2.1 Introduction

In the past two decades, a huge effort has been devoted to analysis and exploitation of the properties of the almost-cyclostationary (ACS) processes. In fact, almost-all modulated signals adopted in communications can be modeled as ACS (Gardner 1994), (Gardner and Spooner 1994), (Spooner and Gardner 1994), (Gardner et al. 2006). For an ACS process, multivariate statistical functions are almost-periodic functions of time and can be expressed by (generalized) Fourier series expansions whose coefficients depend on the lag shifts of the processes and whose frequencies, referred to as cycle frequencies, do not depend on the lag shifts. Almost-cyclostationarity properties have been widely exploited for analysis and synthesis of communications systems. In particular, they have been exploited to develop signal selective detection and parameter estimation algorithms, blind-channel-identification and synchronization techniques, and so on (Gardner 1994), (Gardner et al. 2006). Moreover, ACS processes are encountered in econometrics, climatology, hydrology, biology, acoustics, and mechanics (Gardner 1994), (Gardner et al. 2006).

More recently, wider classes of nonstationary processes extending the class of the ACS processes have been considered in (Izzo and Napolitano (1998a), (Izzo and Napolitano (2002a,b, 2003, 2005), (Napolitano 2003, 2007a, 2009), (Napolitano and Tesauro 2011).

In (Izzo and Napolitano 1998a), the class of the generalized almost-cyclostationary (GACS) processes was introduced and characterized. Processes belonging to this class exhibit multivariate statistical functions that are almost-periodic functions of time whose (generalized) Fourier series expansions have coefficients and frequencies, referred to as lag-dependent cycle frequencies, that can depend on the lag shifts of the processes. The class of the GACS processes includes the class of the ACS processes as a special case, when the cycle frequencies do not depend on the lag shifts. Moreover, chirp signals and several angle-modulated and time-warped communication signals are GACS processes. In (Izzo and Napolitano 1998a), the higher-order characterization in the strict-and wide-sense of GACS signals is provided. Generalized cyclic moments and cumulants are introduced in both time and frequency domains. As examples of GACS signals, the chirp signal and a nonuniformly sampled signal are considered and their generalized cyclic statistics are derived. In (Izzo and Napolitano 2002a) and (Izzo and Napolitano 2002b), it is shown that several time-variant channels of interest in communications transform a transmitted ACS signal into a GACS one. In particular, in (Izzo and Napolitano 2002b) it is shown that the GACS model can be appropriate to describe the output signal of Doppler channels when the input signal is ACS and the channel introduces a quadratically time-variant delay. Therefore, the GACS model is appropriate to describe the received signal in presence of relative motion between transmitter and receiver with nonzero relative radial acceleration (Kelly and Wishner 1965), (Rihaczek 1967). In (Izzo and Napolitano 1998a), (Izzo and Napolitano 2002a), and (Izzo and Napolitano 2002b), it is also shown that communications signals with slowly time-varying parameters, such as carrier frequency or baud rate, should be modeled as GACS, rather than ACS, if the data-record length is such that the parameter time variations can be appreciated. In (Izzo and Napolitano 2003), the problem of sampling a GACS signal is addressed. It is shown that the discrete-time signal constituted by the samples of a continuous-time GACS signal is a discrete-time ACS signal. Moreover, it is shown that starting from the sampled signal, the ACS or GACS nature of the continuous-time signal can only be conjectured, provided that analysis parameters such as sampling period, padding factor, and data-record length are properly chosen. In (Izzo and Napolitano 1998a, 2002a,b, 2003), the signal analysis is carried out in the fraction-of-time probability or nonstochastic approach, where probabilistic parameters are defined through infinite-time averages of functions of a single time-series rather than through expected values (ensemble averages) of a stochastic process (Chapter 6), (Gardner 1987d), (Lekow and Napolitano 2006). A survey of the GACS signals in the nonstochastic approach is provided in (Izzo and Napolitano 2005). In (Hanin and Schreiber 1998), the problem of consistently estimating the time-averaged autocorrelation function for a class of processes with periodically time-variant autocorrelation function is addressed, and it is shown that some results are still valid if the period depends on the lag parameter. This class of processes, therefore, is a subclass of the GACS processes.

