almost-periodic functions are the deterministic signals in the FOT probability framework. All the other signals are the random signals (see also Section II.B). Note that in this work ‘‘random’’ is not synonymous with ‘‘stochastic.’’ In fact, the adjective stochastic is adopted, as usual, when an ensemble of realizations or sample paths exist, whereas the adjective random is referred to a single function of time, namely, a signal or a system impulse-response function. Therefore, to avoid ambiguities, when necessary, deterministic and random signals or systems in the FOT probability sense will be referred to as FOT deterministic and FOT random, respectively. Analogously, a second-order characterization for the real-valued time series x(t) can be obtained by using the almost-periodic component extraction operator as the expectation operator. Specifically, the function of x1 and x2 fG g

t FxðtþtÞxðtÞ ðx1 ; x2 Þ ≜ Efag fUðx1 xðt þ tÞÞUðx2 xðtÞÞg

ð1:20Þ

is a valid second-order joint cumulative distribution function for any fixed t and t, except for the right-continuity property with respect to x1 and x2, where Gt is the set of frequencies (for any (x1, x2)) of the almost-periodic component of the function of t Uðx1 xðt þ tÞÞUðx2 xðtÞÞ. Moreover, the

GENERALIZED ALMOST-CYCLOSTATIONARY SIGNALS

111

fG g

t second-order derivative with respect to x1 and x2 of FxðtþtÞxðtÞ ðx1 ; x2 Þ, say fGt g fxðtþtÞxðtÞ ðx1 ; x2 Þ, is a valid second-order joint probability density function. Furthermore, it results that the autocorrelation function Efag fxðt þ tÞxðtÞg can be expressed as

fag

E

fxðt þ tÞxðtÞg ¼ ¼ ¼

R

fG g

R X 2

t x1 x2 fxðtþtÞxðtÞ ðx1 ; x2 Þ dx1 dx2

Rxx ða; tÞej2pat

a2At X

ð1:21Þ

j2pan ðtÞt RðnÞ xx ðtÞe

n2I

where At is a countable set defined analogously to Eq. (1.5), Rxx ða; tÞ ≜ hxðt þ tÞxðtÞe j2pat it

ð1:22Þ

is the (nonstochastic) (two-variable) cyclic autocorrelation function, and
 j2pan ðtÞt ð1:23Þ RðnÞ xx ðtÞ ≜ xðt þ tÞxðtÞe t is the (nonstochastic) generalized cyclic autocorrelation function. If for some t the set At contains not only the element a ¼ 0, then the time series is GACS (or, in particular, ACS) in the wide sense. From these time series sine waves can be generated by using quadratic or higher-order nonlinear time-invariant transformations. In the special case of ACS time series, it has been shown that algorithms based on the theory of generated sine waves turn out to be asymptotically unaVected by both noise and interference and then are highly tolerant to noise and interference in practice. Finally, note that the almost-periodic component of a time series (or a lag product time series) can be extracted by exploiting the synchronized averaging identity (Gardner, 1988a, 1990) or the algorithms proposed by Sethares and Staley (1999). C. Review on Higher-Order Cyclostationarity The theory of GACS signals presented in the following sections extends the theory of higher-order cyclostationarity that was introduced in the FOT probability framework by Gardner and Spooner (1994) and Spooner and Gardner (1994) and subsequently developed in Flagiello et al. (2000), Izzo and Napolitano (1996c, 1997a, 1998a), Napolitano (1995), and Napolitano and Spooner (2000, 2001). The following brief review introduces the necessary terminology. A continuous-time complex-valued time series x(t) is said to exhibit Nth-order wide-sense cyclostationarity with cycle frequency a 6¼ 0, for a given conjugation configuration, if the Nth-order cyclic temporal moment function (CTMF)

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Rax ðtÞ≜ RaxðÞ1 ;...;xðÞN ðtÞ * + N Y ðÞn j2pat ≜ x ðt þ tn Þe n¼1

ð1:24Þ

t

exists and is not zero for some delay vector t (Gardner and Spooner, 1994). In (1.24), t ≜ ½t1 ; . . . ; tN ⊤ and x ≜ ½xðÞ1 ; . . . ; xðÞN ⊤ are column vectors, and (*)n represents the nth optional complex conjugation. The N-fold Fourier transform S ax ð f Þ of the CTMF, which is called the Nth-order cyclic spectral moment function (CSMF), can be written as S ax ð f Þ ¼ Sxa ð f 0 Þdð f ⊤ 1 aÞ

ð1:25Þ

where d(
) is Dirac delta function, 1 ≜ ½1; . . . ; 1⊤ , and the prime sign denotes the operator that transforms v ¼ ½v1 ; . . . ; vN ⊤ into v0 ¼ ½v1 ; . . . ; vN 1 ⊤ . The function Sxa ð f 0 Þ, referred to as the Nth-order reduced-dimension CSMF (RD-CSMF), can be expressed as the (N 1)-fold Fourier transform of the Nth-order reduced-dimension CTMF defined as Rax ðt0 Þ ≜ Rax ðtÞjtN ¼0 :

ð1:26Þ

For N ¼ 2 and conjugation configuration [xx*], the RD-CTMF is coincident with the cyclic autocorrelation function Raxx ðt1 Þ, whereas, for conjugation configuration [xx] it is coincident with the conjugate cyclic autocorrelation function Raxx ðt1 Þ that can be nonzero only if the signal is noncircular (Picinbono, 1994). Gardner and Spooner (1994) showed that the RD-CSMF Sxb ( f 0 Þ can contain impulsive terms if the vector f with PN 1 fN ¼ b n¼1 fn lies on the b-submanifold, that is, if there exists at least one partition {m1,


, mp} of {1,


, N} with p > 1 such that each sum P ami ¼ n2mi fn is an jmijth-order cycle frequency of x(t), where jmij is the number of elements in mi. A well-behaved frequency-domain function that characterizes a signal’s higher-order cyclostationarity can be obtained starting from the Nth-order cyclic temporal cumulant function (CTCF), that is, the coeYcient


 Cbx ðtÞ ≜ cum xðÞn ðt þ tn Þ; n ¼ 1; . . . ; N e j2pbt t ð1:27Þ of the Fourier-series expansion of the Nth-order temporal cumulant function (Gardner and Spooner (1994). Its N-fold Fourier transform is the Nthorder cyclic spectral cumulant function P bx ð f Þ, which can be written as P bx ð f Þ ¼ Pbx ð f 0 Þdð f ⊤ 1 bÞ, where the Nth-order cyclic polyspectrum (CP) Pbx ð f 0 Þ is the (N 1)-fold Fourier transform of the reduced-dimension CTCF (RD-CTCF) Cxb ðt0 Þ obtained by setting tN ¼ 0 into Eq. (1.27). The CP turns out to be a well-behaved function (i.e., it does not contain impulsive terms) under the mild assumption that the time series x(t) and

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113

x(t þ t) are asymptotically (j t j! 1) independent. Moreover, except on a b-submanifold, it is coincident with the RD-CSMF Sxb ð f 0 Þ. Second- and higher-order cycle frequencies are obtained as linear combinations of parameters such as the baud rate and the carrier frequency. Therefore, cyclostationarity-based signal-processing algorithms are potentially signal selective, the selectivity being obtained by a suitable choice of the working cycle frequency, provided that the useful and disturbance signals exhibit at least one diVerent cycle frequency.

D. Outline Section II introduces the class of the GACS time series (Izzo and Napolitano,1998b, 2002b). Time series belonging to this class are characterized by multivariate statistical functions that are almost-periodic functions of time whose Fourier series expansions can exhibit coeYcients and frequencies depending on the lag shifts of the time series. Moreover, the union over all the lag shifts of the lag-dependent frequency sets is not necessarily countable. ACS time series turn out to be the subclass of GACS time series for which the frequencies do not depend on the lag shifts and the union of the previously-mentioned sets is countable. The higher-order characterization of GACS time series in the strict and wide sense is provided in the nonstochastic (or FOT probability) framework. Generalized cyclic moment and cumulant functions (in both time and frequency domains) are introduced and relationships among them are stated. Section III addresses the problem of the linear time-variant (LTV) filtering of GACS signals in the FOT probability framework (Izzo and Napolitano, 2002b,c). The adopted approach is particularly useful as an alternative to the classical stochastic one, when stochastic systems transform ergodic input signals into nonergodic output signals, as it happens with several channel models encountered in the practice. Systems are classified as deterministic or random in the FOT probability framework. Moreover, the new concept of expectation in the FOT probability sense of the impulse-response function of a system is introduced. For the LTV systems, the higher-order system characterization in the time domain is provided in terms of the system temporal moment function, which is the kernel of the operator that transforms the finite-strength additive sine wave components contained in the input lag product into the finite-strength additive sine wave components contained in the output lag product. The higher-order characterization in the frequency domain is also provided and input/output relationships are derived in terms of temporal and spectral moment and cumulant functions. The countability of the set of the output cycle frequencies is studied with

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reference to LTV systems for both ACS and GACS not containing any ACS component input signals. Thus, the linear almost-periodically time-variant filtering and the product modulation are considered in detail. Moreover, several Doppler channel models are analyzed. In all these examples, it is shown that the FOT probability approach allows characterization of the system and its output in terms of statistical functions that can be measured by a single time series. Furthermore, the usefulness of considering the linear filtering problem within the class of the GACS signals is clarified and several pitfalls arising from continuing to adopt for the observed time series the ACS model when the increasing of the data-record length makes the GACS model more appropriate are pointed out. Section IV addresses, the problem of sampling a continuous-time GACS signal (Izzo and Napolitano, 2003). It is shown that the discrete-time signal constituted by the samples of a GACS signal is a discrete-time ACS signal. Thus, discrete-time ACS signals can arise not only from the sampling of continuous-time ACS signals, but also from the sampling of a wider class of nonstationary signals, that is, the continuous-time GACS signals. Relationships between generalized cyclic statistics of a continuous-time GACS signal and cyclic statistics of the discrete-time ACS signal constituted by its samples are derived. The problem of aliasing in the domain of the cycle frequencies is considered and a condition ensuring that the cyclic temporal moment function of the discrete-time signal can be obtained by sampling that of the continuous-time signal is determined. Finally, it is shown that, starting from the sampled signal, the GACS or ACS nature of the continuous-time signal can be conjectured, provided that the analysis parameters such as the sampling period, padding factor, and data-record length are properly chosen. Section V expresses time-frequency representations for GACS signals in terms of generalized cyclic statistics (Izzo and Napolitano, 1997b). The Wigner–Ville distribution and the ambiguity function are examined in detail and the special case of ACS signals is considered. Moreover, the problem of signal feature extraction based on a single-record estimation is addressed.

II. HIGHER-ORDER CHARACTERIZATION A. Introduction This section deals with time series belonging to a class wider than that of ACS time-series. Specifically, it deals with continuous-time time series whose multivariate statistical functions are almost-periodic functions of time

GENERALIZED ALMOST-CYCLOSTATIONARY SIGNALS

115

exhibiting Fourier series expansions with coeYcients and frequencies that can depend on the lag shifts of the time series. Moreover, the union over all the lag shifts of the lag-dependent frequency sets is not necessarily countable. Time series belonging to the class under consideration are called therein the generalized almost-cyclostationary time series. The class of GACS time series includes, as a subclass, the ACS time series that are obtained when the cycle frequencies do not depend on the lag shifts. Examples of GACS time series not belonging to the subclass of ACS time series arise from some linear time-variant transformations of ACS time series, such as channels introducing a time-variant delay (Izzo and Napolitano, 1995b, 1996b, 1998b, 2002b,c). Chirp signals and several angle-modulated and time-warped communication signals are further examples. The signal analysis framework utilized is that of the FOT probability, where statistical parameters are defined through infinite-time averages of a single time series rather than ensemble averages of a stochastic process (see Section I.B). In Section II.B, GACS time series are introduced and characterized in the strict sense. Moreover, some results for ACS time series derived in Gardner and Brown (1991) are extended to the GACS case. Section II.C.1. shows that for GACS time series the cyclic temporal moment functions (Gardner and Spooner, 1994) are not necessarily continuous functions of the lag vector even if the temporal moment function is continuous. In particular, it is shown that those GACS time series not belonging to the class of ACS time series exhibit a time-averaged autocorrelation function discontinuous in the origin. Moreover, it is pointed out that the spectral characterization in terms of cyclic spectral moment functions (Gardner and Spooner, 1994) can result inadequate. Therefore, generalized cyclic moments in both time and frequency domains are introduced to characterize GACS time series in the wide sense. Furthermore, it is shown that, under mild conditions, the Nth-order temporal moment function can be expressed as a sum of complex sinusoids whose amplitudes and frequencies are called Nth-order generalized cyclic temporal moment functions and (moment) lag-dependent cycle frequencies, respectively. Section II.D introduces a similar representation for the Nthorder temporal cumulant function, which can be expressed in terms of generalized cyclic temporal cumulant functions and (cumulant) lag-dependent cycle frequencies. Then, starting from such a representation, the characterization in the frequency domain in terms of generalized cyclic spectral cumulant functions is presented. Furthermore, relationships among the introduced functions are derived. Section II.E briefly addresses the problem of estimation of the introduced generalized cyclic statistics. Finally, Section II.F considers two examples of GACS time series. It is worth noting that the adopted definition of GACS time series is in agreement with that of almost-periodically correlated signal given in Dehay

116

IZZO AND NAPOLITANO

and Hurd (1994) and Hurd (1991), with reference to second-order statistics. Note that, however, in Dehay and Hurd (1994), and Hurd (1991), the entire theory is practically limited to GACS signals with second-order (reduced dimension) cyclic temporal moment functions that are continuous functions of the lag parameter, that is, second-order ACS signals. B. Strict Sense Characterization This section is aimed at characterizing in the strict sense time series in the FOT probability framework. Such a characterization generalizes that proposed by Gardner and Brown (1991) to the case in which the set of frequencies of the joint probability density function depends on the lag shifts of the time series. Let xðtÞ ≜ xr ðtÞ þ jxi ðtÞ; t 2 R, be a continuous-time complex-valued finite-power time series. If the set Gt,j of all frequencies of the finite-strength additive sine wave components contained in the function of t U x ð1t þ t; jÞ ≜

N Y

Uðxrn xr ðt þ tn ÞÞ Uðxin xi ðt þ tn ÞÞ

ð2:1Þ

n¼1

is countable for each of the column vectors t ≜ ½t1 ; . . . ; tN T 2 RN and j ≜ j r þ jj i ≜ ½xr1 þ jxi1 ; . . . ; xrN þ jxiN T 2 CN , and, moreover, also the set Gt ≜ [j2CN Gt;j is countable for each t 2 RN , then the time series is said to be Nth-order generalized almost cyclostationary in the strict sense (Izzo and Napolitano, 1998b). Following the proof given in Gardner and Brown (1991) with reference to almost-cyclostationary time series, it can be shown that the function fG g

t Fxðtþt ðjÞ ≜ EfGt g fU x ð1t þ t; jÞg 1 Þ


xðtþtN Þ X ¼ Fxg ðt; jÞej2pgt

ð2:2Þ

g2Gt

is a valid almost-periodically time-varying joint cumulative distribution function for each fixed value of t, except for the right continuity property with respect to each xrn and xin variable. In Eq. (2.2), F gx ðt; jÞ ≜ hU x ð1s þ t; jÞe j2pgs is

ð2:3Þ

and EfGt g f
g is the almost-periodic component extraction operator, that is, the operator that extracts all the finite-strength additive sine wave components (with frequencies ranging in the set Gt) present in its argument. The operator EfGt g f
g in the following will be denoted by Efag f:g if not indicating the set Gt does not generate ambiguity. It plays, in the FOT

GENERALIZED ALMOST-CYCLOSTATIONARY SIGNALS

117

probability framework, the same role played by the expectation operator in the stochastic process framework. Furthermore, for a stochastic process X(t) with almost-periodic joint probability density function and cycloergodicity properties, the expected value of any function of X(t), say g(X(t)), is coincident with the almost-periodic component Efag fgðxðtÞÞg for almost all sample-paths x(t) of X (t) (Gardner, 1988a, 1994; Gardner and Brown, 1991). Thus, from Eqs. (2.1) and (2.2) it follows that in the FOT probability framework the joint cumulative distribution function is introduced analogously as in the stochastic process framework by considering the expected value of the indicator of the event ft 2 R : xr ðt þ tn Þ  xrn; xi ðt þ tn Þ  xin ; n ¼ 1; . . . ; Ng. From Eq. (2.2) it follows that the function Eq. (2.1) can be decomposed into the sum of its almost-periodic component (the deterministic component) and a residual term ‘U ðt; t; jÞ not containing any finite-strength additive sine wave component: fG g

t U x ð1t þ t; jÞ ¼ Fxðtþt ðjÞ þ ‘U ðt; t; jÞ 1 Þ...xðtþtN Þ

ð2:4Þ

with 1 lim T!1 T

Z

T=2 T=2

‘U ðt; t; jÞe j2pgt dt 0

8g 2 R:

ð2:5Þ

In the special case where the set G ≜ [ Gt t2RN

ð2:6Þ

is countable, the time series x(t) is said to be Nth-order almost cyclostationary in the strict sense, the sum in Eq. (2.2) can be extended to the set G, and fGg

Fxðtþt1 Þ...xðtþtN Þ ðjÞ ≜ EfGg fU x ð1t þ t; jÞg

ð2:7Þ

is a valid cumulative distribution function except for the right continuity property with respect to each xrn and xin variable (Gardner and Brown, 1991). The 2Nth-order derivative (in the sense of generalized functions) of the cumulative distribution function fG g

@ 2N fGt g Fxðtþt ðjÞ 1 Þ...xðtþtN Þ @xr1 @x . . . @x @x rN iN (i1 ) N Y ð2:8Þ fGt g dðxrn xr ðt þ tn ÞÞdðxin xi ðt þ tn ÞÞ ¼E n¼1 P g j2pgt ¼ g2Gt f x ðt; jÞe

t fxðtþt ðjÞ ≜ 1 Þ...xðtþtN Þ

turns out to be a valid almost-periodically time-varying joint probability density function for each fixed value of t. In Eq. (2.8)

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IZZO AND NAPOLITANO

* f gx ðt; jÞ ≜

N Y

+ dðxrn xr ðv þ tn ÞÞ dðxin xi ðv þ tn ÞÞ e

j2pgv

n¼1

ð2:9Þ v

is the 2Nth-order derivative of the corresponding Fourier coeYcient [Eq. (2.3)] of the joint cumulative distribution function [Eq. (2.2)]. Let us note that if the time series x(t) is almost periodic, then the function Eq. (2.1), which is a memoryless nonlinear transformation of x(t), is in turn almost periodic and hence it is coincident with its almost-periodic component Eq. (2.2) [‘U ðt; t; jÞ 0 in Eq. (2.4)]. Therefore, the probability density function Eq. (2.8) can be expressed as N Y fGt g fxðtþt ðjÞ ¼ dðxrn xr ðt þ tn ÞÞ dðxin xi ðt þ tn ÞÞ ð2:10Þ 1 Þ...xðtþtN Þ n¼1

that is, almost-periodic time series are deterministic in the FOT probability framework. All the signals that are not almost-periodic functions (and, hence, also the GACS and ACS signals) are the random signals in the FOT probability framework. Signals not containing any almost-periodic component (that is, any finite-strength additive sine wave component) are the FOT zero-mean signals and are said FOT purely random signals. The following lag-shift invariance property for the set Gt holds: ð2:11Þ Gtþ1D Gt : In fact, for each D 2 R it results that fG

g

tþ1D Fxðtþt ðjÞ 1 þDÞ...xðtþtN þDÞ X
 ≜ U x ð1v þ t þ 1D; jÞ e j2pgv v e j2pgt

g2Gtþ1D

¼

X

g2Gtþ1D [Gt

¼


 U x ð1v þ t þ 1D; jÞ e j2pgv v e j2pgt

ð2:12Þ

X
 U x ð1ðv þ DÞ þ t; jÞe j2pgðvþDÞ vþD e j2pgðtþDÞ

g2Gt fG g

t ¼ FxðtþDþt ðjÞ: 1 Þ...xðtþDþtN Þ

fG g

t Moreover, from Eq. (2.12) it follows that Fxðtþt ðjÞ is a function 1 Þ...xðtþtN Þ of 1t þ t. Consequently, from Eq. (2.4) it follows that also ‘U (t, t; j) is a function of 1t þ t and, hence, in the following will be denoted by ‘U (1t þ t; j). Finally, the time series x(t) and x(t þ t) are said asymptotically (jtj ! 1) independent in the FOT probability sense if

fG

g

fG g

fG g

½t;0 0 FxðtþtÞxðtÞ ðjÞ ! FxðtþtÞ ðxr1 ; xi1 ÞFxðtÞ0 ðxr2 ; xi2 Þ

ð2:13Þ

GENERALIZED ALMOST-CYCLOSTATIONARY SIGNALS

119

as jtj ! 1 (see Brown, 1987; Gardner, 1988a, and Gardner and Brown, 1991, for the case of ACS signals). C. Generalized Cyclic Moments In this section, generalized cyclic moments in both time and frequency domains are introduced to characterize GACS time series in the wide sense (Izzo and Napolitano, 1996a, 1998b, 2002b). 1. Temporal Parameters In the FOT probability framework, a continuous-time possibly complexvalued time series x(t) is said to exhibit Nth-order wide-sense cyclostationarity with cycle frequency a 6¼ 0, for a given conjugation configuration, if the Nth-order cyclic temporal moment function * + N Y a ðÞn j2pat Rx ðtÞ ≜ x ðt þ tn Þe ð2:14Þ n¼1

t

is not zero for some t (Gardner and Spooner, 1994). In Eq. (2.14), the convergence of the infinite averaging with respect to t is assumed in the temporal mean-square sense (see Appendix A). The more general convergence in the sense of distributions (generalized functions) is discussed in PfaVelhuber (1975), with reference to stationary time series and the results can be extended with minor changes to time series exhibiting cyclostationarity. Note that, for N ¼ 2, (*)1 absent, (*)2 present, and t2 ¼ 0 the CTMF reduces to the cyclic autocorrelation function Eq. (1.22). If the set At ≜ fa 2 R : Rax ðtÞ 6¼ 0g

ð2:15Þ

is countable for each t, then the time series is said to be Nth-order generalized almost-cyclostationary in the wide sense (for the considered conjugation configuration) and the almost-periodic function ( ) N Y fAt g ðÞn Rx ðt; tÞ ≜ E x ðt þ tn Þ ð2:16Þ X n¼1 ¼ Rax ðtÞej2pat a2At

which is called the temporal moment function, is a valid moment function, that is, it can be expressed as Z N Y fGt g Rx ðt; tÞ ¼ ðxrn þ jxin ÞðÞn fxðtþt ðjÞdj r dj i : ð2:17Þ 1 Þx...ðtþtN Þ R2N n¼1

120

IZZO AND NAPOLITANO

In fact, accounting for Eq. (2.9), one has Z Y N ðxrn þ jxin ÞðÞn fxg ðt; jÞdj r dj i R2N

n¼1

¼

Z Y N n¼1

R2N

*

N



Y d xrn xr ðv þ tn Þ d xin xi ðv þ tn Þ e j2pgv

R

* ¼

ðxrn þ jxin ÞðÞn

n¼1

R

2N



N

Y ðxrn þ jxin ÞðÞn d xrn xr ðv þ tn Þ

+ dj r dj i v

n¼1

E :d xin xi ðv þ tn Þ dj r dj i e j2pgv v * + N

ðÞn Y ¼ e j2pgv xr ðv þ tn Þ þ jxi ðv þ tn Þ ¼

n¼1 g Rx ðtÞ

v

ð2:18Þ whose derivation assumes the order of integration and time averaging can be interchanged and exploits the sampling property of Dirac’s delta function. From Eqs. (2.8) and (2.18), Eq. (2.17) easily follows. Note that Eq. (2.17) generalizes to GACS time series the theorem of almost-periodic component extraction, which was stated in Gardner and Brown (1991) with reference to ACS time series. From the previous discussion it follows that the Nth-order lag product can be expressed as a sum of its almost-periodic component (i.e., its deterministic component in the FOT probability framework) and a residual term not containing any finite-strength additive sinewave component, that is, Lx ð1t þ tÞ ≜

N Y xðÞn ðt þ tn Þ n¼1

ð2:19Þ

¼ Rx ðt; tÞ þ ‘x ðt; tÞ with h‘x ðt; tÞe j2pat it 0;

8a 2 R:

ð2:20Þ

The almost-periodic component of a time series (or a lag product time series) can be extracted by exploiting the synchronized averaging identity (Gardner, 1988a, 1990) or the algorithms proposed by Sethares and Staley (1999).

