To illustrate an important difference between stochastic process models and FOT-Probability models of non-stochastic time series, a topic introduced in the chapter [BkC4] of the book [Bk5], is briefly summarized here. There is a class of stochastic processes called Asymptotically-Mean Cyclostationary (AMCS) processes introduced in [JP11]. These processes are nonstationary in the general sense that, while their time-varying (nonstationary) cumulative probability distribution functions (CDFs) are not almost periodic or periodic functions of time, cyclic components of the CDFs exist, even if for some subset of processes in this class they are zero for all non-zero cycle frequencies. Assuming only that the constant component (the time average) exists and is non-zero (while making no assumptions about cyclic components), these processes have been called Asymptotically-Mean Stationary (AMS). If we require that the class of interest has no non-zero cycle frequencies, it should be called Purely AMS to distinguish it from the class AMS which has been shown—somewhat surprisingly—to be identical to the class AMCS [JP11].
AMCS processes exhibit cycloergodic properties [JP11]. As a result, sample paths of an AMCS process are valid models of time series that exhibit cyclostationarity; that is, time series that are almost cyclostationary, or poly-cyclostationary, or purely cyclostationary in the FOT-Probability sense. Consequently, it can be seen in this admittedly round-about manner, that time series that exhibit cyclostationarity are considerably more general than the sample paths of stochastic processes that are cyclostationary or poly cyclostationary or almost cyclostationary. This is so, as explained above, because such time series can exhibit non-cyclic nonstationary behavior for which the impact on the cyclostationary CDFs vanishes in the limit as the averaging time for calculating the CDFs approaches infinity. Similarly, for finite-time empirical CDFs, the non-cyclic nonstationary contributions to the CDFs can be negligibly small for sufficiently long time-series. As an example, the impact on the FOT CDFs of all transient behavior in a time series is small for averaging times that greatly exceed the lengths of transients and vanishes in the limit as averaging time approaches infinity.
When AMCS stochastic process models that exhibit non-cyclic nonstationarity as well as cyclic nonstationarity are used, the cyclic parameters, such as cyclic mean, cyclic covariance, etc. require not only calculating expected values, but also require calculating the cyclic (sinusoidal) components of the time-varying expected values. In contrast, when time-series models are used, only the cyclic component calculations are required. More discussion of the excess baggage carried by stochastic processes is provided on page 3.
In addition to the generalization from ACS stochastic processes to AMCS stochastic processes described on page 5.1, there are other generalizations of time series and ACS stochastic processes. On this page, the classes of Generalized ACS time series and Generalized ACS stochastic processes are described and reference is made to treatments of two other classes of generalizations of ACS stochastic processes, the Spectrally Correlated Processes and the Oscillatory ACS Processes.
Introduction
Relative motion between transmitter and receiver introduces a time-varying delay in the received signal. Consequently, the stationarity or nonstationarity properties of the transmitted signal are modified [10, Sec. 7.1]. Almost-cyclostationary (ACS) signals, which are appropriate nonstationary models for modulated signals used in communications, radar, sonar, and telemetry, are transformed by Doppler channels into generalized almost-cyclostationary (GACS) signals, or spectrally correlated (SC) signals, or oscillatory almost cyclostationary (OACS) signals [12, Chap. 11]. These three classes of signals generalize the class of ACS signals that can be seen to be a special case of each of these classes.
The most appropriate model of nonstationarity for the received signal depends on several transmitter, channel, and receiver properties involving the length of the observation interval, the transmitted signal bandwidth, and motion parameters such as the relative radial acceleration and the relative radial speed. Conditions on these properties, under which each of the models is valid, are summarized in [12, Sec. 11.2].
Up to now, only the GACS signals have been determined to admit a fraction-of-time (FOT) probability model [3], [4], [5]. For the SC and OACS signals, an FOT model has not yet been found and might not exist.
Generalized Almost-Cyclostationary Signals
In this section, the class of GACS signals is presented. GACS signals have autocorrelation functions that are almost periodic with respect to time with (generalized) Fourier series expansion having both Fourier coefficients and Fourier frequencies that are dependant on the lag parameter [3], [4], [5], [6], [8], [9], [10, Chap. 2]. GACS signals are an appropriate model for the received signal under the narrow-band condition [10, Sec. 7.5] when there is non-zero relative radial acceleration between transmitter and receiver (in communications), or between target and transmitter/receiver (in active radar/sonar), or between source and receiver (in passive radar/sonar) [10, Sec. 2.2.6.1, 7.5.2, 7.8].
Let be the almost-periodic component extraction operator, that is, the operator that extracts all the finite-strength additive sine-wave components of its argument [2]. For GACS signals, this operator can be adopted as expectation operator exactly as is done for ACS signals, and a complete FOT higher-order theory can be thereby developed in [3].
The finite-average-power zero-mean second-order signal is said to be generalized almost-cyclostationary in the wide-sense if, for each fixed , its autocorrelation function
is a uniformly almost-periodic function of in the sense of Besicovitch [1, chap. 1]. That is, for each fixed , the autocorrelation can be written in the two following equivalent forms [3], [4]:
(1), (2)
In (1), the cycle frequencies range in a lag-dependent countable set and the functions
(3)
are referred to as cyclic autocorrelation functions in the terminology adopted for the ACS signals.
