Almost-cyclostationary Research Articles

Restoring statistical cyclicity in rotating machinery vibrations using the oscillatory almost-cyclostationary model

Speed fluctuations will affect the statistical characteristics of rotating machinery vibrations, resulting in fuzzy fault features that are difficult to extract. This type of feature ambiguity can be interpreted as the vibrations are warped in time, and the related signal model is described as a time-warped almost-cyclostationary (ACS) process. By employing this sophisticated model in a de-warping procedure, the regular statistical properties of the signal are restored. In this paper, we exploit the de-warping method of Napolitano for restoring fault features of rotating machinery under fluctuating speed conditions. Notice that the time-warped ACS process belongs to a subclass of the oscillatory ACS signal model. More specifically, under moderate speed fluctuation conditions, rotating machinery signals are further characterized as modulated cyclical signals. Therefore, the de-warping can be achieved by demodulating the time-varying autocorrelation function. The approach avoids directly estimating the warping function by the complex non-convex optimization problem. Meanwhile, the accuracy of the solution is guaranteed. More clustered features lead to a more efficient diagnosis. In addition, the rotating speed can also be derived from the warping function without using a tachometer. Case studies of the experiments on a conventional bearing and a turbopump bearing with a large DN-value confirm the feasibility of the oscillatory ACS signal model. Furthermore, the superiority of the de-warping method is corroborated by the enhanced fault features in the classical cyclic spectrum analysis.

Joint Detection of Almost-Cyclostationary Signals and Estimation of Their Cycle Period

We propose a technique that jointly detects the presence of almost-cyclostationary (ACS) signals in wide-sense stationary noise and provides an estimate of their cycle period. Since the cycle period of an ACS process is not an integer, the approach is based on a combination of a resampling stage and a multiple hypothesis test, which deal separately with the fractional part and the integer part of the cycle period. The approach requires resampling the signal at many different rates, which is computationally expensive. For this reason, we propose a filter bank structure that allows us to efficiently resample a signal at many different rates by identifying common interpolation stages among the set of resampling rates.

Time-Warped Almost-Cyclostationary Signals: Characterization and Statistical Function Measurements

Time-warped (TW) almost-cyclostationary (ACS) processes are obtained from ACS processes by linear time-variant transformations that modify the time scale. TW-ACS processes are shown to be a subclass of the oscillatory ACS processes, a recently introduced class of signals extending that of the ACS ones. 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 are other examples. The problem of statistical function estimation is addressed and two estimation techniques are proposed. The theoretical results are corroborated by numerical experiments on simulated and real data.

Generalizations of Cyclostationarity: A New Paradigm for Signal Processing for Mobile Communications, Radar, and Sonar

The relative motion between transmitter and receiver introduces nonstationarity to stationary signals and modifies the nonstationarity of nonstationary signals. Several Doppler channel models of interest transform almost-cyclostationary (ACS) processes, which are an appropriate nonstationary model for modulated signals adopted in communications, radar, sonar, and telemetry, into generalized ACS (GACS) processes or spectrally correlated (SC) processes. Both classes of processes include the ACS processes as special cases. A review of these new classes of nonstationary processes is presented. The problems of probabilistic characterization, statistical function estimation, and determining the effects of time-sampling on these models are addressed. The impact of these models on signal detection and source location problems is discussed.

Almost-Periodic Higher Order Statistic Estimation

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.

Sampling theorems for Doppler-stretched wide-band signals

The paper deals with Doppler-stretched wide-band signals occurring in mobile communications and in radar/sonar problems in the presence of moving targets in the case of constant relative radial speeds. Loève bifrequency spectrum and cross-spectrum are used to characterize in the spectral domain the received signal and to jointly characterize the transmitted and received signals. It is shown that, even if both the transmitted and received signals are singularly almost-cyclostationary (ACS), they are not jointly ACS but, rather, jointly spectrally correlated. The problem of uniformly sampling these nonstationary signals is addressed. 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 these signals. Lower bounds on the sampling frequency are derived to avoid these aliasing effects.