121

GENERALIZED ALMOST-CYCLOSTATIONARY SIGNALS

Let us note that from Eq. (2.18) it follows immediately that, for each t, At  Gt :

ð2:21Þ

Therefore, the countability of Gt ensures that of At. In the special case where the set A ≜ [ At

ð2:22Þ

t2RN

is countable, the time series x(t) is said to be Nth-order almost cyclostationary in the wide sense. Note that this set is countable for N ¼ 2 if the function Rx ðt; tÞjt2 ¼0 is uniformly continuous in t and t1 (Dehay and Hurd, 1994). A useful characterization can be introduced for the GACS time series observing that, accounting for the countability of At for each t, the support in the (a, t) space of the CTMF is constituted by a set of N-dimensional manifolds (Choquet-Bruhat and DeWitt-Morette, 1982) defined by the implicit equations Fz0 ða; tÞ ¼ 0;

z0 2 W 0 ;

ð2:23Þ

where W 0 is a countable set. That is, it results that supp fRax ðtÞg ≜ clfða; tÞ 2 At  RN : Rax ðtÞ 6¼ 0g ¼ cl 0 [ fða; tÞ 2 R  RN : Fz0 ða; tÞ ¼ 0; Rax ðtÞ 6¼ 0g z 2W 0

¼ cl [ fða; tÞ 2 R  Dz : a ¼ az ðtÞg z2W

ð2:24Þ where cl denotes closure, W is a countable set and, in the last equality, each manifold described by an implicit equation Fz0 ða; tÞ ¼ 0 has been decomposed into a countable set of manifolds each described by the explicit equation a ¼ az(t), where each function az(t), called Nth-order (moment) lag-dependent cycle frequency, is defined in Dz  RN and is not necessarily a continuous function of t. In Eq. (2.24), Dz1 \ Dz2 ¼ ; for z1 6¼ z2. Therefore, set At can be written as At ¼ [ fa 2 R : a ¼ az ðtÞg z2W

ð2:25Þ

where the Nth-order (moment) lag-dependent cycle frequencies az(t) are such that for each ða; tÞ 2 At  RN there exists at most one z 2 W such that a ¼ az(t). Starting from Eq. (2.25), the temporal moment function Eq. (2.16) can be expressed in terms of a Fourier series expansion where the sum ranges over a set not depending on t [as in Eq. (2.16)], whereas the frequencies depend on t:

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IZZO AND NAPOLITANO

Rx ðt; tÞ ¼

X

Rx;z ðtÞe j2paz ðtÞt

ð2:26Þ

z2W

where the functions Rx,z(t), which are called the generalized CTMFs (GCTMFs), are given by 8* + N > < Y ðÞn x ðt þ tn Þe j2paz ðtÞt ; 8t 2 Dz ð2:27Þ Rx;z ðtÞ ≜ n¼1 > t : 0; elsewhere and are not necessarily continuous functions of t. In representation Eq. (2.25) we have that 8t it results that az1(t) 6¼ az2(t) for z1 6¼ z2. However, if more functions az1(t), . . . , azk(t) are defined in K (not necessarily coincident) neighborhoods, say I1, . . . , IK, of the same point t0, only one of them is defined in t 0 and, moreover, it results that lim

Dt!0 t0 þDt2Ik

azk ðt0 þ DtÞ ¼ a0 ;

k ¼ 1; . . . ; K

ð2:28Þ

then it is convenient to consider in Eq. (2.25) az1 ðt0 Þ ¼ . . . ¼ azK ðt 0 Þ ¼ a0 and to put Rx;zk ðt 0 Þ ≜

lim

Dt!0 t 0 þDt2Ik

Rx;zk ðt 0 þ DtÞ;

k ¼ 1; . . . ; K:

ð2:29Þ

The set of points t 0 such that Eq. (2.28) holds for some zk 2 W will be denoted by Dx and is assumed to be countable. With convention Eq. (2.29), it can be shown that CTMFs and GCTMFs are related by the following relationship: X Rax ðtÞ ¼ Rx;z ðtÞda az ðtÞ : ð2:30Þ z2W

Rax ðt1 Þej2pat

represents the sum of all the finite-strength Thus, the function sinewaves with frequency a contained in Eq. (2.26) when t ¼ t1. Moreover, 8t 2 Dz Dx Rx;z ðtÞ ¼ Rax ðtÞja¼az ðtÞ :

ð2:31Þ

For GACS time series that are not ACS, even if the set At and the temporal moment function Rx(t, t) are continuous functions of the lag vector t, the CTMFs are not necessarily continuous functions of t. Specifically, according to Eq. (2.30), they can result to be constituted by sums of Kronecker’s delta functions depending on t. It is well known that for every finite-power time series x(t) the conventional time-averaged autocorrelation function R0xx ðtÞ is continuous in t ¼ 0

123

GENERALIZED ALMOST-CYCLOSTATIONARY SIGNALS

if and only if the cross-correlation function R0xy ðtÞ is continuous for any t 2 R and for any finite-power time series y(t) (Lee, 1967; pp. 74–78). Therefore, if one defines the time series 2 3 6Y 7 6 N ðÞ 7 n ðt þ t Þ7 e j2pat ya ðtÞ ≜ 6 x n 7 6 4 n¼1 5

ð2:32Þ

n6¼k

then the Nth-order CTMF of x(t) can be written as Rax ðtÞ ¼ R0xðÞk ya ðtk Þ:

ð2:33Þ

Consequently, the time-averaged autocorrelation function R0xx ðtÞ is continuous in t ¼ 0 if and only if the CTMFs are continuous in t for all Nth-order cycle frequencies a and for all orders N. Therefore, all GACS time series that are not ACS exhibit time-averaged autocorrelation functions discontinuous in t ¼ 0. In particular, accounting for the property R0xx ðtÞ ¼ R0xx ð tÞ, it follows that R0xx ðtÞ contains the additive term x2 dt , where x2 is the time-averaged power of x(t). Such a property should not be confused with that examined in Hurd (1974), where the discontinuity of the time-varying autocorrelation function is considered. Note that the class of time series considered here is coincident with that in Izzo and Napolitano (2002b,c; 2003) and is wider than that analyzed in Izzo and Napolitano (1998b), where the continuity with respect to t for both Rx ðt; tÞ and At was assumed and, consequently, lag-dependent cycle frequencies az(t) and GCTMFs Rx;z ðtÞ continuous with respect to t were obtained. Accounting for Eqs. (2.11) and (2.21), the following lag-shift invariance property holds for any D 2 R: Atþ1D At :

ð2:34Þ

Consequently, for all real numbers D, from Eq. (2.25) it follows that az ðt þ 1DÞ ¼ az ðtÞ:

ð2:35Þ

Thus, it can be easily shown that Rx;z ðt þ 1DÞ ¼ Rx;z ðtÞe j2paz ðtÞD : Therefore, accounting for Eqs. (2.26) and (2.36), it results that X Rx;z ðt þ 1tÞ Rx ðt; tÞ ¼ z2W

ð2:36Þ

ð2:37Þ

124

IZZO AND NAPOLITANO

In other words, the temporal moment function Rx ðt; tÞ of a GACS time series is a function of 1t þ t and hence in the following, with a little abuse of notation, it will be denoted by Rx ð1t þ tÞ: Rx ðt; tÞ Rx ð1t þ tÞ:

ð2:38Þ

Consequently, from Eq. (2.19), we will also have that ‘x ðt; tÞ ‘x ð1t þ tÞ:

ð2:39Þ

Note that Eqs. (2.35) and (2.36) suggest that the dimensions of the lagdependent cycle frequencies and of the GCTMFs can be reduced without information loss. In fact, if one defines the Nth-order reduced-dimension (moment) lag-dependent cycle frequencies
a z ðt0 Þ ≜ az ðtÞjtN ¼0

ð2:40Þ

and the Nth-order reduced-dimension GCTMFs (RD-GCTMFs) Rx;z ðt0 Þ ≜ Rx;z ðtÞ ≜ jtN ¼0 ;

ð2:41Þ

it can be easily shown that the Nth-order lag-dependent cycle frequencies and the Nth-order GCTMFs can be expressed as az ðtÞ ¼ az ðt 1tN Þ ¼
a z ðu0 Þju0 ¼t0 1tN

ð2:42Þ

and 0

Rx;z ðtÞ ¼ Rx;z ðu0 Þej2p
a z ðu ÞtN ju0 ¼t0 1tN ;

ð2:43Þ

respectively. For N ¼ 2, (*)1 absent, and (*)2 present, the RD-GCTMF in Eq. (2.41) is called the generalized cyclic autocorrelation function [see Eq. (1.23)]. Moreover, Eq. (2.36) generalizes to GACS time series the result derived in Gardner and Spooner (1994) for the case of ACS time series (see Eq. 46 in Gardner and Spooner, 1994). By substituting Eqs. (2.42) and (2.43) into Eq. (2.26), the following expression for the temporal moment function is obtained X 0 Rx ð1t þ tÞ ¼ Rx;z ðt0 1tN Þej2p
a z ðt 1tN ÞðtþtN Þ : ð2:44Þ z2W

Let us note that for the special case of ACS time series the functions az(t) are independent of t and then there exists a one-to-one correspondence between the elements z belonging to W and the cycle frequencies a belonging to the countable set A. Moreover, for each a and z such that az(t) ¼ a, it results that Rx;z ðtÞ ¼ Rax ðtÞ;

ð2:45Þ

GENERALIZED ALMOST-CYCLOSTATIONARY SIGNALS

125

that is, for ACS time series, the generalized cyclic temporal moment functions are coincident with the cyclic temporal moment functions. Furthermore, for ACS signals, Eq. (2.44) specializes to X Rx ð1t þ tÞ ¼ Rax ðt0 1tN Þej2pa:ðtþtN Þ : ð2:46Þ a2A

Finally, we say that the GACS signal x(t) contains an ACS component if there exists at least one lag-dependent cycle frequency az(t) that assumes a constant value within a set with nonzero Lebesgue measure in RN . 2. Spectral Parameters The almost-cyclostationary time series can be characterized in the frequency domain by the Nth-order reduced-dimension CSMF Sxa ð f 0 Þ [see Eq. (1.25)], which is the (N 1)-fold Fourier transform of the reduced-dimension CTMF. Such a spectral characterization, however, is not appropriate for those GACS time series that do not belong to the class of ACS time series, that is, when the set A is not countable. In fact, in such a case, the expression of the reduced-dimension CTMFs can contain Kronecker delta functions depending on t0 [see Eq. (2.30) with tN ¼ 0]. Consequently, the reduceddimension CSMFs can be infinitesimal. In this section, a useful characterization in the spectral domain is provided for those GACS time series that are not necessarily ACS. In the following, all the Fourier transforms are assumed to exist at least in the sense of distributions (generalized functions) (Zemanian, 1987). The N-fold Fourier transform of the Nth-order GCTMF of a time series x(t) Z ⊤ S x;z ð f Þ ≜ Rx;z ðtÞe j2pf t dt ð2:47Þ RN

is called the Nth-order generalized CSMF (GCSMF). Moreover, accounting for Eq. (2.43), it can be expressed as Z

0⊤ 0 S x;z ð f Þ ¼ Rx;z ðt0 Þd
a z ðt0 Þ f ⊤ 1 e j2pf t dt0 : ð2:48Þ RN 1

Let us now consider the Nth-order spectral moment function defined by ( ) N

Y ðÞn fL0 g XT t; ð Þn fn S x ð f Þ ≜ lim E ð2:49Þ T!1

n¼1

where ( )n denotes an optional minus sign to be considered only when the optional conjugation (*)n is present,

126

IZZO AND NAPOLITANO

Z XT ðt; f Þ ≜

tþT=2

xðuÞe j2pfu du

ð2:50Þ

t T=2

and L0 is the set of possible Nth-order cycle frequencies of the time series XT(t, f) when T ! 1. In Appendix B, it is shown that L 0 contains only the element a ¼ 0 and, moreover, the spectral moment function can be expressed in terms of GCSMFs by the relationship X Sxð f Þ ¼ S x;z ð f Þ: ð2:51Þ z2W

Furthermore, accounting for Eqs. (2.47) and (2.26) with t ¼ 0, from Eq. (2.51) it follows that Z ⊤ Sxð f Þ ¼ Rx ðtÞe j2pf t dt: ð2:52Þ RN

Note that, as follows immediately from Eq. (2.45), for the ACS time series the GCSMFs are coincident with the CSMFs. Moreover, in the following, it is shown that the GCSMFs of GACS time series not containing any ACS component are not impulsive, unlike those of the ACS time series. For an ACS time series x(t), there is a one-to-one correspondence between the elements z 2 Wx and the cycle frequencies a 2 Ax and, moreover, the GCSMFs are coincident with the cyclic spectral moment functions, which, can be expressed as in Gardner and Spooner (1994) [see Eq. (1.25)]. S x;z ðlÞ S ax ðlÞ ¼ Sxa ðl0 Þdðl⊤ 1 aÞ 2 3 p X X Y bm 4 5 Pxmii ðl0mi Þdðbmi l⊤ ¼ mi 1Þ P

ð2:53Þ

b⊤ 1¼a i¼1

where P is the set of distinct partitions of {1, . . . , N}, each constituted by the subsets {mi : i ¼ 1, . . . , p}, jmij is the number of elements in mi, and xmi is the jmij-dimensional vector whose components are those of x having indices in mi. bm In Eq. (2.53), Pxmii ðl0mi Þ is the jmijth-order cyclic polyspectrum at cycle frequency bmi of {x(*)n(t), n 2 mi} and b ≜ ½bm1 ; . . . ; bmp ⊤ . Thus, the GCSMFs of ACS time series are impulsive. By considering the N-fold Fourier transform of both sides of Eq. (2.36) with D ¼ t, one has S x;z ð f Þe j2pf where

⊤

1t

¼ S x;z ð f Þ ! Az ðt; f Þ f

ð2:54Þ

GENERALIZED ALMOST-CYCLOSTATIONARY SIGNALS

Az ðt; f Þ ≜

R RN

e j2paz ðtÞt e j2pf

⊤

t

dt

127 ð2:55Þ


z ðt; f 0 Þdð f ⊤ 1Þ: ¼A

¯ z(t, f 0 ) is In Eq. (2.55), !f denotes N-fold convolution with respect to f, A 0 Þt j2p
a ðt z the (N 1)-fold Fourier transform of e with respect to t 0 and the last equality has been obtained accounting for Eq. (2.42). Finally, by substituting Eq. (2.55) into Eq. (2.54), one obtains that S x;z ð f Þe j2pf

⊤

1t


z ðt; f 0 Þdð f ⊤ 1Þ ¼ S x;z ð f Þ !½A f

ð2:56Þ

from which it follows that the GCSMF S x;z ð f Þ can be impulsive if and only if both the functions convolved in the right-hand-side are impulsive, that is,
z ðt; f 0 Þ is impulsive in f 0 , which occurs only if if and only if the function A 0
a z ðt Þ is constant in a set of values of t0 with nonzero Lebesgue measure in RN 1 , i.e., if x(t) contains an ACS component (see Section II.C.1.). D. Generalized Cyclic Cumulants In this section, generalized cyclic cumulants in both time and frequency domains are introduced to characterize GACS time series in the wide sense (Izzo and Napolitano, 1996a, 1998b, 2002b). 1. Temporal Parameters The definition of the Nth-order temporal cumulant function of a continuoustime complex-valued time series x(t) given in [Spooner and Gardner (1994)] for ACS time series, can be extended to GACS time series:

Cx ð1t þ tÞ ≜ cum xðÞn ðt þ tn Þ; n ¼ 1; . . . ; N @N ≜ ð jÞN @o( 1 . . . @o ð2:57Þ "N # ) N X  f Gt g ðÞn loge E exp j on x ðt þ tn Þ  : v¼0

n¼1

It can be expressed as Cx ð1t þ tÞ ¼

X P

" ð 1Þ

p 1

ð p 1Þ!

p Y

# Rxmi ð1t þ tmi Þ

ð2:58Þ

i¼1

where v ≜ ½o1 ; . . . ; oN ⊤ , P is the set of distinct partitions of {1, . . . , N}, each constituted by the subsets {mi : i ¼ 1, . . . , p}, jmij is the number of elements in mi, xmi is the jmij-dimensional vector whose components are those of x having indices in mi, and, according to definition Eq. (2.16),

128

IZZO AND NAPOLITANO

( Rxmi ð1t þ tmi Þ ¼ E

Y

fGtmi g

) ðÞn

x

ðt þ tn Þ :

ð2:59Þ

n2mi

Moreover, according to that shown in Gardner and Spooner (1994) for ACS time series, " # p X Y Rx ð1t þ tÞ ¼ Cxmi ð1t þ tmi Þ ð2:60Þ P

i¼1

where

Cxmi ð1t þ tmi Þ ≜ cum xðÞn ðt þ tn Þ; n 2 mi :

ð2:61Þ

The function Cx ð1t þ tÞ is a valid cumulant function, that is, it satisfies all the properties of the ordinary cumulant function in the stochastic process framework. For example, if the time series x(t) and x(t þ t) are asymptotically (jtj ! 1) independent in the FOT probability sense (see Section II.B.), then, for tn, n 6¼ k fixed, it results limjtk j!1 Cx ð1t þ tÞ ¼ 0. Since, for GACS time series, Gt is a countable set, from Eq. (2.57) it follows that the temporal cumulant function is almost periodic in t and can be expressed as X Cx ð1t þ tÞ ¼ Cbx ðtÞej2pbt ð2:62Þ b2Bt

where Cbx ðtÞ ≜ hCx ð1t þ tÞe j2pbt it is the Nth-order cyclic temporal cumulant function and

Bt ≜ b 2 R : Cbx ðtÞ 6¼ 0

ð2:63Þ

ð2:64Þ

is a countable set 8t 2 R . The support in the (b,t) space of the CTCF is constituted by a set of N-dimensional manifolds that can be described by the explicit equations N

b ¼ bx ðtÞ; x 2 WC

ð2:65Þ

where WC is a countable set and the functions bx(t), which are called the Nth-order (cumulant) lag-dependent cycle frequencies, are not necessarily continuous functions of t. Thus, for each t, Bt ¼ [ fb 2 R : b ¼ bx ðtÞg x2WC

and the almost-periodic function of time Cx ð1t þ tÞ can be written as

ð2:66Þ

GENERALIZED ALMOST-CYCLOSTATIONARY SIGNALS

Cx ð1t þ tÞ ¼

X

Cx;x ðtÞej2pbx ðtÞt

129 ð2:67Þ

x2WC

where the functions Cx;x ðtÞ, which are called the Nth-order generalized cyclic temporal cumulant functions (GCTCFs), are given by 
 Cx ð1t þ tÞe j2pbx ðtÞt t ; 8t 2 Dx Cx;x ðtÞ ≜ ð2:68Þ 0; elsewhere where Dx is the domain of bx(t). In representation (2.66) we have that 8t it results that bx1(t) 6¼ bx2(t) for x1 6¼ x2. However, if more functions bx1(t), . . . , bxK(t) are defined in K (not necessarily coincident) neighborhoods, say I1, . . . , IK, of the same point t 0, only one of them is defined in t 0 and, moreover, it results that lim

Dt!0 t0 þDt2Ik

bxk ðt0 þ DtÞ ¼ b0 ; k ¼ 1; . . . ; K;

ð2:69Þ

then it is convenient to consider in Eq. (2.66) bx1(t0) ¼ . . . ¼ bxK (t0) ¼ b0 and to put Cx;xk ðt0 Þ ≜

lim

Dt!0 t0 þDt2Ik

Cx;xk ðt 0 þ DtÞ; k ¼ 1; . . . ; K:

ð2:70Þ

The set of points t0 such that Eq. (2.69) holds for some xk 2 WC is denoted by D0x and assumed to be countable. With convention Eq. (2.70), it can be shown that CTCFs and GCTCFs are related by the following relationship: X Cbx ðtÞ ¼ Cx;x ðtÞdb bx ðtÞ : ð2:71Þ x2Wc

Moreover, 8t 2 Dx D0x , Cx;x ðtÞ ¼ Cbx ðtÞjb¼bx ðtÞ :

ð2:72Þ

For any real number D, accounting for Eq. (2.11), from Eq. (2.57) and Eq. (2.62) it follows that Bt þ 1D Bt

ð2:73Þ

bx ðt þ 1DÞ ¼ bx ðtÞ

ð2:74Þ

and hence

from which it results that

130

IZZO AND NAPOLITANO

Cx;x ðt þ 1DÞ ¼ Cx;x ðtÞej2pbx ðtÞD :

ð2:75Þ

Moreover, if one introduces the Nth-order reduced-dimension GCTCF (RD-GCTCF) Cx;x ðt 0 Þ ≜ Cx;x ðtÞjtN ¼0

ð2:76Þ

one has


0

Cx;x ðtÞ ¼ Cx;x ðu0 Þej2pb x ðu ÞtN ju0 ¼t0 1tN

ð2:77Þ


ðt0 Þ ≜ b ðtÞj b x x tN ¼0

ð2:78Þ

where are the reduced-dimension cumulant lag-dependent cycle frequencies. Moreover, it can be easily shown that
ðu0 Þj 0 0 bx ðtÞ ¼ bx ðt 1tN Þ ¼ b x u ¼t 1tN and, by substituting Eq. (2.77) and Eq. (2.79) into Eq. (2.67) X
0 Cx ð1t þ tÞ ¼ Cx;x ðt0 1tN Þej2pb x ðt 1tN ÞðtþtN Þ :

ð2:79Þ

ð2:80Þ

x2Wc

The Nth-order GCTCF can be expressed in terms of GCTMFs of order less than or equal to N by substituting Eq. (2.58) into Eq. (2.68) and accounting for Eq. (2.30): " # p X XY p 1 Cx;x ðtÞ ¼ ð 1Þ ðp 1Þ! Rxmi ;zi ðtmi Þdbx ðtÞ az ðtÞ⊤ 1 ð2:81Þ z2W i¼1

P

where zi 2 Wmi ði ¼ 1; . . . ; pÞ; z ≜ ½z1 ; . . . ; zp ⊤ ; W ≜ ½Wm1 ; . . . ; Wmp ⊤ ; az ðtÞ ≜ ½az1 ðtm1 Þ; . . . ; azp ðt mp Þ⊤ , and, according to Eq. (2.27), 8* + > < Y ðÞn j2pazi ðtmi Þt x ðt þ tn Þe ; t mi 2 D z i ð2:82Þ Rxmi ;zi ðtmi Þ ¼ > t : n2mi 0; elsewhere where Dzi is set in which azi(tmi) is defined. Furthermore, the Nth-order GCTMF can be expressed in terms of GCTCFs of order less than or equal to N, by substituting Eq. (2.60) into Eq. (2.27), in which the lag product has been replaced by Rx ð1t þ tÞ: " # p X XY Cxmi ;xi ðt mi Þdaz ðtÞ bj ðtÞ⊤ 1 Rx;z ðtÞ ¼ ð2:83Þ P

j2W c i¼1

where WC ≜ [WCm1 , . . . ,WCmp ]⊤ and Eq. (2.71) has been accounted for.

GENERALIZED ALMOST-CYCLOSTATIONARY SIGNALS

131

In Gardner and Spooner (1994), it is shown that for ACS time series such that x(t) and x(t þ t) are asymptotically (jtj ! 1) independent, the reduceddimension cyclic temporal cumulant function Cxb ðt 0 Þ is summable whereas, in general, the reduced-dimension cyclic temporal moment function Rax ðt0 Þ is not. In the following, a similar result will be found for GACS time series. From Eq. (2.83) with tN ¼ 0, accounting for Eq. (2.77) and assuming that each partition is ordered so that mp always contains N as its last element, it results that " Y X X p 1 0 0 Rx;z ðt Þ ¼ Cx;z ðt Þ þ P p6¼1

j2W C i¼1


ðt 0 1tr Þtr Þ Cxmi ;xi ðt 0mi 1tri Þ exp ð j2pb xi mi i  i :Cxm ;xp ðt0m Þd 0 Pp 1 0
p

p


a z ðt Þ

i¼1

ð2:84Þ

bxi ðtmi Þ b xp ðtmp Þ

where, for each partition, tri is the last element in mi. Therefore, the RDGCTMF is the sum of two contributions: the RD-GCTCF, which, under mild conditions, converges to zero as kt0 k ! 1 and the remaining term, which is not convergent as kt0 k ! 1. In fact, under the assumption that the time series x(t) and x(t þ tn) (n ¼ 1,. . ., N 1) are asymptotically (jtnj ! 1) independent in the FOT probability sense, accounting for the well-known independence property of cumulants, it results that Cx,z(t0 ) ! 0 as kt 0 k ! 1. Moreover, for each fixed partition, when kt 0 k ! 1 with the constraints tri ! 1, tm0 i ¼ 1tri (i ¼ 1, . . . p 1) and tm0 p finite, one has lim

p 1 Y

kt0 k!1 i¼1 t0mi ¼1tri ; t 0mp finite

Cxmi xi ; ðt0mi 1tri Þ




ðt0 1tr Þtr Cx x ðt0 Þ :exp j2pb x i mi i i mp; i mp ¼

ð2:85Þ

p 1

Y
ð0Þtr ; Cxmi ;ji ð0ÞCmp ;jp ðt0mp Þ lim exp j2pb xi i tri !1

i¼1

which is not convergent. Consequently, for each t it results that Rx;z ðt0 Þ 0

) C ðt0 Þ 0 (/ x;z

ð2:86Þ

that is, fbx ðtÞgx2Wc  faz ðtÞgz2W

ð2:87Þ

132

IZZO AND NAPOLITANO

and hence Bt  At :

ð2:88Þ

In other words, the set of the moment lag-dependent cycle frequencies includes, as a subset, the one of the cumulant lag-dependent cycle frequencies. Finally, let us note that if two time series x1(t) and x2(t) are statistically independent in the FOT probability sense (see Section II.B.), then, reasoning as in the stochastic process framework, for the time series y(t) ¼ x1(t) þ x2(t) it can be shown that Cy ð1t þ tÞ ¼ Cx1 ð1t þ tÞ þ Cx2 ð1t þ tÞ

ð2:89Þ

Cy;x ðtÞ ¼ Cx1 ;x ðtÞ þ Cx2 ;x ðtÞ:

ð2:90Þ

and hence

2. Spectral Parameters In this section, all the Fourier transforms are assumed to exist at least in the sense of distributions (generalized functions) (Zemanian, 1987). The N-fold Fourier transform of the Nth-order GCTCF of a time series x(t) Z ⊤ P x;x ð f Þ ≜ Cx;x ðtÞe j2pf t dt ð2:91Þ RN

is called the Nth-order generalized cyclic spectral cumulant function (GCSCF) which, accounting for Eq. (2.77), can be expressed as Z

0
ðt0 Þ f ⊤ 1 e j2pf ⊤ dt0 : P x;x ð f Þ ¼ Cx;x ðt 0 Þd b ð2:92Þ x RN 1

Moreover, the (N 1)-fold Fourier transform of Cx,x(t0 ) is called the Nthorder generalized cyclic polyspectrum, which, for ACS time series, is coincident with the cyclic polyspectrum. Note that, under the assumption of asymptotic independence for the time series x(t), if there exists an e > 0 such that jCx,x(t 0 )j ¼ o(kt0 k Nþ1 e) as kt 0 k ! 1, then the RD-GCTCF is absolutely integrable and, therefore, Fourier transformable in the ordinary sense. Let us now consider the Nth-order spectral cumulant function n o ðÞ P x ð f Þ ≜ lim cum XT n ðt; ð Þn fn Þ; n ¼ 1; . . . ; N : ð2:93Þ T!1

133

GENERALIZED ALMOST-CYCLOSTATIONARY SIGNALS

It can be written as

n o ðÞ P x ðf Þ ¼ lim cum0 XT n ðt; ð Þn fn Þ; n ¼ 1; . . . ; N T!1 " * +# p X Y ðÞ Y p 1 n ¼ lim ð 1Þ ðp 1Þ! XT ðt; ð Þn fn Þ T!1

¼

X P

"

P

ð 1Þp 1 ðp 1Þ!

i¼1

n2mi

p Y S xmi ð f mi Þ

#

t

ð2:94Þ

i¼1

where in the first equality cum0 denotes the cumulant evaluated by substituting E{Gt}{
} with h
it in definition Eq. (2.57). By considering the N-dimensional Fourier transform of the right-hand side of Eq. (2.58) with t ¼ 0 and comparing the result with Eq. (2.94), one obtains that Z ⊤ P x ðf Þ ¼ Cx ðtÞe j2pf t dt: ð2:95Þ RN

Moreover, taking into account the N-dimensional Fourier transform of the right-hand side of Eq. (2.67) with t ¼ 0, the Nth-order spectral cumulant function can be expressed in terms of Nth-order GCSCFs by the relationship X Pxð f Þ ¼ P x;x ð f Þ: ð2:96Þ x2Wc

Finally, let us note that for the ACS time series the GCSCFs are coincident with the cyclic spectral cumulant functions.

E. Estimation of the Generalized Cyclic Statistics Appropriate estimators for the cyclic statistics of ACS time series have been proposed in Gardner (1988a), Spooner and Gardner (1994) within the FOT probability framework and in Dandawate´ and Giannakis (1994, 1995), Dehay and Hurd (1994), Hurd (1991), and Hurd and Les´kow (1992) in the classical stochastic process framework. In this section, the problem of estimating the higher-order (generalized) cyclic statistics of those GACS time series not belonging to the class of ACS time series is briefly considered in the FOT probability framework (see also Appendix A). Results on the estimation problem in the stochastic process framework are presented in Napolitano (2004). Assuming that t 2 Dz and t 2 = Dz0 for any z0 6¼ z (see Eq. (2.27)) and the lag-dependent cycle frequency az (t) is known, the function

134

IZZO AND NAPOLITANO

RxT ;z ðt0 ; tÞ ≜

1 T

Z Y N

rect

R n¼1

u t 0 xðÞn ðu þ tn Þe j2paz ðtÞu du T

ð2:97Þ

where rect(t) ¼ 1 for jtj  1/2 and rect(t) ¼ 0 otherwise and T is the observation time, turns out to be an estimate of the GCTMF, as follows immediately observing that lim RxT ;z ðt0 ; tÞ ¼ Rx;z ðtÞ

T!1

ð2:98Þ

where the convergence is in the temporal mean-square sense. In Appendix A, this kind of convergence is discussed and the definition of both bias and variance of estimators in the nonstochastic approach is given. For t 2 Dx and t 2 = Dx0 for any x0 6¼ x [see Eq. (2.68)], the function " # p X XY p 1 CxT ;j ðt0 ; tÞ ≜ ð 1Þ ðp 1Þ! RxTmi ;zi ðt0 ; t mi Þdbx ðtÞ az ðtÞ⊤ 1 z2W i¼1

P

ð2:99Þ converges to the theoretical GCTCF as T ! 1: lim CxT ;x ðt0 ; tÞ ¼ Cx;x ðtÞ:

T!1

ð2:100Þ

When the set of the lag-dependent cycle frequencies {az(t)}z2W is unknown, accounting for Eq. (2.24), it can be estimated starting from the support supp fRaxT ðt0 ; tÞg ≜ clfða; tÞ 2 R  RN : RaxT ðt0 ; tÞ 6¼ 0g

ð2:101Þ

where RaxT ðt0 ; tÞ ≜

1 T

Z Y N R n¼1

rect

u t 0 xðÞn ðu þ tn Þe j2pau du T

ð2:102Þ

is the estimate of the Nth-order CTMF given in Spooner and Gardner (1994). In fact, in the limit for T ! 1, the support Eq. (2.101) has zero measure in RNþ1 and is constituted by a countable set of manifolds described by the implicit equations a ¼ az (t), z 2 W. Moreover, estimates az,T(t) of the functions az(t) are, for each fixed value of t, the estimated frequencies of the almost-periodic component contained in the lag product, which can be obtained, for example, by exploting the algorithm proposed in Dehay and Hurd (1996). Note that estimated lag-dependent cycle frequencies can be

GENERALIZED ALMOST-CYCLOSTATIONARY SIGNALS

135

substituted to the true values in the estimator Eq. (2.97) of the GCTMF, provided that the maximum absolute error in the estimate is much smaller than the cycle frequency resolution 1/T, that is, max jaz;T ðtÞ az ðtÞj % t2R

N

1 : T

ð2:103Þ

In Napolitano (2004) it is shown that, in the stochastic process framework, for N ¼ 2 and t2 ¼ 0 the function Eq. (2.102) turns out to be a meansquare consistent estimator of the (conjugate) cyclic autocorrelation function as a function of (a, t1), provided that some mixing conditions expressed in terms of summability of the second and fourth-order cumulants of x(t) are satisfied. Such conditions generally hold for stochastic processes with finite or approximately finite memory. As regards the parameters in the frequency domain, the function " # Z N ⊤ 1 t0 þZ=2 Y ðÞn S xT ;z ðt0 ; f Þz ≜ XT ðt; ð Þn fn Þ ! Az ðt; f Þ ej2pf 1t dt ð2:104Þ Z t0 Z=2 n¼1 f where Az(t, f ) is defined in Eq. (2.55), is an estimator for the GCSMF, since it can be shown that lim lim S xT ;z ðt0 ; f Þz ¼ S x;z ð f Þ:

T!1 Z!1

ð2:105Þ

F. Examples of GACS Signals In this section, two examples of GACS signals are presented (Izzo and Napolitano, 1998b). The first one is the chirp signal, which is widely considered in various application fields (e.g., physics, sonar, radar, communications). The second one, is a nonuniformly sampled signal. Further examples are given in Section III. 1. Chirp Signal Let us consider the chirp signal xðtÞ ≜ expð jpct2 Þ

ð2:106Þ

where the nonzero real parameter c is the chirp rate. The cyclic temporal moment function at the cycle frequency a can be obtained by substituting Eq. (2.106) into Eq. (2.14):

136

IZZO AND NAPOLITANO



Rax ðtÞ ¼ exp jpc1ð Þ⊤ tð2Þ da c1ð Þ⊤ t d1ð Þ⊤ 1

ð2:107Þ

where 1ð Þ ≜ ½ð Þ1 1; . . . ,ð ÞN 1⊤ and tð2Þ ≜ ½t21 ; . . . ; t2N ⊤ : Equation (2.107) shows that the CTMF is nonzero only when the number of conjugated and unconjugated entries of the vector x is the same and, moreover, only on the hyperplane in the (a, t) space defined by a ¼ c1ð Þ⊤ t:

ð2:108Þ

Therefore, since the set of t’s such that Eq. (2.108) holds has zero Lebesgue measure in RN, the CSMFs are infinitesimal (see Section II.C.2). A spectral characterization for the chirp signal can be obtained by utilizing the introduced generalized cyclic spectral moment functions. In fact, by comparing Eq. (2.107) with Eq. (2.30), it results that the set W contains just one element and, moreover, az ðtÞ ¼ c1ð Þ⊤ t

ð2:109Þ



Rx;z ðtÞ ¼ exp jpc1ð Þ⊤ tð2Þ d1ð Þ⊤ 1

ð2:110Þ

and

which is nonzero only when N is even and the number of conjugated and unconjugated entries of the vector x is the same. Therefore, the GCSMF is given by p

1 S x;z ð f Þ ¼ N=2 exp j 1ð Þ⊤ f ð2Þ d1ð Þ⊤ 1 ð2:111Þ c jcj where f ð2Þ ≜ ½ f12 ; . . .; fN2 ⊤ : Figure 3a shows the real part and Fig. 3b the support in the (a, t) plane of the second-order RD-CTMF Rax ðtÞ with (*)1 absent and (*)2 present (i.e., the cyclic autocorrelation function), for a chirp signal with c ¼ 0:002=Ts2 , where Ts is the sampling period, as estimated by 512 samples (for a discussion on the aliasing issue, see Section IV.C). The linear behavior of the reduced-dimension lag-dependent cycle frequency
a z ðtÞ ¼ ct is quite evident. The function obtained by setting a ¼ ct is just the real part of the second-order RD-GCTMF Rx,z(t), that is, the generalized cyclic autocorrelation function. Note that in Section IV.D it is explained that the GACS nature of a continuous-time signal can only be conjectured starting from the discrete-time signal constituted by its samples. As regards the GCTCFs, accounting for Eq. (2.110) and Eq. (2.81), it results that

GENERALIZED ALMOST-CYCLOSTATIONARY SIGNALS

137

FIGURE 3. (a) Graph of real part and (b) support in the (a, t) plane of the cyclic autocorrelation function Raxx ðtÞ for a chirp signal with chirp rate c ¼ 0:002=Ts2 .

