4.1 Introduction

It is well known that wide-sense stationary (WSS) stochastic processes do not exhibit correlation between spectral components at distinct frequencies. That is, by passing a WSS process throughout two bandpass filters with nonoverlapping passbands, and then frequency shifting the two output processes to a common band, one obtains two uncorrelated stochastic processes (Gardner 1987d). Equivalently, for WSS processes, the Loève bifrequency spectrum (Loève 1963) (also called dual-frequency spectrum (Øigård et al. 2006) or cointensity spectrum (Middleton 1967)) has support contained in the main diagonal of the bifrequency plane. The presence of spectral correlation between spectral components at distinct frequencies is an indicator of the nonstationarity of the process (Loève 1963), (Gardner 1987d), (Hurd and Gerr 1991), (Genossar 1992), (Dehay and Hurd 1994), (Napolitano 2003), (Dmochowski et al. 2009). When correlation exists only between spectral components that are separated by quantities belonging to a countable set of values, the process is said almost-cyclostationary (ACS) or almost-periodically correlated (Gardner 1985, 1987d), (Gardner et al. 2006), (Hurd and Miamee 2007). The values of the separation between correlated spectral components are called cycle frequencies and are the frequencies of the (generalized) Fourier series expansion of the almost-periodically time-variant statistical autocorrelation function (Gardner 1987d), (Gardner 1991b), (Dehay and Hurd 1994). For ACS processes, the Loève bifrequency spectrum has the support contained in lines parallel to the main diagonal of the bifrequency plane and its values on such lines are described by the spectral correlation density functions (Hurd 1989a, 1991), (Hurd and Gerr 1991).

A new class of nonstationary stochastic processes, the spectrally correlated (SC) processes, has been introduced and characterized in (Napolitano 2003). SC processes exhibit Loève bifrequency spectrum with spectral masses concentrated on a countable set of support curves in the bifrequency plane. Thus, ACS processes are obtained as a special case of SC processes when the support curves are lines with unit slope. In communications, SC processes are obtained when ACS processes pass through Doppler channels that induce frequency warping on the input signal (Franaszek and Liu 1967). For example, let us consider the case of relative motion between transmitter and receiver in the presence of moving reflecting objects. If the involved relative radial speeds are constant within the observation interval and the product signal-bandwidth times data-record length is not much smaller than the ratios of the medium propagation speed and the radial speeds, then the resulting multipath Doppler channel introduces a different complex gain, time-delay, frequency shift, and nonunit time-scale factor for each path (Napolitano 2003). Since the time-scale factors are nonunit, when an ACS process passes through such a channel, the output signal is SC (Napolitano 2003). Therefore, the SC model is appropriate in high mobility scenarios in modern communication systems where wider and wider bandwidths are considered to get higher and higher bit rates and, moreover, large data-record lengths are used for blind channel identification or equalization algorithms or for detection techniques in highly noise-and interference-corrupted environments. Further situations where nonunit time-scale factors should be accounted for can be encountered in radar and sonar applications (Van Trees 1971, pp. 339–340), communications with wide-band and ultra wide-band (UWB) signals (Jin et al. 1995), (Schlotz 2002), space communications (Oberg 2004), and underwater acoustics (Munk et al. 1995). In all these cases, SC processes are appropriate models for the signals involved (Napolitano 2003). In this chapter, it is shown that SC processes with nonlinear support curves in the bifrequency plane occur in spectral analysis with nonuniform frequency spacing (Oppenheim et al. 1971), (Oppenheim and Johnson 1972), (Braccini and Oppenheim 1974), or signal processing algorithms exploiting frequency warping techniques as those considered in (Makur and Mitra 2001), (Franz et al. 2003). It is also shown that jointly SC processes occur in the presence of some linear time-variant systems as those described in (Franaszek 1967), (Franaszek and Liu 1967), (Liu and Franaszek 1969), (Claasen and Mecklenbräuker 1982). In (Øigård et al. 2006), it is shown that fractional Brownian motion (fBm) processes have Loève bifrequency spectrum with spectral masses concentrated on three lines of the bifrequency plane. Finally, in discrete-time, ACS processes and their multirate processed versions turn out to be jointly SC (Napolitano 2010a).

SC processes are characterized at second-order in the frequency domain in terms of Loève bifrequency spectrum. The amount of spectral correlation existing between two separate spectral components is characterized by the bifrequency spectral correlation density function, that is, the density of the Loève bifrequency spectrum on its support curves. Then the estimation problem for such a density is considered.

