Nonstationary White Noise Research Articles

An efficient simulation algorithm for non-Gaussian nonstationary processes

A novel algorithm is proposed for simulating univariate non-Gaussian nonstationary processes (NNP) with the specified evolutionary power spectral density (EPSD)/nonstationary auto-correlation function (NACF) and first four-order time-varying marginal moments (TVMMs). The sample realizations of the target NNP are generated as the outputs from a specific time-varying auto-regressive (TVAR) model via filtering the non-Gaussian and nonstationary white noise inputs. These white noise inputs are also non-Gaussian and nonstationary, and their first four-order TVMMs are predetermined using an approach developed herein according to the specified EPSD/NACF and first four-order TVMMs of the outputs. The conventional Johnson transformation is updated to accommodate the nonstationary cases for producing desired white noise inputs. This algorithm is developed from the linear filtering method (LFM), and inherits the simplicity and high efficiency from LFM. It fills the gaps in LFM-based algorithms for simulating NNP. Two numerical examples, i.e., a ground motion acceleration and a downburst velocity, are presented to fully demonstrate the capabilities of the proposed algorithm by comparing the simulation statistics with the targets.

EVALUATION OF INFLUENCE OF UNINTENTIONAL NON-STATIONARY INTERFERENCE ON FUNCTIONING OF SATELLITE COMMUNICATION SYSTEM WITH MULTIFREQUENCY TIME-DIVISION MULTIPLE ACCESS

The article considers the effect of non-Stationary unintentional interference on the earth station of a satellite communications system with Multi-Frequency Time-Division Multiple Access. The influence of non-stationary interference and white Gaussian noise on the earth station reception channel is compared using simulation modeling. The simulation model takes into account the effect of unintentional interference on the synchronization subsystem and the allocation of the boundaries of the transmitted packet of the receiver of the earth station of the satellite communication system. The probability of a bit error was used as a criterion for the quality of the satellite channel operation. The article shows the dependence of the protective ratio on the ratio of the preamble length to the packet length under the influence of pulsed interference. The obtained dependence can be used to assess the noise immunity of an earth station in a satellite communications system under the influence of pulsed interference.

Spatial time-frequency distribution of cross term-based direction-of-arrival estimation for weak non-stationary signal

In the radar array signal processing direction of arrival (DOA), the estimation of weak non-stationary signal is an important and difficult problem when both strong and weak signals are coexisting particularly because the weak non-stationary signals are often submerged in noise. In this paper, we proposed a novelty method to estimate the direction of arrival (DOA) of weak non-stationary signal in scenario for strong non-stationary interference signals and Gaussian white noise. The method utilizes spatial time-frequency distribution (STFD) of cross terms rather than suppressing cross terms in time-frequency analysis. The STFD of cross terms are introduced as an alternative matrix, which is similar to data covariance matrix in multiple signal classification (MUSIC), for the DOA estimation of a weak non-stationary signal. The cross-term amplitude of the strong and weak signals is usually above the noise and is easier to use than the auto-term of the weak signal. In the cross term, the information of the weak signal is included, and the auto-term of these weak signals is difficult to extract directly. The ability to incorporate the STFD of cross terms empowers information about a weak non-stationary signal for DOA estimation, leading to improved signal estimates for direction finding. The method based on the STFD of cross terms for DOA estimation of the weak non-stationary signal is revealed to outperform the time-frequency MUSIC and traditional MUSIC algorithm by simulation, respectively. This method has the advantages of the time-frequency direction finding method and also deals with the situation of weak signals. When the strong and weak signals exist at the same time and the two angles are similar, the cross-terms can be used to perform DOA estimation on the weak signal.