Cbx ðtÞ ¼

X P

" ð
1Þp
1 ðp
1Þ!

p XY z2W i¼1


 ð
Þ⊤ ð2Þ P exp jpc1mi tmi d1ð
Þ⊤ 1 db
c p 1ð
Þ⊤ t m mi m i¼1 mi
 i i ð
Þ⊤ ð2Þ ¼ exp jpc1 t db
c1ð
Þ⊤ t " # p X Y p
1 : ð
1Þ ðp
1Þ! d1ð
Þ⊤ 1 P

i¼1

mi

mi

ð2:112Þ

138

IZZO AND NAPOLITANO

where 1mi is the jmij-dimensional vector [1, . . . , 1]⊤. Therefore, the set WC contains just one element and, moreover, bx ðtÞ ¼ c1ð Þ⊤ t

ð2:113Þ

and

Cx;x ðtÞ ¼ exp jpc1

ð Þ⊤ ð2Þ

t

X

"

P

ð Þ

p 1

p Y ðp 1Þ! d1ð Þ⊤ 1 i¼1

mi

mi

# :

ð2:114Þ

Let us note that for the chirp signal the RD-GCTCFs do not converge to zero as kt0 k ! 1. In fact, the time series x(t) and x(t þ tn) are not asymptotically independent: xðt þ tn Þ ¼ xðtÞdðt; tn Þ

ð2:115Þ

t2n Þ

where d(t, tn) is the sinewave exp½ jpcð2ttn þ which, for any tn, is a deterministic signal in the FOT probability sense. Finally, it is noteworthy that the chirp signal, as each signal which is random in the FOT probability framework and deterministic in the stochastic process framework, is not ergodic for the cumulants (and hence for the GCTCFs and GCSCFs), since all stochastic cumulants (N  2) are identically zero. 2. Nonuniformly Sampled Signal Uniformly sampled signals can be modeled as the product of a continuoustime signal by a train of impulses with constant period. In such a case, when the continuous-time signal is strictly bandlimited and ACS, its higher-order cyclostationarity features can be easily determined provided that the sampling rate is suYciently high (Napolitano, 1995; Izzo and Napolitano, 1996c). A more realistic model considers nonuniformly spaced impulse trains, that is, with a ‘‘period’’ not constant with time. Let us consider the nonuniformly sampled signal xðtÞ ≜ wðtÞsðtÞ;

ð2:116Þ

where sðtÞ ≜

þ1 X

dðt kTp ðtÞÞ

ð2:117Þ

k¼ 1

is an impulse train whose ‘‘period’’ Tp(t) is a slowly varying function of t and w(t) is a finite-power time series exhibiting Nth-order wide-sense stationarity (WSS) (i.e., such that Raw ðtÞ ≢ 0 only for a ¼ 0).

GENERALIZED ALMOST-CYCLOSTATIONARY SIGNALS

139

According to definition Eq. (2.14), by using the Poisson’s sum formula þ1 þ1 X X 1 dðt kTp ðuÞÞ ¼ e j2pmt=Tp ðuÞ ð2:118Þ T ðuÞ p m¼ 1 k¼ 1 specialized for u ¼ t, it results that * " !#+ N N Y X X a Rs ðtÞ ¼ fp ðt þ tn Þ exp j2p ð Þn mn fp ðt þ tn Þ : ðt þ tn Þ at n¼1

m2ZN

n¼1

ð2:119Þ where fp(t) ≜ 1/Tp(t) and m ≜ [m1, . . . , mN]⊤. In general, from this equation the analytical expression of the lag-dependent cycle frequencies cannot be easily derived. However, the lag-dependent cycle frequencies can be estimated following the approach presented in Section II.E. Moreover, according to the WSS exhibited by w(t), the lag-dependent cycle frequencies of x(t) are the same as those of s(t) and, furthermore, the GCTMFs of x(t) are given by [see Eq. (3.136)] Rx;z ðtÞ ¼ R0w ðtÞRs;z ðtÞ:

ð2:120Þ

Note that, when the maximum variation of the lag-dependent cycle frequencies is not greater than the cycle frequency resolution 1/T (with T the datarecord length), then the sampling ‘‘period’’ can be considered practically constant and hence the sampled signal can be modeled as ACS rather than GACS. Figure 4a shows the magnitude and Figure 4b the support in the (a, t) plane of the cyclic autocorrelation function Raxx ðtÞ of a signal x(t) obtained by uniformly sampling a signal w(t) exhibiting WSS and with power spectral 0 ð f Þ ¼ ð1 þ f 2 =B2 Þ 4 with B ¼ 0.0025/T . The period of the density Sww s impulse train has been fixed at Tp0 ¼ 4Ts and the estimate has been obtained on the basis of 128 samples. The almost-cyclostationary behavior of x(t) is evident since the support of Raxx ðtÞ is contained in lines parallel to the t axis. Figure 5a shows the magnitude and Figure 5b the support of Raxx ðtÞ with reference to a nonuniformly spaced impulse train with Tp(t) ¼ Tp0/(1 þ 0.5 cos(2p f0t)), where f0 ¼ 0.0005/Ts, and 128 samples have been processed. In such a case, the dependence of the cycle frequencies on t cannot be appreciated. However, when the sample size is increased to 16384, the generalized cyclostationary nature of the sampled signal x(t) becomes evident (see Figure 6). Finally, it is worthwhile to underline that the continuous-time nonuniformly sampled signal x(t) is GACS. On the contrary, in Section IV.C it is shown that the discrete-time signal constituted by the samples of a continuous-time GACS signal is always ACS.

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FIGURE 4. (a) Graph of magnitude and (b) support in the (a, t) plane of the cyclic autocorrelation function Raxx ðtÞ for a signal obtained by sampling a colored signal exhibiting WSS by an ideal impulse train with period Tp0 ¼ 4Ts . A sample size of 128 samples has been used.

G. Summary In this section, the higher-order characterization of the generalized almostcyclostationary time series has been addressed in the nonstochastic framework. For such a class of time series, multivariate statistical functions are almost-periodic functions of time whose Fourier series expansions can exhibit coeYcients and frequencies depending on the lag shifts of the time

GENERALIZED ALMOST-CYCLOSTATIONARY SIGNALS

141

FIGURE 5. (a) Graph of magnitude and (b) support in the (a, t) plane of the cyclic autocorrelation function Raxx ðtÞ for a signal obtained by sampling a colored signal exhibiting WSS by an impulse train with ‘‘period’’ Tp ðtÞ ¼ Tp0 =ð1 þ 0:5 cosð2pf0 tÞÞ, where Tp0 ¼ 4Ts and f0 ¼ 0:0005=Ts . A sample size of 128 samples has been used.

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FIGURE 6. (a) Graph of magnitude and (b) support in the (a, t) plane of the cyclic autocorrelation function Raxx ðtÞ for a signal obtained by sampling a colored signal exhibiting WSS by an impulse train with ‘‘period’’ Tp ðtÞ ¼ Tp0 =ð1 þ 0:5 cosð2pf0 tÞÞ, where Tp0 ¼ 4Ts and f0 ¼ 0:0005=Ts . A sample size of 16384 samples has been utilized.

GENERALIZED ALMOST-CYCLOSTATIONARY SIGNALS

143

series. Moreover, the union over all the lag shifts of the lag-dependent frequency sets is not necessarily countable and the time-averaged autocorrelation function can be discontinuous in the origin. Almost-cyclostationary time series are the subclass of GACS time series for which the frequencies do not depend on the lag shifts and the union of the previously-mentioned sets is countable. The GACS time series have been characterized in the strict sense and some known results for the subclass of ACS time series have been extended to the GACS case. Generalized cyclic moments in both time and frequency domains have been defined and their properties have been discussed. Moreover, generalized cyclic temporal and spectral cumulants have also been introduced. For ACS time series, the generalized cyclic statistics have been shown to reduce to the corresponding (nongeneralized) cyclic statistics.

III. LINEAR TIME-VARIANT TRANSFORMATIONS

OF

GACS SIGNALS

A. Introduction In several problems of interest in signal processing and communications, stochastic processes are processed by linear time-variant systems. Depending on the nature of the system, it can be modeled as deterministic or stochastic. Deterministic systems operate only on the time variable of the input stochastic process and treat as a constant the variable belonging to the outcome space (Papoulis, 1991, paragraph 10–2). Thus, two identical sample paths of the input give rise to two identical sample paths of the output. Hence, deterministic systems transform deterministic input signals into deterministic output signals. On the contrary, stochastic systems operate on both time and variable belonging to the outcome space. Thus, in general, they transform deterministic signals into stochastic processes. Many communication channels can be physically modeled as stochastic LTV systems. In Middleton (1967a), a statistical theory of reverberation and related first-order scattered fields is developed by describing the scattering mechanism by a LTV filter response; in Middleton (1967b), the second-order statistics of the reverberation (nonstationary) processes are determined in detail. In Bello (1963), a statistical characterization of stochastic LTV channels is carried out in terms of correlation functions of system functions defined in both time and frequency domains. In Tsao (1984), the problem of LTV filtering is addressed by stochastic diVerential equations.

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In all these classical papers, the stochastic LTV systems are described by an ensemble of impulse-response functions. Thus, the system functions are defined through ensemble averages of quantities related to the impulseresponse function, to its mono- or bi-dimensional Fourier transform, or products of time-and/or frequency-shifted versions of them. For such systems, even if the input stochastic process possesses amenable ergodicity properties, the output process, in general, is not ergodic (Gardner, 1990), and, therefore, neither the output process statistical functions nor the system functions can be estimated by a single sample path of the input and output stochastic processes. In this section, the problem of LTV filtering is addressed in the FOT probability framework. Thus, such a problem is treated in a new perspective with respect to the classical one of the stochastic process framework. This new approach is motivated by the necessity of characterizing in a useful way the output signal of randomly fluctuating linear channels which, in the stochastic process framework, in general, provide nonergodic outputs. On the contrary, in the FOT probability approach adopted here, measurements based on a finite data-record length asymptotically approach, as the datarecord length approaches infinity, the theoretical statistical functions, provided that the system impulse-response function and the input signal are suYciently regular to assure the considered infinite-time averages exist in the temporal mean-square sense. The (linear and nonlinear) systems are classified as deterministic or random in the FOT probability framework. Deterministic systems are those that map deterministic (i.e., constant, periodic, or almost-periodic) inputs into deterministic outputs. They include the linear almost-periodically timevariant (LAPTV) systems, as well as the systems that perform a time scale changing. Random systems are all transformations that cannot be classified as deterministic. The random linear systems include chirp modulators, modulators whose carrier is a pseudo-noise sequence, channels introducing timevarying delays, and systems that perform a time windowing. Note that throughout this section ‘‘random’’ is not synonymous with ‘‘stochastic.’’ In fact, the adjective stochastic is adopted, as usual, when an ensemble of realizations or sample paths exist, whereas the adjective random is referred to a single function of time, namely a signal or a system impulse-response function. Therefore, to avoid ambiguities, when necessary, deterministic and random systems in the FOT probability sense will be referred to as FOT deterministic and FOT random systems, respectively. The LTV systems are decomposed into the parallel connection of their FOT deterministic and purely random components and the concept of expectation (in the FOT sense) of the impulse-response function is introduced. The

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145

characterization in terms of higher-order statistics of the LTV systems is performed by analyzing the way in which periodicities present in the input Nth-order lag product are transformed into periodicities contained in the output Nth-order lag product. Such a characterization turns out to be particularly useful for GACS signals, since their Nth-order lag product can be decomposed into an almost-periodic component, which is called the Nthorder temporal moment function, and a residual term (see Section II.C.1). The choice of operating in the GACS context is useful in practice since ACS signals processed by LTV systems can generate GACS signals that are not necessarily ACS. Moreover, communication signals with parameters, such as the baud rate or the carrier frequency, slowly varying with time can be modeled as GACS and can be thought as obtained by linear (not periodically) time-variant transformations of ACS signals. It is shown that the Nth-order lag product of the system impulse-response function can be decomposed into two terms. The former, which will be referred to as the system temporal moment function, is by definition the kernel of the linear (with respect to the lag product) operator that transforms the almost-periodic component of the input lag product (i.e., the input temporal moment function), into the almost-periodic component of the output lag product. The latter is the kernel of the operator that transforms any almost-periodic component of the input lag product into a component of the output lag product not containing any almost-periodic component. The proposed decomposition of the Nth-order lag product of the system impulse-response function allows one to easily describe the behavior of the LTV systems in terms of input/output relations involving generalized cyclic statistics. Therefore, input/output relations in terms of generalized cyclic temporal and spectral moment functions are derived with reference to both FOT deterministic and FOT random linear systems. Moreover, for FOT deterministic systems, the input/output relations in terms of moment functions and cumulant functions are shown to be the same. The lack of ergodicity of the output signal of a stochastic LTV system in the stochastic process framework is discussed with reference to moment and cumulant functions. Then, the countability of the set of the output cycle frequencies is studied. Such a problem turns out to be relevant in the FOT probability approach since, in this framework, the almost-periodic component extraction operator is the analogous of the stochastic expectation operator in the classical stochastic process framework. Thus, the output-signal FOT moments and cumulants exist only if the output lag product contains finitestrength additive sinewave components, that is, a countable set of output cycle frequencies exists. If such a set is empty or more than countable, we have identically zero or divergent, respectively, generalized cyclic statistics.

146

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The countability of the set of the output cycle frequencies is studied with reference to the general case of LTV systems for both ACS and GACS not containing any ACS component input signals. Moreover, the special case of FOT deterministic LTV systems is analyzed in more detail. Thus, a characterization of the output signal is provided. Furthermore, several special cases of LTV systems are analyzed in detail. Specifically, LAPTV systems, systems performing product modulation, and several kinds of Doppler channels are considered. These results allow to properly characterize the output of several classes of LTV channels of interest in communications in terms of statistical functions that can be estimated by a single data record. Such a characterization is important as a first step to properly identify or equalize a channel. In the considered examples, it is shown that in practical situations diVerent system or signal models should be adopted depending on the data-record length. For example, for large data-record length possible time variations of timing parameters of the signals must be taken into account, making the GACS model more appropriate than the ACS model. Moreover, it is shown that GACS signals not containing any ACS component, filtered by a LAPTV system, give rise to signals with identically zero (generalized) cyclic statistics. Then, filtered versions of such signals exhibit estimated (generalized) cyclic statistics that are asymptotically zero when the data-record length approaches infinity. Multipath Doppler channels are also analyzed. They generate at the output several replicas of the input signal, each characterized by a diVerent complex amplitude, delay, time scaling factor, and frequency shift. Such channels are encountered very often in radar and mobile communication problems. In radar applications, they account for the presence of multiple slow fluctuating point scatterers. In mobile communications, such models are appropriate when the mobile station receives the signal transmitted by the base station through multiple trajectories, each characterized by a diVerent complex attenuation, length, and radial speed of the mobile station with respect to the base station. It is shown that multipath Doppler channels can be modelled as LAPTV systems or, more generally, as FOT deterministic LTV systems, depending on the length of the data-record adopted to observe the input and output signals and the bandwidth of the input signal.

B. FOT Deterministic and Random Linear Systems In this section, systems are classified as deterministic or random in the FOT probability sense. Then, a characterization is provided for both deterministic and random linear systems (Izzo and Napolitano, 1999, 2002b).

GENERALIZED ALMOST-CYCLOSTATIONARY SIGNALS

147

1. FOT Deterministic and Random Systems In the FOT probability framework, a deterministic system is defined as a possibly complex (and not necessarily linear) system that for every deterministic (i.e., almost periodic) input time series delivers a deterministic output time series (Izzo and Napolitano, 1995a). Systems that are not FOT deterministic will be referred to as FOT random systems. Therefore, for the input time series xðtÞ ¼ e j2plt

ð3:1Þ

a FOT deterministic system delivers the output almost-periodic time series X yðtÞ ¼ G0
e j2p
t ð3:2Þ
2El

where, for each fixed l, El ≜ f
1 ðlÞ; . . . ;
n ðlÞ; . . .g is the countable set of the output frequencies (depending on the input frequency l) and the G
0 are complex coeYcients. The set of points ð
; lÞ 2 El  R such that G0
6¼ 0 is constituted by a set of not necessarily continuous curves defined by the implicit equations Fs0 ð
; lÞ ¼ 0;

s0 2 O0

ð3:3Þ

where O0 is a countable set. That is, it results that clfð
; lÞ 2 El  R : G0
6¼ 0g 0 ¼ cl [ 0 fð
; lÞ 2 R  R : Fs0 ð
; lÞ ¼ 0; G
6¼ 0g s0 2O

ð3:4Þ

¼ cl [ fð
; lÞ 2 R  D’s ðlÞ :
¼ ’s ðlÞg s2O

where O is a countable set and, in the last equality, each curve described by the implicit equation Fs0 ð
; lÞ ¼ 0 has been decomposed into a countable set of curves, each described by the explicit equation
¼ ’s ðlÞ, where ’s(
) can always be chosen among the monotonic real functions and such that ’s ðlÞ 6¼ ’s0 ðlÞ for s 6¼ s0 , and D’s is the domain of ’s(
). Therefore, defined the functions  0 G
j
¼’s ðlÞ ; l 2 D’s Gs ðlÞ ≜ ð3:5Þ 0; elsewhere the output almost-periodic time series Eq. (3.2) can be written as X Gs ðlÞe j2p’s ðlÞt : yðtÞ ¼

ð3:6Þ

s2O

Note that, for a given system, the functions ’s(
) and Gs (
) are not univocally determined, since, in general, for each curve described by an implicit

148

IZZO AND NAPOLITANO

equation, several decompositions into curves described by explicit equations are possible. Moreover, if more functions ’s1 ðlÞ; . . . ; ’sK ðlÞ are defined in K (not necessarily coincident) neighborhoods of the same point l0, all have the same limit, say ’0, for l ! l0, and only one is defined in l0, then it is convenient to assume all the functions defined in l0 with ’s1 ðl0 Þ ¼


¼ ’sK ðl0 Þ ¼ ’0 and, consequently, to define Gsi ðl0 Þ ≜ lim Gsi ðlÞ; l!l0

i ¼ 1; . . . ; K

ð3:7Þ

where, for each i, the limit is made with l ranging in the neighborhood of l0 where the function ’si(l) is defined. Note that a mild regularity assumption on the system is that a small change in the input frequency l gives rise to small changes in the output frequencies
n ðlÞ 2 El . Thus, the
n(l) are continuous functions (not necessarily invertible) of the input frequency l, that is, the set El is continuous with respect to l. In such a case, the functions Fs0 (
, l) and, hence, the functions ’s(l) are continuous in their domain. Finally, let us observe that the FOT deterministic systems are those called ‘‘stationary’’ by Claasen and Mecklenbr€auker in Claasen and Mecklenbr€ auker (1982). 2. FOT Deterministic Linear Systems Let us now consider a linear time-variant system. Its input/output relationship is given by Z yðtÞ ¼ hðt; uÞxðuÞ du ð3:8Þ R

where h(t, u) is the system impulse-response function. By Fourier transforming both sides of Eq. (3.8), one obtains the input/output relationship in the frequency domain Z Yð f Þ ≜ yðtÞe j2pft dt R Z ð3:9Þ ¼ Hð f ; lÞX ðlÞ dl R

where the transmission function H( f, l) [10] (also referred to as Zadeh’s bifrequency function (Bello, 1963)) is the double Fourier transform of the impulse-response function: Z Hð f ; lÞ ≜ hðt; uÞe j2pð ft luÞ dt du: ð3:10Þ R2

GENERALIZED ALMOST-CYCLOSTATIONARY SIGNALS

149

In Eqs. (3.9) and (3.10), and in the following, the Fourier transforms areassumed to exist in the sense of distributions (generalized functions) (Zemanian, 1987). Since the transmission function Eq. (3.10) represents the Fourier transform of the output y(t) corresponding to the input Eq. (3.1), for FOT deterministic linear systems the transmission function can be obtained by Fourier transforming the right-hand side of Eq. (3.6): X Hð f ; lÞ ¼ Gs ðlÞdð f ’s ðlÞÞ ð3:11aÞ s2O

¼

X

Hs ð f Þdðl cs ð f ÞÞ

ð3:11bÞ

s2O

where the functions cs(
), referred to as the frequency mapping functions, are the inverse functions of ’s(
), and the functions Gs(
) and Hs(
), accounting for the fact that dð f ’s ðlÞÞ ¼ jc_ s ð f Þjdðl c_ s ð f ÞÞ and dðl cs ð f ÞÞ ¼ j’_ s ðlÞjdð f ’s ð f ÞÞ [Zemanian, 1987; Section 1.7], are linked by the relationships Hs ð f Þ ¼ jc_ s ð f ÞjGs ðcs ð f ÞÞ

ð3:12Þ

and ð3:13Þ Gs ðlÞ ¼ j’_ s ðlÞjHs ð’s ðlÞÞ _ with cs ð
Þ and ’_ s ð
Þ denoting the derivative of cs(
) and ’s(
), respectively. Moreover, accounting for Eq. (3.10), from Eqs. (3.11a) and (3.11b) it follows that the impulse-response function of FOT deterministic linear systems can be expressed as XZ hðt; uÞ ¼ Gs ðlÞej2p’s ðlÞt e j2plu dl ð3:14aÞ s2O

¼

R

XZ s2O

R

Hs ð f Þe j2pcs ð f Þu e j2pft df :

ð3:14bÞ

By substituting Eqs. (3.11a) and (3.11b) into Eq. (3.9), one obtains the input/output relationship for FOT deterministic linear systems in the frequency domain: XZ Yð f Þ ¼ Gs ðlÞdð f ’s ðlÞÞX ðlÞ dl ð3:15aÞ s2O

¼

X s2O

R

Hs ð f ÞX ðcs ð f ÞÞ

ð3:15bÞ

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IZZO AND NAPOLITANO

from which it follows that yðtÞ ¼

XZ s2O

¼

X

R

Gs ðlÞX ðlÞej2p’s ðlÞt dl

hs ðtÞ ! xcs ðtÞ

s2O

where ! denotes convolution and Z xcs ðtÞ ≜ X ðcs ð f ÞÞej2pft df : R

ð3:16aÞ ð3:16bÞ

ð3:17Þ

In other words, the output of FOT deterministic LTV systems is constituted by frequency compressed or stretched and filtered versions of the input. It can be shown that the parallel and cascade concatenation of FOT deterministic LTV systems is still a FOT deterministic LTV system. The class of FOT deterministic LTV systems includes that of the LAPTV systems which, in turn, includes, as special cases, both linear periodically time-variant and linear time-invariant (LTI) systems. For LAPTV systems, the frequency mapping functions cs ( f ) are linear with unitary slope, that is, cs ð f Þ ¼ f s;

s2O

and then the impulse-response function can be expressed as X hðt; uÞ ¼ hs ðt uÞej2psu :

ð3:18Þ

ð3:19Þ

s2O

Moreover, the systems performing time-scale changing are FOT deterministic. In such a case, the impulse-response function is given by hðt; uÞ ¼ dðu stÞ

ð3:20Þ

where s 6¼ 0 is the scale factor, the set O contains just one element, cs ð f Þ ¼

f s

ð3:21Þ

1 : jsj

ð3:22Þ

and Hs ð f Þ ¼

Furthermore, decimators and interpolators are FOT deterministic discretetime systems (Izzo and Napolitano, 1998a). The subclass of FOT deterministic LTV systems obtained by considering O containing only one element was studied, in the stochastic process framework, in Franaszek (1967) and Franaszek and Liu (1967) with reference to

GENERALIZED ALMOST-CYCLOSTATIONARY SIGNALS

151

the continuous-time case and in Liu and Franaszek (1969) with reference to the discrete-time case. The most important property of these systems, as evidenced in Franaszek (1967), Franaszek and Liu (1967), and Liu and Franaszek (1969) is that they preserve in the output the wide-sense stationarity of the input random process. In Claasen and Mecklenbr€auker (1982) the class of LTV systems called here FOT deterministic are analyzed considering a concept of stationarity with reference to a single time series. Linear time-variant systems that cannot be modeled as FOT deterministic systems include chirp modulators, modulators whose carrier is a pseudonoise sequence, channels introducing a time-varying delay, and systems performing time windowing. In fact, all these systems do not deliver an almost-periodic function when they are excited by a sinewave. Finally, note that, on the basis of the introduced terminology, a deterministic system in the stochastic process framework can be classified either as FOT deterministic or FOT random in the FOT probability sense, depending on the behavior of its impulse-response function. Moreover, a stochastic system can exhibit an impulse-response function whose sample paths are either FOT deterministic or FOT random. 3. Impulse-Response Function Decomposition for FOT Random LTV Systems In the FOT probability framework, a GACS time series x(t) can be decomposed into its deterministic (i.e., almost periodic) and purely random components: xðtÞ≜ Efag fxðtÞg þ xr ðtÞ ¼ xp ðtÞ þ xr ðtÞ

ð3:23Þ

where xp(t) is the almost-periodic component of x(t) and xr(t) is the purely random component such that hxr ðtÞe j2pat it 0;

8a 2 R:

ð3:24Þ

The decomposition [Eq. (3.23)] in the FOT probability framework is analogous to the decomposition, in the stochastic process framework, of a stochastic process into its statistical average and (zero mean) residual term. Moreover, in the stochastic process framework, a similar decomposition into deterministic and purely random components can be considered for the impulse-response function of a LTV system (Bello, 1963), since the statistical expectation operator can be applied in the same way to stochastic processes and stochastic system impulse-response functions. In fact, in such