In (2), is a countable set and the Fourier series expansion has both coefficients and frequencies depending on : The generalized cyclic autocorrelation functions
(4)
and the lag-dependent cycle frequencies .
ACS signals are obtained as the special case of GACS signals for which the lag-dependent cycle frequencies are constant w.r.t. and are coincident with the cycle frequencies. In such a case the generalized cyclic autocorrelation functions are coincident with the cyclic autocorrelation functions.
A more general definition of GACS processes is given in [10, Sec. 2.2.2] by considering the almost-periodicity property in a generalized sense [10, Sec. 1.2].
In [6], it is shown that continuous-time GACS signals do not have a discrete-time counterpart. That is, discrete-time GACS signals do not exist. In addition, the discrete time signal obtained by uniformly sampling a continuous-time GACS signal is a discrete-time ACS signal. Also, starting from the sampled ACS signal, the GACS or ACS nature of the underlying continuous-time signal can only be conjectured, provided that the analysis parameters such as sampling period, padding factor, and data-record length are properly chosen.
In the case of a Doppler channel with transmitter and receiver having constant relative radial acceleration, the transmitted signal experiences a quadratically time-variant delay. If the narrow-band condition [10, Sec. 7.5] is satisfied, the received signal is a chirp-modulated version of the transmitted signal . Consequently, if the transmitted signal is ACS, the received signal is GACS with lag-dependent cycle frequencies that are linear functions of the lag parameter with common slope equal to the chirp rate [8], [10, Sec. 7.8].
Spectrally Correlated Signals and Oscillatory Almost-Cyclostationary Signals
Spectrally correlated signals have a Loève bifrequency spectrum with spectral masses concentrated on a countable set of support curves in the bifrequency plane [7], [10, Chap. 4]. The ACS signals are obtained as the special case for which the support curves are lines with unity slope.
In [7], [10, Chap. 4], a stochastic-process model is adopted for SC signals. It is shown that the density of spectral correlation on the support curves can be consistently estimated only if the loci of the support curves are perfectly known. That is, only if the exact form of the departure from stationarity is exactly known. In contrast, if the location of the support curves is unknown, then the spectral correlation density can be estimated only with some uncertainty, meaning it will not converge to the true spectral correlation density.
Oscillatory almost-cyclostationary signals have autocorrelation functions which are the superposition of amplitude- and angle-modulated sine waves [11, Sec. 6], [12, Chap. 14]. They can be modeled as a linear time-variant transformation of an underlying almost-cyclostationary signal. The ACS signals are obtained as special cases when the sine waves in the autocorrelation function are not modulated.
References
[1] A. S. Besicovitch, Almost Periodic Functions. London: Cambridge UniversityPress, 1932, (New York: Dover Publications, Inc., 1954).
[2] W. A. Gardner and W. A. Brown, “Fraction-of-time probability for time-series that exhibit cyclostationarity,” Signal Processing, vol. 23, pp. 273–292, June 1991.
[3] L. Izzo and A. Napolitano, “The higher-order theory of generalized almost-cyclostationary time-series,” IEEE Transactions on Signal Processing, vol. 46, no. 11, pp. 2975–2989, November 1998.
[4] L. Izzo and A. Napolitano, “Linear time-variant transformations ofgeneralized almost-cyclostationary signals. Part I: Theory and method,” IEEETransactions on Signal Processing, vol. 50, no. 12, pp. 2947–2961, December2002.
[5] L. Izzo and A. Napolitano, “Linear time-variant transformations of generalized almost-cyclostationary signals. Part II: Development and applications,” IEEE Transactions on Signal Processing, vol. 50, no. 12, pp. 2962–2975, December 2002.
[6] L. Izzo and A. Napolitano, “Sampling of generalized almost-cyclostationarysignals,” IEEE Transactions on Signal Processing, vol. 51, no. 6, pp. 1546–1556,June 2003.
[7] A. Napolitano, “Uncertainty in measurements on spectrally correlatedstochastic processes,” IEEE Transactions on Information Theory, vol. 49, no. 9,pp. 2172–2191, September 2003.
[8] A. Napolitano, “Estimation of second-order cross-moments of generalizedalmost-cyclostationary processes,” IEEE Transactions on Information Theory,vol. 53, no. 6, pp. 2204–2228, June 2007.
[9] A. Napolitano, “Discrete-time estimation of second-order statistics ofgeneralized almost-cyclostationary processes,” IEEE Transactions on SignalProcessing, vol. 57, no. 5, pp. 1670–1688, May 2009.
[10] A. Napolitano, Generalizations of Cyclostationary Signal Processing: Spectral Analysis and Applications. John Wiley & Sons Ltd – IEEE Press, 2012.
[11] A. Napolitano, “Cyclostationarity: Limits and generalizations,” SignalProcessing, vol. 120, pp. 323–347, March 2016.
[12] A. Napolitano, Cyclostationary Processes and Time Series: Theory,Applications, and Generalizations. Elsevier, 2019.