The statistical characterization of heavy-tailed data with hidden irregular periodicities is introduced. Specifically, processes generated by the interaction of random phenomena with heavy-tailed distribution and almost-periodic phenomena with possibly irregular or disturbed peridicities are characterized in terms of fractional lower-order moments. For this wide class of processes, fractional lower-order moments are expressed as the superposition of amplitude- and angle-modulated sine waves. The model introduced in the paper extends the previously introduced one for heavy-tailed almost cyclostationary (ACS) processes to the case of irregular periodicities. Moreover, it introduces for the class of the oscillatory ACS processes a characterization in terms of fractional lower-order moments. For the new class, the problem of statistical function estimation is addressed. The effectiveness of the proposed methodology is corroborated by the analysis of simulated alpha-stable time-warped ACS processes and of real acoustic helicopter data. • The statistical characterization of heavy-tailed data with hidden irregular periodicities is introduced. • Fractional lower-order moments are expressed as the superposition of amplitude- and angle- modulated sine waves. • The problem of statistical function estimation is addressed. • Simulated alpha-stable time-warped almost-cyclostationary processes and real acoustic helicopter data are analyzed.

In the above article[1], expression (A128), in the proof of Theorem 17 on page 672, has a missed term. Here, the missed term is considered and its convergence to zero is proved. The (conjugate) cyclic correlogram$R_{x }^{(T)}(\alpha,\tau)$is nonzero only for$\tau \in [-T,T]$. Thus, from (42b) and (43), we have the following corrected version of (A128) (normalized bias of the frequency-smoothed (conjugate) cyclic periodogram with estimated (conjugate) cycle frequency)

For almost-cyclostationary processes, sufficient conditions are derived, such that estimates of the second-order cyclic probabilistic functions with estimated cycle frequencies are mean-square consistent and asymptotically complex normal. Cyclic (conjugate) autocorrelation functions and cyclic (conjugate) spectra are considered. Under the derived conditions, asymptotically, as the data-record length approaches infinity, the estimates of cyclic statistics with estimated cycle frequencies have the same complex normal distribution as the case of exactly known cycle frequencies. The results are applied to the detection of a moving cyclostationary source in the presence of strong Doppler effect and for low values of SNR.

A passive localization algorithm is proposed for moving sources emitting man-made signals. The method models the received signals on two sensors as (singularly) almost-cyclostationary. It is not based on the usual so called narrowband assumption that limits bandwidth, data-record length, and relative radial speed between source and sensors. Thus, unlike previous techniques, it statistically characterizes the signals on the two sensors as jointly spectrally correlated rather than as jointly almost cyclostationary. The algorithm estimates time-scale ratio, frequency-difference-of-arrival, and time-delay-of-arrival of the source signal impinging on the two sensors. It is highly tolerant to noise and interference and outperforms classical cyclostationarity-based techniques for large data-record lengths.

In this paper, stochastic processes with higher order statistical functions decomposable into an almost-periodic function plus a residual term not containing finite-strength additive sinewave components are considered. These processes arise in mobile communications when almost-cyclostationary (ACS) processes pass through time-varying channels. They include as special case the generalized almost-cyclostationary processes which, in turn, include the ACS processes. In the paper, the problem of estimating the Fourier coefficients of the (generalized) Fourier series expansion of the almost-periodic component of higher order statistical functions is addressed. Estimators are proposed for cyclic temporal cross moment and cumulants. They are proved to be mean square consistent and asymptotically complex Normal under mild assumptions on the memory of the processes expressed in terms of summability of cross cumulants. Numerical results confirm the theoretical results and the derived rate of convergence to zero of bias and standard deviation of the estimators.

In this paper, the problem of estimating the spectral correlation density of spectrally correlated stochastic processes is addressed. These processes have Loeve bifrequency spectrum with spectral masses concentrated on a countable set of support curves in the bifrequency plane. The almost-cyclostationary processes are obtained as a special case when the support curves are lines with unit slope. Spectrally correlated processes find application in wide-band or ultrawideband mobile communications. It is shown that the cross-periodogram frequency smoothed along a known support curve and properly normalized provides a mean-square consistent estimator of the spectral correlation density of the Loeve bifrequency spectrum along that curve.

In this paper, the problem of estimating second-order cross-moments of generalized almost-cyclostationary (GACS) processes is addressed. GACS processes have statistical functions that are almost-periodic functions of time whose (generalized) Fourier series expansions have both frequencies and coefficients that depend on the lag shifts of the processes. The class of such nonstationary processes includes the almost-cyclostationary (ACS) processes which are obtained as a special case when the frequencies do not depend on the lag shifts. ACS processes filtered by Doppler channels and communications signals with time-varying parameters are further examples. It is shown that the second-order cross-moment of two jointly GACS processes is completely characterized by the cyclic cross-correlation function. Moreover, it is proved that the cyclic cross-correlogram is an asymptotically normal, mean-square consistent, estimator of the cyclic cross-correlation function. Furthermore, it is shown that well-known consistency results for ACS processes can be obtained by specializing the results of this paper.

In this letter, the cyclostationarity properties of a general class of long-code direct-sequence spread-spectrum (DS-SS) signals are analyzed, and it is shown that these properties can be exploited in long-code DS code-division multiple-access systems. Specifically, it is shown that a continuous-time DS-SS signal exhibits cyclostationarity at cycle frequencies related to both the symbol rate and the chip rate. Moreover, it is pointed out that the discrete-time signal obtained by uniformly sampling a continuous-time long-code DS-SS signal exhibits cyclostationarity, provided that at least two samples per chip are taken. Finally, it is shown how such a cyclostationarity can be suitably exploited for blind signal-parameter estimation

The problem of blindly estimating the parameters of a Doppler channel for noncircular transmissions in multiple-access communication systems is addressed. A nondata-aided algorithm based on the cyclostationarity features of the received signal is proposed to estimate amplitude, phase, time delay, and frequency shift of each user. Under mild assumptions on the disturbance and user signals, the proposed method provides estimates of the unknown parameters that are mean-square consistent. Moreover, the proposed algorithm is asymptotically near-far resistant, and is not based on the usual assumption of white and/or Gaussian noise.

In this paper, the class of the spectrally correlated stochastic processes is introduced. Processes belonging to this class exhibit a Loe/spl grave/ve (1963) 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 the almost-cyclostationary (ACS) processes that are obtained as a special case when the separation between correlated spectral components assumes values only in a countable set. In such a case, the support curves are lines with unit slope. For the spectrally correlated processes, 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 Loe/spl grave/ve 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 bifrequency spectral correlation density function in those points of the bifrequency plane where the slope of the support curves is not too far from unity. Moreover, there exists a tradeoff between the departure of the nonstationarity from the almost-cyclostationarity and the reliability of spectral correlation measurements obtainable by a single sample-path. Furthermore, in general, the estimate accuracy cannot be improved as wished by increasing the data-record length and the spectral resolution.

A blind algorithm for the estimation of amplitude, phase, and relative time delay of each user in multiple-access communication systems is proposed. This algorithm is based on the cyclostationarity features of the received signals and provides estimates of the unknown parameters that are intrinsically immune to the effects of noise and interference, provided that a cycle frequency of the user signals exists which is not shared with the disturbance terms. The proposed estimators are, under mild assumptions, asymptotically unbiased and consistent, Moreover, the proposed algorithm is asymptotically near-far resistant and is not based on the usual assumption of white and/or Gaussian noise.