152

IZZO AND NAPOLITANO

a framework, the randomness of both processes and systems is due to the dependence on the variable belonging to the outcome space, which is not linked to the time variables. In the FOT probability framework, instead, the almost-periodic extraction operator is not able to provide the deterministic component also for the systems, since in this framework the randomness is a consequence of the time behavior of the functions. As an example, let us consider the impulse-response function hðt þ t; tÞ ¼ dðð1 sÞt stÞ of a system performing a time-scale changing [see Eq. (3.20)]. Even though the system is FOT deterministic, by applying the almost-periodic component extraction operator to hðt þ t; tÞ to obtain its deterministic component, one would obtain the incongruous result that the deterministic component of hðt þ t; tÞ should be identically zero, unless the system were LTI (s ¼ 1). In the FOT probability framework, a useful and congruous decomposition of a LTV system into the parallel concatenation of two subsystems, that will be referred to as its FOT deterministic and FOT purely random components, can be obtained by writing the impulse-response function in the following way: hðt; uÞ ≜ hD ðt; uÞ þ hR ðt; uÞ:

ð3:25Þ

In Eq. (3.25), hD(t, u) denotes the impulse-response function of the subsystem that for any almost-periodic input delivers an almost-periodic output. Its analytic expression is therefore given by Eqs. (3.14a) or (3.14b) and it will be referred to as the impulse-response function of the FOT deterministic component of the LTV system. The function hR(t, u) is the impulse-response function of the subsystem that for any almost-periodic input delivers an output signal not containing any finite-strength additive sinusoidal component. Such a system will be referred to as the FOT purely random component of the LTV system. Thus, for FOT deterministic LTV systems, it results that hðt; uÞ hD ðt; uÞ:

ð3:26Þ

hðt; uÞ hR ðt; uÞ

ð3:27Þ

Systems for which

will be referred to as FOT purely random systems. A summary of the meanings of FOT deterministic and FOT random for both signals and systems is reported in Table 1. By substituting Eqs. (3.23) and (3.25) into the input/output relationship Eq. (3.8), one obtains that the almost-periodic component of the output y(t) can be expressed as

GENERALIZED ALMOST-CYCLOSTATIONARY SIGNALS

153

TABLE 1 Summary of the Meanings of FOT Deterministic and FOT Random for Both Signals and Systems

Signals Systems

FOT deterministic

FOT random

FOT purely random

Almost-periodic (AP) functions Transform AP functions into AP functions

Not FOT deterministic Not FOT deterministic

Not containing any finite-strength additive sine wave Transform AP functions into signals not containing any finite-strength additive sine wave

Efag fyðtÞg ¼

Z hD ðt; uÞ xp ðuÞ du Z  hR ðt; uÞ xp ðuÞ du þ Efag ZR  þ Efag hD ðt; uÞ xr ðuÞ du ZR  fag þE hR ðt; uÞ xr ðuÞ du :

R

ð3:28Þ

R

The second term in the right-hand side of Eq. (3.28) is identically zero since, by definition, hR(t, u) transforms almost-periodic inputs into signals not containing finite-strength additive sinusoidal components. Moreover, under the mild assumption that the deterministic component hD(t, u) is suYciently regular so that the functions Gs(l) do not contain impulsive terms in l, from (3.14a) it follows that ( ) Z  XZ fag fag j2p’s ðlÞt hD ðt; uÞ xr ðuÞ du ¼ E Gs ðlÞ Xr ðlÞe dl ¼ 0 E R

s2O

R

ð3:29Þ where the absence of impulsive terms in the Fourier transform Xr( f ) of xr(t) has been accounted for [see Eq. (3.24)]. Furthermore, let us observe that the fourth term in the right-hand side of Eq. (3.28) could give nonzero contribution if and only if there exists statistical dependence, in the FOT probability sense, between the input signal and the system. This occurrence, however, means that functional dependence should exist between the functions hRð
;
Þ and xr ð
Þ, which is in contradiction with the linearity assumption for the system. Then, unlike the stochastic process framework, in the FOT probability framework linear systems cannot be statistically dependent on

154

IZZO AND NAPOLITANO

the input signal and, hence, also the fourth term in the right-hand side of Eq. (3.28) is zero, which leads to Z Efag fyðtÞg ¼ hD ðt; uÞ xp ðuÞ du: ð3:30Þ R

fag

Note that, since xp ðtÞ ¼ E fxðtÞg and Efag fyðtÞg are the expectations (in the FOT sense) of the input and output signals, respectively, by analogy with the stochastic process framework, hD(t, u) can be interpreted as the expectation (in the FOT sense) of the impulse-response function h(t, u). It is worthwhile to note that the expectation of h(t, u) cannot be obtained, in general, by extracting the almost-periodic component of h(t, u) as it happens in the case of the signals. In the case where a functional dependence between x(t) and h(t, u) exists, then Eq. (3.30) does not hold. For example, if xðtÞ xr ðtÞ 2 R (e.g., any zero-mean real GACS signal) and hðt; uÞ hRðt; uÞ ¼ xr ðt þ tÞ dðu tÞ, it results that Z  Efag fyðtÞg ¼ Efag hR ðt; uÞ xr ðuÞ du R

¼ Efag fxr ðt þ tÞxr ðtÞg X ¼ Rxr ;z ðtÞej2paz ð½t;0Þt

ð3:31Þ

z

where the functions Rxr ;z ðtÞ are the generalized cyclic autocorrelation functions of xr(t). It is worthwhile to note that, in the stochastic process framework, when a stochastic process passes through a LTV system with random impulseresponse function (i.e., which is in turn a stochastic process), a result analogous to Eq. (3.30) can be obtained, provided that the almost-periodic component extraction operator is substituted with the statistical expectation (i.e., the ensemble average) operator and the FOT deterministic component of the impulse-response function is substituted with the ensemble average of the stochastic impulse-response function and, moreover, the input signal and the system are statistically independent. The time-varying ensemble average of the output stochastic process, however, can be estimated by a single sample path only if appropriate ergodicity conditions involving both the input stochastic process and the stochastic LTV system are verified. The result of Eq. (3.30), instead, allows one to determine the time-varying FOT expectation of y(t), i.e., the finite-strength additive sine wave components of y(t), starting from a single available time series, independently of the possible existence of underlying (ergodic or not) stochastic processes.

GENERALIZED ALMOST-CYCLOSTATIONARY SIGNALS

155

As an example of application of the previous results, let us evaluate the almost-periodic component of the pulse-amplitude–modulated (PAM) signal xPAM ðtÞ ¼ xd ðtÞ ! qðtÞ

ð3:32Þ

where q(t) is a finite-energy pulse and xd(t) is the ideal sampled signal xd ðtÞ ¼

þ1 X

xðkÞ dðt kTs Þ:

ð3:33Þ

k¼ 1

The signal xPAM(t) can be interpreted as a LTI (and, hence, deterministic) transformation of the random signal xd(t) statistically independent of the LTI system. Therefore, according to Eq. (3.30), it results that Efag fxPAM ðtÞg ¼ Efag fxd ðtÞg ! qðtÞ þ1 X Ef~ag fxðkÞg qðt kTs Þ ¼

ð3:34Þ

k¼ 1 f~ ag

where E f
g denotes the discrete-time almost-periodic component extraction operator, and the second equality is proved in Appendix C. Note that Eq. (3.34) is a result formally analogous to that obtained in the stochastic process framework (Franks, 1969; Section 8.3). C. Higher-Order System Characterization in the Time Domain In this section, the system temporal moment function is introduced to provide the higher-order characterization of LTV systems in the time domain. Moreover, input/output relationships for LTV systems excited by GACS time series are derived in terms of temporal moment and cumulant functions (Izzo and Napolitano, 1999, 2002b). The Nth-order lag product of the output y(t) of a LTV system, accounting for Eqs. (2.19) and (3.8), is given by Ly ð1t þ tÞ ≜

N Y yðÞn ðt þ tn Þ n¼1

Z ¼

N Y hðÞn ðt þ tn ; t þ sn ÞLx ð1t þ sÞ ds

ð3:35Þ

RN n¼1

where s ≜ ½s1 ; . . . ; sN ⊤ . Therefore, the hidden periodicities of the input signal that are regenerated by the input Nth-order lag product are transformed into periodicities of the output Nth-order lag product by a linear (with respect to the lag product) operator whose kernel is the Nth-order lag product of the impulse-response function.

156

IZZO AND NAPOLITANO

Let us consider the following decomposition of the Nth-order lag product of the impulse-response function: N Y

hðÞn ðt þ tn ; un Þ ≜ Rh ð1t þ t; uÞ þ ‘h ð1t þ t ; uÞ

ð3:36Þ

n¼1

where u ≜ ½u1 ; . . . ; uN T . In Eq. (3.36), the function Rh (1t þ t, u), which will be referred to as the system temporal moment function, is by definition the kernel of the linear (with respect to the lag product) operator that transforms the almost-periodic component of the input lag product, that is, the input temporal moment function, into an almost-periodic component of the output lag product (which, at this point, cannot yet be recognized to be the whole almost-periodic component, that is the output temporal moment function). Moreover, the function ‘h (1t þ t, u) is the kernel of the operator that transforms any almost-periodic component of the input lag product into a FOT purely random component (i.e., not containing any additive finite-strength sinusoidal component) of the output lag product. In order to obtain a powerful expression of the system temporal moment function, let us consider the input lag product T

Lx ð1t þ sÞ Rx ð1t þ sÞ ¼ ej2pl

ð1tþsÞ

ð3:37Þ

where l ≜ ½l1 ; . . . ; lN T . The almost-periodic component of the corresponding output lag product Ly;l ð1t þ tÞ can be written as X Efag fLy;l ð1t þ tÞg ¼ G0
ðtÞe j2p
t ð3:38Þ
2El;t

where, for each fixed (l, t), El,t is a countable set and G0
ðtÞ are complex functions. Moreover, note that either El,t or G0
ðtÞ depend on the choice of the conjugation configuration of the factors of the output lag product. Since the set El,t is countable, the set of points ð
; lÞ 2 El;t  RN such 0 that G
ðtÞ 6¼ 0 is constituted by a countable set of N-dimensional manifolds (Choquet-Bruhat and DeWitte-Morette, 1982) that can be described by explicit equations. Therefore, it results that cl fð
; lÞ 2 El;t  RN : G0
ðtÞ 6¼ 0g ¼ cl [ fð
; lÞ 2 R  Dy ðtÞ :
¼ ’y ðl; tÞg

ð3:39Þ

y2Y

where Y is a countable set, ’y ð
;
Þ are real functions (which depend on the choice of the conjugation configuration of the factors of the lag products) such that ’y ðl; tÞ 6¼ ’y0 ðl; tÞ for y 6¼ y0 , and Dy(t) is the domain of ’y(
, t). Thus, defined the functions

GENERALIZED ALMOST-CYCLOSTATIONARY SIGNALS

 Gy ðl; tÞ ≜

G0
ðtÞj
¼’y ðl;tÞ ; l 2 Dy ðtÞ 0; elsewhere

157 ð3:40Þ

the almost-periodic component of the output lag product can be written as X Efag fLy;l ð1t þ tÞg ¼ Gy ðl; tÞe j2p’y ðl;tÞt : ð3:41Þ y2Y

Furthermore, since the input lag product [Eq. (3.37)] does not contain the residual term ‘x(1t þ s), from the definition of system temporal moment function it results that Z ⊤ Efag fLy; l ð1t þ tÞg ¼ Rh ð1t þ t; uÞe j2pl u du ð3:42Þ RN

from which, accounting for Eq. (3.41), it follows that the system temporal moment function can be expressed as XZ ⊤ Rh ð1t þ t; uÞ ¼ Gy ðl; tÞe j2p’y ðl;tÞt e j2pl u dl: ð3:43Þ y2Y

RN

Note that, for a given system, the functions ’y(
,
) and Gy(
,
) are not univocally determined since, in general, for each manifold several representations by explicit equations are possible. Furthermore, if there exist vectors l0 such that more functions ’yi ðl; tÞ i ¼ 1; . . . ; K have the same limit, say ’0(t), as l ! l0 , then it is convenient to assume all the functions defined in l0 with ’yi ðl0 ; tÞ ¼ ’0 ðtÞ and consequently to define Gyi ðl0 ; tÞ ¼ lim Gyi ðl; tÞ; l!l0

i ¼ 1; . . . ; K

ð3:44Þ

where, for each i, the limit is made with l ranging in the neighborhood of l0 where ’yi(l, t) is defined. If the functions ’y(l, t) are constant with respect to l for any y 2 Y, then the function Rh ð1t þ t; uÞ is almost periodic in t. Finally, if for N ¼ 2 Rh ð1t þ t; uÞ is periodic in t, then it is the FOT counterpart of the system intercorrelation function introduced in Duverdier et al. (1999) for the class of (stochastic) cyclostationary systems. It can be shown that for all real numbers D the following lag-shift invariance property holds El;tþ1D El;t

ð3:45Þ

G0
ðt þ 1DÞ ¼ G0
ðtÞej2p
D ;

ð3:46Þ

and, moreover,

158

IZZO AND NAPOLITANO

from which it follows that ’y ðl; t þ 1DÞ ¼ ’y ðl; tÞ

ð3:47Þ

Gy ðl; t þ 1DÞ ¼ Gy ðl; tÞej2p’y ðl;tÞD

ð3:48Þ

and

where, in the derivation of Eq. (3.48), Eq. (3.40) has been accounted for. Since in the FOT probability framework the almost-periodic component extraction operator is the expectation operator, for N ¼ 2 the function Efag fLy; l ð1t þ tÞg is the correlation between the output signals corresponding to the inputs ej2pl1 t and ej2pl2 t [see Eqs. (3.37) and (3.41)]. Moreover, from Eqs. (3.45)–(3.48) it follows that the correlation function Efag fLy; l ð1t þ tÞg depends on t1 t2. A similar property, however, does not hold in general for the dependence on l1 and l2. In the special case where, for all y 2 Y, Gy ðl1 ; l2 ; t1 ; t2 Þ ¼ Gy ðDl; DtÞ

ð3:49Þ

’y ðl1 ; l2 ; t1 ; t2 Þ ¼ ’y ðDl; DtÞ

ð3:50Þ

where Dl ≜ l1 l2 and Dt ≜ t1 t2 , the function defined in Eq. (3.41) is the counterpart in the FOT probability framework of the spaced-frequency spaced-time correlation function defined in the stochastic process framework (Proakis, 1995). Moreover, its Fourier transform with respect to the variable Dt evaluated for Dl ¼ 0 is the FOT counterpart of the Doppler power spectrum of the channel defined in the stochastic process framework. Furthermore, the double Fourier transform with respect to Dl and Dt of the right-hand-side of Eq. (3.41) is the FOT counterpart of the scattering function of the channel (Proakis, 1995). Let us observe that for N ¼ 1 the system temporal moment function is coincident with the FOT deterministic component of the impulse-response function, that is, Rh ðt; uÞ hD ðt; uÞ:

ð3:51Þ

Moreover, the set Y is coincident with O, the functions Gy(l1, t1) and ’y (l1, t1) do not depend on t1 and are coincident with the functions Gs(l1) and ’s ðl1 Þðs ¼ yÞ, respectively, and the functions ’s(
) can be chosen among the monotonic functions. The output temporal moment function of a LTV system, by substituting Eqs. (2.19) and (3.36) into Eq. (3.35), can be written as

GENERALIZED ALMOST-CYCLOSTATIONARY SIGNALS

Efag fLy ð1t þ tÞg ¼

159

Z Rh ð1t þ t; 1t þ sÞRx ð1t þ sÞ ds Z  ‘h ð1t þ t; 1t þ sÞRx ð1t þ sÞ ds þ Efag RN Z  þ Efag Rh ð1t þ t; 1t þ sÞ ‘x ð1t þ sÞ ds RN Z  fag ‘h ð1t þ t; 1t þ sÞ‘x ð1t þ sÞ ds : þE RN

ð3:52Þ

RN

The second term in the right-hand side of Eq. (3.52) is identically zero since ‘h ð1t þ t; 1t þ sÞ is by definition the kernel of the operator that transforms the almost-periodic component of the input lag product into a FOT purely random component of the output lag product. Moreover, under the mild assumption that the system temporal moment function is suYciently regular so that the functions Gy(l, t) do not contain impulsive terms in l, accounting for Eq. (3.43), one has Z  fag Rh ð1t þ t; 1t þ sÞ‘x ð1t þ sÞds E RN ( ) XZ ð3:53Þ fag j2p’y ðl;tÞt ¼E Gy ðl; tÞLx ðlÞe dl y2Y

RN

¼0 where Lx(l) is the Fourier transform of the function ‘x(u) and, hence, does not contain impulsive terms in l [in fact, Eq. (2.20) with ‘x (t, t) ‘x (1t þ t) holds for any t]. Furthermore, also the fourth term in the right-hand side of Eq. (3.52) is zero since it could give nonzero contribution only in the case of functional dependence between the functions ‘h(
,
) and ‘x(
), that is, only in the case of functional dependence between the functions hR(
,
) and xr(
), which is in contrast with the linearity assumption (see Section III.B.3). Therefore, the temporal moment function of the output y(t) can be written as Efag fLy ð1t þ tÞg Ry ð1t þ tÞ Z ¼ Rh ð1t þ t; 1t þ sÞ Rx ð1t þ sÞds:

ð3:54Þ

RN

Equation (3.54) is the input/output relationship in terms of temporal moment functions for LTV systems and is formally analogous to that obtained in the stochastic process framework. Its advantage with respect to the corresponding formula in the stochastic process framework is that it is the asymptotic result of a time average measure without the necessity of

160

IZZO AND NAPOLITANO

ergodicity assumptions. Note that, since Rx (1t þ t) and Ry (1t þ t) are the expectations (in the FOT sense) of the input and output lag products, respectively, by analogy with the stochastic process framework, Rh (1t þ t, 1t þ s) can be interpreted as the expectation (in the FOT sense) of the impulse-response function lag product and this justifies its name ‘‘system temporal moment function.’’ Furthermore, Rh (1t þ t, 1t þ s) cannot be obtained, in general, by extracting the almost-periodic component of the impulse-response function lag product as it happens in the case of signals. Let us observe that, in general, for N > 1, the Nth-order lag product of the FOT deterministic component hD (t, u) of the impulse-response function is not coincident with the system temporal moment function Rh (1t þ t, u). However, in the special case of FOT deterministic LTV systems, accounting for Eqs. (3.14a) and (3.26), one has N Y hðÞn ðt þ tn ; un Þ n¼1

¼

X Z

s2ON

N Y

RN n¼1

ð Þ

j2pws n GðÞ sn ðð Þn ln Þe

ðlð Þ Þ⊤ ð1tþtÞ j2pl⊤ u

e

ð3:55Þ dl

ð Þ

where ws ðlð Þ Þ ≜ ½ð Þ1 ’s1 ðð Þ1 l1 Þ;


ð ÞN jsN ðð ÞN lN Þ⊤ : Therefore, by comparing Eq. (3.43) with Eq. (3.55), we get Rh ð1t þ t; uÞ ¼

N Y hðÞn ðt þ tn ; un Þ n¼1 N Y

ðÞ hD n ðt

ð3:56Þ þ t n ; un Þ

n¼1

which is a result formally analogous to that obtained in the stochastic process framework. Moreover, in such a case there exists a one-to-one correspondence between the elements y 2 Y and the vectors s ≜ [s1,


, sN]⊤ 2 ON, ð Þ ⊤ ’y ðl; tÞ ¼ wð Þ Þ 1 ðindependent of tÞ s ðl

ð3:57Þ

and Gy ðl; tÞ ¼

N Y ð Þ j2pws ðlð Þ Þ⊤ t n GðÞ : sn ðð Þn ln Þe

ð3:58Þ

n¼1

Finally, in Appendix C it is shown that for FOT deterministic LTV systems the input/output relationship in terms of temporal cumulant functions is the same as that given in Eq. (3.54) in terms of temporal moment functions, that is,

161

GENERALIZED ALMOST-CYCLOSTATIONARY SIGNALS

Z Cy ð1t þ tÞ ¼

RN

Rh ð1t þ t; 1t þ sÞ Cx ð1t þ sÞds:

ð3:59Þ

Such a result is analogous to that for deterministic LTV systems in the stochastic process framework. Moreover, Eq. (3.59) generalizes the result derived in Napolitano (1995) for LAPTV systems excited by ACS signals. D. Higher-Order System Characterization in the Frequency Domain In this section, random and deterministic LTV systems in the FOT probability sense are characterized in the frequency domain. Moreover, input/ output relationships in terms of spectral moment and cumulant functions are derived for LTV systems excited by GACS time series (Izzo and Napolitano, 2002b). By assuming that all Fourier transforms considered in the following exist at least in the sense of distributions (Zemanian, 1987), one has that the N-fold Fourier transform of both sides of Eq. (3.54) gives the input/output relationship in terms of spectral moment functions: Z Sy ð f Þ ¼ S h ð f ; lÞS x ðlÞ dl ð3:60Þ RN

where the function

Z

S h ð f ; lÞ ≜

R

2N

Rh ðt; uÞe j2pð f

⊤

t l⊤ uÞ

dtdu

ð3:61Þ

referred to as the system spectral moment function, accounting for Eq. (3.43), can be expressed as X Gy ðl; f Þ ð3:62Þ S h ð f ; lÞ ¼ y2Y

with

Z Gy ðl; f Þ ≜ ¼

Gy ðl; tÞe j2pf

R

N

R

N 1

Z

⊤

t

dt ð3:63Þ

Gy ðl; ½t 0 ; 0Þ 0

dð’y ðl; ½t0 ; 0Þ f ⊤ 1Þe j2pf

0⊤ 0 t

dt0

where in the derivation of the last equality, Eq. (3.48) has been accounted for. In the special case of FOT deterministic LTV systems, by substituting Eqs. (3.57) and (3.58) into Eq. (3.63), one obtains that

162

IZZO AND NAPOLITANO

Gy ðl; f Þ ¼

N

Y ðÞ Gsn n ðð Þn ln Þd fn ð Þn ’sn ðð Þn ln Þ

ð3:64aÞ

n¼1

¼

N

Y n ðð Þ f Þd l ð Þ c ðð Þ f Þ HsðÞ n n sn n n n n n

ð3:64bÞ

n¼1

where, in the derivation of Eq. (3.64b), Eq. (3.12) has been accounted for. Moreover, from Eq. (3.62), accounting for Eqs. (3.11a) and (3.11b), it follows that N Y S h ð f ; lÞ ¼ H ðÞn ðð Þn fn ; ð Þn ln Þ: ð3:65Þ n¼1

Finally, let us note that since for FOT deterministic LTV systems the input/output relationship in terms of temporal cumulant functions is the same as that in terms of temporal moment functions (see Appendix C), accounting for Eq. (3.60), one obtains that the input/output relationship in terms of spectral cumulant functions can be expressed as Z Pyð f Þ ¼ S h ð f ; lÞP x ðlÞdl: ð3:66Þ RN

E. Ergodicity of the Output Signal of a LTV System In this section, the lack of ergodicity of the output signal of a stochastic LTV system in the stochastic process framework is discussed. Such a discussion suggests one of the most important motivations to consider the LTV filtering in the FOT probability framework (Izzo and Napolitano, 2002b). A stochastic process is said to be ergodic (Gardner, 1994) in the stochastic TMF [TCF] if it is equal to the TMF [TCF] of almost every sample path. The stochastic TMF is defined as the statistical expectation of the lag product, whereas the stochastic TCF can be expressed by Eq. (2.58) in which, however, the involved moment functions are stochastic. For stochastic systems in the stochastic process framework, in general the input/output relations in terms of stochastic TMFs and TCFs can be diVerent from those in the FOT probability framework, that is, those in terms of their possible asymptotic estimators (i.e., TMFs and TCFs, respectively). In other words, in general such systems transform ergodic input stochastic processes into nonergodic output stochastic processes. In the stochastic process framework, stochastic systems, whose sample paths of the impulse-response function are impulse-response functions of deterministic systems in the FOT probability sense, destroy the input

163

GENERALIZED ALMOST-CYCLOSTATIONARY SIGNALS

ergodicity properties. In fact, let us consider a stochastic LTV system characterized by the impulse-response function H (t, u) and excited by a stochastic process X(t) statistically independent of the system. The Nth-order stochastic temporal moment and cumulant functions of the output process Y(t) can be written as ( ) N Y ðÞ
R Y ð1t þ tÞ ≜ E Y n ðt þ tn Þ Z ¼

n¼1(

E RN

N Y n¼1

) HðÞn ðt þ tn ; t þ sn Þ

( E

N Y

) X ðÞn ðt þ sn Þ ds

n¼1

ð3:67Þ
Y ð1t þ tÞ ≜ cumfY ðÞn ðt þ tn Þ; n ¼ 1; . . . ; Ng C ( ) Z X" p Y Y p 1 ðÞn ¼ ð 1Þ ðp 1Þ! E H ðt þ tn ; t þ sn Þ RN P n2mi i¼1 3 q XY
X ð1t þ sm ;n Þ5ds C mi ;vj i j Pmi j¼1

ð3:68Þ where E{
} denotes statistical expectation, the overbar has been adopted to denote stochastic statistical functions, Pmi is the set of distinct partitions of the elements of mi, each constituted by the subsets {nj : j ¼ 1, . . ., q}, jnjj is the number of elements in nj, and Xmi,nj is the jnjj-dimensional vector whose components are those (possibly conjugate) of Xmi having indices in nj. From Eqs. (3.67) and (3.68), it follows that the input/output relations in terms of stochastic TMFs and TCFs are diVerent. However, since almost every sample path of the considered stochastic system is a deterministic system in the FOT probability sense, then the input/output relations in terms of TMFs and TCFs (which are candidate asymptotic estimators of their stochastic counterparts) are the same (see Appendix C). Therefore, such a system transforms in a diVerent way the stochastic TMFs and TCFs and their candidate asymptotic estimators and, hence, it transforms ergodic stochastic processes into nonergodic stochastic processes. Let us note that, with reference to the moments, a stochastic system with impulse-response function with FOT deterministic sample-paths transforms an ACS ergodic input stochastic process X(t) into a nonergodic output process Y(t). In fact, for any ACS ergodic input it results that
X ð1t þ tÞ ¼ Rx ð1t þ t) for almost all sample-paths x(t) (Dandawate´ and R Giannakis¸, 1995; Gardner, 1994), but for the systems under consideration

164

IZZO AND NAPOLITANO

( E

N Y

) H

ðÞn

ðt þ tn ; t þ sn Þ

6¼

n¼1

N Y

hðÞn ðt þ tn ; t þ sn Þ

ð3:69Þ

n¼1

for almost all sample paths h(t, u). Hence, the kernel of the integral that transforms Rx ð1t þ tÞ into Ry ð1t þ t) [see Eq. (3.54) with Eq. (3.56) sub
X ð1t þ t) into R
Y ð1t þ tÞ stituted into] is diVerent from that transforming R
Y ð1t þ tÞ 6¼ Ry ð1t þ t). [see Eq. (3.67)] and, consequently, R Examples of stochastic systems with the previously-mentioned behavior in the stochastic process framework are the stochastic LTV systems with impulse-response functions whose sample paths are deterministic in the FOT probability sense and whose randomness is due to the presence of random parameters. A notable case is the widely adopted model for the noncoherent channel, that is, the stochastic system with impulse-response function H (t, u) ¼ R(t) ejY(t) d(u t), where, for a fixed t, R(t) is a Rayleigh random variable, Y(t) is a uniformly distributed random variable, and the dependence on t in R(t) and Y(t) can be neglected in the observation interval. Further examples are carrier modulators introducing random phase shifts and the Doppler channel that transforms a sine wave with a fixed frequency into one whose frequency is a random variable (Papoulis, 1991, paragraph 10–3). Stochastic linear systems destroying ergodicity whose randomness is not due to a random parameter in the impulse-response function are, for example, the stochastic bandlimited LTI systems, that is, those systems whose impulseresponse function sample paths are almost all bandlimited functions. A further drawback of considering the random filtering problem in the stochastic process framework is evident if one considers the input signal modeled as a deterministic signal x(t) (with possibly unknown, but nonrandom, parameters). In such a case, the input/output relationships for random channels Eqs. (3.67) and (3.68) become ( ) Z N N Y Y ðÞ
Y ð1t þ tÞ ¼ R E H n ðt þ tn ; t þ sn Þ xðÞn ðt þ sn Þ ds ð3:70Þ RN

C
Y ð1t þ tÞ ¼

Z

X

RN

Y

P ðÞj

x

n¼1

n¼1

" ð 1Þ

p 1

ðp 1Þ!

