Nonstationary Stochastic Signals Research Articles

Relaxed evolutionary power spectral density functions: A probabilistic approach to model uncertainties of non-stationary stochastic signals

The identification of patterns and underlying characteristics of natural or engineering time-varying phenomena poses a challenging task, especially in the scope of simulation models and accompanying stochastic models. Because of their complex nature, time-varying processes such as wind speed, seismic ground motion, or vibrations of machinery in the presence of degradation oftentimes lack a closed-form description of their underlying Evolutionary Power Spectral Density (EPSD) function. To overcome this issue, a wide range of measurements exist for these types of processes. This opens up the path to a data-driven stochastic representation of EPSD functions. Rather than solely relying on time–frequency transform methods like the familiar short-time Fourier transform or wavelet transform for EPSD estimation, a probabilistic representation of the EPSD can provide valuable insights into the epistemic uncertainty associated with these processes. To address this problem, the evolutionary EPSD function is relaxed based on multiple similar data to account for these uncertainties and to provide a realistic representation of the time data in the time–frequency domain. This results is the so-called Relaxed Evolutionary Power Spectral Density (REPSD) function, which serves as a modular probabilistic representation of the time–frequency content of stochastic signals. For this purpose, truncated normal distributions and kernel density estimates are used to determine a probability density function for each time–frequency component. The REPSD function enables the sampling of individual EPSD functions, facilitating their direct application to the simulation model through stochastic simulation techniques like Monte Carlo simulation or other advanced methods. Even though the accuracy is highly dependant on the data available and the time–frequency transformation method used, the REPSD representation offers a stochastic representation of characteristics used to describe stochastic signals and can reduce epistemic uncertainty during the modelling of such time-varying processes. The method is illustrated by numerical examples involving the analysis of dynamic behaviour under random loads. The results show that the method can be successfully employed to account for uncertainties in the estimation of the EPSD function and represent the accuracy of the time–frequency transformation used.

Rolling Bearing Fault Diagnosis by Aperiodic Stochastic Resonance Under Variable Speed Conditions

The fault characteristic of rolling bearings under variable speed condition is a typical non-stationary stochastic signal. It is difficult to extract due to the interference of strong background noise makes the applicability of traditional noise reduction methods less. In this paper, an aperiodic stochastic resonance (ASR) method is proposed to study the fault diagnosis of rolling bearings under variable speed conditions. Based on numerical simulation, the effect of noise intensity and damping coefficient on the ASR of the second-order underdamped system is discussed, and an appropriate damping coefficient is found to reach the optimal ASR. The proposed method enhances the fault characteristic information of bearing fault simulation signal. Corresponding to rising-stationary and the stationary-declining running conditions, the method is verified by both simulated and experimental signals. It provides reference for fault diagnosis under variable speed condition.

Novel Second-Order Statistics of the Chirp Cyclostationary Signals

In communications and radar/sonar systems, one of the most popular nonstationary stochastic signal models is the chirp cyclostationary (CCS) signal. Recently, the second-order statistics of complex CCS signals are defined based on the conjugate correlation function, which have shown to be more proper in linear canonical transform domain than in the frequency domain. However, the information provided by unconjugate correlation function is missing, which leads to an incomplete study of second-order CCS signals. In this letter, the unified conjugate and unconjugate correlation function of CCS signals is researched. In detail, the definitions of the novel second-order statistics, chirp cyclic correlation and chirp cyclic spectrum, are expounded. Then the properties of these statistics are investigated. Finally, the additional information provided by the correlation function with conjugate operation and the usefulness of the proposed statistics are introduced and explained by simulations.

New Statistics of the Second-Order Chirp Cyclostationary Signals: Definitions, Properties and Applications

The chirp-cyclostationary (CCS) process is the generalized form of the cyclostationary (CS) process and it can describe the nonstationary stochastic signal models more exactly in practical applications. Basically, the statistics of the CCS signals have been defined and studied based on the correlation function. However, when the measurement is chosen as the above defined statistics, the output of a linear-variant filter is a zero-power signal for any input pure CCS signal. This is a barrier to further analyze the properties of the filters by the second-order statistics. In this paper, firstly two new complementary correlation function-based statistics and their basic properties are proposed and discussed. These two new statistics are connected by the generalized cyclic Wiener-Khinchin theorem. Then, we examine that the output of the filter is non-zero power under these new measurements. Based on this, the theories about the matched filter and system identification are developed, whose advantages are analyzed in theory. Finally, by proposing the estimators to the two statistics via a single record, the advantages of the proposed statistics over the classical second-order statistics are examined by the communication signals and the electrocardiogram (ECG) signals.

