Stationary Gaussian White Noise Research Articles

Efficient path integral approach via analytical asymptotic expansion for nonlinear systems under Gaussian white noise

In this paper an efficient formulation of the Path integral (PI) approach is developed for determining the response probability density functions (PDFs) and first-passage statistics of nonlinear oscillators subject to stationary and time-modulated external Gaussian white noise excitations. Specifically, the evolution of the response PDF is obtained in short time steps, by using a discrete version of the Chapman-Kolmogorov equation and assuming a Gaussian form for the conditional response PDF. Next, the technique involves proceeding to treating the problem via an analytical asymptotic expansion procedure, namely the Laplace’s method of integration. In this manner, the repetitive double integrals involved in the standard implementation of the PI approach are evaluated in a closed form, while the response and first-passage PDFs are obtained by mundane step-by-step application of the derived approximate analytical expression. It is shown that the herein proposed formulation can drastically decrease the associated computational cost by several orders of magnitude, as compared to both the standard PI technique and Monte Carlo solution (MCS) approach. A number of nonlinear oscillators are considered in the numerical examples. Notably, for these systems both response PDFs and first-passage probabilities are presented, whereas comparisons with pertinent MCS data demonstrate the efficiency and accuracy of the technique.

Structural health monitoring with non-linear sensor measurements robust to unknown non-stationary input forcing

Bayesian filtering based structural health monitoring algorithms typically assume stationary white Gaussian noise models to represent an unknown input forcing. However, typical structural damages occur mostly under the action of extreme loading conditions, like earthquake or high wind/waves, which are characteristically non-stationary and non-Gaussian. Clearly, this invalidates this basic assumption, causing these algorithms to perform poorly under non-stationary noise conditions. This paper extends an existing interacting filtering algorithm to efficiently estimate structural damages while being robust to unknown non-stationary non-Gaussian input forcing. Furthermore, this approach is generalized beyond linear measurements to encompass the case of non-linear measurements such as strains. The joint estimation of state and parameters is performed by combining Ensemble Kalman filtering, for non-linear system state estimation, and Particle filtering to estimate changes in the structural parameters. The robustness against input forcing is achieved through an output injection approach embedded in the state filter equation. Numerical simulations for two kinds of response measurements (acceleration and strain) are performed on a 3D frame structure under different damage location and severity scenarios. The sensitivity with respect to noise and the impact of different sensor combinations have also been investigated.

Variance analysis of noisy LFM signal in linear canonical Cohen's class

Exact expression for the discrete-time linear canonical Cohen's class's average variance of signals in additive stationary white circular Gaussian noise processes is derived. It is shown that the average variance depends on signal form, kernel type and linear canonical transform (LCT) free parameters. For a kind of special analytic signal form, linear frequency-modulated signals, the effect of kernels and LCT free parameters on the average variance is observed. By focusing on two kinds of minimum variance kernels, the frequency marginal constrained kernel and the time marginal, frequency marginal and time support constrained kernel, a correlation analysis between the average variance and LCT free parameters is further conducted.

Variance analysis of linear canonical Wigner distribution

Exact expression for the linear canonical Wigner distribution's variance of deterministic signals added with stationary white circular Gaussian noise is derived. It is shown that only one of the linear canonical transform free parameters, the parameter b, affects the value of variance.

First-passage time study of a stochastic growth model

In this paper, a stochastic nonlinear growth model is proposed, which can be considered a generalization of the stochastic logistic model. It can be applied to the growth of a generic population as well as to the propagation of the fracture in engineering materials. The excitation is assumed to be a stationary Gaussian white noise stochastic process, which affects the system parametrically. A preliminary study of the stochastic differential equation governing the model reveals that there is a phase transition when the nonlinearity parameter c reaches one: when $$c<1$$ , the system tends to the stationary state, while it is never stationary when $$c\ge 1$$ . Then, attention is focused on the first-passage time problem, which is of crucial importance for dynamical systems. The first-order stochastic differential equation (SDE) that describes the model is transformed into an Ito’s SDE by adding the Wong–Zakai–Stratonovich corrective term. For the last equation, the backward Kolmogorov equation is formulated. By solving it with appropriate initial and boundary conditions, the probability of survival is obtained, that is, the probability of not exceeding a given threshold. The solution is looked for three cases $$c 1$$ . In any case, the numerical analyses show that the survival probability decays fast.