#

p Y i¼1

( E

Y n2mi

) ðÞn

H

ðt þ tn ; t þ sn Þ

ð3:71Þ

ðt þ sj Þ ds:

j2mi

Consequently, in the stochastic process framework, for the same input signal x(t), one obtains diVerent output statistical functions depending on the choice of modeling such a signal as a sample path of a stochastic process [Eqs. (3.67) and (3.68)] or as a deterministic (with possible unknown

GENERALIZED ALMOST-CYCLOSTATIONARY SIGNALS

165

parameters) signal Eqs. (3.70) and (3.71)]. Such a choice, of course, influences the possible match between the output signal statistical functions and their asymptotic estimators. It is worthwhile to note that a similar drawback is not present in the FOT approach adopted here. F. Countability of the Set of the Output Cycle Frequencies In this section, the countability of the set of the output cycle frequencies of a LTV system excited by a GACS signal is discussed (Izzo and Napolitano, 1995a,b, 2002c). Such a problem turns out to be relevant in the FOT probability approach since, in such a framework, the almost-periodic component extraction operator is the analogous of the stochastic expectation operator in the classical stochastic process framework (see Section II.B). Let us consider an LTV system excited by a time series x(t). By substituting the expression in Eq. (3.43) of the system temporal moment function into the input/output relationship [Eq. (3.54)] and accounting for Eqs. (2.26) and (2.36), one obtains the following expression for the Nth-order temporal moment function of the output time series y(t) XXZ Ry ð1t þ tÞ ¼ Gy ðl; tÞej2p’y ðl;tÞt S x;z ðlÞdl ð3:72Þ RN

y2Y z2Wx

from which, by taking the coeYcient of the periodic component at frequency a, one obtains the output CTMF:
 Ray ðtÞ ≜ Ry ð1t þ tÞe j2pat t XXZ ð3:73Þ ¼ Gy ðl; tÞ d’y ðl;tÞ a S x;z ðlÞdl: y2Y z2Wx

RN

Thus, the set of the Nth-order output cycle frequencies for each value of t is given by Ay;t ≜ fa 2 R : Ray ðtÞ 6¼ 0g   Z ¼ [ [ a2R: Gy ðl; tÞ d’y ðl;tÞ a S x;z ðlÞdl 6¼ 0 : y2Y z2Wx

R

ð3:74Þ

N

An alternative expression for the output CTMF can be obtained by substituting Eq. (3.43) into Eq. (3.54), taking the sine wave component at frequency a, and accounting for Eqs. (2.48) and (2.51): Z XXZ 0 a Ry ðtÞ ¼ Gy ð½l0 ;
a z ðu0 Þ l ⊤ 1; tÞRx;z ðu0 Þ N 1 N 1 R ð3:75Þ y2Y z2Wx R 0⊤ 0 0 j2pl u 0 d’y ð½l0 ;
a z ðu0 Þ l0 ⊤ 1;tÞ a du dl : e

166

IZZO AND NAPOLITANO

In the following, the general case of LTV systems and the special case of FOT deterministic LTV systems will be considered in order to discuss the countability of the cycle frequency set of the output lag-product waveforms. To this end, two subclasses of GACS input signals will be considered. The former is the class of the GACS signals not containing any ACS component, that is, such that none of the lag-dependent (moment) cycle frequencies az(t) is constant in a set of values of t with nonzero Lebesgue measure in RN (see Section II.C.1). The latter is the subclass of the ACS signals, that is, such that all the lag-dependent cycle frequencies are constant with t. 1. Analysis of LTV Systems a. GACS Input Not Containing ACS Components. Let us assume that the input signal x(t) is GACS not containing ACS components and that the LTV system is suYciently regular so that the functions Gy(l, t) do not contain impulses in l. Such an assumption on the system means that we are not considering ideal resonator systems, that is, systems such that the output lag product contains finite-strength additive sine waves only in correspondence of particular values of frequencies in the input lag product. Since x(t) does not contain ACS components, S x,z(l) does not contain impulsive terms in l (see Section II.C.2) and, hence, nonempty sets in the right-hand side of Eq. (3.74) can be obtained only if the supports of the integrand functions have nonzero Lebesgue measure in RN. Consequently, the following inclusion relationship holds: Ay;t  [ Ayt

ð3:76Þ

y2Y

where Ayt ≜ fa 2 R : measN fl 2 RN : ’y ðl; tÞ ¼ ag 6¼ 0g

ð3:77Þ

with measN{
} denoting the Lebesgue measure in R . For fixed values of y and t, ð3:78Þ ’y ðl; tÞ ¼ a N

is the equation of an N-dimensional manifold in the (N þ 1)-dimensional space (a, l). Therefore, for each set L  RN such that measN {l 2 L : ’y(l, t) ¼ a} > 0, the corresponding set of values of a satisfying Eq. (3.78) must be at most countable. In other words, each set Ayt is at most countable and hence, accounting for Eq. (3.76), the set Ay,t is at most countable, that is, the output time series y(t) is GACS (or possibly a zero-power signal if, for N ¼ 2, (*)1 absent, and (*)2 present, Ay,t is an empty set for all t 2 R2). b. ACS input. In the case where the input signal x(t) is ACS, there exists a one-to-one correspondence between the elements z 2 Wx and the cycle

GENERALIZED ALMOST-CYCLOSTATIONARY SIGNALS

167

frequencies g 2 Ax and, moreover, the GCSMF S x;z ðlÞ can be expressed by Eq. (2.53), which substituted into Eq. (3.73), leads to XXX X Z Ray ðtÞ ¼ Gy ðvm ðl; bÞ; tÞ d’y ðvm ðl;bÞ;tÞ a N p y2Y g2Ax P b⊤ 1¼g R p Y bm Pxmii ðl0mi Þdl0m1 . . . dl0mp i¼1

ð3:79Þ

where vm ðl; bÞ ≜ ½l0m1; bm1 l0m1 ⊤ 1; . . . ; l0mp; bmp l0mp ⊤ 1 with m ≜ ½m1 ; . . .; mp ⊤ and b ≜ ½bm1; . . .; bmp ⊤ . Moreover, under the mild assumption that the time series x(t) and x(t þ t) are asymptotically (jtj ! 1) independent in the FOT probability sense (see Section II.B), the cyclic polyspectra bm Pxmii ðl0mi Þ are not impulsive (Gardner and Spooner, 1994) and, hence, the following inclusion relationship holds: Ay;t  [

[

[

[

y2Y g2Ax m2P b2Bm1 


Bmp

Aðy;g;m;bÞ t

ð3:80Þ

where Bmi denotes the set of jmijth-order cumulant cycle frequencies bmi of {x(*)n(t), n 2 mi}, and n n ðy;g;m;bÞ At ≜ a 2 R : measN p ½l0m1 ;


; l0mp  2 RN p : o o ð3:81Þ ’y ðvm ðl; bÞ; tÞ ¼ a; b⊤ 1 ¼ g > 0 : Thus, reasoning as in the case of GACS input, it follows that each ðy;g;m;bÞ is empty or countable and, hence, the set Ay,t is in turn empty set At or countable, that is, the output time series y(t) is GACS (or possibly zero power). As an example, let us consider the Doppler channel existing between a transmitter and a receiver with nonzero relative acceleration, that is, a purely random LTV system [see Eq. (3.27)] characterized by the impulseresponse function hðt; uÞ ¼ dðu t þ DðtÞÞ

ð3:82Þ

where DðtÞ ≜ d0 þ d1 t þ d2 t2 ;

d2 6¼ 0:

ð3:83Þ

The almost-periodic component of the output lag product when the input ⊤ lag product is e j2pl (1tþs) is given by [see Eq. (3.41)] n o ð Þ⊤ ⊤ ð Þ Efag fLy; l ð1t þ tÞg ¼ e j2pl ð1tþtÞ Efag e j2pl D ð1tþtÞ ð3:84Þ

168

IZZO AND NAPOLITANO

where Dð Þ ð1t þ tÞ ≜ ½ð Þ1 Dðt þ t1 Þ; . . . ; ð ÞN Dðt þ tN Þ⊤ . Moreover, in the special case of D(t) given by Eq. (3.83), it results that n o ð Þ ⊤ ð Þ ⊤ ð Þ ð Þð2Þ Efag e j 2pl D ð1tþtÞ ¼ e j2pl ½d0 1 þd1 t þd2 t  ð3:85Þ ð Þ ⊤ ð Þ e j2pl ½d1 1 þ2d2 t t dl⊤ 1ð Þ d2 where 1ð Þ ≜ ½ð Þ1 1; . . . ; ð ÞN 1⊤ ; tð Þ ≜ ½ð Þ1 t1 ; . . . ; ð ÞN tN ⊤ ; tð Þð2Þ ≜ ½ð Þ1 t21 ; . . . ; ð ÞN t2N ⊤ , and the fact that Z þ1 cosðat þ bt2 Þ dt; b 6¼ 0 0

is finite, and, hence

D E 2 e j 2pðatþbt Þ ¼ da db

ð3:86Þ

t

has been accounted for. From Eqs. (3.84) and (3.85) and accounting for Eq. (3.41), it follows that for the LTV system with impulse-response function Eq. (3.82), Eq. (3.83), the set Y contains only one element and, for d2 6¼ 0, Gy ðl; tÞ ¼ e j2pl

ð Þ⊤

½ð1 d1 Þt d2 tð2Þ 

’y ðl; tÞ ¼ 2d2 lð Þ⊤ t;

ð3:87Þ

dlð Þ⊤ 1

for lð Þ⊤ 1 ¼ 0:

ð3:88Þ

Thus, accounting for Eq. (3.43), it results that Z ð Þ ⊤ Rh ð1t þ t; uÞ ¼ e j2pl ½u F ðt;tÞ dlð Þ⊤ 1 dl RN

ð3:89Þ

where Fð Þ ðt; tÞ ≜ ½ð Þ1 Fðt; t1 Þ; . . . ; ð ÞN Fðt; tN Þ⊤ ¼ ð1 d1 Þtð Þ d2 tð Þð2Þ 2d2 tð Þ t

ð3:90Þ

Fðt; tÞ ≜ ð1 d1 Þt d2 t2 2d2 tt:

ð3:91Þ

with ⊤

ð Þ

Note that, even if F( )(t, t) is not function of 1t þ t; ej2pl F ðt;tÞ dlð Þ⊤ 1 is. By substituting the expression of the system temporal moment function Eq. (3.89) into the input/output relationship Eq. (3.54), one obtains that Z ⊤ ð Þ Ry ð1t þ tÞ ¼ ej2pl F ðt;tÞ dlð Þ⊤ 1 S x ðlÞ dl: ð3:92Þ RN

The presence of the Kronecker delta in the integrand function implies that the output TMF can contain finite-strength additive sine wave components

169

GENERALIZED ALMOST-CYCLOSTATIONARY SIGNALS

only if the input spectral moment function S x ðlÞ contains impulsive terms, that is, if the input signal contains an ACS component (see Section II.C.2). In particular, if the input signal is ACS, it results that X X S x ðlÞdlð Þ⊤ 1 ≜ S ax ðlÞdlð Þ⊤ 1 ¼ Sxa ðl0 Þdða l⊤ 1Þdlð Þ⊤ 1 ð3:93Þ a2Ax

a2Ax

Sxa ðl0 Þ

where is the reduced-dimension cyclic spectral moment function (Gardner and Spooner, 1994), which can be impulsive and is given by Eq. (2.53). If all the optional conjugations are absent, from Eq. (3.93) and accounting for the identity dða l⊤ 1Þdl⊤ 1 ¼ dðl⊤ 1Þda

ð3:94Þ

S x ðlÞdl⊤ 1 ¼ Sx0 ðl0 Þdðl⊤ 1Þ:

ð3:95Þ

it follows that

Thus, by substituting Eq. (3.95) into Eq. (3.92) and accounting for Eq. (2.53), one obtains that Ry ð1t þ tÞ ¼

R

ej2pl

0⊤

½F0 ðt;t 0 Þ 1Fðt;tN Þ

9 > 3> > > p 1 = X X bmp Y b m 4 5 dl0 P0x ðl0 Þ þ Pxmp ðl0mp Þ Pxmii ðl0mi Þdðbmi l⊤ 1Þ mi > > > > i¼1 b⊤ 1¼0 > > P > > ; : p6¼1 8 > > > > <

RN 1

2

ð3:96Þ where, in the derivation, the sampling property of Dirac’s delta function has been accounted for and it is assumed, without loss of generality, that each partition is ordered such that mp always contains N as its last element. Thus, by indicating with ri the last element of each subset mi (i ¼ 1, . . ., N 1) and performing the inverse Fourier transforms in Eq. (3.96), one has 0 0 Ry ð1t þ tÞ ¼ Cx0 ðF 2 ðt; t Þ 1Fðt; tN ÞÞ X X bmp 4 þ Cxmp ðF0mp ðt; t 0mp Þ 1Fðt; tN ÞÞ P p6¼1

b⊤ 1¼0

3

p 1 Y bm e j2pbmi ½Fðt;tri Þ Fðt;tN Þ Cxmii ðF0mi ðt; t 0mi Þ 1Fðt; tri ÞÞ5 i¼1

ð3:97Þ

170

IZZO AND NAPOLITANO

where Cxb ðt0 Þ is the reduced-dimension cyclic temporal cumulant function, which is the inverse Fourier transform of the cyclic polyspectrum Pbx ð f 0 Þ (Gardner and Spooner, 1994); (see also Section I.C). Finally, by substituting the expression of F(t, t) given by Eq. (3.91) into Eq. (3.97) with tN ¼ 0 and taking the coeYcient of the finite-strength additive sinewave component at frequency a, one obtains the expression for the Nth-order reduced-dimension CTMF Ray ðt 0 Þ. For example, in the case of a zero mean signal x(t), that is, a signal not containing any finite-strength additive sine wave component and, moreover, such that x(t) and x(t þ t) are asymptotically (jtj ! 1) independent, for N ¼ 4, one has X 0 Rayyyy ðt1 ; t2 ; t3 Þ ¼ Cxxxx ð0; 0; 0Þdt1 dt2 dt3 da þ Rbxx2 ð0ÞRbxx1 ð0Þ ⊤

b 1¼0 h j2pb1 ½ð1 d1 Þt2 d2 t22  daþb1 2d2 t2 dt3 dt1 t2 e

ð3:98Þ

þ dt2 dt1 t3 ej2pb1 ½ð1 d1 Þt3 d2 t3  daþb1 2d2 t3 i 2

þ dt1 dt2 t3 ej2pb1 ½ð1 d1 Þt3 d2 t3  daþb1 2d2 t3 2

where, in the derivation of Eq. (3.98), the fact that the equality in Eq. (3.97) is in the temporal mean-square sense and the fact that, for an asymptotically independent time series x(t) it results that, for any order N, lim Cxb ðt0 tÞ ¼ 0;

jtj!1

8t0 6¼ 0

ð3:99Þ

have been accounted for. Note that, by comparing Eq. (3.98) with Eq. (2.30), it follows that the GCTMFs are discontinuous functions of t1, t2, and t3. Furthermore, for N ¼ 2 and t2 ¼ 0 one has Fðt; t1 Þ Fðt; t2 Þjt2 ¼0 ¼ ð1 d1 Þt1 d2 t21 2d2 t1 t

ð3:100Þ

from which we have 0 ðFðt; t Þ Fðt; 0ÞÞ Ryy ð1t þ ½t1 ; 0Þ ¼ Cxx 1 0 ðð1 d Þt d t2 2d t tÞ ¼ Cxx 1 1 2 1 2 1 0 ð0Þd d Rayy ðt1 Þ ¼ Cxx a t1

ð3:101Þ ð3:102Þ

which are coincident with the autocorrelation function and the cyclic autocorrelation function, respectively, in the case of x(t) real signal. In order to corroborate the eVectiveness of the previously presented theoretical results, a simulation experiment has been realized. In this experiment (and in those presented in the following), time has been discretized with sampling increment Ts ¼ T/M, where T is the data-record length and M is the number of samples (for a discussion on the aliasing issue, see Section IV.C). The parameters of the Doppler channel have been assumed to be d0 ¼ 0Ts,

GENERALIZED ALMOST-CYCLOSTATIONARY SIGNALS

171

d1 ¼ 4.5, and d2 ¼ 1.52676
10 3/Ts. Moreover, it has been selected, as input signal to the Doppler channel, the double side-band (DSB) signal xðtÞ ¼ sðtÞ cosð2pf0 tÞ

ð3:103Þ

where f0 ¼ 0.06/Ts and s(t) is a colored fourth-order wide-sense stationary signal that has been obtained by passing fourth-order wide-sense stationary white noise through the LTI filter H0( f ) ¼ (1 þ jf/B0) 8, with B0 ¼ 0.00035/Ts. It is well known that x(t) is wide-sense cyclostationary with fourth-order cyclic temporal moment functions nonzero only in correspondence of the cycle frequencies a ¼ )2f0, a ¼ )4f0 (and a ¼ 0) (Napolitano, 1995), Thus, the support in the (a, t1) plane of a slice with t2 ¼ t1 and t3 ¼ 0 of the reduced-dimension CTMF of x(t) is confined on the five lines with equations a ¼ 0, a ¼ )2f0, and a ¼ )4f0. Figure 7 shows (a) the magnitude and (b) the support in the (a, t1) plane of the slice for t2 ¼ t1, t3 ¼ 0 of the fourth-order reduced-dimension (i.e., t4 ¼ 0) CTMF of the input DSB signal x(t) estimated by M ¼ 211 samples. The estimated magnitude and support of the slice of the reduced-dimension CTMF of the output signal y(t) are reported in Figures 8a and b, respectively. The generalized cyclostationary nature of y(t) is evident. Note that, in Section IV.D it is explained that the GACS nature of a continuous-time signal can only be conjectured starting from the discrete-time signal constituted by its samples. 2. The Special Case of FOT Deterministic LTV Systems Let us now assume that the LTV system is FOT deterministic and suYciently regular so that the functions Gy(l, t) do not contain impulses in l. By substituting into Eq. (3.72) the expressions from Eqs. (3.57) and (3.58) of ’y(l, t) and Gy (l, t) for FOT deterministic LTV systems, one obtains N 



X Z Y   n ’ Ry ð1t þ tÞ ¼ ’_ sn ð Þn ln HsðÞ sn ð Þn ln n N ð3:104Þ s2ON R n¼1 ð Þ

S x ðlÞej2pws

ðlð Þ Þ⊤ ð1tþtÞ

dl

where Eqs. (2.51) and Eqs. (3.13) have been accounted for. Moreover, by using the change of variables ( )n ln ¼ csn( fn), with csn(
) being the inverse function of ’sn(
) (see Section III.B.2), Eq. (3.104) becomes X Ry ð1t þ tÞ ¼ Bx;s ð1t þ tÞ ð3:105Þ s2On

where

172

IZZO AND NAPOLITANO

FIGURE 7. Graph of the (a) magnitude and (b) support in the (a, t1) plane of the slice for t2 ¼ t1 and t3 ¼ 0 of the fourth-order reduced-dimension CTMF Raxxxx ðt1 ; t2 ; t3 Þ of the input DSB signal x(t).

GENERALIZED ALMOST-CYCLOSTATIONARY SIGNALS

173

FIGURE 8. Graph of the (a) magnitude and (b) support in the (a, t1) plane of the slice for t2 ¼ t1 and t3 and t3 ¼ 0 of the fourth-order reduced-dimension CTMF Rayyyy ðt1 ; t2 ; t3 Þ of the output y(t) to the purely random Doppler channel defined in Eqs. (3.82) and (3.83) and excited by the DSB signal x(t).

174

IZZO AND NAPOLITANO

Z Bx;s ð1t þ tÞ ≜

N



Y ⊤ ðÞ Hsn n ð Þn fn S x c ð Þ f ð Þ ej2pf ð1tþtÞ df s

RN n¼1

ð3:106Þ ð Þ with c ð Þ Þ ≜ ½ð Þ1 cs1 ðð Þ1 f 1 Þ; . . . ; ð ÞN csN ðð ÞN fN Þ⊤ . s ðf

a. GACS Input not Containing ACS Components. If hs ðtÞ 2 L2 ðRÞ 8s 2 O, then also Hs ð f Þ 2 L2 ðRÞ 8s 2 O. In addition, if x(t) is GACS not containing any ACS component so that the SMF S x ðlÞ is not impulsive in l (see ðÞn Section II.C.2) and, moreover, is measurable and bounded, then PN n¼1 Hsn N N ð Þ ð Þ ðð Þn fn ÞS x ðc s ð f ÞÞ 2 L2 ðR Þ and, hence, Bx;s ð1t þ tÞ 2 L2 ðR Þ (as a function of t). Consequently, if Bx;s ð1t þ tÞ is regular for ktk ! 1, then it is infinitesimal as ktk ! 1 and, hence, as j t j! 1. Therefore, Bx;s ð1t þ tÞ, as function of t, is a function with zero power so that the product Bx;s1 ð1t þ t1 ÞBx;s2 ð1t þ t2 Þ does not contain any finite-strength additive sinewave component. Thus, the TMF Ry ð1t þ tÞ is zero in the temporal mean-square sense. The same result is found if Hs ð f Þ 2 L1 ðRÞ and S x ðlÞ is measurable and bounded. Therefore, in general, FOT deterministic systems excited by GACS signals not containing any ACS component deliver signals with zero power. Such a result could not hold if some hs ðtÞ 2 = L2 ðRÞ ore some Hs ð f Þ 2 = L1 ðRÞ. In Section III.I the case in which hs(t) is impulsive is treated in detail [see also the example of Eq. (3.113)]. b. ACS Input. In the case where the signal x(t) is ACS, by substituting the expression of the GCSMF [Eq. (2.53)] valid for ACS signals into Eq. (3.73) specialized to the case of FOT deterministic systems [i.e., with Eqs. (3.57) and (3.58) substituted into] and accounting for the sampling property of Dirac’s delta function, one obtains that X XX X Z a Ry ðtÞ ¼ N g2A t s2O0

P b⊤ 1¼g

RN p

p Y 0⊤ ri @GðÞ sri ðð Þri ðbmi lmi 1ÞÞ i¼1 ej2pKm;s ðl;tÞ d

p Y

Y n2mi fri g

1 A n GðÞ sn ðð Þn ln Þ

ð3:107Þ

Km;s ðl;1Þ a

bm 0 0 Pxmii ðlmi Þdlm1

0

. . . dlmp

i¼1

with Km;s ðl; tÞ ≜

p X i¼1

0 0 ð Þ ⊤ 0 ½ð Þri ’sr ð Þri ðbmi lm⊤i 1Þ tri þ ws0 ðlð Þ mi Þ tmi  ð3:108Þ i

mi

ð Þ

where ri denotes the last element in mi and wsmi ðlð Þ mi Þ is the jmij-dimensional

GENERALIZED ALMOST-CYCLOSTATIONARY SIGNALS

175

vector whose elements are ð Þn ’sn ðð Þn ln Þ with n 2 mi. Moreover, under the mild assumption that the time series x(t) and x(t þ t) are asymptotically bm ðjtj ! 1Þ independent (see Section II.B.), the cyclic polyspectra Pxmii ðl0mi Þ are not impulsive (Gardner and Spooner, 1994) and, hence, the following inclusion relationship holds: Ay;t  [

[

[

[

s2ON g2Ax m2P b2Bm1 ...Bmp

where ðs;g;m;bÞ

At

Aðs;g;m;bÞ t

n n ≜ a 2 R : measN p ½l0m1 ; . . . ; l0mp  2 RN p: o o Km;s ðl; 1Þ ¼ a; b⊤ 1 ¼ g > 0

ð3:109Þ

ð3:110Þ

which is independent of t and in the following will be denoted by A(s,g,m,b). In Eq. (3.110), the set corresponding to the partition with p ¼ N can be not empty only if x(t) contains finite-strength additive sine wave components. In fact, for p ¼ N it results that ( ) N

X ⊤ ðs;g;m;bÞ A ≜ a2R: ð Þn ’sn ð Þn bn ¼ a; b 1 ¼ g ð3:111Þ n¼1 ⊤

where m ¼ ½1; . . . ; N and b ¼ ½b1 ; . . . ; bN ⊤ with bn frequencies of the additive almost-periodic component in x(t). From the countability of the bn’s, the countability of the set A(s,g,m,b) follows immediately. With regard to the sets corresponding to partitions with p < N, Eq. (3.110) can be written as ( ( Aðs;g;m;bÞ ¼

a 2 R : measjmi j 1 l0mi 2 Rjmi j 1 :

0 ð Þ ð Þri ’sr ð Þri ðbmi l0mi ⊤ 1Þ þ ws0m ðlð Þ Þ⊤ 1 ¼ ki ; m i i i ) ) p X ki ¼ a b⊤1 ¼ g > 0; i ¼ 1; . . . ; p;

ð3:112Þ

i¼1

from which it follows that a suYcient condition to ensure that some set A(s,g,m,b) is nonempty is that some function ’s (
) contains a linear part with unit slope, that is, that the system contains a LAPTV component. In the special case where s ¼ s01, it is not necessary that the linear part has unit slope; that is, it is suYcient that the system contains a time-scale changing component. Finally, note that since the sets A(s,g,m,b) do not depend on t, if the output y(t) has finite power, then it is ACS. As an example, let us consider the Doppler channel existing between a transmitter and a receiver with constant relative speed, that is, an FOT

176

IZZO AND NAPOLITANO

deterministic LTV system characterized by the impulse-response function hðt; uÞ ¼ dðu t þ ðd0 þ d1 tÞÞ:

ð3:113Þ

Such a system introduces a linearly time-varying delay D(t) ¼ d0 þ d1t, that is, it performs a time-scale changing [see Eqs. (3.20) through (3.22)] and introduces a constant delay in the output signal: yðtÞ ¼ xðð1 d1 Þt d0 Þ ¼ xðð1 d1 ÞtÞ ! dðt d0 =ð1 d1 ÞÞ:

ð3:114Þ

By assuming, for the sake of simplicity, d0 ¼ 0 and by substituting Eqs. (3.20) through (3.22) into Eqs. (3.105) and (3.106), it results that Ry ð1t þ tÞ ¼ Rx ðsð1t þ tÞÞ

ð3:115Þ

Ray ðtÞ ¼ Ra=s x ðstÞ

ð3:116Þ

where s ≜ 1 d1 . Therefore, if the input signal is the DSB signal defined in Eq. (3.103), then the output signal y(t) is in turn DSB with cycle frequencies )2f0s and )4f0s. Figure 9 shows (a) the magnitude and (b) the support in the (a, t1) plane of the slice for t2 ¼ t1 and t3 ¼ 0 of the reduced-dimension fourth-order CTMF of the output DSB signal y(t) estimated by M ¼ 211 samples. The input DSB signal has been assumed to be the same as that considered in the experiment described in Section III.F.1. and the Doppler channel parameters have been fixed at d0 ¼ 0Ts and d1 ¼ 1:7. The simulation results are in accordance with the theoretical results. Finally, it is worthwhile to note the diVerent behavior of the two Doppler channels Eqs. (3.82) and (3.83) and (3.113)] in the presence of an ACS input signal. Specifically, the Doppler channel [Eqs. (3.82) and (3.83)] is a FOT purely random LTV system and, then, the output signal is a GACS signal. On the contrary, the Doppler channel Eq. (3.113) is a FOT deterministic LTV system and, hence, the output signal is an ACS signal. G. LAPTV Filtering This section considers the LAPTV filtering of GACS signals (Izzo and Napolitano, 2000a, 2002c). Let us consider an LAPTV system, that is, an FOT deterministic LTV system with impulse-response function X hðt; uÞ ¼ hs ðt uÞe j2psu ð3:117Þ s2O

where O is the countable set of the frequency shifts introduced by the system.

GENERALIZED ALMOST-CYCLOSTATIONARY SIGNALS

177

FIGURE 9. Graph of the (a) magnitude and (b) support in the (a, t1) plane of the slice for t2 ¼ t1 and t3 ¼ 0 of the fourth-order reduced-dimension CTMF Rayyyy ðt1 ; t2 ; t3 Þ of the output y(t) to the FOT deterministic Doppler channel defined in Eq. (3.113) and excited by the DSB signal x(t).