The design of signal processing algorithms can require the estimation of second-order statistics. Statistical functions of GACS signals need to be estimated, for example, in channel identification and/or equalization problems if the channel model is linear time-variant but not almost-periodically time variant (e.g., when the channel introduces quadratically time-varying delays) (Izzo and Napolitano 2002a,b).

In this chapter, the problem of estimating second-order cross-moments of complex-valued jointly GACS processes is addressed. Estimators of autocorrelation function and conjugate autocorrelation function of a single process are obtained as special cases. The second-order cross-moment of jointly GACS processes can be expressed by a Fourier series expansion with lag-dependent cycle frequencies and lag-dependent coefficients referred to as generalized cyclic cross-correlation functions. Equivalently, it can be expressed by a (generalized) Fourier series expansion with constant cycle frequencies ranging in a countable set depending on the lag shift and lag-dependent coefficients referred to as cyclic cross-correlation functions. Thus, the second-order cross-moment is completely characterized by the cyclic cross-correlation function as a function of the two variables lag shift and cycle frequency. Such a function has support contained in a countable set of curves in the lag-shift cycle-frequency plane which are described by the lag-dependent cycle frequencies. For ACS processes these curves reduce to lines parallel to the lag-shift axis.

The cyclic cross-correlogram is proposed in (Napolitano 2007a) as an estimator of the cyclic cross-correlation function of jointly GACS processes. It is shown that, for GACS stochastic processes satisfying some mixing conditions expressed in terms of summability of cumulants, the cyclic cross-correlogram, as a function of the two variables lag shift and cycle frequency, is a mean-square consistent and asymptotically complex Normal estimator of the cyclic cross-correlation function. Furthermore, in the limit as the data-record length approaches infinity, the region of the cycle-frequency lag-shift plane where the cyclic cross-correlogram is significantly different from zero becomes a thin strip around the support curves of the cyclic cross-correlation function, that is, around the lag-dependent cycle frequency curves. Thus, the proved asymptotic complex Normality result can be used to establish statistical tests for presence of generalized almost-cyclostationarity.

The discrete-time cyclic cross-correlogram of the discrete-time jointly ACS processes obtained by uniformly sampling two continuous-time jointly GACS processes is considered in (Napolitano 2009) as an estimator for the continuous-time cyclic cross-correlation function. It is shown that for GACS processes no simple condition on the sampling frequency can be stated as for band-limited wide-sense stationary or ACS processes in order to avoid or limit aliasing. However, the discrete-time cyclic cross-correlogram is shown to be a mean-square consistent and asymptotically Normal estimator of the continuous-time cyclic cross-correlation function as the data-record length approaches infinity and the sampling period approaches zero, provided that some mild regularity conditions are satisfied. The well-known result for ACS processes that the cyclic correlogram is a mean-square consistent and asymptotically Normal estimator of the cyclic autocorrelation function (Hurd 1989a, 1991), (Hurd and Lekow 1992b), (Dehay and Hurd 1994), (Genossar et al. 1994), (Dandawaté and Giannakis 1995) is obtained as a special case of the results established for GACS processes.

The discrete-time counterparts of the asymptotic results of (Napolitano 2007a) are not straightforward. In fact, in (Izzo and Napolitano 2003), it is shown that uniformly sampling a continuous-time GACS process gives rise to a discrete-time ACS process and a discrete-time counterpart of continuous-time GACS processes does not exist. Furthermore, the GACS or ACS nature of an underlying continuous-time process can only be conjectured starting from the analysis of the discrete-time ACS process. In addition, since GACS processes cannot be strictly band-limited (Izzo and Napolitano 1998a), (Napolitano 2007a), unlike the case of ACS or stationary processes, a minimum value of the sampling frequency to completely avoid aliasing in the discrete-time cyclic statistics does not exist. This constitutes a complication for the estimation of the cyclic autocorrelation function of a GACS process starting from the discrete-time process of its samples.

