Time-variant Autocorrelation Function 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.

Period Determination in Cyclo-Stationary Signals by Autocorrelation and Ramanujan Subspaces

Period determination in a periodic-like signal is a challenging process, if the signal is contaminated with a noise or noise-like interference signals. In this work, multiple period determination was considered in aforementioned signals. Recently, cyclostationary properties of the periodic-like signals were utilized to determine the time-varying autocorrelation function (TVAC). First we proved that TVAC can be expressed in terms of Ramanujan sums, then we used TVAC in the periodicity metric to identify the periods. Periodicity metric provides energy distribution of each periodic component as a function of block folding index and reaches a maximum at the points representing the hidden periods in the signal. Proposed method was verified by artificial signals and mean estimation errors versus signal to noise ratio were illustrated. Finally, hidden periods in the noisy respiration signals were estimated by the proposed method successfully.

A 3D Non-Stationary Channel Model with Moving Mobile Station in Rectangular Tunnel

This paper presents a three-dimensional (3D) geometry-based stochastic model (GBSM) for capturing the non-stationarity of radio channel at 1.8 GHz in a rectangular tunnel. A time-variant (TV) complex channel gain is derived for obtaining the statistical properties in time, frequency, and spatial domains such as the time-variant autocorrelation function (TV-ACF), the time-variant Doppler power spectral density (TV-DPSD), and the time-variant spatial cross-correlation function (TV-CCF), respectively. Then the TV channel statistical properties at different time instants can be extracted and the non-stationary channel characteristics caused by the TV scattering environment are thoroughly discussed. Furthermore, three cases including “approach”, “arrival”, and “away” are set to allow a comprehensive study on how the DPSD behaves with the relative position between transmitter and receiver. The reliability of proposed 3D GBSM is highlighted by a good agreement with the measured result in terms of the correlation function.

3D Non-Stationary Wideband Tunnel Channel Models for 5G High-Speed Train Wireless Communications

High-speed train (HST) communications in tunnels have attracted more and more research interests recently, especially within the framework of the fifth generation (5G) wireless networks. In this paper, based on cuboid-shape, three-dimensional (3D) non-stationary wideband geometry-based stochastic models (GBSMs) for HST tunnel scenarios are proposed. By considering the influence of the tunnel walls, a theoretical channel model is first established, which assumes clusters with an infinite number of scatterers randomly distributed on the tunnel walls. The corresponding simulation model is then developed and the method of equal areas is employed to obtain the discrete parameters, such as the azimuth and elevation angles. We derive and investigate the most important channel statistical properties of the proposed 3D GBSMs, including the time-variant autocorrelation function, spatial cross-correlation function, and Doppler power spectrum density. It is indicated that all statistical properties of the simulation model, verified by simulation results, can match very well with those of the theoretical model. Furthermore, a validation is presented by comparing the stationary regions of our proposed tunnel channel model to those of relevant measurement data.

A geometry-based non-stationary MIMO channel model for vehicular communications

This paper derives a non-stationary multiple-input multiple-output(MIMO) from the one-ring scattering model. The proposed channel model characterizes vehicular radio propagation channels with considerations of moving base and mobile stations, which makes the angle of arrivals (AOAs) along with the angle of departures (AODs) time-variant. We introduce the methodology of including the time-variant impacts when characterizing non-stationary radio propagation channels through the geometrical channel modelling approach. We analyze the statistical properties of the proposed channel model including the local time-variant autocorrelation function (ACF) and the space cross-correlation functions (CCFs). We show that the model developed in this paper for non-stationary scenarios includes the existing one-ring wide-sense stationary channel model as its special case.