An Extended Neo-Fuzzy Neuron and its Adaptive Learning Algorithm

A modification of the neo-fuzzy neuron is proposed (an extended neo-fuzzy neuron (ENFN)) that is characterized by improved approximating properties. An adaptive learning algorithm is proposed that has both tracking and smoothing properties and solves prediction, filtering and smoothing tasks of non-stationary noisy stochastic and chaotic signals. An ENFN distinctive feature is its computational simplicity compared to other artificial neural networks and neuro-fuzzy systems.

EXTENDED NEO-FUZZY NEURON IN THE TASK OF IMAGES FILTERING

The paper describes a modification of the neo-fuzzy neuron called as «extended neo-fuzzy neuron» (ENFN) that characterized by improved approximating properties. The adaptive learning algorithm for ENFN is proposed, that has both following and smoothing properties and allows to solve problems of prediction, filtering and smoothing of non-stationary disturbed stochastic and chaotic signals. A distinctive feature of ENFN is its implementation computational simplicity compared with artificial neural networks and neuro-fuzzy systems. These properties of the proposed neo-fuzzy neuron make it very effective in suppressing noise in image filtering.

Quantum neural network-based EEG filtering for a brain-computer interface.

A novel neural information processing architecture inspired by quantum mechanics and incorporating the well-known Schrodinger wave equation is proposed in this paper. The proposed architecture referred to as recurrent quantum neural network (RQNN) can characterize a nonstationary stochastic signal as time-varying wave packets. A robust unsupervised learning algorithm enables the RQNN to effectively capture the statistical behavior of the input signal and facilitates the estimation of signal embedded in noise with unknown characteristics. The results from a number of benchmark tests show that simple signals such as dc, staircase dc, and sinusoidal signals embedded within high noise can be accurately filtered and particle swarm optimization can be employed to select model parameters. The RQNN filtering procedure is applied in a two-class motor imagery-based brain-computer interface where the objective was to filter electroencephalogram (EEG) signals before feature extraction and classification to increase signal separability. A two-step inner-outer fivefold cross-validation approach is utilized to select the algorithm parameters subject-specifically for nine subjects. It is shown that the subject-specific RQNN EEG filtering significantly improves brain-computer interface performance compared to using only the raw EEG or Savitzky-Golay filtered EEG across multiple sessions.

Design of RLS fixed-lag smoother using covariance information in linear discrete stochastic systems

This paper newly designs the recursive least-squares fixed-lag smoother using the covariance information in linear discrete-time stochastic systems. It is assumed that the signal is observed with additive white observation noise and the signal is uncorrelated with the observation noise. The fixed-lag smoother uses the covariance function of the signal in the semi-degenerate kernel form and the variance of the observation noise. The proposed fixed-lag smoother is suitable for the estimations of stationary or non-stationary stochastic signals generally.

RLS fixed-lag smoother using covariance information in linear continuous stochastic systems

This paper newly designs the recursive least-squares fixed-lag smoother using the covariance information in linear continuous-time stochastic systems. It is assumed that the signal is observed with additive white observation noise and the signal is uncorrelated with the observation noise. The fixed-lag smoother uses the covariance function of the signal in the semi-degenerate kernel form and the variance of the observation noise. The proposed fixed-lag smoother is appropriate for the estimations of stationary or non-stationary stochastic signals generally.

Single channel nonstationary stochastic signal separation using linear time-varying filters

Separability of signal mixtures given only one mixture observation is defined as the identification of the accuracy to which the signals can be separated. The paper shows that when signals are separated using the generalized Wiener filter, the degree of separability can be deduced from the signal structure. To identify this structure, the processes are represented on an general spectral domain, and a sufficient solution to the Wiener filter is obtained. The filter is composed of a term independent of the signal values, corresponding to regions in the spectral domain where the desired signal components are not distorted by interfering noise components, and a term dependent on the signal correlations, corresponding to the region where components overlap. An example of determining perfect separability of modulated random signals is given with application in radar and speech processing.