Discrete-Time Estimation of Second-Order Statistics of Generalized Almost-Cyclostationary Processes

In this paper, the problem of estimating second-order statistics of continuous-time generalized almost-cyclostationary (GACS) processes is addressed. GACS processes in the wide sense have autocorrelation function almost-periodic in time whose generalized Fourier series expansion has both frequencies and coefficients that depend on the lag shifts. Almost-cyclostationary (ACS) processes 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 continuous-time GACS processes do not have a discrete-time counterpart. The discrete-time cyclic cross-correlogram of the discrete-time ACS processes obtained by uniformly sampling GACS processes is considered as estimator of samples of the continuous-time cyclic cross-correlation function. The asymptotic performance analysis is carried out by resorting to the hybrid cyclic cross-correlogram which is partially continuous-time and partially discrete-time. Its mean-square consistency and asymptotic complex Normality as the number of data-samples approaches infinity and the sampling period approaches zero are proved under mild conditions on the regularity of the Fourier series coefficients and the finite or practically finite memory of the processes expressed in terms of summability of cumulants. It is shown that the asymptotic properties of the hybrid cyclic cross-correlogram are coincident with those of the continuous-time cyclic cross-correlogram. Hence, discrete-time estimation does not give rise to any loss in asymptotic performance with respect to continuous-time estimation.

Estimation of Second-Order Cross-Moments of Generalized Almost-Cyclostationary Processes

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.

Uncertainty in measurements on spectrally correlated stochastic processes

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.

Sampling of generalized almost-cyclostationary signals

In this paper, the problem of sampling a continuous-time generalized almost-cyclostationary (GACS) signal is addressed. The class of such nonstationary signals includes, as a special case, the almost-cyclostationary (ACS) signals. ACS signals filtered by some linear time-variant channels are further examples. 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 from the sampling of a wider class of nonstationary signals as well, 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, and a condition assuring that the cyclic temporal moment function of the discrete-time signal can be obtained by sampling that of the continuous-time signal is determined. Finally, it is shown that, starting from the sampled signal, the GACS or ACS nature of the continuous-time signal can be conjectured, provided that the analysis parameters such as the sampling period, padding factor, and data-record length are properly chosen.

Linear time-variant transformations of generalized almost-cyclostationary signals.II. Development and applications

For pt. I see ibid., vol.50, no.12, p.2947-61 (2000). In Part I, the problem of the linear time-variant (LTV) filtering is addressed in the fraction-of-time (FOT) probability framework. The adopted approach, which is an alternative to the classical stochastic one, provides a statistical characterization of the systems in terms of functions that can be estimated by a single time-series. The analysis is carried out with reference to the wide class of the generalized almost-cyclostationary (GACS) signals, which includes, as a special case, the class of the almost-cyclostationary (ACS) signals. Examples of applications and developments of the theory introduced in Part I are presented here in Part II. Specifically, the countability of the set of the output cycle frequencies is studied with reference to linear time-variant systems for both ACS and GACS not containing any ACS component input signals. Thus, the linear almost-periodically time-variant filtering and the product modulation are considered in detail. Moreover, several Doppler channel models are analyzed. In all these examples, it is shown that the FOT probability approach allows one to characterize the system and its output in terms of statistical functions that can be measured by a single time-series. Furthermore, the usefulness of considering the linear filtering problem within the class of the GACS signals is clarified, and several pitfalls arising from continuing to adopt for the observed time-series the ACS model when an increase in the data-record length makes the GACS model more appropriate are pointed out.

The higher order theory of generalized almost-cyclostationary time series

In this paper, the class of generalized almost-cyclostationary (GACS) time series is introduced. Time series belonging to this class are characterized by multivariate statistical functions that are almost-periodic functions of time whose Fourier series expansions can exhibit coefficients and frequencies depending on the lag shifts of the time series. Moreover, the union over all the lag shifts of the lag-dependent frequency sets is not necessarily countable. 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 and the union of the above-mentioned sets is countable. The higher order characterization of GACS time series in the strict and wide sense is provided. It is shown that the characterization in terms of cyclic moment and cumulant functions is inadequate for those GACS time series that are not ACS. Then, generalized cyclic moment and cumulant functions (in both the time and frequency domains) are introduced. Finally, the problem of estimating the introduced generalized cyclic statistics is addressed, and two examples of GACS time series are considered.