178

IZZO AND NAPOLITANO

Accounting for Eqs. (3.12) and (3.18), it results that ’s ðlÞ ¼ l þ s;

s2O

ð3:118Þ

and ð3:119Þ Gs ðlÞ ¼ Hs ðl þ sÞ; s 2 O where the functions Hs(
) are Fourier transforms of the functions hs(
). Then, by specializing Eqs. (3.57) and (3.58), one has ’y ðl; tÞ ¼ l⊤ 1 þ sð Þ⊤ 1

ð3:120Þ

and Gy ðl; tÞ ¼

N Y * + j2pðlþsð Þ Þ⊤ t n ð Þ l þ s HsðÞ n e n n n

ð3:121Þ

n¼1

where s( ) ≜ [( )1 s1, . . ., ( )N sN]⊤. The case of LAPTV systems excited by ACS signals is widely treated in Napolitano (1995). The general result, valid for any FOT deterministic LTV system, that the output TMF is zero in the temporal mean-square sense when the input is a GACS signal not containing any ACS component (see Section III.F.2) can be analyzed in more detail in the case of LAPTV systems. Let us denote by Dx,y the union of the two sets Dx and Dy, where Dx [Dy] is the set of the points t 2 RN such that some of the diVerent input [output] lag-dependent cycle frequencies assume the same value (see Section II.C.1). Thus, for all t not belonging to Dx,y, it results that the GCTMF of the output time series y(t) can be written as Ry;
ðtÞ ≜ hLy ð1t þ tÞe j2pb
ðtÞt it X ¼ hBx;s ð1t þ tÞe j2pb
ðtÞt it N s2O # *" + N Y X X j2psð Þ⊤ ð1tþtÞ ðÞn j2pb
ðtÞt ¼ Rx;z ð1t þ tÞe ! hsn ðtn Þ e t

s2ON z2Wx

¼

X h

s2O

Rb s x

ð Þ⊤1

ðtÞe

j2psð Þ⊤ t

i

! t

N

n¼1

  ðÞn hsn ðtn Þ

N Y n¼1

t

b¼b
ðtÞ

ð3:122Þ which, in the case of LTI systems, reduces to Ry;
ðtÞ ¼

Rbx ðtÞ ! t

N Y n¼1

hðÞn ðt

   n Þ

b¼b
ðtÞ

:

ð3:123Þ

GENERALIZED ALMOST-CYCLOSTATIONARY SIGNALS

179

In Eqs. (3.122) and (3.123), !t denotes N-dimensional convolution with respect to t and, in the derivation of the second and fourth equality in Eq. (3.122), Eqs. (3.105) (specialized to LAPTV systems) and (2.30), respectively, have been accounted for. Moreover, for all t 2 Dx;y it results that Ry;
ðtÞ ¼

lim

Dt!0 tþDt2RN Dx;y

Ry;
ðt þ DtÞ:

ð3:124Þ

Taking into account Eq. (2.30) and the formal relationship dt t0 ! dðtÞ ¼ dt t0

ð3:125Þ

Equations (3.122) through (3.124) reveal that the output GCTMFs can be not identically zero only if the input time series contains ACS components (in which case the output time series is ACS) unless some function hsn(
) (or h (
)) contains impulsive terms, as it occurs in the case of systems introducing constant time delays or frequency shifts. Therefore, every LAPTV (or, in particular, LTI) filtering of a GACS signal not containing ACS components delivers a signal with all the GCTMFs identically zero, that is, with zero power. Indeed, GACS signals not containing ACS components exhibit timeaveraged autocorrelation function Rxx ðt1 ;0Þ containing the additive term x
2 dt1 (see Section II.C.1), so that the power x
2 is uniformly spread over an infinite bandwidth. Then, the output of every filter Hs( f ) with finite bandwidth is a signal with zero power. To corroborate such a result by an example, let us consider the output y(t) of a LTI filter with transfer function Hð f Þ ¼ ð1 þ jð f fh Þ=Bh Þ 4 þ ð1 þ jð f þ fh Þ=Bh Þ 4 , where fh ¼ 0:015=Ts and Bh ¼ 0:005=Ts : The input GACS signal x(t) is that obtained at the output of the Doppler channel considered in Section III.F.1. The estimate of the magnitude of the cyclic a autocorrelation function Ryy* (t1) of y(t) is reported in Figure 10 and as a function of the lag t1 and the cycle frequency a, for a data-record length of 211 and 214 points, respectively. The approach of the estimates to the identically zero value as the data-record length increases is evident. Finally, let us observe that in several estimation problems signals can be modeled as ACS or GACS depending on the data-record length. This fact puts some limitations in the performances obtainable with some signal processing algorithms adopted in communication applications where ACS signals are processed by LAPTV systems (see, e.g., the cyclic Wiener filtering for interference removal; Gardner, 1994). In fact, if the data-record length is increased too much in order to gain a better immunity against the eVects of noise and interference, it can happen that the ACS model for the input signal is not appropriate anymore but, rather, a GACS model needs to be considered since possible time variations of timing parameters of the signals (not

180

IZZO AND NAPOLITANO

FIGURE 10. Graph of magnitude of the cyclic autocorrelation function Rayy ðt1 Þ of the output y(t) to a LTI system excited by a GACS input signal, estimated by a data-record length of (a) 211 and (b) 214 samples.

evidenced with smaller data-record length) must be taken into account (see, e.g., the example in Section II.F.2). Consequently, increasing the datarecord length too much does not have, for example, the beneficial eVect of improving the reliability of the output-signal cyclic statistic estimates but, rather, gives rise to cyclic statistics (and generalized cyclic statistics) that are asymptotically zero. Therefore, there exists an upper limit to the maximum

GENERALIZED ALMOST-CYCLOSTATIONARY SIGNALS

181

usable data-record length and, consequently, there exists a limit to the minimum acceptable signal-to-noise ratio for cyclostationarity-based algorithms which are, in principle, intrinsically immune to the eVects of noise and interference, provided that the data-record length approaches infinity. The identically zero (generalized) cyclic statistics of the LAPTV filtered GACS signals are consequence of the properties of the single observed time series (e.g., the possible time variation of a timing parameter, such as carrier frequency or baud rate). In contrast to this, statistical functions of a stochastic process can be identically zero as a consequence of the presence, in the stochastic process model, of a random parameter whose eVect is to make the stochastic expectations equal to zero. In such a case, however, in general the stochastic process is not ergodic and, hence, the FOT framework is more appropriate. H. Product Modulation Let us consider the LTV system that performs a product modulation of the input GACS time series x(t), that is, whose input/output relationship is yðtÞ ¼ cðtÞxðtÞ

ð3:126Þ

where c(t) is a GACS time series statistically independent of x(t) in the FOT probability sense and with Nth-order TMF X Rc ð1t þ tÞ ¼ Rc;x ðtÞe j2pgx ðtÞt : ð3:127Þ x2Wc

The system impulse-response function is hðt; uÞ ¼ cðtÞdðt uÞ:

ð3:128Þ

Then, the system is purely random [see Eq. (3.27)] if c(t) does not contain finite-strength additive sine waves. Accounting for Eqs. (3.126) and (3.127), the almost-periodic component ⊤ of the output lag product corresponding to the input lag product e j2pl (1tþs) is given by X ⊤ ⊤ Rc;x ðtÞe j2pl t e j2p½gx ðtÞþl 1t : ð3:129Þ Efag fLy;l ð1t þ tÞg ¼ x2Wc

Then, by comparing Eq. (3.129) with Eq. (3.41), it results that there is a one-to-one correspondence between the elements y of the set Y and the elements x of Wc and, moreover, ’y ðl; tÞ ¼ gx ðtÞ þ l⊤ 1

ð3:130Þ

182

IZZO AND NAPOLITANO ⊤

Gy ðl; tÞ ¼ Rc;x ðtÞej 2pl t :

ð3:131Þ

Furthermore, from Eq. (3.43) it follows that the temporal moment function of the product modulation transformation can be written as Rh ð1t þ t; uÞ ¼ Rc ð1t þ tÞdð1t þ t uÞ

ð3:132Þ

where d(u) is N-dimensional Dirac’s delta function. Hence, accounting for Eq. (3.54), the input/output relationship in terms of TMFs is Ry ð1t þ tÞ ¼ Rc ð1t þ tÞRx ð1t þ tÞ

ð3:133Þ

which is formally analogous to that obtained in the stochastic process framework. Note that, Eq. (3.132) confirms the interpretation (see Section III.C) of Rh(1t þ t, u) as the expectation (in the FOT sense) of the lag product of the impulse-response function. In fact, Rc(1t þ t) is just the almost-periodic component (that is, the expectation in the FOT sense) of the lag product of the signal c(t). By substituting Eq. (3.127) into Eq. (3.133) and taking into account that the expression of the Nth-order TMF of x(t) is X Rx ð1t þ tÞ ¼ Rx;z ðtÞe j 2paz ðtÞt ; ð3:134Þ z2Wx

one obtains that the potential lag-dependent cycle frequencies of the output GACS time series are b
ðtÞ ¼ az ðtÞ þ gx ðtÞ;

ðz; xÞ 2 Wx  Wc

ð3:135Þ

Ry;
ðtÞ ≜ hRy ð1t þ tÞe j2pb
ðtÞt it X X ¼ Rx;z ðtÞRc;x ðtÞdb
ðtÞ ½az ðtÞþgx ðtÞ

ð3:136Þ

and, moreover,

z2Wx x2Wc

for all t 2 = Dx [ Dy . Note that, in the special case in which x(t) and c(t) are both ACS time series, Eq. (3.136) reduces to Eq. (29) of Spooner and Gardner (1994). Finally, let us observe that when the time series c(t) is almost periodic, the transformation Eq. (3.126) is LAPTV (and hence FOT deterministic) and therefore, according to Eq. (3.56), it results that Rc ð1t þ tÞ ¼

N Y cðÞn ðt þ tn Þ: n¼1

ð3:137Þ

GENERALIZED ALMOST-CYCLOSTATIONARY SIGNALS

183

Moreover, accounting for Eqs. (3.59) and (3.133), the input/output relationship in terms of temporal cumulant functions is the same as that in terms of temporal moment functions: Cy ð1t þ tÞ ¼ Rc ð1t þ tÞCx ð1t þ tÞ:

ð3:138Þ

I. Multipath Doppler Channels Let us consider the reflection of a transmitted signal xðtÞ ¼ Ref~ xðtÞe j2pfc t g

ð3:139Þ

on a target, embedded in a homogeneous medium, that can be described by the slowly fluctuating point target model (Van Trees, 1971). The received signal is given by yðtÞ ¼ Ref~ yðtÞej2pfc t g ¼ Refb~ xðt DðtÞÞej2pfc ðt DðtÞÞ g

ð3:140Þ

where Re{
} denotes real part, b is the complex attenuation, and D(t) is the round-trip time-varying delay introduced by the channel: , 2 DðtÞ DðtÞ ¼ R t : ð3:141Þ c 2 In Eq. (3.141), R(t) is the time-varying distance between the transmitter and the target at time t and c is the medium propagation speed. In the case of constant radial speed w of the target with respect to the transmitter, i.e., if R(t) ¼ R0 þ wt, then the delay D(t) depends linearly on t, say D(t) ¼ d0 þ d1t, where d0 ≜ 2R0/(c þ w) and d1 ≜ 2w/(c þ w). Thus, the ~ðtÞ by the LTV transformation with impulsesignal ~ yðtÞ is obtained from x response function hðt; uÞ ¼ a dðu st þ d0 Þe j2pnt

ð3:142Þ

where s ≜ 1 d1 is the time-scale factor, n ≜ d1 fc is the Doppler shift introduced by the channel, and a ≜ be j2pfd0. In the special case where fc ¼ 0 and a ¼ 1, this channel reduces to that considered in the example in Section III.F.2. Moreover, in the case of multiple slow fluctuating point scatterers, the LTV system is a multipath Doppler channel with impulse-response function hðt; uÞ ¼

K X

ak dðu sk t þ dk Þej2pnk t

k¼1

where K is the number of the channel paths.

ð3:143Þ

184

IZZO AND NAPOLITANO

By double Fourier transforming the right-hand side of Eq. (3.143), one obtains the transmission function defined in Eq. (3.10) Hð f ; lÞ ¼

K X ak e j2pldk dð f nk lsk Þ

ð3:144Þ

k¼1

from which, accounting for Eq. (3.11a), it follows that the multipath Doppler channel is an FOT deterministic LTV system with O ¼ {1, . . . , K}, ’k ðlÞ ¼ sk l þ nk ; Gk ðlÞ ¼ ak e j2pldk ;

k ¼ 1; . . . ; K

ð3:145Þ

k ¼ 1; . . . ; K:

ð3:146Þ

Equivalently, accounting for Eq. (3.12), we get 1 ck ð f Þ ¼ ð f nk Þ; k ¼ 1; . . . ; K sk Hk ð f Þ ¼

ak j2pð f nk Þdk =sk e ; jsk j

k ¼ 1; . . . ; K


ð3:147Þ ð3:148Þ

Therefore, by substituting Eqs. (3.145) and (3.146) into Eqs. (3.57) and (3.58) and then the result into Eq. (3.43), one obtains the following expression for the system temporal moment function of the multipath Doppler channel (Izzo and Napolitano, 2000b, 2002c): N

X Y ð Þ⊤ ðÞ Rh ð1t þ t; uÞ ¼ akn n e j2pnk ð1tþtÞ dðsk + ð1t þ tÞ d k uÞ k2IKN

n¼1

ð3:149Þ ð Þ

where k ≜ ½k1 ; . . . ; kN ⊤ , sk ≜ ½sk1 ; . . . ; skN ⊤ , d k ≜ ½dk1 ; . . . ; dkN ⊤ , nk ≜ ½ð Þ1 nk1 ; . . . ; ð ÞN nkN ⊤ , IK ≜ {1, . . . , K}, and + denotes the Hadamard matrix product. By substituting Eq. (3.149) into Eq. (3.54), one obtains the TMF of the output signal N

X Y ð Þ⊤ ðÞ R~y ð1t þ tÞ ¼ akn n ej2pnk ð1tþtÞ Rx~ ðsk + ð1t þ tÞ d k Þ ð3:150Þ k2IKN

n¼1

where the equality must be intended in the temporal mean-square sense (see Appendix A). In general, the time-scale factors of diVerent paths are diVerent, that is, sh 6¼ sk for h 6¼ k. Therefore, since the equality in Eq. (3.150) is in the temporal mean-square sense, not all terms in the right-hand-side of Eq. (3.150) give a contribution to the almost-periodic component of the ~(t) Nth-order output lag product. In fact, under the mild assumption that x

GENERALIZED ALMOST-CYCLOSTATIONARY SIGNALS

185

~(t þ t) are asymptotically (jtj ! 1) independent in the FOT probabiland x ity framework (see Section II.B), it results that if sh 6¼ sk for h 6¼ k, then, for t ¼ 1t0, jt0j ! 1, one has Rx~ ðs + ð1t þ tÞÞ !

N Y Efag f~ xðÞn ðsn t þ sn tn Þg

ð3:151Þ

n¼1

~(*)h(sht þ shth) and x ~(*)k(skt þ sktk) turn since for any h 6¼ k the time series x out to be time shifted of infinite quantities one another. Moreover, note that the same factorization of the TMF is obtained for jtj ! 1 since also in this ~(*)h(sht þ sh th) and x ~(*)k(skt þ sk tk) are time shifted of case the time series x infinite quantities one another. If at least one of the sk is diVerent from all the others, it results that even if not all the entries sk are diVerent, at least two of them are not equal, so that in the asymptotic (jtj ! 1) factorization of Rx~ ðs + ð1t þ tÞÞ there is at least one term of the kind E{a}{~ x(*)n (snt þ sntn)}. ~(t) does not contain any finite-strength additive sine wave compoThus, if x nent, then lim Rx~ ðs + ð1t þ tÞÞ ¼ 0:

jtj!1

ð3:152Þ

Consequently, when at least one of the sk is diVerent from all the others, the function of t Rx~ ðs + ð1t þ tÞÞ has zero power and, hence, the product Rx~ ðs1 + ð1t þ t1 ÞÞ
Rx~ ðs2 + ð1t þ t2 ÞÞ does not contain finite-strength additive sine wave components. Therefore, by assuming in (3.150) that at least one of the sk is diVerent from all the others, the terms with s 6¼ 1sn (that is, with k 6¼ 1kn) are zero in the temporal mean-square sense, so that Eq. (3.150) becomes R~y ð1t þ tÞ ¼

N

X Y ð Þ⊤ ðÞ ak n e j2pnk 1 ð1tþtÞ Rx~ ðsk ð1t þ tÞ 1dk Þ k2IK

ð3:153Þ

n¼1

where the equality is in the temporal mean-square sense and the fact that s1kn ¼ 1skn has been accounted for. Finally, from Eq. (3.153) we have Ra~y ðtÞ ¼

N

X Y ð Þ⊤ ðÞ ak n ej2pnk 1 t k2IK

X

zk 2Wx

n¼1

Rx;zk ðsk t 1dk Þdaz

k

ðsk tÞ ða 1ð Þ⊤ 1nk Þ=sk

N

X Y ð Þ⊤ ðÞ ¼ ak n e j2pnk 1 t k2IK

n¼1

ða 1ð Þ⊤ 1nk Þ=sk Rx~ ðsk t

1dk Þ

ð3:154Þ

186

IZZO AND NAPOLITANO

where 1ð Þ ≜ ½ð Þ1 1;


; ð ÞN 1⊤ and, in the derivation of the last equality, Eq. (2.30) has been accounted for. ~(t) is GACS so is ~yðtÞ and, moreover, From Eq. (3.154) it follows that if x the lag-dependent cycle frequencies of ~ yðtÞ are given by bz;k ðtÞ ¼ sk az ðtÞ þ 1ð Þ⊤ 1nk ;

z 2 Wx ; k 2 IK :

ð3:155Þ

~ðtÞ is ACS, y~ðtÞ is in turn ACS. Furthermore, in the special case in which x Let us assume that the observation interval for the estimation of the (generalized) cyclic statistics is [ T/2, T/2]. Thus, equation Eq. (3.150) or Eq. (3.153) must be used to describe the output behavior, depending on the value of the product bandwidth data-record length. In fact, for the signal ~ðtÞ with bandwidth W, it results that x ,, -2w ~ðstÞ ¼ x ~ 1 ~ðtÞ x t ’x ð3:156Þ cþw provided that the condition WT % 1 þ

c w

ð3:157Þ

is satisfied (Van Trees, 1971). Therefore, if such a condition holds for each of the K channel paths, then the time-scale factors sk can be considered unitary and, hence, the functions ’k(l) [see Eq. (3.145)] are linear with unit slope, that is, the multipath Doppler channel can be modeled as a LAPTV system. Consequently, all terms in Eq. (3.150) give nonzero contribution in the temporal mean-square sense and the CTMF Ra~y ðtÞ of the output signal is given by Ra~y ðtÞ ¼

N

ð Þ⊤ X Y ð Þ⊤ a n 1 ðÞ akn n e j2pnk t Rx~ k ðt d k Þ: k2IKN

ð3:158Þ

n¼1

On the contrary, when Eq. (3.157) is not satisfied, Eq. (3.153) holds and, hence, the CTMFs of the output signal are related to those of the input signal by Eq. (3.154). To corroborate the fact that a diVerent behavior of the (generalized) cyclic statistics can be predicted by Eq. (3.154) or Eq. (3.158) depending on the data-record length adopted for the CTMF estimates, the following experiment has been carried out. Let us consider the output of a two-path Doppler channel characterized by a1 ¼ 1, d1 ¼ 0, s1 ¼ 1, n1 ¼ 0, a2 ¼ 1, d2 ¼ 32Ts, s2 ¼ 0.9998, and v2 ¼ 2.49975
10 5 / Ts, where Ts is the sampling period, and the second path is obtained by considering a relative radial speed v2/c ¼ 10 4. The channel is excited by a binary pulse-amplitude modulated signal with bit rate 1/16Ts and full-duty-cycle rectangular pulse. The estimate

GENERALIZED ALMOST-CYCLOSTATIONARY SIGNALS

187

of the magnitude of the cyclic autocorrelation function of y(t) is reported in Figure 11a and b as a function of the lag t1 and the cycle frequency a, for a data-record length of 213 and 218 points, respectively. For T ¼ 213 Ts it results WT  (1 þ c/w2), then the terms with k 6¼ 1kn must be accounted

FIGURE 11. Graph of magnitude of the cyclic autocorrelation function Ryy a ðt1 Þ of the output y(t) to a multipath Doppler channel excited by a pulse-amplitude–modulated input signal, estimated by a data-record length of (a) 213 and (b) 218 samples.

188

IZZO AND NAPOLITANO

for according to Eq. (3.158) and hence, the channel can be modeled as LAPTV (see Figure 11a). On the contrary, according to Eq. (3.154), such terms disappear for T ¼ 218Ts since, in such a case, WT ’ 1.6 (1 þ c/w2) (see Figure 11b). Briefly, both the above analysis and the previous experiment show that if the product bandwidth data-record length is suYciently small, then the multipath Doppler channel can be modeled as an LAPTV system, since the eVects of the time scaling factors can be neglected. On the contrary, if the data-record length is increased (e.g., to obtain a high noise immunity in cyclostationarity-based algorithms) the time scaling factors must be taken into account and the system cannot be modeled as LAPTV but, rather, as FOT deterministic. In Napolitano (2003), for the output of a multipath Doppler channel with ACS input, the second-order spectral characterization is performed in the stochastic process framework. It is shown that, if at least one of the sk is diVerent from the others, then the output can be modeled as a spectrally correlated stochastic process whose Loe`ve bifrequency spectrum has the support in the bifrequency plane constituted by lines with slopes sk1/sk2. Moreover, if the values of the sk’s are unknown, then only the density of the Loe`ve bifrequency spectrum on lines with unit slope (sk1 ¼ sk2) can be consistently estimated. Finally, let us note that if the attenuation introduced by the channel cannot be assumed constant, the appropriate model is the Doppler-spread channel (or time-selective fading channel) for which the output corresponding to the input signal Eq. (3.139) is given by yðtÞ ¼ Ref~ yðtÞej 2pfc t g ¼ RefAðt DðtÞ=2Þ~ xðt DðtÞÞej2pfc ðt DðtÞÞ g

ð3:159Þ

(Van Trees, 1971), where A(t) is the time-varying attenuation. Therefore, the ~ðtÞej2pfc t into ~ LTV channel that transforms x yðtÞe j2pfct has impulse-response function hðt; uÞ ¼ Aðt DðtÞ=2Þdðu t þ DðtÞÞ:

ð3:160Þ

Consequently, the output statistics can be evaluated considering first the eVects of the time-varying delay (see the examples of Sections III.F.1 and 2) and then the eVects of the product modulation introduced by A(t D(t)/2) (see Section III.H). The model [Eq. (3.159)] was studied in Blanco and Hill (1979) in the stochastic process framework. Moreover, the special case of A(t) and D(t) periodic functions with the same period was considered in Duverdier and Lacaze (1996).

GENERALIZED ALMOST-CYCLOSTATIONARY SIGNALS

189

J. Summary In this section, the problem of linear time-variant filtering of GACS signals has been addressed in the nonstochastic framework. Systems have been classified as deterministic or random in the fraction-of-time probability framework. Moreover, the higher-order characterization of linear time-variant systems has been provided in the time domain by the system temporal moment function, which is the operator that transforms the almost-periodic component of the input lag product into the almost-periodic component of the output lag product. The linear systems have also been characterized in the frequency domain and input/output relationships have been provided in terms of both temporal and spectral moment and cumulant functions. The usefulness of the proposed approach has been demonstrated by the statistical characterization of those systems that in the stochastic approach are modelled as random. In fact, such systems, which include, for example, the fading communications channels, in general transform ergodic input signals into nonergodic output signals. The countability of the cycle frequency set of the output signal of a LTV system has been studied. Then, it has been shown that, unless the output signal is zero power, general LTV systems deliver GACS signals if they are excited either by ACS signals or GACS signals not containing any ACS component. The FOT deterministic LTV systems deliver ACS signals when excited by ACS signals, provided that they contain a LAPTV or a time-scale changing component. Moreover, they deliver an output zero-power signal when excited by GACS signals not containing any ACS component, unless the impulse-response function contains impulsive terms. The special cases of LAPTV filtering, product modulation, and Doppler channel filtering have been analyzed in detail. In the considered examples, it has been shown that in practical situations diVerent system or signal models should be adopted depending on the data-record length. With reference to LAPTV filtering, it has been shown that there exists an upper limit in the maximum usable data-record length and, consequently, there exists a limit to the minimum acceptable signal-to-noise ratio for cyclostationarity-based algorithms when the increasing of the data-record length makes for the input signal the GACS model more appropriate than the ACS one. With regard to the multipath Doppler channels, it has been shown that they can be modeled as LAPTV or FOT deterministic systems depending on the value of the product bandwidth data-record length. If the product bandwidth data-record length is suYciently small, then the multipath Doppler channel should be modeled as an LAPTV system, whereas, if the data-record length is not small (e.g., to obtain a high-noise immunity in cyclostationarity-based algorithms) the system cannot be modeled as LAPTV but, rather, as FOT deterministic. As a consequence, the output

190

IZZO AND NAPOLITANO

estimated (generalized) cyclic statistics obey to two diVerent models, depending on the available data-record length for the estimates. Simulation results for the described applications have been presented.

IV. SAMPLING

OF

GACS SIGNALS

A. Introduction The proper statistical characterization of the nonstationary signal at the output of a time-varying channel is the first step to properly perform detection, equalization, and demodulation. Thus, since in most of signal-processing applications continuous-time signals are subject to sampling operations, the link between continuous- and discrete-time is of great interest. This section addresses the problem of sampling a continuous-time GACS signal. It is shown that the discrete-time signal constituted by the samples of a continuous-time GACS signal is a discrete-time ACS signal, whether the continuous-time signal is ACS or not. Moreover, relationships between higher-order generalized cyclic statistics of a GACS signal and higher-order cyclic statistics of the discrete-time signal of its samples are determined and some results derived in Izzo and Napolitano (1996c) and Napolitano (1995) for the case of ACS signals are extended to the more general case of GACS signals. The problem of aliasing in the domain of the cycle frequencies is considered and a condition ensuring that the cyclic temporal moment function of the discrete-time signal can be obtained by sampling that of the continuous-time signal is determined. Spectral parameters of sampled GACS signals have not been considered here since a continuous-time GACS signal that is not ACS exhibits cyclic spectral moment functions that are infinitesimal (see Section II.C.2). Spectral parameters of sampled ACS signals are considered in Izzo and Napolitano (1996c) and Napolitano (1995). The continuous-time GACS signals, ACS or not, give rise to a sampled discrete-time ACS signal. Thus, the nonstationarity type of the sampled signal does not allow us to determine if the underlying continuous-time GACS signal is ACS or not unless further a priori information is available. However, it is shown how the GACS or ACS nature of the continuous-time signal can be conjectured from the behavior of the cyclic temporal moment function of the sampled ACS signal. Specifically, it is shown that, under some regularity conditions, the analysis parameters (e.g., sampling frequency, data-record length, padding factor) can be chosen such that the lag-dependent cycle frequency variations within a sampling period are

GENERALIZED ALMOST-CYCLOSTATIONARY SIGNALS

191

suYciently small so that a slice of the cyclic temporal moment function of a continuous-time GACS signal over a discrete grid for both lags and cycle frequencies can appear as if its support were piecewise continuous. That is, it can appear as if the lag-dependent cycle frequencies were varying with continuity with respect to the lag parameters. Consequently, if the lagdependent cycle frequencies are piecewise constant, then the underlying continuous-time signal should be conjectured to be ACS, otherwise, it should be conjectured to be GACS but not ACS. B. Discrete-Time ACS Signals In this section, some results on the strict and wide sense characterization in the FOT probability framework of discrete-time ACS signals are provided since they will be used in the sequel. Further results can be found in Izzo and Napolitano (1998a). For a treatment in the stochastic approach, see Giannakis (1998). Let us consider a discrete-time complex-valued finite-power time series ~ m;j of all frequencies of the finitex(k) ≜ xr(k) þ jxi(k), k 2 Z. If the set G strength additive sinewave components contained in the function of k N Y ~ x ð1k þ m; jÞ ≜ U Uðxrn xr ðk þ mn ÞÞUðxin xi ðk þ mn ÞÞ ð4:1Þ n¼1

is countable for each of the column vectors m ≜ ½m1 ; . . . ; mN ⊤ 2 ZN and j ≜ j r þ jj i ≜ [xr1 þ jxi1, . . ., xrN þ jxiN]⊤ 2 CN and, moreover, also the set ~m ≜ [ N G ~ m;j is countable for each m 2 ZN, then the time series is said to G j2C be Nth-order almost-cyclostationary in the strict sense. Note that, whereas in the continuous-time case the set G defined in Eq. (2.6) can be countable or ~ ≜ [m2ZN G ~ m is always not, in the discrete-time case, the corresponding set G countable. Consequently, all the discrete-time signals for which the function Eq. (4.1) contains finite-strength additive sinewave components are ACS in the strict sense. Therefore, unlike in the continuous-time case, in the discretetime case a class of GACS (in the strict sense) signals extending that of the ~ uncountable. ACS ones cannot be introduced by considering G It can be shown that the function F~ xðkþm1 Þ


xðkþmN Þ ðjÞ ≜ Ef~ag fU x ð1k þ m; jÞg X ~g ¼ F~ x ðm; jÞe j2p~gk

ð4:2Þ

~m ~g2G

is a valid joint cumulative distribution function for each fixed value of m except for the right-continuity property with respect to each xrn and

192

IZZO AND NAPOLITANO

~ . In xin variable. Moreover, the sum can be equivalently extended to G Eq. (4.2), K X 1 ~ x ð1k þ m; jÞe j2p~gk U k!1 2K þ 1 K¼ K

~ ~g ðm; jÞ ≜ lim F x

ð4:3Þ

and Ef~ag f
g is the almost-periodic component extraction operator for discrete-time series. Consequently, the 2Nth-order derivative of the joint cumulative distribution function Eq. (4.2) with respect to xr1, xi1, . . . , xrN, xiN turns out to be a valid joint probability density function almost periodic in k: f~xðkþm1 Þ


xðkþmN Þ ðjÞ ≜

@ 2N ~ xðkþm Þ


xðkþm Þ ðjÞ F 1 N @ xr1 @ xi1


@ xrN dxiN X ~g ~ fx ðm; jÞe j2p~gk ¼

ð4:4Þ

~m ~g2G

~g where each Fourier coeYcient f~x ðm; jÞ is the 2Nth-order derivative of the corresponding Fourier coeYcient [Eq. (4.3)] of the joint cumulative distribution function shown in Eq. (4.2). In the FOT probability framework, a discrete-time finite-power possibly complex-valued time series x(k) is said to exhibit Nth-order wide-sense cyclo-stationarity with cycle frequency ~ a2 = Z, for a given conjugation configuration, if the Nth-order CTMF K Y N X 1 xðÞn ðk þ mn Þe j2p~ak k!1 2K þ 1 k¼ K n¼1

~ ~a ðmÞ ≜ lim R x

ð4:5Þ

exists and is not zero for some column vector m ≜ ½m1 ;


; mN ⊤ 2 ZN . In Eq. (4.5), x ≜ [x(*)1 (k),


, x(*)N(k)]⊤, and the convergence of the infinite averaging with respect to k is assumed in the temporal mean-square sense or, more generally, in the sense of distributions (generalized functions). If the set ~ m ≜ f~ A a 2 ½ 1=2; 1=2½:

~ ~a ðmÞ 6¼ 0g R x

ð4:6Þ

is countable for each m, then the Nth-order lag product can be expressed as a sum of its almost-periodic component and a residual term not containing any finite-strength additive sine wave component, that is, ~ x ð1k þ mÞ ≜ L

N Y xðÞn ðk þ mn Þ n¼1

~ x ð1k þ mÞ þ ~‘x ð1k þ mÞ: ¼R In Eq. (4.7), the almost-periodic function

ð4:7Þ

GENERALIZED ALMOST-CYCLOSTATIONARY SIGNALS

( ~ x ð1k þ mÞ ≜ E R

f~ag

X

¼

~m ~a2A

N Y xðÞn ðk þ mn Þ

193

) ð4:8Þ

n¼1

~ ~a ðmÞe j2p~ak R x

which is called the temporal moment function, is a valid moment function and the residual term is such that K X 1 ~‘x ð1k þ mÞe j2p~ak 0; K!1 2K þ 1 k¼ K

lim

8~a 2 ½ 1=2; 1=2½:

ð4:9Þ

Moreover, it can be shown that ~ x ð1k þ mÞ ¼ R

R R

2N

N Y ðxrn þ jxin ÞðÞn

ð4:10Þ

n¼1

f~xðkþm1 Þ


xðkþmN Þ ðjÞdj r dj i : Note that, whereas in the continuous-time case the set A defined by Eq. (2.22) can be countable or not, in the discrete-time case, the corresponding set ~≜ [ A ~m A

ð4:11Þ

m2ZN

is always countable. Consequently, all the discrete-time signals for which the Nth-order lag product contains finite-strength additive sine wave components are ACS in the wide sense and the sum in Eq. (4.8) can be extended ˜. over the set A The Nth-order temporal cumulant function of a discrete-time complexvalued ACS signal x(k) can be expressed as " # p X Y p 1 ~ ~ Cx ð1k þ mÞ ¼ ð 1Þ ðp 1Þ! Rx ð1k þ mm Þ : ð4:12Þ mi

P

i

i¼1

Moreover, it turns out to be an almost-periodic function of k that can be written as X b~ ~ ~ ðmÞe j2pbk C~x ð1k þ mÞ ¼ ð4:13Þ C x ~ B ~ b2

where ~

K X 1 ~ ~x ð1k þ mÞe j2pbk C K!1 2K þ 1 k¼ K

b C~x ðmÞ ≜ lim

ð4:14Þ

194

IZZO AND NAPOLITANO

and the sets ~≜ [ B ~m B

ð4:15Þ

  ~ ~ 2 ½ 1=2; 1=2½: C ~b ðmÞ 6¼ 0 ~m ≜ b B x

ð4:16Þ

m2ZN

are countable.