Guaranteeing the Filtering Accuracy for Non-Stationary Processes with Bounded Dispersions of Derivatives

The feasibility of guaranteeing the accuracy at filtering with substantial prior uncertainty in a class of input signals is analyzed. The class γ2 containing nonstationary stochastic signals with the only known prior information - upper bounds of some derivatives is considered. The procedure of estimating the extreme attainable error dispersion at filtering the input signals of the class γ2 at the noise with known correlation function is given. The proposed methods are applicable in automatic control and system analysis.

L/sup 2/(R) nonstationary processes and the sampling theorem

In a previous paper W. Gardner (see IEEE Trans. Inform. Theory, vol.IT-18, p.808-9, 1972) developed a sampling theorem for nonstationary random processes, under the condition that the two-dimensional (2-D) power spectrum (2DPS) of the process has compact support. In this letter, it is shown that, for L/sup 2/(R) processes, only a one-dimensional (1-D) restriction on the marginal along time of the time-frequency distribution is necessary to guarantee the compactness of the 2DPS in the 2-D plane. As a direct consequence, it is observed that under mild conditions, a nonstationary autocorrelation function of a bandpass L/sup 2/(R) process is nearly stationary in small time intervals, The influence of this result in real-time detection of nonstationary stochastic signals is discussed.

The Effects of Digitization on Nonstationary Stochastic Signals with Applications to Pulsar Signal Baseband Recording

ABSTRACTThe process of digitizing a stochastic signal introduces systematic distortions into the resulting digitized data. Further processing of these data may result in the appearance of unwanted artifacts, especially when the input signal was generated by a nonstationary stochastic process. In this paper the magnitude of these distortions are calculated analytically and the results are applied to a specific example found in pulsar signal processing. A pulsar signal is an excellent example of a nonstationary stochastic process. When analyzing pulsar data, the effects of interstellar medium (ISM) dispersion must be removed by digitally filtering the received signal. The distortions introduced through the digitization process cause unwanted artifacts to appear in the final “dedispersed” signal. These artifacts are demonstrated using actual 2‐bit (4‐level) digitized data of the pulsar PSR B0833−45 (Vela). Techniques are introduced that simultaneously minimize these artifacts and maximize the signal‐to‐noise ratio of the digitized data. The distortion analysis and artifact removal techniques described in this paper hold for an arbitrary number of input digitization thresholds (i.e., number of bits). Also presented are tables of the optimum digitizer thresholds for both uniform and nonuniform input threshold digitizers.

Neural filtering method for stationary signal processes

In this paper a new method of neural filtering using artificial neural network systems is presented for the filtering problems of linear and nonlinear, stationary and nonstationary stochastic signals. The neural filter (denoted neurofilter) developed in this paper has either finite impulse response (FIR) structure or infinite impulse response (IIR) structure. The neurofilter differs from the conventional linear digital FIR and IIR filters because the artificial neural network system used in the neurofilter has a nonlinear structure due to the sigmoid function. Numerical studies for the estimation of a second-order Butterworth process are performed by changing the structures of the neurofilter in order to evaluate the performance indices under changes of the output noises or disturbances. The results obtained from these studies verified the capabilities which are essentially necessary for on-line filtering of various stochastic signals.

A hardware-oriented method for simulating service loads

This is the third paper in a series which includes A Computer Method for Simulating Service Loads' (pp. 42–46) and ‘A Spectral-Analysis Method for Nonstationary Field Measurements’ (pp. 47–55). This paper presents a hardware method to digitally simulate nonstationary stochastic signals which are analyzed using the procedures outlined in the first paper. The method of analysis hypothesizes that the nonstationary signal consists of two stationary signals which belong to different populations occurring consecutively according to a suitable probabilistic model. The analysis procedure involves the following steps: segmenting the time history, estimating the population of each segment, estimating the power spectrum of each segment, averaging the power spectra which belong to each population, presenting the power spectra via parameters of digital filters which shape white-noise sequences into sequences with the measured power spectra, and measuring the parameters of the probabilistic method.

The design of on-line linear least-squares estimators given covariance specifications via an imbedding method

The method of invariant imbedding is employed to develop on-line algorithms for all types of linear least-squares estimators (filter; fixed-point, fixed-lag, and fixed-interval smoothers; fixed-point, fixed-lead, and fixed-interval predictors) using covariance specifications for nonstationary stochastic signals in the presence of white Gaussian plus colored noise. Two design procedures are given. One is based on the initial-value solution of the function I y ( t) which is the solution to a Fredholm integral equation of the second kind, and the other on that of the function h( t,s) which is the impulse response function obeyed by a generalized Wiener-Hopf integral equation.