Spectral estimation techniques have been developed for stationary, ACS, and some classes of nonstationary processes. For WSS processes, the power spectrum can be consistently estimated by the frequency-smoothed periodogram, provided that some mixing assumptions regulating the memory of the process are satisfied (Brillinger 1981), (Thomson 1982). The spectral-correlation density estimation problem has been addressed in the case of ACS processes in (Gardner 1986a), (Gardner 1987d), (Hurd 1989a), (Hurd and Lekow 1992a), (Dandawatacute; and Giannakis 1994), (Dehay and Hurd 1994), (Genossar et al. 1994), (Gerr and Allen 1994), (Dehay and Lekow 1996), (Sadler and Dandawatacute; 1998). The proposed techniques are based on time-or frequency smoothing the cyclic periodogram and provide consistent estimators of the spectral correlation density function under very mild conditions (e.g., the finite or approximately finite memory of the stochastic process). These estimators are the extensions to ACS signals of the well known power-spectrum estimators for WSS signals (Brillinger 1981), (Thomson 1982).

The correlation existing between separated spectral components, that in the following will be referred to as the spectral correlation property, is adopted by several authors in detecting and characterizing nonstationary processes. In (Gardner 1988a), (Gardner 1991c), it is shown that spectral correlation measurements can be reliable (small bias and variance) only if the nonstationarity is of almost-periodic nature (almost-cyclostationarity) or of known form. The former case is considered in (Gardner 1987d), (Hurd 1991), (Hurd and Gerr 1991), (Gerr and Allen 1994), and (Varghese and Donohue 1994). To the latter case belong the techniques of the radial and Doppler smoothing presented in (Allen and Hobbs 1992) and the estimator proposed in (Lii and Rosenblatt 2002). They provide consistent estimators of the spectral correlation density function for processes whose Loève bifrequency spectrum has the support on known lines with not necessarily unit slopes. The case of transient processes is treated in (Hurd 1988).

The problem of estimating the bifrequency spectral correlation density function is addressed at first when the location of the spectral masses is unknown. The bifrequency cross-periodogram is shown to be an estimator of the bifrequency spectral correlation density function asymptotically biased with nonzero asymptotic variance and the bias can be compensated only if the location of the spectral masses in the bifrequency plane is known. The time-smoothed cross-periodogram is considered as an estimator of the bifrequency spectral correlation density function since it represents the finite-time-averaged cross-correlation between finite-bandwidth spectral components of the process. It is shown that time-smoothing the cross-periodogram is the only practicable way of smoothing when the location of the spectral masses is unknown. In contrast, the frequency-smoothing technique can be adopted only if such a location is known (Allen and Hobbs 1992; Lii and Rosenblatt 2002).

Bias and variance of the time-smoothed bifrequency cross-periodogram are derived in the case of finite bandwidth of the spectral components and finite observation interval. Moreover asymptotic results as the bandwidth approaches zero and the observation interval approaches infinity are derived. It is shown that the bifrequency spectral correlation density function of SC processes that are not ACS can be estimated with some degree of reliability only if the departure of the nonstationarity from the almost-cyclostationarity is not too large. In particular, if in the neighborhood of a given point of the bifrequency plane the slope of the support curve of the Loève bifrequency spectrum is not too far from unity, then the smoothing product can be large enough to obtain a small variance and maintaining, at the same time, small bias. Then, a trade-off exists between the departure of the nonstationarity from the almost-cyclostationarity and the reliability of spectral correlation measurements obtainable by a single sample path. Moreover, in general, the estimate accuracy cannot be improved as wished by increasing the data-record length and the spectral resolution.

In (Gardner 1988a, 1991b), it is shown that spectral correlation measurements can be reliable only if the nonstationarity is almost cyclostationarity or of known form. Consequently, accordingly with the results in (Napolitano 2003), for SC processes that are not ACS, reliable estimates of spectral correlation density cannot be obtained if the location of the support curves is unknown.

For ACS processes, frequency smoothing the cross-periodogram along the unit-slope support lines of the bifrequency Loève spectrum leads to the frequency-smoothed cyclic periodogram which is a mean-square consistent and asymptotically Normal estimator of the cyclic spectrum (Gardner 1986a, 1987d), (Alekseev 1988), (Hurd 1989a), (Hurd 1991), (Hurd and Gerr 1991), (Hurd and Lekow 1992a), (Dandawatacute; and Giannakis 1994), (Dehay and Hurd 1994), (Gerr and Allen 1994), (Genossar et al. 1994), (Dehay and Lekow 1996), (Sadler and Dandawatacute; 1998).