C. Sampling of GACS Signals In this section, the sampling of GACS signals is considered and the link between higher-order generalized cyclic statistics of a continuous-time GACS signal and higher-order cyclic statistics of the discrete-time ACS signal constituted by its samples is established (Izzo and Napolitano, 2001, 2003). Let us consider the discrete-time signal x(k) constituted by the samples of the continuous-time GACS signal xc(t): xðkÞ ≜ xc ðtÞjt¼kTs

k 2 Z;

ð4:17Þ

where Ts denotes the sampling period. Accounting for Eq. (2.2), the sampled version of Eq. (2.4) can be written as ~ x ð1k þ m; jÞ ≜ U x ð1t þ t; jÞj U c t¼kTs ;t¼mTs X ¼ F gxc ðmTs ; jÞe j2pgkTs

ð4:18Þ

g2GmTs

þ‘U ð1kTs þ mTs ; jÞ where U xc ð1t þ t; jÞ and Fxgc ðt; jÞ are defined by Eqs. (2.1) and (2.3), respectively, and the fact that ‘U(t, t;j ) ‘U(1t þ t;j ) was used [see the observation following Eq. (2.12)]. The sampled residual term ‘U(1kTs þ mTs; j) does not contain any finite-strength additive (discrete-time) sine wave component, that is (see Appendix D), K X 1 ‘U ð1kTs þ mTs ; jÞ e j2p~gk 0 K!1 2K þ 1 k¼ K

lim

8~g 2 ½ 1=2; 1=2½: ð4:19Þ

Thus, the almost-periodic function (with respect to k) in Eq. (4.18) is ~ x ð1k þ m; jÞ, that is, coincident with the almost-periodic component of U ~ xðkþm Þ


xðkþm Þ ðjÞ defined by (4.2). with F 1

N

GENERALIZED ALMOST-CYCLOSTATIONARY SIGNALS

195

The Fourier coeYcients of the joint cumulative distribution function of xc(t) are related to those of the joint cumulative distribution function of the discrete-time signal x(k) by the following relationship (see Appendix D): ~ ~g ðm; jÞ ¼ F x

þ1 X

Fxð~gc þpÞfs ðmTs ; jÞ

ð4:20Þ

p¼ 1

where fs ≜ 1/Ts is the sampling frequency. Furthermore, by taking the 2Nthorder derivative with respect to xr1, xi1, . . . , xrN, xiN of both sides of Eq. (4.20), one obtains the relationship between the Fourier coeYcients of the joint probability density function of xc(t) and those of the joint probability density function of the discrete-time signal x(k): ~g f~x ðm; jÞ ¼

þ1 X

fxð~cgþpÞfs ðmTs ; jÞ

ð4:21Þ

p¼ 1

where fxgc ðt; jÞ is defined by (2.9). The sampled signal x(k) is ACS in the strict sense whether the continuoustime GACS (in the strict sense) signal xc(t) is ACS or not, that is, independently of the fact that the set G ≜ [t 2 R Gt is countable or not. Such a result follows from the relation (proved in Appendix D) X ~g ~ x ð1k þ m; jÞ ¼ ~ ðm; jÞ ej2p~gk þ ‘U ð1kTs þ mTs ; jÞ U F ð4:22Þ x ~m ~g2G

where ~ m ≜ f~g 2 ½ 1=2; 1=2½: F ~ ~g ðm; jÞ 6¼ 0 for some jg G x

ð4:23Þ

~ m. ~≜[ NG and the sum in Eq. (4.22) can be equivalently made over G m2Z To enlighten the wide-sense cyclostationarity properties of the sampled signal x(k) and their relationship with the generalized cyclostationarity properties of the subsumed GACS signal xc(t), it is appropriate to start by considering the sampled version of Eq. (2.19), that is ~ x ð1k þ mÞ ¼ Lxc ð1t þ tÞj L t¼kTs ;

t¼mTs

~ 0 ð1k þ mÞ þ ~‘0 ð1k þ mÞ ¼R x x

ð4:24Þ

where, accounting for Eq. (2.20), the residual term ~‘0 ð1k þ mÞ ≜ ‘x ð1t þ tÞj c t¼kTs ; x

t¼mTs

ð4:25Þ

does not contain any finite-strength (discrete-time) additive sine wave component (the proof is similar to that provided for Eq. (4.19) in Appendix D). ~ 0 ð1k þ mÞ, accounting for Eqs. (2.16) and (2.26), can be The function R x

196

IZZO AND NAPOLITANO

written as ~ 0 ð1k þ mÞ ≜ Rx ð1t þ tÞj R c t¼kTs ; t¼mTs x ¼

X

Raxc ðmTs Þe j 2pakTs

ð4:26aÞ ð4:26bÞ

a2AmTs

¼

X

Rxc ;z ðmTs Þe j2paz ðmTs ÞkTs :

ð4:26cÞ

z2W 0

~ ð1k þ mÞ ≜ is an almost-periodic function of k. Furthermore, since Thus, R x ~‘0 ð1k þ mÞ ≜ does not contain any finite-strength additive sine wave comx ~ 0 ð1k þ mÞ is coincident with R ~ x ð1k þ mÞ, which is the alponent, then R x ~ x ð1k þ mÞ (see (4.7)). most-periodic component contained in L Let us note that the set AmTs in Eq. (4.26b), accounting for Eqs. (2.15) and (2.25), can be written as AmTs ≜ fa 2 R : Raxc ðmTs Þ 6¼ 0g

[ fa 2 R : a ¼ az ðmTs Þg z2W

~0 g ¼ [ fa 2 R : a ¼ ð~ a þ pÞfs ; ~a 2 A m

ð4:27Þ

p2Z

~ 0 is defined by where, A m ~ 0 ≜ f~ a 2 ½ 1=2; 1=2½: ~ a ¼ ða=fs Þ mod 1; a 2 AmTs g: A m

ð4:28Þ

In Eq. (4.28), mod b denotes the modulo b operation with values in [ b/2, b/2[, that is,  a Mod b a Mod b 2 ½0; b=2½ a mod b ≜ ð4:29Þ ða Mod bÞ b a Mod b 2 ½b=2; b½ with Mod b being the usual modulo b operation with values in [0, b[. The CTMFs of the sampled signal x(k) are linked to the CTMFs of the continuous-time signal xc(t) by the relationship (see Appendix D) ~ ~a ðmÞ ¼ R x

þ1 X

aþpÞfs Rð~ ðmTs Þ xc

ð4:30Þ

p¼ 1

which extends the result derived in Napolitano (1995) for ACS signals to the case of GACS signals. Then, accounting for Eq. (4.27), Eq. (4.26b) can be written as ~ x ð1k þ mÞ ¼ R

þ1 X X ~ 0 p¼ 1 ~a2A m

aþpÞfs Rð~ ðmTs Þej2p~afs kTs xc

ð4:31aÞ

197

GENERALIZED ALMOST-CYCLOSTATIONARY SIGNALS

X

¼

0

~ ~a2A m

~ ~a ðmÞej2p~ak R x

ð4:31bÞ

where, in the second equality, Eq. (4.30) has been accounted for. The sampled signal x(k) is ACS in the wide sense whether the continuoustime GACS (in the wide sense) signal S xc(t) is ACS or not, that is, independently of the fact that the set A ≜ t2R At is countable or not. In fact, from ~0 A ~ m and, comparison of Eq. (4.8) with Eq. (4.31b), it results that A m moreover, the last sum in Eq. (4.31b) can be equivalently extended to the elements of the set ~m ~ ≜ [ A A m2ZN

¼ [ m2Z

¼ [

N

m2RN

[ f~ a 2 ½ 1=2; 1=2½: ~ a ¼ ða=fs Þ mod 1; Raxc ðmTs Þ 6¼ 0g

a2AmTs

[ f~ a 2 ½ 1=2; 1=2½: ~ a ¼ az ðmTs ÞTs mod 1; Rxc ;z ðmTs Þ 6¼ 0g

z2W

ð4:32Þ ˜ is countwhere Eqs. (4.26b) and (4.26c) have been considered. Note that A ˜ m for m ranging able since it is obtained as the union of the countable sets A in the countable set ZN. Therefore, unlike in the continuous-time case, in the discrete-time case a class of GACS (in the wide sense) signals extending that ˜ uncountable. of the ACS ones cannot be introduced by considering A The CTMFs of a discrete-time ACS signal obtained by sampling a continuous-time GACS signal can be expressed in terms of its GCTMFs by the relationship [obtained by substituting Eq. (2.30) into Eq. (4.30)] X ~ ~a ðmÞ ¼ R Rxc ;z ðmTs Þ x ðz;mÞ2X~a

¼

X

Rxc ;z ðmTs Þd½az ðmTs ÞTs ~a mod 1

ð4:33Þ

z2W

where, accounting for Eq. (4.32), a mod 1; Rxc ;z ðmTs Þ 6¼ 0g
X~a ≜ fðz; mÞ 2 W  ZN : az ðmTs ÞTs ¼ ~

ð4:34Þ

Let us note that, although the discrete-time TMF is the sampled version of the continuous-time TMF [see Eq. (4.26a)], in general for the CTMFs it results that ~ ~a ðmÞ 6¼ Ra ðtÞj R t¼mTs ;a¼~afs xc x

ð4:35Þ

due to the presence of aliasing in the cycle frequency domain [see Eq. (4.30)]. Moreover, from Eq. (4.30) it follows that there is no aliasing in considering

198

IZZO AND NAPOLITANO

the sampled version of the Nth-order CTMF at a given cycle frequency a if and only if there are no cycle frequencies of xc(t) that diVer from a for an integer multiple of the sampling frequency fs. A necessary and suYcient condition ensuring the absence of aliasing in the whole cycle frequency domain is that the sampling frequency fs is suYciently high so that all the cycle frequencies of the continuous-time signal xc(t) belong to the interval [ fs/2, fs/2]. In fact, in such a case, only the term with p ¼ 0 in the right-hand side of Eq. (4.30) can give nonzero contribution in the base support region and, moreover, it results that ~ ~a ðmÞ ¼ Ra ðtÞj R a 2 ½ 1=2; 1=2½; 8m 2 ZN : x xc t¼mTs ;a¼~afs 8~

ð4:36Þ

Note that, in the special case of continuous-time ACS signals, the necessary and suYcient condition on the cycle frequencies assuring no aliasing for the Nth-order CTMF in the whole cycle frequency domain is verified when the signal is strictly bandlimited with bandwidth less than fs/2N since, in this case, all the Nth-order cycle frequencies are less than fs/2 (Napolitano, 1995). In the more general case of GACS signals, from Eq. (4.33) it follows that a suYcient condition assuring no aliasing in the whole cycle frequency domain is jaz ðmTs ÞTs j 

1 8m 2 ZN and 8z 2 W such that Rxc ;z ðmTs Þ 6¼ 0: ð4:37Þ 2

With regard to the GCTCFs of the discrete-time signal x(k) constituted by the samples of the continuous-time signal xc(t), in Appendix D it is shown that þ1 X

~

b C~x ðmÞ ¼

~

pÞfs Cðxbþ ðmTs Þ: c

ð4:38Þ

p¼ 1

Moreover, by reasoning as for the GCTMFs, one obtains that the CTCFs of the discrete-time ACS signal obtained by sampling a continuous-time GACS signal can be expressed in terms of its GCTCFs by the relationship X ~ b C~x ðmÞ ¼ Cxc ;x ðmTs Þ ðx;mÞ2Yb~

¼

X

ð4:39Þ

Cxc ;x ðmTs Þd½bx ðmTs ÞTs b ~ mod 1

x2WC

where ~ mod 1; Cx ;x ðmTs Þ 6¼ 0g: ð4:40Þ Yb~ ≜ fðx; mÞ 2 WC  ZN : bx ðmTs ÞTs ¼ b c

GENERALIZED ALMOST-CYCLOSTATIONARY SIGNALS

199

D. Conjecturing the Nonstationarity Type of the Continuous-Time Signal In the previous section it is shown that both ACS and GACS that are not ACS continuous-time signals, by sampling, give rise to ACS discrete-time signals. Thus, the nonstationarity type of the subsumed continuous-time signal cannot be derived from that of the discrete-time signal of its samples. In this section, it is shown how the possible ACS nature of a continuous-time GACS signal can be conjectured from the behavior of the support of the CTMF of the sampled ACS signal (Izzo and Napolitano, 2002a, 2003). The support in [ 1/2, 1/2 [ZN of the CTMF (as a function of (~a, m) ) of the discrete-time signal Eq. (4.17), accounting for Eq. (4.33), is given by ~ a

~ ðmÞ ¼ [ fð~ supp ½R a; mÞ 2 ½ 1=2; 1=2½ZN : x z2W

ð4:41Þ

~ a ¼ az ðmTs ÞTs mod 1; Rxc ;z ðmTs Þ 6¼ 0g that is, is contained in the countable set of manifolds described by the equations ~ a ¼ az (mTs)Ts mod 1, z 2 W. Furthermore, by using a M-point DFT based algorithm with a zero-padding factor F for evaluating the CTMF in Eq. (4.41) as a function of ~ a and m, the cycle frequency step will be D~ a ¼ 1/MF. Consequently, if the parameters Ts, M, and F are such that the cycle frequency variations within a sampling period are smaller than the (normalized) cycle frequency step, that is, if for all z 2 W the condition jaz ðm1 Ts Þ az ðm2 Ts Þj  2D~a=Ts

ð4:42Þ

is satisfied for all m1, m2 2 Z , such that N

jm1n m2n j  1;

n ¼ 1; . . . ; N;

Rxc ;z ðm1 Ts Þ 6¼ 0; Rxc ;z ðm2 Ts Þ 6¼ 0

ð4:43Þ

then, due to the interpolation made by the graphic software, the support of the slice of the CTMF as a function of (~ a, mh) in [ 1/2, 1/2[ Z will appear to be piecewise continuous in correspondence of the continuous tracts of the functions az(t). Thus, in the case of piecewise constant functions az(mTs), according to the considerations following Eq. (2.44), the continuous-time signal should be conjectured to be ACS. On the contrary, when the az(mTs) are not piecewise constant, the continuous-time GACS signal should be conjectured to be not ACS. Let us note that the above-described conjecturing procedure can result to be not adequate when the relationship involving analysis and signal parameters is not appropriate. For example, if jaz ðm1 Ts Þ az ðm2 Ts Þj > 2D~a=Ts

ð4:44Þ

200

IZZO AND NAPOLITANO

for some m1, m2 2 ZN and some z 2 W such that m1n ¼ m2n for n 6¼ h; jm1h m2h j ¼ 1; Rxc ;z ðm1 Ts Þ 6¼ 0; Rxc ;z ðm2 Ts Þ 6¼ 0

ð4:45Þ

then the support of the slice of the CTMF as a function of (~a, mh) will appear discontinuous also in correspondence of the continuous tracts of the functions az(t). Specifically, it will be constituted by small piecewise constant tracts and, hence, the continuous-time signal should be conjectured to be anyway ACS. To illustrate how diVerent relationships among analysis and signal parameters can lead to diVerent conjectures on the nonstationarity type of the continuous-time signal, two simulation experiments have been conducted. In the experiments, time has been discretized with sampling increments Ts ¼ T/M, where T is the data-record length and M ¼ 211 is the number of samples. Moreover, a padding factor F ¼ 2 has been adopted in the DFTs. The subsumed GACS signal xc(t) is that obtained by passing through a channel introducing a time-varying delay the binary-phase shift keyed (BPSK) signal x0 ðtÞ ¼

þ1 X

ak pðt kTp Þ cosð2pf0 tÞ

ð4:46Þ

k¼ 1

where f0 ¼ 0.06/Ts, Tp ¼ 64Ts, ak 2 {)1} are equiprobable symbols, and p(t) is a Tp-width full-duty-cycle rectangular pulse. The linear time-variant system introducing the time-varying delay D(t) has impulse-response function hðt; uÞ ¼ dðu t þ DðtÞÞ

ð4:47Þ

DðtÞ ≜ d0 þ d1 t þ d2 t2 :

ð4:48Þ

where

The fourth-order (N ¼ 4) reduced-dimension (t4 ¼ 0) lag-dependent cycle frequencies of the (real-valued) signal xc(t) are [see Eq. (3.98)] 8 > < b1 2d2 t2 dt3 dt1 t2 ð4:49Þ az ðt1 ; t2 ; t3 ; 0Þ ¼ b1 2d2 t3 dt2 dt1 t3 > : b1 2d2 t3 dt1 dt2 t3 with b1 ¼ )2nf0 þ kTp, n 2 {0, 1}, k 2 Z being the second-order cycle frequencies of the BPSK signal x0(t). Thus, the slice for t1 ¼ t2 and t3 ¼ 0 of the fourth-order reduced-dimension CTMF of xc(t) has a support in the (a, t1) plane confined on the lines with equations a ¼ 2d2b1t1.

GENERALIZED ALMOST-CYCLOSTATIONARY SIGNALS

201

In the first experiment, the channel parameters are set to d0 ¼ 0, d1 ¼ 4.5, and d2 ¼ 0.0015/Ts, so that Eqs. (4.42) and (4.43) hold. In Figure 12, the magnitude and the support of the slice for m2 ¼ m1 and m3 ¼ 0 of the reduced-dimension (m4 ¼ 0) CTMF of the sampled signal are represented as functions of ~ a ¼ aTs and m1 ¼ t1/Ts. From this figure, according to the fact that Eqs. (4.42) and (4.43) hold, the non ACS nature of the underlying continuous-time GACS signal xc(t) can be conjectured. In the second experiment, the channel parameters are set to d0 ¼ 7705Ts, d1 ¼ 16.7Ts, and d2 ¼ 0.0076/Ts and, hence, (4.44) and (4.45) are satisfied. In Figure 13 the magnitude and the support of the same slice considered in Figure 12 of the reduced-dimension CTMF of the sampled signal are reported. From such a figure, it is evident that xc(t) should be conjectured to be ACS. E. Summary In this section, the problem of sampling a continuous-time GACS signal has been addressed. It has been shown that the discrete-time signal constituted by the samples of a GACS signal is a discrete-time ACS signal. Thus, discrete-time ACS signals can arise from ACS and nonACS continuous-time GACS signals. Relationships between generalized cyclic statistics of a continuous-time GACS signal and cyclic statistics of the ACS discrete-time signal constituted by its samples have been derived. Probability density functions and temporal moment and cumulant functions have also been considered. The problem of aliasing in the domain of the cycle frequencies has been considered and a condition ensuring that the cyclic temporal moment function of the discrete-time signal can be obtained by sampling that of the continuous-time signal has been determined. Moreover, it has been shown how, starting from the sampled signal, the possible ACS nature of the continuous-time GACS signal can be conjectured, provided that the analysis parameters are chosen so that the lag-dependent cycle frequency variations within a sampling period are suYciently small.

V. TIME-FREQUENCY REPRESENTATIONS

OF

GACS SIGNALS

A. Introduction This section deals with time-frequency representations of second-order GACS signals (Izzo and Napolitano, 1997b). A brief introduction (Section V.B.) on GACS signals is reported here to introduce notation since for

202

IZZO AND NAPOLITANO

FIGURE 12. Graph of the (a) magnitude and (b) support, as functions of ð~a; m1 Þ  ~ ~a ðm1 ; ðaTs ; t1 =Ts Þ, of the slice for m2 ¼ m1 and m3 ¼ 0 of the reduced-dimension CTMF R xxxx m2 ; m3 ; 0Þ of the sampled signal in the first experiment.

GENERALIZED ALMOST-CYCLOSTATIONARY SIGNALS

203

FIGURE 13. Graph of the (a) magnitude and (b) support, as functions of ð~a; m1 Þ  ~ a~ ðm1 ; ðaTs ; t1 =Ts Þ, of the slice for m2 ¼ m1 and m3 ¼ 0 of the reduced-dimension CTMF R xxxx m2 ; m3 ; 0Þ of the sampled signal in the second experiment.

204

IZZO AND NAPOLITANO

time-frequency representations a symmetric version of the autocorrelation is more appropriate. In Section V.C the Cohen’s general class of time-frequency distributions (Cohen, 1995; Hlawatasch and Boudreaux-Bartels, 1992) of GACS signals is considered and it is shown that any representation belonging to this class is expressed as sum of two terms. The first one involves the generalized cyclic statistics of the signal; the second one is related to the residual term obtained by subtracting to the second-order lag product its almost-periodic component (the time-varying autocorrelation function). The Wigner–Ville distribution is examined in detail. Moreover, the ambiguity function is considered. It is shown that it can be expressed as a sum of impulsive terms related to the generalized cyclic statistics of the signal and a non impulsive component related to the previously-mentioned residual term. Furthermore, the subclass of the ACS signals is examined. In Section V.D, the problem of signal feature extraction is considered. It is shown that the estimation of the cyclic autocorrelation as a function of cycle frequency and lag parameter allows one to determine the lag-dependent cycle frequencies and the generalized cyclic autocorrelation functions annihilating the eVect of the residual term when the collect time increases. Then, an estimate of the time-varying autocorrelation function can be derived. On the contrary, in general, in time-frequency representations the component related to the residual term cannot be separated by the components related to the generalized cyclic statistics. Finally, let us note that the time-frequency distributions and the ambiguity function were originally defined with reference to finite-energy signals. Moreover, finite-power signals can be considered by using Dirac’s delta functions (see, e.g., sine waves and chirp signals in Cohen, 1995). The approach adopted here follows this line since GACS time series have finite power. However, it is note worthy that a diVerent approach is adopted in Gardner (1988a), where for a time-windowed ACS signal, the Wigner– Ville distribution is related to the cyclic periodograms and the ambiguity function is recognized to be equal, but for a scale factor, to the cyclic correlogram.