The asymptotic statistical analysis of the discrete-time cyclic correlogram of the ACS process obtained by uniformly sampling a continuous-time GACS process is carried out when the number of data samples approaches infinity (to get consistency) and the sampling period approaches zero (to counteract aliasing). It is shown that the discrete-time cyclic correlogram of the discrete-time process obtained by uniformly sampling a continuous-time GACS process is a mean-square consistent estimator of samples of the aliased continuous-time cyclic autocorrelation function of the GACS process as the number of data-samples approaches infinity. Moreover, it is pointed out that the discrete-time cyclic correlogram has the drawback that, when the sampling period approaches zero, the unnormalized lag parameter approaches zero and the unnormalized cycle frequency approaches infinity. It is shown that such a drawback is also present if the discrete-time cyclic correlogram is adopted to estimate the continuous-time cyclic autocorrelation function of ACS processes which are not strictly band-limited so that the sampling period should approach zero to reduce aliasing. Furthermore, the same problem is encountered with the correlogram estimate of the autocorrelation function of non strictly band-limited wide-sense stationary processes. This problem does not occur if continuous-and discrete-time estimation problems are treated separately, and the aliasing problem arising from sampling is not addressed (Brillinger and Rosenblatt 1967).

A procedure to carry out the asymptotic analysis of the discrete-time cyclic cross-correlogram as the number of data-samples approaches infinity and the sampling period approaches zero is proposed in (Napolitano 2009) by resorting to the hybrid cyclic correlogram. It is called “hybrid” since some parameters are continuous-time and others are discrete-time. Specifically, data-samples are discrete-time and lag parameter and cycle frequency are those of the continuous-time cyclic correlogram and are assumed to be constant with respect to the sampling period. It is shown that the hybrid cyclic correlogram is a mean-square consistent and asymptotically Normal estimator of the cyclic autocorrelation function when the number of samples approaches infinity and the sampling period approaches zero in such a way that the overall data-record length approaches infinity. Thus, it is shown that the mean-square error between the discrete-time cyclic correlogram and samples of the continuous-time cyclic autocorrelation function can be made arbitrarily small provided that the number of data samples is sufficiently large and the sampling period is sufficiently small. Moreover, it is shown that the asymptotic bias, covariance and distribution of the hybrid cyclic correlogram are coincident with those of the continuous-time cyclic correlogram. That is, the discrete-time analysis, asymptotically, does not introduce performance degradation with respect to continuous-time analysis. The proposed discrete-time asymptotic analysis can be applied also to the case of non strictly band-limited continuous-time ACS and stationary processes.

For continuous-time ACS processes continuous-time estimators are proposed in (Hurd 1989a, 1991), (Hurd and Lekow 1992a,b), (Dehay 1994), and references therein, and for discrete-time ACS processes, discrete-time estimators are proposed in (Genossar et al. 1994), (Dandawaté and Giannakis 1995), (Schell 1995). Aliasing and estimation issues have not been previously considered together. The results in this chapter jointly treat the problem of reducing aliasing and getting consistency.

The chapter is organized as follows. In Section 2.2 the class of the GACS processes is characterized and motivations and examples are provided. In Section 2.3, the problem of linear time-variant filtering of GACS processes is addressed. In Section 2.4.1, the cyclic cross-correlogram is proposed as an estimator for the cyclic cross-correlation function of two jointly GACS processes. Moreover, its expected value and covariance are derived for finite data-record length. In Section 2.4.2, the mean-square consistency of the cyclic cross-correlogram is established and its asymptotic expected value and covariance are derived. In Section 2.4.3, the cyclic cross-correlogram is shown to be asymptotically complex Normal. In Section 2.5, the problem of sampling continuous-time GACS processes is treated. The discrete-time estimation of the cyclic cross-correlation function of continuous-time jointly GACS processes is addressed in Section 2.6. Specifically, in Section 2.6.1 the discrete-time cyclic cross-correlogram is defined and its mean and covariance for finite number N of data-samples and finite sampling period T_s are derived. Results of consistency and asymptotic complex Normality as are derived in Section 2.6.2. Results for and are derived in Section 2.6.3. Numerical results to corroborate the effectiveness of the theoretical results are reported in Section 2.7. A discussion on the stated results is made in Section 2.7.5. A Summary is given in Section 2.8. Proofs of the results presented in Sections 2.4.1, 2.4.2, 2.4.3, are reported in Sections 3.4, 3.5, and 3.6, in Chapter 3, respectively. Proofs of results of Sections 2.6.1, 2.6.2, and 2.6.3 are reported in Sections 3.9, 3.10, and 3.11, respectively. Some issues concerning complex processes are addressed in Sections 3.7 and 3.13.