Synthesis of Cross-Correlation Functions of Partial Responses and the Extended Kalman Filter Approach for Structural Damage Detection Under Ambient Excitations

It is still necessary to investigate the detection of structural damage under ambient excitations since the excitations are random and unmeasured while measurement noises are inevitable. In this paper, a method based on the synthesis of cross-correlation functions of partial structural responses and the extended Kalman filter (EKF) approach is proposed for the identification and damage detection of structures under ambient excitations, in which both independent stationary and non-stationary white noise excitations in the product models are discussed. First, the equations of cross-correlation functions of structural responses are established when the ambient excitations are independent stationary white noise processes. Then, the EKF approach is utilized to identify structural parameters and cross-correlation functions using partial measurements of structural acceleration responses. Structural damage is detected based on the degradations of the identified structural element stiffness parameters. Finally, the proposed method is extended to deal with independent non-stationary white noise excitations in the product models. The numerical simulation examples of the ASCE structural health monitoring benchmark building subject to ambient excitation, a moment resisting frame model under white noise excitation, and a cantilever beam model under multiple independent non-stationary excitations are used to validate the feasibility of the proposed method. It is shown that the method is not sensitive to measurement noises. Also, a lab experimental study of the identification of a multi-story shear structure is investigated to further illustrate the applicability of the proposed method.

Improvement of decentralized random decrement technique for data processing in wireless sensor network

The Random Decrement Technique (RDT), based on decentralized computing approaches implemented in wireless sensor networks (WSNs), has shown advantages for modal parameter and data aggregation identification. However, previous studies of RDT-based approaches from ambient vibration data are based on the assumption of a broad-band stochastic process input excitation. The process normally is modeled by filtered white or white noise. In addition, the choice of the triggering condition in RDT is closely related to data communication. In this project, research has been conducted to study the nonstationary white noise excitations as the input to verify the random decrement technique. A local extremum triggering condition is chosen and implemented for the purpose of minimum data communication in a RDT-based distributed computing strategy. Numerical simulation results show that the proposed technique is capable of minimizing the amount of data transmitted over the network with accuracy in modal parameters identification.

Ambient modal identification using non-stationary correlation technique

The topic of this paper is the modal identification from non-stationary ambient response data by applying correlation technique. It is shown theoretically that by assuming the ambient excitation to be non-stationary white noise in the form of a product model, the non-stationary response signals can be converted into free vibration data via the correlation technique. Previous studies have showed that the practical problem of insufficient data samples available for evaluating non-stationary correlation can be approximately resolved by first extracting the amplitude-modulating function from the response and then transforming the non-stationary responses into stationary ones. However, the errors involved in the approximate free-decay response would generally lead to a distortion in the modal identification. In the present paper, we propose that, if the ambient excitation can be represented by a product model with slowly time-varying function, without any additional treatment of transforming the original nonstationary responses, the non-stationary responses of the system can be treated approximately as a stationary random process; then, the nonstationary cross correlation functions of structural response evaluated at an arbitrary, fixed time instants of structural response are of the same mathematical form as that of free vibration of a structure, from which modal parameters of the original system can thus be identified. Numerical simulations, including one example of using the practical earthquake data served as the excitation input acting on a linear two-dimensional model of one-half of a railway vehicle, confirm the validity of the proposed method for identification of modal parameters from non-stationary ambient response data only.

Target Tracking in 3-D Using Estimation Based Nonlinear Control Laws for UAVs

This paper presents an estimation based backstepping like control law design for an Unmanned Aerial Vehicle (UAV) to track a moving target in 3-D space. A ground-based sensor or an onboard seeker antenna provides range, azimuth angle, and elevation angle measurements to a chaser UAV that implements an extended Kalman filter (EKF) to estimate the full state of the target. A nonlinear controller then utilizes this estimated target state and the chaser’s state to provide speed, flight path, and course/heading angle commands to the chaser UAV. Tracking performance with respect to measurement uncertainty is evaluated for three cases: (1) stationary white noise; (2) stationary colored noise and (3) non-stationary (range correlated) white noise. Furthermore, in an effort to improve tracking performance, the measurement model is made more realistic by taking into consideration range-dependent uncertainties in the measurements, i.e., as the chaser closes in on the target, measurement uncertainties are reduced in the EKF, thus providing the UAV with more accurate control commands. Simulation results for these cases are shown to illustrate target state estimation and trajectory tracking performance.