A Frequency Correcting Method Combining Bilateral Correction With Weighted Average for Vortex Flow Sensor Signal

The sinusoidal signal outputted by the vortex flow sensor is often superimposed with the white noise, which will affect the accuracy of spectrum correction. When the signal is truncated by rectangular window, a method (hereinafter referred to as RWBWFCR) combining bilateral correction with weighted average on the basis of Fourier coefficients ratio is proposed so as to improve the antinoise performance of the frequency correction. The correction principle of this method is presented. The effects on the frequency correction of changing the initial phase, the frequency deviation, the calculation points of fast Fourier transform, and the signal-to-noise ratio (SNR) are analyzed. The theoretical accuracy of this method when there is stationary Gaussian white noise is deduced and verified. This method has some advantages, such as less computation load, small storage space, high correction precision, strong anti-interference ability, and is easy to be implemented by a microcontroller in real time. RWBWFCR method is applied to the low-power and antivibration type vortex flowmeter with single sensor to improve the reliability of the digital signal processing method based on the frequency correction.

Exact Closed-Form Fractional Spectral Moments for Linear Fractional Oscillators Excited by a White Noise

Abstract In the last decades, the research community has shown an increasing interest in the engineering applications of fractional calculus, which allows to accurately characterize the static and dynamic behavior of many complex mechanical systems, e.g., the nonlocal or nonviscous constitutive law. In particular, fractional calculus has gained considerable importance in the random vibration analysis of engineering structures provided with viscoelastic damping. In this case, the evaluation of the dynamic response in the frequency domain presents significant advantages, once a probabilistic characterization of the input is provided. On the other hand, closed-form expressions for the response statistics of dynamical fractional systems are not available even for the simplest cases. Taking advantage of the residue theorem, in this paper the exact expressions of the spectral moments of integer and complex orders (i.e., fractional spectral moments of linear fractional oscillators driven by acceleration time histories obtained as samples of stationary Gaussian white noise processes are determined.

Speech Denoising in White Noise Based on Signal Subspace Low-rank Plus Sparse Decomposition

In this paper, a new subspace speech enhancement method using low-rank and sparse decomposition is presented. In the proposed method, we firstly structure the corrupted data as a Toeplitz matrix and estimate its effective rank for the underlying human speech signal. Then the low-rank and sparse decomposition is performed with the guidance of speech rank value to remove the noise. Extensive experiments have been carried out in white Gaussian noise condition, and experimental results show the proposed method performs better than conventional speech enhancement methods, in terms of yielding less residual noise and lower speech distortion.

Harvesting Energy from Vibration Absorber under Random Excitations

Abstract A dynamic vibration absorber is a passive device used often to reduce the vibrations in host structures. The potential of harvesting energy out of the motion of vibration absorber has shown promise under harmonic excitations. To achieve this goal, vibration absorber is supplemented with a piezoelectric stack for both vibration confinement and energy harvesting. The primary goal is to control the vibration of the host structure and the secondary goal is to harvest energy out of the dynamic vibration absorber simultaneously. The harvested energy can be used to power low-powered wireless sensor systems used for infrastructural health monitoring. In field applications the excitation frequency may not be harmonic and may have randomness. This manuscript looks into the power harvested by these devices under random excitations. Analytical studies are carried out considering stationary Gaussian white noise excitation within the frequency band of interest. The analytical results show the promise of broadband energy harvesting using the device combined with vibration reduction in the primary structure.

Mean square stability of a second-order parametric linear system excited by a colored Gaussian noise

The stochastic stability of a second-order linear parametric oscillator whose stiffness is perturbed by a stationary zero-mean colored Gaussian process is investigated in this paper referring to the stability in the second response moments (mean square stability). The stochastic perturbation is the output of a linear filter excited by a stationary Gaussian white noise stochastic process, and the hypothesis of weak excitation is not formulated. The first assumption allows using Itô׳s stochastic differential calculus. The moment equations of the response are written till the second order by means of Itô׳s differential rule. They form an infinite hierarchy, that is the equations for the moments of order r contain moments of order larger than r. In order to close the hierarchy, the cumulant neglect closure method is applied herein. The moment equation set is closed by neglecting the cumulants of third order. Nonlinear moment equations result: they are linearized, and the limit of stability is searched by studying the eigenvalues of the matrix of the coefficients of the linearized moment equations. Several numerical analyses are performed to evaluate the critical mean square value of the excitation when a system parameter is varied for the cases of both first and second-order parametric excitation. A first-order excitation, an Ornstein–Uhlenbeck process, does not cause stochastic resonance, while a second-order excitation does. The first result was not known previously.