B. Second-Order GACS Signals In the FOT probability framework, a continuous-time complex-valued finitepower time series x(t) is said to exhibit second-order wide-sense cyclostationarity with cycle frequency a 6¼ 0 if the symmetric cyclic autocorrelation function Raxx ðtÞ ≜ hxðt þ t=2Þx  ðt t=2Þe j2pat it

ð5:1Þ

GENERALIZED ALMOST-CYCLOSTATIONARY SIGNALS

205

exists and is not zero for some t (Gardner, 1988a). Analogously, the time series is said to exhibit second-order wide-sense conjugate cyclostationarity if the symmetric conjugate cyclic autocorrelation function Raxx ðtÞ ≜ hxðt þ t=2Þxðt t=2Þe j2pat it

ð5:2Þ

exists and is not zero for some t. In the following, we deal with time series exhibiting cyclostationarity. The consideration of time series exhibiting conjugate cyclostationarity will require some obvious minor changes. If the set At ≜ fa 2 R : Raxx ðtÞ 6¼ 0g

ð5:3Þ

is countable for each t, then the time series is said to be secondorder generalized almost-cyclostationary in the wide-sense, and the almostperiodic function Rxx ðt; tÞ ≜ EfAt g fxðt þ t=2Þx ðt t=2Þg X ¼ Raxx ðtÞej2pat

ð5:4Þ

a2At

is referred to as the time-varying symmetric autocorrelation function. Then, the lag product x(t þ t/2)x*(t t/2) can be expressed as the sum of its almost-periodic component and a residual term not containing finitestrength additive sine wave components: xðt þ t=2Þx ðt t=2Þ ≜ Rxx ðt; tÞ þ ‘xx ðt; tÞ

ð5:5Þ

h‘xx ðt; tÞe j2pat it 0; 8a 2 R:

ð5:6Þ

where

The support in the (a, t) plane of the symmetric cyclic autocorrelation function can be written as suppfRaxx ðtÞg ≜ clfða; tÞ 2 At  R : Raxx ðtÞ 6¼ 0g ¼ cl [ fða; tÞ 2 R  Dz : a ¼
a z ðtÞg

ð5:7Þ

z2W

where W is a countable set and the functions
a z ðtÞ, z 2 W, are the (reduceddimension) lag-dependent cycle frequencies whose domains are denoted by Dz, z 2 W. By reasoning as in Section II.C.1, it can be shown that the time-varying autocorrelation function in Eq. (5.4) can be expressed as X Rxx ðt; tÞ ¼ Rxx;z ðtÞej2p
a z ðtÞt ð5:8Þ z2W

where the functions

206

IZZO AND NAPOLITANO

 Rxx;z ðtÞ ≜

hxðt þ t=2Þx ðt t=2Þe j2p
a z ðtÞt it ; 0;

8t 2 Dz elsewhere

ð5:9Þ

are the symmetric generalized cyclic autocorrelation functions. The Fourier transform of the generalized cyclic autocorrelation function Z Rxx;z ðtÞe j2pf t dt ð5:10Þ Sxx;z ð f Þ ≜ R

is called the symmetric generalized cyclic spectrum. In the special case of ACS time series, it is coincident with the cyclic spectrum. C. Time-Frequency Representations of GACS Signals All time-frequency distributions for a complex-valued time series x(t) can be obtained from Z Cxx ðt; f Þ ¼ xðu þ t=2Þx ðu t=2Þ fðy; tÞ e j2pyðt uÞ e j2pf t du dt dy R3

ð5:11aÞ Z ¼

R

xðt þ t=2Þx ðt t=2Þ ! Fðt; tÞe j2pf t dt t

where

Z Fðt; tÞ ≜

fðy; tÞe j2pyt dy

R

ð5:11bÞ

ð5:12Þ

and ! denotes convolution with respect to t. The kernel function f(y, t) t determines the distribution and its properties (Cohen, 1995). By substituting Eq. (5.5) into Eq. (5.11b) and accounting for Eqs. (5.8) and (5.10), the expression of the generic time-frequency distribution in terms of generalized cyclic statistics can be obtained: X F Cxx ðt; f Þ ¼ Sxx;z ð f Þ ! AF ð5:13Þ z ðt; f Þ þ Lxx ðt; f Þ f

z2W

where

Z AF z ðt; f Þ ≜

and

Z LF xx ðt; f Þ ≜

ej2p
a z ðtÞt ! Fðt; tÞe j2pf t dt

ð5:14Þ

‘xx ðt; tÞ ! Fðt; tÞ e j2pf t dt:

ð5:15Þ

R

R

t

t

GENERALIZED ALMOST-CYCLOSTATIONARY SIGNALS

207

In other words, all time-frequency distributions of GACS time series can be expressed as the sum of two contributions. The first one is the sum of all generalized cyclic spectra each spread (in the frequency domain) by the time-varying function AF z ðt; f Þ depending on the corresponding (reduceddimension) lag-dependent cycle frequency
a z ðtÞ and the kernel function. The second one is related to the residual term ‘xx* (t, t). Note that by taking N ¼ 2 in (2.55) and F(t, t) ¼ d(t) in Eq. (5.14), it
results AF z ð t; f Þ ¼ A z ðt; f Þ. By adopting the kernel function f(y, t) ¼ 1 in (5.11a) and, hence, F(t, t) ¼ d(t) in Eq. (5.11b), one obtains the Wigner–Ville distribution (Cohen, 1995) Z Wxx ðt; f Þ ≜ xðt þ t=2Þx ðt t=2Þ e j 2pf t dt ð5:16Þ R

which, accounting for Eqs. (5.13) through (5.15), can be expressed as X Sxx;z ð f Þ ! F fej2p
a z ðtÞt g þ F f‘xx ðt; tÞg ð5:17Þ Wxx ðt; f Þ ¼ f t!f

z2W

t!f

(Izzo and Napolitano, 1997b), where F t!f denotes the Fourier transform operator from the t domain to the f domain. In the special case of ACS time series the (reduced dimension) lagdependent cycle frequencies are constant and, hence, Eq. (5.13) reduces to X a Cxx ðt; f Þ ¼ Sxx ð f Þ ! Cða; f Þej2pat þ LF ð5:18Þ xx ðt; f Þ f

a2A

where

Z Cða; f Þ ≜

R2

Fðt; tÞ e j 2pðatþf tÞ dt dt

ð5:19Þ

and (2.45) has been accounted for. Moreover, for the Wigner–Ville distribution F(t, t) ¼ d(t), that is, C (a, f) ¼ d(f) and, hence, X a Wxx ðt; f Þ ¼ Sxx ð f Þ ej2pat þ F f‘xx ðt; tÞg ð5:20Þ a2A

t!f

which is just the result derived in Gournay and Nicolas (1995) except for the component depending on ‘xx*(t, t). The absence of such a term in Gournary and Nicolas (1995) stems from the fact that for ACS signals, in the stochastic process framework [adopted in Gournay and Nicolas (1995)], ‘xx*(t, t) is dropped out by the statistical expectation operation. However, let us note that such a residual term is present also in the stochastic process approach when asymptotically mean ACS (AMACS) processes (Boyles and Gardner, 1983) are considered. Moreover, it is worthwhile to underline that the

208

IZZO AND NAPOLITANO

residual term is always present in the single sample-path–based estimate of the Wigner–Ville distribution that, for both ACS and AMACS processes, asymptotically approaches expression Eq. (5.20) for almost all sample paths when the collect time increases, provided that appropriate ergodicity properties are satisfied (Boyles and Gardner, 1983; Gardner, 1994). Multiplerecord estimates of the Wigner–Ville distribution lead to a zero residual term. However, they can be singled out only when the signal is cyclostationary (i.e., all the cycle frequencies are multiple of a fundamental one) and the period of cyclostationarity is known (Ko¨ nig and Bo¨ me, 1994). The ambiguity function Z Axx ðn; tÞ ≜ xðt þ t=2Þx ðt t=2Þ e j2pnt dt ð5:21Þ R

for GACS time series, accounting for Eqs. (5.5) and (5.8), can be expressed in terms of generalized cyclic statistics (Izzo and Napolitano, 1997b): X Rxx;z ðtÞdðn az ðtÞÞ þ F f‘xx ðt; tÞg: ð5:22Þ Axx ðn; tÞ ¼ z2W

t!n

Equation (5.22) shows that the ambiguity function of GACS signals is the sum of some impulsive terms whose supports are curves described by the lagdependent cycle frequencies and whose amplitudes are the generalized cyclic autocorrelation functions and an aperiodic component that, accounting for Eq. (5.6), does not contain impulses. Finally, let us note that, in the special case of ACS time series, Eq. (5.22) specializes to X Axx ðn; tÞ ¼ Raxx ðtÞdðn aÞ þ F f‘xx ðt; tÞg: ð5:23Þ a2A

t!n

D. Signal Feature Extraction In problems of signal feature extraction for GACS time series, in general, no a priori knowledge exists on the possible cyclostationary nature of the signal. Therefore, single sample-path–based estimators of time-frequency distributions and generalized cyclic statistics must be used. The estimators are obtained directly by applying the definitions where, however, integrals and time averages are performed over a finite data record. Then, they asymptotically approach the theoretical values when the observation time increases. Once the lag-dependent cycle frequencies and/or the generalized cyclic statistics have been estimated, the time-varying autocorrelation function

GENERALIZED ALMOST-CYCLOSTATIONARY SIGNALS

209

can be reconstructed, signal parameters can be estimated, and signals can be classified on the basis of their diVerent generalized cyclic statistic characteristics. It is worthwhile to note that, in general, in time-frequency representations of GACS time series, the component related to the residual term ‘xx*(t, t) cannot be separated from the component related to the cyclic statistics [see Eq. (5.13)]. In the special case of ACS time series, however, from Eq. (5.18) it follows that the component related to the cyclic statistics is almost periodic and, hence, algorithms for estimating amplitude and frequencies of almostperiodic signals embedded in noise can be exploited to obtain estimates of the cyclic parameters of interest. The role played by the residual term can be illustrated by an example. Specifically, let us consider the signal Z xðtÞ ¼ hðt; uÞsðuÞdu ð5:24Þ R

where sðtÞ ¼ wðtÞ expð j2pf0 tÞ

ð5:25Þ

hðt; uÞ ¼ dðu t þ DðtÞÞ

ð5:26Þ

and

is the impulse-response function of a channel introducing a time-varying delay D(t). Figure 14a shows the magnitude of the cyclic autocorrelation function Raxx ðtÞ, for the signal x(t), as a function of a and t, as estimated by 256 samples. It has been assumed that f0 ¼ 0.04/Ts, where Ts is the sampling period (see Section IV.C for the aliasing issue), and the real signal w(t) is 0 ð f Þ ¼ ð1 þ f 2 =B2 Þ 8 wide-sense stationary with power spectral density Sww  with B ¼ 0.015/Ts. Moreover, a time-varying delay D(t) ¼ d1t þ d2t2 with d1 ¼ 0.25 and d2 ¼ 0.02/Ts has been considered. The support of Raxx ðtÞ is constituted by curves described by the reduced-dimension lag-dependent cycle frequencies
a z ðtÞ; on each of them, the cyclic autocorrelation function is just equal to the corresponding generalized cyclic autocorrelation function Rxx; zðtÞ. Figure 14b shows the magnitude of the Wigner–Ville distribution Wxx ðt; f Þ for the same signal. The presence of a component related to the residual term is evident. Finally, let us observe that the estimates of the cyclic autocorrelation function and the ambiguity function diVer only for a scaling factor (Gardner, 1988a). Therefore, the above cyclic autocorrelation function– based estimation procedure can also be interpreted in terms of ambiguity function.

210

IZZO AND NAPOLITANO

FIGURE 14. Magnitude of (a) the cyclic autocorrelation function Raxx ðtÞ as a function of a and t and (b) the Wigner–Ville distribution Wxx ðt; f Þ as a function of t and f.

GENERALIZED ALMOST-CYCLOSTATIONARY SIGNALS

211

APPENDICES Appendix A In this appendix, the convergence in the temporal mean-square sense of time averages is briefly discussed. A more comprehensive treatment can be found in Weiner (1930) for stationary time series and in Brown (1987), Gardner (1988a), for almost-cyclostationary time series. The more general convergence in the sense of distributions (generalized functions) of statistical functions defined starting from a single time series is treated in PfaVelhuber (1975). In all the sections, all time averages are assumed to exist and to be convergent in the temporal mean-square sense (t.m.s.s.). That is, given a time series z(t) (possibly obtained as lag product of another time series) and defined Z 1 tþT=2 zb ðtÞT ≜ zðuÞ e j2pbu du ð2:A:1Þ T t T=2 Z 1 þT=2 zb ≜ lim zðuÞ e j2pbu du ð2:A:2Þ T!1 T T=2 it is assumed that lim zb ðtÞT ¼ zb

T!1

that is, lim

T!1

ðt:m:s:s:Þ;

D E jzb ðtÞT zb j2 ¼ 0; t

8b 2 R

ð2:A:3Þ

8b 2 R:

ð2:A:4Þ

It can be shown that if the time series z(t) has finite average-power (i.e., 2 hjz(t)j it < 1), then the set B ≜ {b 2 R : zb 6¼ 0} is countable, the series P 2 jz b2B b j is summable (Brown, 1987), and, accounting for Eq. (2.A.3), it follows that X X lim zb ðtÞT e j2pbt ¼ zb e j2pbt ðt:m:s:s:Þ: ð2:A:5Þ T!1

b2B

b2B

The magnitude and phase of zb are the amplitude and phase of the finitestrength additive complex sine wave with frequency b contained in the time series z(t). Moreover, the right-hand side in Eq. (2.A.5) is just the almostperiodic component contained in the time series z(t). The function zb(t)T is an estimator of zb based on the observation {z(u), u 2 [t T/2, t þ T/2]}. It is worthwhile to underline that, in the FOT probability framework probabilistic functions are defined in terms of the almost-periodic

212

IZZO AND NAPOLITANO

component extraction operation, which plays the same role played by the statistical expectation operation in the stochastic process framework (Gardner, 1988a, 1994) (see also Section II.B). Therefore, biasfzb ðtÞT g ≜ Efag fzb ðtÞT g zb
 ’ zb ðtÞT t zb n o varfzb ðtÞT g ≜ Efag jzb ðtÞT Efag fzb ðtÞT gj2 D E ’ jzb ðtÞT hzb ðtÞT it j2

ð2:A:6Þ

ð2:A:7Þ

t

where the approximation becomes exact equality in the limit as T ! 1. Thus, unlike the stochastic process framework, where the variance accounts for fluctuations of the estimates over the ensemble of sample paths, in the FOT probability framework the variance accounts for the fluctuations of the estimates in the time parameter t (e.g., the central point of the finite-length time series segment adopted for the estimation). Therefore, the assumption that the estimator asymptotically approaches the true value (the infinite-time average) in the mean-square sense is just equivalent to the statement that the estimator is mean-square consistent in the FOT probability sense. In fact, from Eqs. (2.A.4), (2.A.6), and (2.A.7), it follows that D E jzb ðtÞT zb j2 ’ varfzb ðtÞT g þ jbiasfzb ðtÞT gj2 ð2:A:8Þ t

and this approximation becomes exact as T ! 1. In such a case, estimates obtained by using different time segments asymptotically do not depend on the central point of the segment. Appendix B In this appendix, it is shown that for T ! 1 the set L0 of the Nth-order cycle frequencies of the time series XT (t, f ) [see Eq. (2.50)] contains only the element a ¼ 0. Moreover, the derivation of Eq. (2.51) is presented. Let us consider the function * + N Y ðÞn a j2pat Mx ð f Þ ≜ lim XT ðt; ð Þn f n Þe ð2:B:1Þ T!1

n¼1

t

whose magnitude and phase are amplitude and phase of the finite-strength sine component with frequency a contained in the product QN wave ðÞn X ðt; ð Þ n¼1 T n fn Þ when T ! 1. By substituting Eq. (2.50) into Eq. (2.B.1) and accounting for Eqs. (2.14) and (2.30), one has

GENERALIZED ALMOST-CYCLOSTATIONARY SIGNALS

Max ð f Þ ¼

XZ z2W

R

N

Rx;z ðtÞdaz ðtÞ a f ⊤ 1 e j2pf

⊤

t

dt:

213 ð2:B:2Þ

Moreover, substituting Eqs. (2.42) and (2.43) into Eq. (2.B.2) and accounting for Eq. (2.48), one obtains that X Max ð f Þ ¼ S x;z ð f Þda ð2:B:3Þ z2W

from which it follows immediately that the set L0 contains only the element a ¼ 0. Therefore, the spectral moment function can be expressed as ( ) N Y ðÞn fL0 g S x ð f Þ ≜ lim E XT ðt; ð Þn fn Þ T!1 n¼1 * + N Y ðÞn ð2:B:4Þ ¼ lim X ðt; ð Þ fn T!1

¼

X

T

n¼1

S x;z ð f Þ

n

t

z2W

which is just Eq. (2.51). Appendix C This appendix presents proofs of some results shown in Section III. Proof of Eq. (3.34) With reference to the ideal sampled signal xd(t) defined in Eq. (3.33), we have that for any 0 < E < Ts, it results Z 1 T=2 xd ðtÞ e j2pat dt lim T!1 T T=2 Z þ1 1 T=2 X ¼ lim xðkÞ dðt kTs Þ e j2pat dt T!1 T T=2 k¼ 1 Z KTs þE X þK 1 xðkÞ dðt kTs Þ e j2pat dt ¼ lim K!1 ð2K þ 1ÞTs KT E s k¼ K ð3:A:1Þ Z þ1 þK X 1 1 j2pat lim xðkÞ dðt kTs Þ e dt ¼ Ts K!1 2K þ 1 k¼ K 1 þK X 1 1 lim xðkÞ e j2pakTs ¼ Ts K!1 2K þ 1 k¼ K  1 ~a  ¼ x Ts ~a¼aTs where

214

IZZO AND NAPOLITANO K X 1 xðkÞe j2p~ak : K!1 2K þ 1 k¼ K

x~a ≜ lim

ð3:A:2Þ

In the first equality in Eq. (3.A.1), we used the fact that if the limit of f(T) as T ! 1, T 2 Rþ, exists, then it is equal to the limit made on any extracted sequence f~ðKÞ ¼ f ðKTs Þ as K ! 1, K 2 N, where K ¼ ⌊T/Ts⌋, with ⌊
⌋ denoting the integer part. Moreover, by denoting with Ef~ag f
g the discrete-time almost-periodic component extraction operator, we have þ1 X

Ef~ag fxðkÞg dðt kTs Þ k¼ 1 " # þ1 X X ~a j2p~ak ¼ x e dðt kTs Þ ¼

~ k¼ 1 ~a2A þ1 X X ~a

x

~ ~a2A

¼

X

, e j2p~ak Ts d

k¼ 1

x~a e j2p~at=Ts

~ ~a2A

¼

X

x~a e j2p~at=Ts

~ ~a2A

þ1 X

t k Ts

-

dðt kTs Þ

ð3:A:3Þ

k¼ 1 þ1 X

1 e j2ppt=Ts Ts p¼ 1

þ1 1 X ~a X x e j2pð~aþpÞt=Ts Ts ~ p¼ 1 ~a2A 1 X aTs j2pat ¼ x e Ts a2A

¼

where ~ ≜ f~ A a 2 ½ 1=2; 1=2½: x~a 6¼ 0g

ð3:A:4Þ

A ≜ fa 2 R : a ¼ ~ a=Ts þ p; p 2 Z; x~a 6¼ 0g

ð3:A:5Þ

and Poisson’s sum formula þ1 X k¼ 1

dðt kTs Þ ¼

þ1 1 X ej2ppt=Ts Ts p¼ 1

ð3:A:6Þ

has been accounted for. Therefore, from Eqs. (3.A.1) and (3.A.3) it follows that the almostperiodic component of the signal xd(t) can be expressed as

GENERALIZED ALMOST-CYCLOSTATIONARY SIGNALS

1 X aTs j2pat x e Ts a2A þ1 X ¼ Ef~ag fxðkÞgdðt kTs Þ

215

Efag fxd ðtÞg ¼

ð3:A:7Þ

k¼ 1

and, hence, the second equality in Eq. (3.34) easily follows. Proof of Eq. (3.59) The Nth-order temporal cumulant function of the output y(t) of a LTV system excited by a GACS time series x(t), accounting for Eqs. (2.58) and (3.54), is given by " # p X Y p 1 Cy ð1t þ tÞ ¼ ð 1Þ ðp 1Þ! Rxmi ð1t þ t mi Þ i¼1 P Z X" p Y ¼ ð 1Þp 1 ðp 1Þ! ð3:A:8Þ RN P i¼1 # Rhmi ð1t þ t mi ; 1t þ smi ÞRxmi ð1t þ smi Þ ds where

( Rxmi ð1t þ smi Þ ≜ Efag

Y

) xðÞk ðt þ sk Þ

ð3:A:9Þ

k2mi

and Rhmi (1t þ t mi, 1t þ smi) is the jmijth-order temporal moment function of the LTV system which, according to Eq. (3.43), can be written as Rhmi ð1t þ tmi ; 1t þ smi Þ X Z ⊤ ¼ Gymi ðlmi ; t mi Þ e j2p’ymi ðlmi ;tmi Þt e j2plmi ð1tþsmi Þ dsmi : ymi 2Ymi

ð3:A:10Þ

Rjmi j

By assuming that the LTV system is FOT deterministic, there is a oneto-one correspondence between the elements ymi 2 Ymi and the vectors smi 2 Ojmij and, according to Eqs. (3.57) and (3.58), it results that ð Þ ⊤ ’ym ðlmi ; t mi Þ ¼ wð Þ smi ðlmi Þ 1 i

and Gymi ðlmi ; t mi Þ ¼

Y



ð Þ ð Þ ⊤ k ð Þ l GðÞ ej2pwsmi ðlmi Þ tmi : k sk k

ð3:A:11Þ

ð3:A:12Þ

k2mi

Therefore, by substituting Eqs. (3.A.11) and (3.A.12) into Eq. (3.A.10), it results that

216

IZZO AND NAPOLITANO

p Y Rhmi ð1t þ tmi ; 1t þ smi Þ i¼1

¼

X Z s2ON

R

N

N

Y ð Þ j 2pws ðlð Þ Þ⊤ ð1tþtÞ j 2pl⊤ ð1tþsÞ n ð Þ l e dl GðÞ sn n n e

ð3:A:13Þ

n¼1

¼ Rh ð1t þ t; 1t þ sÞ where the fact that the sets mi are disjoint has been accounted for. Finally, Eq. (3.59) immediately follows by substituting Eq. (3.A.13) into (3.A.8) and taking into account Eq. (2.58). Appendix D In this appendix, proofs of some results presented in Section IV.C, are reported. Proof of Eq. (4.19) For any 0 < E < Ts the argument of the limit in Eq. (4.19) can be written as K X 1 ‘U ð1t þ t; jÞjt¼kTs ;t¼mTs e j2p~gk 2K þ 1 k¼ K Z KTs þE þ1 X 1 ¼ ‘U ð1t þ mT s ; jÞ dðt kTs Þ e j2p~gt=Ts dt ð4:A:1Þ 2K þ 1 KTs E k¼ 1 Z KTs þE þ1 X 1 ‘U ð1t þ mTs ; jÞ e j2pð~g pÞt=Ts dt ¼ ð2K þ 1ÞT s KT E s p¼ 1

where, in the second equality, the Poisson sum formula [Eq. (3.A.6)] has been accounted for. Taking the limit for K ! 1 in Eq. (4.A.1) and accounting for Eq. (2.5), one obtains Eq. (4.19). Proof of Eq. (4.20) By substituting Eq. (4.18) into Eq. (4.3) and accounting for Eq. (4.19), one has ~ ~g ðm; jÞ ¼ F x

X

K X 1 e j2pð~g gTs Þk : K!1 2K þ 1 k¼ K

Fxgc ðmTs ; jÞ lim

g2GmTs

ð4:A:2Þ

Finally, by using the limit K þ1 X X 1 e j2p~ak ¼ d~aþp K!1 2K þ 1 p¼ 1 k¼ K

lim

into Eq. (4.A.2), Eq. (4.20) easily follows.

ð4:A:3Þ

GENERALIZED ALMOST-CYCLOSTATIONARY SIGNALS

Let us define the set

Proof of Eq. (4.22) ~0 G m

217

≜ f~g 2 ½ 1=2; 1=2½: ~g ¼ ðg=fs Þ mod 1; g 2 GmTs g;

ð4:A:4Þ

where mod b denotes the modulo b operation with values in [ b/2, b/2[ [see Eq. (4.29)]. From Eq. (4.A.4), accounting for Eqs. (2.2) and (4.18), it results that g GmTs ≜ fg jg [2 R : Fxc ðmTs ; jÞ 6¼ 0 for some ~ 0 ; F g ðmT s ; jÞ 6¼ 0 for some jg: ¼ fg 2 R : g ¼ ð~g þ pÞfs ; ~g 2 G m xc p2Z

ð4:A:5Þ Therefore, from Eq. (4.18) it follows that ~ x ð1k þ m; jÞ ¼ U

þ1 X X ~0 ~g2G m

¼

X 0

~ ~g2G m

Fxð~gc þpÞfs ðmTs ; jÞej2pð~gþpÞfs kTs þ ‘U ð1kTs þ mTs ; jÞ

p¼ 1

~ ~g ðm; jÞej2p~gk þ ‘U ð1kTs þ mTs ; jÞ F x ð4:A:6Þ

where, in the second equality, Eq. (4.20) has been used. Thus, by comparing Eq. (4.A.6) with Eq. (4.2) and accounting for Eq. (4.19), it results that the set ~ 0 is coincident with the set G ~ m defined by Eq. (4.23) and, hence, Eq. (4.22) G m is proved. Proof of Eq. (4.30) By taking the Fourier coefficient at frequency a in both sides of Eq. (2.17), one obtains that Z N Y a Rxc ðtÞ ¼ ðxrn þ jxin ÞðÞn fxac ðt; jÞ dj r dj i : ð4:A:7Þ R2N n¼1

Analogously, by taking the Fourier coefficient at frequency ~a in both sides of Eq. (4.10), one has Z N Y ~a ~ a ~ Rx ðmÞ ¼ ðxrn þ jxin ÞðÞn f~x ðm; jÞ dj r dj i ¼

R2N n¼1 þ 1 Z X

p¼ 1

N Y

R2N n¼1

ð4:A:8Þ ðxrn þ

jxin ÞðÞn fxð~caþpÞfs ðmTs ; jÞ

dj r dj i

where, in the second equality, Eq. (4.21) has been used. Thus, by substituting Eq. (4.A.7) into Eq. (4.A.8), Eq. (4.30) easily follows.

218

IZZO AND NAPOLITANO

Proof of Eq. (4.38) By taking the Fourier coefficient at frequency b in both sides of Eq. (2.58), one has " # p X X Y ami p 1 b ð 1Þ ðp 1Þ! Rxc;mi ðt mi Þ ð4:A:9Þ Cxc ðtÞ ¼ P

a⊤ 1¼b i¼1

where the second summation ranges over all vectors a ≜ ½am1 ; . . . ; amp ⊤ with am ami such that Rxc;mi i ðt mi Þ ≢ 0, and the fact that the sets mi are disjoint has been accounted for. In Eq. (4.A.9), Z 1 T=2 Y ðÞn ami Rxc;mi ðtmi Þ ≜ lim xc ðt þ tn Þe j2pami t dt: ð4:A:10Þ T!1 T T=2 n2m i

~ in both sides of Analogously, by taking the Fourier coefficient at frequency b Eq. (4.12), one obtains " # p X XY ~ ~ami b p 1 ~ ðmÞ ¼ ~ ðmm Þd ⊤ ~ ð 1Þ ðp 1Þ! R ð4:A:11Þ C xm x i ð~ a 1 bÞmod 1 ~ a

P

i

i¼1

~ ≜ ½~am1 ; . . . ; ~amp ⊤ with where the second summation ranges over all vectors a ~ami ~ ðmm Þ ≢ 0. In Eq. (4.A.11), ~ ami 2 ½ 1=2; 1=2½ such that R xm i i K X Y 1 ~ ~ ami ðmm Þ ≜ lim xðÞn ðk þ mn Þe j2p~ami k : ð4:A:12Þ R xmi i K!1 2K þ 1 k¼ K n2m i

By substituting Eq. (4.30) into Eq. (4.A.11), it results that 2 X XX X ~ b 4ð 1Þp 1 ðp 1Þ! ... C~x ðmÞ ¼ ~ a

P

q1 2Z

qp 2Z

p Y ð~am þqi Þfs Rxc;mii ðmmi Ts Þdð~a⊤ 1 bÞmod ~ 1

#

ð4:A:13Þ

i¼1

~ and qi, (i ¼ 1, . . . , p) give rise to a sum over all where the summations over a cycle frequencies ~ ami 2 ½ 1=2; 1=2½ such that (~ami þ qi Þfs ¼ ami ði ¼ 1; . . . ; pÞ ~ s a⊤ 1Þ mod fs ¼ 0. That is, and ðbf 2 3 p Y X X ~ b a m 4ð 1Þp 1 ðp 1Þ! Rxc;mi i ðmmi Ts Þ5 C~x ðmÞ ¼ P

¼

XX q2Z

P

2

~ s a⊤ 1¼qfs ;q2Zg i¼1 fa:bf

4ð 1Þp 1 ðp 1Þ!

X

3 p Y ami Rxc;mi ðmmi Ts Þ5

~ s qfs i¼1 a⊤ 1¼bf

ð4:A:14Þ from which, using Eq. (4.A.9), Eq. (4.38) immediately follows.

219

GENERALIZED ALMOST-CYCLOSTATIONARY SIGNALS GLOSSARY OF ACRONYMS AND NOTATIONS h
it E{a} {
} or EfAt g f
g d(
) d(
) U(
) cl vT v0 ! ! n

1 * (*) or (*)n ( ) or ( )n 1( ) n( )

Rax ðtÞ Rax ðt0 Þ Rx ð1t þ tÞ Rx;z ðtÞ

continuous time average with respect to t almost-periodic component extraction operator Dirac delta Kronecker delta unit step function closure (of a set) vector transposition v ¼ ½v1 ; . . . ; vN ⊤ ) v0 ¼ ½v1 ; . . . ; vN 1 ⊤ convolution N-fold convolution with respect to v ≜ ½v1 ; . . . ; vN ⊤ [1, . . . , 1]⊤ complex conjugation optional complex conjugation optional minus sign ½ð Þ1 1; . . . ; ð ÞN 1⊤ ½ð Þ1 v1 ; . . . ; ð ÞN vN ⊤ fraction of time linear time variant linear time invariant linear almost periodically time variant almost cyclostationary generalized almost cyclostationary cyclic temporal moment function reduced-dimension cyclic temporal moment function temporal moment function

P x( f )

generalized cyclic temporal moment function reduced-dimension generalized cyclic temporal moment function (moment) lag-dependent cycle frequency reduced-dimension (moment) lag-dependent cycle frequency cyclic spectral moment function reduced-dimension cyclic spectral moment function spectral moment function temporal cumulant function generalized cyclic temporal cumulant function spectral cumulant function

Rh ð1t þ t; uÞ S h( f, l)

system temporal moment function system spectral moment function

Rx;z ðt 0 Þ az(t)
a z ðt 0 Þ S ax ð f Þ Sxa ð f 0 Þ Sx ð f Þ Cx(1t þ t) Cx,x(t)

(1.2) (1.15), (2.2)

(1.16)

FOT LTV LTI LAPTV ACS GACS CTMF RD-CTMF TMF

(1.24), (2.14) (1.26)

GCTMF

(2.16), (2.26), (2.38) (2.27)

RD-GCTMF

(2.41) (2.24), (2.25) (2.40)

CSMF RD-CSMF

(1.25) (1.25)

SMF TCF GCTCF

(2.49), (2.51) (2.58), (2.67) (2.68) (2.93), (2.94), (2.96) (3.43) (3.61)

220

IZZO AND NAPOLITANO

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