Efficient estimation of the peak factor for the stochastic characterization of structural response to non-stationary ground motions

The determination of the peak factor is quite cumbersome in the case of a non-stationary process, due to the necessity of calculating the non-geometric spectral moments of the process and obtaining the threshold level of the first-passage problem corresponding to a certain non-exceedance probability value. In this paper, an efficient approximate new method is presented for practical applications in Earthquake Engineering. The so-called Equivalent Oscillator and Response method consists of transforming the original oscillator excited by a uniformly modulated non-stationary colored noise into an equivalent oscillator excited by a uniformly modulated non-stationary white noise, in such a way that the responses of both oscillators have similar values for the spectral parameters that affect the calculation of the peak factor. A new dimensionless spectral peakedness factor is introduced in order to ensure the matching of the spectral bandwidth of both oscillators. A numerical application of the new method, based on a non-stationary seismological model for the ground accelerations process, is carried out, showing the reliability and robustness of the new method for efficiently and accurately estimating the peak factor of the non-stationary structural response.

Averaging of nonlinearly aligned signal cycles for noise suppression

Abstract Averaging of nonlinearly aligned (time-warped) signal cycles is an important method for suppressing noise of quasi-periodical or event related signals. However, in the paper we show that the operation of time warping introduces unfavorable violation of the requirements that should be satisfied for effective averaging and, as a result, it causes poor suppression of noise. To limit these effects, we redefine the matrix of the alignment costs. To improve results of averaging in cases of variable energy noise, we apply weighting of the summed signal samples. The derived formula gives smaller weights for more noisy signal cycles and this way limits their influence on the constructed template. The proposed modifications caused a significant increase of the Noise Reduction Factor (NRF) in the experiments on the simulated evoked potentials. Whereas the greatest NRF obtained by the reference methods in nonstationary white noise environment was equal to 1.55, for the new method proposed we achieved a value of 4.44. For non-stationary colored noise, the corresponding values were 1.44 and 2.99. Moreover, application of the developed method to ECG signal processing, prior to the measurements of the QT interval, significantly improved the measurements immunity to noise.

The Theory of Compressive Sensing Matching Pursuit Considering Time-domain Noise with Application to Speech Enhancement

Compressive sampling matching pursuit (CoSaMP) is an efficient compressive sensing algorithm holding rigorous estimation error bounds and low computational complexity, when it deals with an additive noise signal model in the observation domain. However, in some applications, e.g., speech enhancement (SE), noise is added to a signal in the time domain, where the conventional CoSaMP cannot be directly applied. In this paper, we establish the theory of CoSaMP to address the time-domain noise, referred to as Tdn-CoSaMP, which extends the canonical theory of CoSaMP. In particular, we prove the existence of a new upper bound of Tdn-CoSaMP, which is found to be larger than that of the conventional CoSaMP by appending two additional terms: a multiplier 1+√{N/s}, where N is the dimension of the signal, and an l1 norm of the noise [1/(√s)]||e||1 scaled by the sparse level s of the signal. We also apply Tdn-CoSaMP to the SE task based on the sequential denoising of overlapped frames in the discrete cosine transform (DCT) domain. The proposed system, CoSaMP-based speech enhancement (CoSaMPSE), has been evaluated in terms of both objective and subjective criteria on various types of noise. Positive results have been achieved for denoising stationary and nonstationary white Gaussian noise (WGN) and are comparable to other SE methods. Moreover, due to its low computational complexity, CoSaMPSE is possible to be combined with optimally modified log-spectrum amplitude estimation (OMLSA) and able to achieve complementary denoising effects in various noisy conditions.

Modal identification from nonstationary ambient response data using extended random decrement algorithm

The topic of this paper is the estimation of modal parameters from nonstationary ambient vibration data by applying the random decrement algorithm. If the ambient excitation can be modeled as a nonstationary white noise in the form of a product model, then the nonstationary cross randomdec signatures of structural response evaluated at any fixed time instant are shown theoretically to be proportional to the nonstationary cross-correlation functions, which is of the same form as the free-vibration decay or impulse response of the original system. The practical problem of insufficient data samples available for evaluating nonstationary randomdec signatures can be approximately resolved by first extracting the amplitude-modulating function from the response and then transforming the nonstationary responses into stationary ones. However, the error involved in the approximate free-decay response would generally lead to a distortion in the modal identification. In the present paper, we also further propose that, if the ambient excitation can be modeled as a zero-mean nonstationary process, without any additional treatment of transforming the original nonstationary responses, the nonstationary cross randomdec signatures of structural response are shown in the same mathematical form as that of free vibration of a structure, from which modal parameters of the original system can thus be identified. Numerical simulations, including one example of using the practical excitation data, confirm the validity of the proposed method for identification of modal parameters from nonstationary ambient response data only.