A Novel 3D Non-Stationary Wireless MIMO Channel Simulator and Hardware Emulator

In this paper, a new WINNER+-based 3-D non-stationary geometry-based stochastic model (GBSM) for multiple-input multiple-output channels is proposed, as well as extended evolving algorithms of time-variant channel parameters. Meanwhile, important statistical properties of the channel model, i.e., time-variant autocorrelation function, time-variant cross-correlation function, and time-variant Doppler power spectrum density are derived and analyzed. Moreover, we propose an efficient hardware implementation method, namely, sum-of-frequency-modulation (SoFM) method, to generate non-stationary channel coefficients. By utilizing compact hardware architecture with SoFM modules, the proposed 3-D non-stationary GBSM is realized on a field-programmable gate array hardware platform. Simulations and hardware measurement results demonstrate that our proposed channel simulator and emulator can get more accurate and realistic Doppler frequency than those of the existing models. In addition, hardware measurements of statistical properties are also well consistent with the corresponding theoretical ones, which verify the correctness of both the hardware emulation scheme and theoretical derivations.

Blind Estimation of Channel Order and SNR for OFDM Systems

In this paper, blind channel order and signal to noise ratio (SNR) estimators based on time-varying autocorrelation function (TVAF) are proposed for orthogonal frequency division multiplexing systems. The TVAF of the received signal reveals clearly the inherent periodicity introduced by cyclic prefix (CP) and carries rich information of channel and additive noise. To investigate the relationships among the TVAF, channel order, and SNR, a newly close-form expression of the TVAF of the received signal is derived. Not only the CP-induced TVAF components but the channel-spread TVAF elements are employed to derive the proposed estimators. Specifically, the two kinds of the TVAF components are separate from the components superimposed by the additive noise, which enables optimal estimation performance even for the low SNR regime. Numerical results show that, in contrast with the alternative methods, the proposed estimators achieve significant performance gain in the low SNR regime, while hold competitive performance in the high SNR regime.

Exploiting HOCS to tackle mutual coupling in a multi-antenna-based cognitive radio receiver

ABSTRACTCognitive radio (CR) has been acknowledged as a probable solution to combat spectrum shortage problem because of diverse wireless applications. Cyclostationary (CS) feature detection is a spectrum hole (i.e. vacant licensed frequency band) detection technique with higher degree of accuracy than the energy detection system. This method takes advantage of time varying autocorrelation function. Higher order CS detector usually exploits more than second-order probabilistic functions of CS signals. In this paper, we propose to detect spectrum holes in multi-antenna-assisted CR using higher order cyclostationarity. The entire detection has been performed under consideration of Gaussian as well as non-Gaussian noise (Gamma distribution). In this context, mutual coupling effects in the receiver section have been considered. Finally, the expressions for false alarm as well as detection probabilities are derived. Numerical results are given in support of the proposed detection technique.

Cyclic spectral analysis of the GPS signal

A detailed analytical model is derived for the GPS-L1 signal which is shown to be second-order cyclostationary. Closed forms for the periodically time-variant autocorrelation function and for the cyclic autocorrelation functions and cyclic spectra are derived for the complex envelope of this signal. Conjugate statistics are also derived. It is shown that different signal models must be considered for different data-record lengths of interest in the applications, including indoor ones. Numerical results are reported to corroborate the theoretical results. As example of application, a cyclostationarity-based parameter estimation method is proposed to operate with long observation intervals when the so-called narrow-band condition is not satisfied. The usefulness of the derived results for time-difference-of-arrival estimation, synchronization, cognitive radio in the Global Navigation Satellite System context, and for anti-jamming techniques is discussed.

Extension of the Capon's spectral estimator to time–frequency analysis and to the analysis of polynomial-phase signals

Incorporation of the linear time-varying filter and its Zadeh's generalized transfer function concepts to the derivation of the Capon's minimum-variance spectral estimator leads to a new, bilinear, cross-term suppressed and alias-free time-frequency representation (TFR) that has a higher resolution than the spectrogram with the same window width. Time-variant autocorrelation function of the nonstationary signal of interest is employed in this proposed TFR. By adopting an approximation for time-variant autocorrelation functions, we obtain another new, bilinear, parameterized TFR related to the spectrogram, the frequency resolution of which can be adjusted by varying its parameter for a fixed window width. We compare resolution and cross-term suppression properties of these proposed TFR's with other basic bilinear TFR's, via simulations on synthesized signals. Then, by incorporating polynomial-phase kernel functions to the Capon's estimator, we propose new bilinear signal representations for the analysis of constant-amplitude polynomial-phase signals and apply them to interference excision in direct-sequence spread-spectrum communications.