Spectral analysis of nonstationary processes is addressed in (Hurd 1988), (Gardner 1991b), (Genossar 1992) with reference to the spectral-coherence estimation problem. For nonstationary processes having the Loève bifrequency spectrum not concentrated on sets of zero Lebesgue measure in the plane, the only possibility to get consistent estimates of the spectral density is to resort to more realizations or sample paths (Lii and Rosenblatt 2002), (Soedjack 2002). For SC processes, by following the main idea of smoothing the periodogram as for WSS and ACS processes, in the special case of known support curves constituted by lines, in (Chiu 1986), (Allen and Hobbs 1992), (Lii and Rosenblatt 1998), and (Lii and Rosenblatt 2002), the cross-periodogram smoothed along a given known support line has been proposed as estimator of the spectral correlation density on this support line. In (Lii and Rosenblatt 2002), the case of finite number of support lines is considered.

The problem of estimation of the density of Loève bifrequency spectrum (or spectral correlation density) of SC processes is addressed in this chapter in the case of known support curves. Specifically, the cross-periodogram frequency smoothed along a known given support curve is proposed as estimator of the spectral correlation density on this support curve. It is shown that the frequency-smoothed cross-periodogram asymptotically (as the data-record length approaches infinity and the spectral resolution approaches zero) approaches the product of the spectral correlation density function and a function of frequency and slope of the support curve. Since the support curve is assumed to be known, such a multiplicative bias term can be compensated. Moreover, it is shown that the asymptotic covariance of the frequency smoothed cross-periodogram approaches zero. Therefore, the properly normalized frequency-smoothed cross-periodogram is a mean-square consistent estimator of the spectral correlation density function. Moreover, it is shown to be asymptotically complex Normal. Thus, such a result can be exploited to design statistical tests for presence of spectral correlation. Finally, it is shown that the well-known result for ACS processes that the frequency-smoothed cyclic periodogram is a mean-square consistent and asymptotically Normal estimator of the cyclic spectrum (Gardner 1986a, 1987d), (Alekseev 1988), (Hurd 1989a, 1991), (Hurd and Gerr 1991), (Hurd and Lekow 1992a), (Dandawatacute; and Giannakis 1994), (Dehay and Hurd 1994), (Gerr and Allen 1994), (Genossar et al. 1994), (Dehay and Lekow 1996), (Sadler and Dandawatacute; 1998) can be obtained as a special case of the results for SC processes.

A definition of band limitedness for nonstationary processes is given and, according to such definition, band-limited (continuous-time) SC processes are characterized. Then, discrete-time SC processes are introduced and characterized. It is shown that a discrete-time SC process can be obtained by uniformly sampling a continuous-time SC process and its Loève bifrequency spectrum is constituted by the superposition of replicas of the Loève bifrequency spectrum of the continuous-time SC process. It is also shown that for strictly band-limited SC processes a sufficient condition to avoid overlapping replicas in the Loève bifrequency spectrum is that the sampling frequency f_s is at least two times the process bandwidth which is the classical Shannon condition. However, more strict conditions on the sampling frequency need to be considered if a mapping between frequencies of the density of Loève bifrequency spectrum of the continuous-time process and frequencies of the density of Loève bifrequency spectrum of the discrete-time process must hold in the whole principal frequency domain (Napolitano 2011).

The chapter is organized as follows. In Section 4.2, SC stochastic processes are defined and characterized and motivating examples are provided. Bias and covariance are determined in Section 4.4 for the bifrequency cross-periodogram and in Section 4.5 for the time-smoothed bifrequency cross-periodogram in the case of unknown support curves. Proofs of lemmas and theorems of Sections 4.4 and 4.5 are reported in Sections 5.2 and 5.3, respectively. Bias and covariance are determined in Section 4.6 for the frequency-smoothed cross-periodogram in the case of known support curves. In Section 4.7, the properly normalized frequency-smoothed cross-periodogram is shown to be mean-square consistent and its asymptotic bias and variance are determined. Moreover, it is shown to be asymptotically complex Normal. Proofs of the results presented in Sections 4.6, 4.7.1, and 4.7.2 are reported in Sections 5.4, 5.5, and 5.6, respectively. Results obtained in the case of assumptions alternative to those made in this chapter are briefly presented in Section 5.7. Some issues concerning complex processes are addressed in Section 5.8. A discussion on the established results is made in Section 4.7.3. In Section 4.8, discrete-time SC processes are defined and characterized. Sampling theorems for SC processes are provided in Section 4.9. Proofs of theorems are reported in Section 5.10. Illustrative examples are presented in Section 4.9.3. Multirate processing of SC processes is treated in Section 4.10 and proofs of the results are reported in Section 5.11. In Section 4.11, discrete-time estimators for the spectral cross-correlation density are considered. In Section 4.12 simulation results on the spectral correlation density estimation are reported. Finally, in Section 4.13, spectral analysis with nonuniform frequency spacing is addressed. A Summary is given in Section 4.14.