A modified random decrement technique for modal identification from nonstationary ambient response data only

Modal identification is considered from response data of structural system under nonstationary ambient vibration. In a previous paper, we showed that by assuming the ambient excitation to be nonstationary white noise in the form of a product model, the nonstationary response signals can be converted into free-vibration data via the correlation technique. In the present paper, if the ambient excitation can be modeled as a nonstationary white noise in the form of a product model, then the nonstationary cross random decrement signatures of structural response evaluated at any fixed time instant are shown theoretically to be proportional to the nonstationary cross-correlation functions. The practical problem of insufficient data samples available for evaluating nonstationary random decrement signatures can be approximately resolved by first extracting the amplitude-modulating function from the response and then transforming the nonstationary responses into stationary ones. Modal-parameter identification can then be performed using the Ibrahim time-domain technique, which is effective at identifying closely spaced modes. The theory proposed can be further extended by using the filtering concept to cover the case of nonstationary color excitations. Numerical simulations confirm the validity of the proposed method for identification of modal parameters from nonstationary ambient response data.

Identification of modal parameters from nonstationary ambient vibration data using the channel-expansion technique

The identification of modal parameters from the response data only is studied for structural systems under nonstationary ambient vibration. In a previous paper by the authors, the modal parameters of a system were identified using the correlation method in conjunction with the curve-fitting technique. This was done by working within the assumption that the ambient excitation is a nonstationary white noise in the form of a product model. In the present paper, the Ibrahim time-domain method (ITD) is extended for modal-parameter identification from the nonstationary ambient response data without any additional treatment of converting the original data into the form of free vibration. The ambient responses corresponding to various nonstationary inputs can be approximately expressed as a sum of exponential functions. In effect, the ITD method can be used in conjunction with the channel-expansion technique to identify the major modes of a structural system. To distinguish the structural modes from the non-structural modes, the concept of mode -shape coherence and confidence factor is employed. Numerical simulations, including one example of using the practical excitation data, confirm the validity and robustness of the proposed method for identification of modal parameters from the nonstationary ambient response.

Approximate calculation method for integral of mean square value of nonstationary response

The response of the structure subjected to nonstationary random vibration such as earthquake excitation is nonstationary random vibration. Calculating method for statistical characteristics of such a response is complicated. Mean square value of the response is usually used to evaluate random response. Integral of mean square value of the response corresponds to total energy of the response. In this paper, a simplified calculation method to obtain integral of mean square value of the response is proposed. As input excitation, nonstationary white noise and nonstationary filtered white noise are used. Integrals of mean square value of the response are calculated for various values of parameters. It is found that the proposed method gives exact value of integral of mean square value of the response.

Identification of Modal Parameters from Nonstationary Ambient Vibration Data Using Correlation Technique

Identification of modal parameters is considered from response data of structural systems under nonstationary ambient vibration. It is shown theoretically that by assuming the ambient excitation to be nonstationary white noise in the form of a product model, the nonstationary response signals can be converted into free-vibration data via the correlation technique. The practical problem of insufficient data samples available for evaluating nonstationary correlation can be approximately resolved by first extracting the amplitude-modulating function from the response and then transforming the nonstationary responses into stationary ones. Modal-parameter identification can then be performed using the Ibrahim time-domain technique, which is effective at identifying closely spaced modes. The theory proposed can be further extended by using the filtering concept to cover the case of nonstationary color excitations. Numerical simulations confirm the validity of the proposed method for identification of modal parameters from nonstationary ambient response data.

