The almost-cyclostationary time-series exhibit statistical parameters that are almost-periodic functions of time whose Fourier series expansions exhibit frequencies not depending on the lag shifts of the time-series. In [C16], [J9], the class of the Generalized Almost-CycloStationary (GACS) time-series is introduced. Time-series belonging to this class are characterized by multivariate statistical functions that are almost periodic function of time whose Fourier series expansions can exhibit coefficients and frequencies depending on the lag shifts of the time-series. Almost cyclostationary (ACS) time-series turn out to be the subclass of GACS time-series for which the frequencies do not depend on the lag shifts. Examples of GACS time-series not belonging to the subclass of ACS time-series arise from several time-variant transformations of ACS signals, such as channels introducing a time-variant delay and multipath Doppler channels [C24], [C25], [C29]. Chirp signals and several angle-modulated and time-warped communication signals are further examples. In [J9], the higher-order characterization of GACS time-series in the strict and wide sense is provided. Generalized cyclic moment and cumulant functions (in both the time and frequency domains) are introduced.
In [J15], the problem of linear time-variant filtering of GACS signals is addressed in the fraction-of-time probability framework. The adopted approach, which is alternative to the classical stochastic one, provides a statistical characterization of the system in terms of time averages of functions of time rather than ensemble averages of stochastic processes. Thus, it is particularly useful when stochastic systems transform ergodic input signals into nonergodic output signals, as it happens with several channel models encountered in the practice. In the paper, systems are classified as deterministic or random in the fraction-of-time probability framework. Moreover, the new concept of expectation in the fraction-of-time probability framework of the impulse-response function of a system is introduced. For the linear time-variant 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 sinewave components contained in the input lag product into the finite-strength additive sinewave components contained in the output lag product. Moreover, 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. Examples of applications and developments of the theory introduced in Part I [J15] are presented in Part II [J16] where several Doppler channel models are analyzed 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.
In [J17], the problem of sampling a continuous-time GACS signal is addressed. 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, namely, the continuous-time GACS signals. In the paper, 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.
The book chapter [BC3] provides a a survey on the GACS signals.
The GACS signals are considered in stochastic approach in [J25], [J27], [J30]. In [J25] it is shown that the cyclic correlogram is an asymptotically Normal and mean-square consistent estimator of the cyclic autocorrelation function of GACS processes, provided that some mixing assumptions expressed in terms of summability of cumulants are satisfied. The problem of the discrete-time estimation of second-order statistics of GACS processes is addressed in [J27]. Sufficient conditions are provided such that the cyclic correlogram of the discrete-time ACS process obtained by uniformly sampling a continuous-time GACS process is an asymptotically Normal and mean-square consistent estimator of samples of the cyclic autocorrelation function of the GACS process as the observation interval approaches infinity and the sampling period approaches zero.
The estimation of cyclic higher-order statistics of GACS processes is treated in [J30].
In [C28], [J18], the class of the Spectrally Correlated (SC) stochastic processes is introduced. Processes belonging to this class exhibit a Loeve bifrequency spectrum with spectral masses concentrated on a countable set of support curves in the bifrequency plane. Thus, such processes have spectral components that are correlated. The introduced class generalizes that of the almost-cyclostationary processes that are obtained as a special case when the separation between correlated spectral components assumes values only in a countable set and the support curves are lines with unitary slope. The amount of spectral correlation existing between two separate spectral components is characterized by the bifrequency spectral correlation density function which is the density of the Loeve bifrequency spectrum on its support curves. It is shown that, in general, when the location of the support curves is unknown, the time-smoothed cross-periodogram can provide a reliable (low bias and variance) single sample-path based estimate of the spectral correlation density function in those points of the bifrequency plane where the slope of the support curves is not too far from unity. In contrast, if the location of the support curves is known, the frequency-smoothed cross-periodogram is a mean-square consistent estimator of the spectral correlation density function [C41].
The usefulness of SC processes in modeling Doppler-stretched wide-band signals is addressed in [J29]. The problem of sampling SC processes is treated in [J31]. It is shown that, unlike the case of the wide-sense stationary signals, several kind of aliasing effects occur for the various spectral (cross-)statistical functions used to characterize SC signals. Lower bounds on the sampling frequency are derived to avoid these aliasing effects.
The theory of GACS and SC processes is presented in the book [B1]. The main problems arising in signal processing such as probabilistic characterization, statistical function estimation, filtering, sampling and link between continuous- and discrete-time processes, are addressed for both classes of processes. An analysis of Doppler channels is carried out to show the usefulness of these new signal models in communications and radar/sonar applications.
In [J37], limits to the applicability of the cyclostationary model in the presence of relative motion between transmitter and receiver are enlightened. The new classes of GACS and SC signals that extend the class of the almost-cyclostationary signals and allow one to overcome these limits are reviewed. Probabilistic characterization, statistical function estimation, and time-sampling are reviewed for both classes of signals and a comprehensive bibliography commented. The impact of these models on classical problems considered in the cyclostationary framework is illustrated. Finally, the new class of the Oscillatory Almost-Cyclostationary (OACS) processes is introduced and characterized. It allows one to describe the output signal of a generic linear time variant system with ACS input. OACS processes have autocorrelation function which is the superposition of amplitude- and angle-modulated complex sinewaves, where the modulating functions, referred to as evolutionary cyclic autocorrelation functions, depend on both time and lag parameter [J37], [C55]. This class of processes includes that of the almost-cyclostationary processes. The problem of statistical function measurements is addressed in [C55] for the special case of amplitude-modulated time-warped almost-cyclostationary processes. These processes are shown to be a suitable model for the electrocardiogram.
Time-warped (TW) almost-cyclostationary (ACS) processes are obtained from ACS processes by linear time-variant transformations that modify the time scale. In [J39], TW-ACS processes are shown to be a subclass of the oscillatory ACS processes. The TW-ACS model is useful to describe signals with hidden imperfect periodicities. An example is the received signal when the transmitted one is ACS in the case of general motion law between transmitter and receiver. Some biological signals as the electrocardiogram are other examples. The problem of statistical function estimation is addressed in [J39] and two estimation techniques are proposed. The theoretical results are corroborated by numerical experiments on simulated and real data.