Spectral estimation of nonstationary ARMA processes using the evolutionary cepstrum

The spectral estimation problem of nonstationary autoregressive moving-average (ARMA) processes is considered and a new method is proposed for the estimation of the time-varying spectral content of such signals. The proposed method can be viewed as an extension of the estimator proposed earlier by the authors to the time-varying case. The AR part of the model is estimated by solving the time-varying modified Yule-Walker equations using an estimated time-varying autocorrelation function. An evolutionary cepstrum estimator is proposed, which is then used in a simple recursion to obtain the MA parameters of the model.

An ECG signal modelling based on the time-frequency kernel

In this paper, we present a method for modelling an electrocardiogram (ECG) signal using time-varying parameters by considering that the signal is generated by a linear, time-varying (LTV) system with a stationary white noise input, then we estimate the time-varying coefficients of the LTV system. Since the ECG signal is considered to be non-stationary, this method is based on the Wold-Cramer representation of a non-stationary signal. Because the relationship between the generalised transfer function of an LTV system and the time-varying coefficients of the difference equation of a discrete-time system is not addressed so far in literature, in this paper we propose a solution to this problem and apply it for modelling a human ECG signal. We first derive a relationship between the system generalised transfer function and the time-varying parameters of the system. Then we develop an algorithm to solve for the system time-varying parameters from the time-frequency kernel of the system output using the time-varying auto-correlation function (TVACF). A comparison analysis between the proposed algorithm and the RLS and RLSL algorithms has been discussed. Computer simulations illustrating the effectiveness of our algorithm are presented when the signal is embedded in noise.

A New Approach to the Generalized Transfer Function and Its Application to Biomedical Signal Analysis

In this paper, we present a method for modeling an electrocardiogram (ECG) signal using time-varying parameters by considering that the signal is generated by a linear, time-varying (LTV) system with a stationary white noise input, and then by estimating the time-varying coefficients of the LTV system. In effect, since the ECG signal is considered to be non-stationary, this method is based on the Wold-Cramer representation of a non-stationary signal. The relationship between the generalized transfer function of an LTV system and the time-varying coefficients of the difference equation of a discrete-time system is not addressed clearly in the literature; in this paper, we propose a new approach to this problem and apply it for modeling a human ECG signal. We first derive a relationship between the system generalized transfer function and the time-varying parameters of the system. Then we develop an algorithm to solve for the system time-varying parameters from the time-frequency kernel of the system output using the time-varying autocorrelation function (TVACF) and by modifying the modified least-square (MLS) and Durbin's approximation algorithms. A comparison between the proposed algorithm and the recursive least-square (RLS) and RLS lattice (RLSL) algorithms is considered. Computer simulations illustrating the effectiveness of our algorithm are presented when the signal is embedded in noise.

Quasar Variability from Microlensing

Over the past 17 years a large scale quasar monitoring programme has been underway at the Royal Observatory, Edinburgh, using COSMOS measures of a long sequence of plates from the UK 1.2 m Schmidt telescope in Australia. 3 or 4 plates were taken each year, and quasar candidates were selected solely on the basis of variability, with a minimum amplitude of 0.35 mag. over the 17 years of the survey. Of the 1000 or so quasars detected in this way, redshifts were available for about 300, which formed the sample to be investigated. Inspection of the light curves suggested a typical timescale of variation of around 5 to 10 years. To investigate the cause of variability, the timescales of subsamples were measured using the timevarying autocorrelation function (ACF).

Two alternative philosophies for estimation of the parameters of time-series

Two alternative philosophical frameworks for the two problems of estimating the time-invariant or time-variant autocorrelation function and its Fourier transform for stationary and cyclostationary time-series are compared. One is based on the stochastic process model, and the other is based on the nonstochastic time-series model. It is then explained that results on estimator bias and variance for these two problems couched within the stochastic process framework have analogs within the nonstochastic framework. The bias and variance results for cyclostationary time-series that are available within these two frameworks are then summarized.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>