Approximate representation of the statistics for extreme responses of single degree-of-freedom system excited by non-stationary processes

The objective of this study is to estimate the statistics of the maximum response of a structural system excited by a non-stationary process. To provide basic information for the design of structures, the response of a single degree-of-freedom (SDOF) system subjected to a non-stationary Gaussian white noise is considered. For this purpose, we first derive the probabilistic properties of the maximum values of a non-stationary and zero-mean white noise, whose standard deviation varies temporally under some limited conditions. Then, using the obtained properties, we propose a method to estimate the stochastic properties of the maximum response of a given SDOF system excited by a non-stationary white noise whose time-varying standard deviation is known. The validity of the obtained results is confirmed through Monte Carlo simulation.

Simulation of Nonstationary Stochastic Processes by Spectral Representation

This paper presents a rigorous derivation of a previously known formula for simulation of one-dimensional, univariate, nonstationary stochastic processes integrating Priestly's evolutionary spectral representation theory. Applying this formula, sample functions can be generated with great computational efficiency. The simulated stochastic process is asymptotically Gaussian as the number of terms tends to infinity. This paper shows that (1) these sample functions accurately reflect the prescribed probabilistic characteristics of the stochastic process when the number of terms in the cosine series is large, i.e., the ensemble averaged evolutionary power spectral density function (PSDF) or autocorrelation function approaches the corresponding target function as the sample size increases, and (2) the simulation formula, under certain conditions, can be reduced to that for nonstationary white noise process or Shinozuka's spectral representation of stationary process. In addition to derivation of simulation formula, three methods are developed in this paper to estimate the evolutionary PSDF of a given time-history data by means of the short-time Fourier transform (STFT), the wavelet transform (WT), and the Hilbert-Huang transform (HHT). A comparison of the PSDF of the well-known El Centro earthquake record estimated by these methods shows that the STFT and the WT give similar results, whereas the HHT gives more concentrated energy at certain frequencies. Effectiveness of the proposed simulation formula for nonstationary sample functions is demonstrated by simulating time histories from the estimated evolutionary PSDFs. Mean acceleration spectrum obtained by averaging the spectra of generated time histories are then presented and compared with the target spectrum to demonstrate the usefulness of this method.

Simulation of non-stationary ground motion processes (I)

This paper presents a spectral representation method for simulation of non-stationary ground motion processes on the basis of Priestley’s evolutionary spectral theory. Following this method, sample processes can be generated using a cosine series formula. It is shown that, these sample processes accurately reflect the prescribed characteristics of the evolutionary power spectral density function when the number of the terms in the cosine series is large enough; and the ensemble expected value and the ensemble autocorrelation function approach the corresponding target functions, respectively, as the sample size increases; and these sample processes are asymptotically normal as the number of the terms in the series tends to infinity. Finally, a few special cases of the formula are discussed, one of which is non-stationary white noise process, and other one is reduced to the formula for simulation of stationary stochastic processes.

Relative Transfer Function Identification Using Speech Signals

An important component of a multichannel hands-free communication system is the identification of the relative transfer function between sensors in response to a desired source signal. In this paper, a robust system identification approach adapted to speech signals is proposed. A weighted least-squares optimization criterion is introduced, which considers the uncertainty of the desired signal presence in the observed signals. An asymptotically unbiased estimate for the system's transfer function is derived, and a corresponding recursive online implementation is presented. We show that compared to a competing nonstationarity-based method, a smaller error variance is achieved and generally shorter observation intervals are required. Furthermore, in the case of a time-varying system, faster convergence and higher reliability of the system identification are obtained by using the proposed method than by using the nonstationarity-based method. Evaluation of the proposed system identification approach is performed under various noise conditions, including simulated stationary and nonstationary white Gaussian noise, and car interior noise in real pseudo-stationary and nonstationary environments. The experimental results confirm the advantages of proposed approach.

Consistent CFAR detection of a linear signal based on partially consistent observations

This letter addresses the problem of consistent constant false-alarm rate (CFAR) detection based on partially consistent observations. The term "partially consistent observations" refers to the fact that the number of unknown parameters increases with the number of samples, and therefore, the number of observed samples is insufficient for consistent estimation of all parameters. Specifically, the problem of detecting a deterministic signal with unknown linear parameters in nonstationary white Gaussian noise is addressed. Two families of CFAR detectors are proposed; their consistency is discussed; and their performance is examined.