Energy harvesting from a rotational transducer under random excitation

This paper evaluates the performance of a proposed device for harvesting energy from the vertical motion of boats and yachts under broadband and band-limited random vibrations. The device comprises a sprung mass coupled to an electrical generator through a ball screw. The mathematical equations describing the dynamics of the system are derived. Then by utilizing the theory of random vibration, the frequency response function of the system is obtained. This is used to derive an expression for the mean power produced by the harvester when it is subjected to broadband and band-limited stationary Gaussian white noise. The power expressions are derived in dimensional form to provide an insightful understanding of the effect of the physical parameters of the system on output power. An expression for the optimum load resistance to harvest maximum power under random excitation is also derived and validated by conducting Monte-Carlo simulation. The discussion presented in the paper provides guidelines for designers to maximize the expected harvested power from a system under broadband and band-limited random excitations. Also, based on the method developed in this paper, the output power of a rotational harvester subjected to the vertical excitation of a sailing boat is obtained.

Probabilistic solution of vibro-impact stochastic Duffing systems with a unilateral non-zero offset barrier

This paper proposes a solution procedure for obtaining an approximate probability density function (PDF) of the response of vibro-impact Duffing systems with a unilateral non-zero offset barrier. The impact model is used with classical impacts with instantaneous velocity jumps and the excitation is modelled by a zero mean stationary Gaussian white noise. First the original vibro-impact system is converted into a system without barriers by adopting the Zhuravlev non-smooth coordinate transformation. Second the PDF of the converted system is obtained by solving the Fokker–Planck (FP) equation with the exponential–polynomial closure (EPC) method. Last the PDF of the original system is achieved using the well-established methodology on seeking the PDF distribution of a function of a random variable. A further parametric study is also conducted to evaluate the effectiveness of the proposed solution procedure. Comparison with the simulated result shows that when lightly inelastic impacts occur (i.e, the restitution coefficient is very close to 1.0), the technique of the Zhuravlev non-smooth coordinate transformation is feasible. The transformed system can adequately represent the original system. Consequently the EPC method can provide a good approximate PDF solution. The tail region of the PDF is also approximated well. The analysis further shows that the non-zero offset significantly affects the shape of the PDF distribution of displacement.

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.

Statistical modelling of a non‐linear high‐power amplifier with memory in multi‐input–multi‐output orthogonal frequency division multiplexing systems

A simple and accurate statistical model (SASM) for a high-power amplifier (HPA) with memory in multi-input multi-output (MIMO) orthogonal frequency division multiplexing (OFDM) systems is proposed. This model is statistically equivalent to the more complex generalised memory polynomial (GMP) model, which is one of the most accurate models. The HPA distortion is modelled using stationary white Gaussian noise sources that pass through finite impulse response (FIR) filters. The filter coefficients and the variance of noise sources are derived analytically from the GMP model parameters by spectral decomposition of the GMP noise power spectrum. The number of FIR filters is equal to the memory length of the GMP model. The HPA effect on the OFDM signal, considered as frequency-dependent attenuation coefficients, is modelled using another FIR filter. These FIR filters with small lengths make the model computationally inexpensive and mathematically tractable. The main characteristics of this model are identical bit error rate (BER) and error vector magnitude of subcarriers with those in the GMP model. The proposed model is the simplest representation of a HPA with memory effects. An excellent agreement between the simulation results of the SASM and GMP model is observed. This verifies the accuracy and efficiency of the proposed method.

LQG Control of Along-Wind Response of a Tall Building with an ATMD

Modern tall buildings use lighter construction materials that have high strength and less stiffness and are more flexible. Although this results in the improvement of structural safety, excessive wind-induced excitations could lead to occupant discomfort. The optimal control law of a linear quadratic Gaussian (LQG) controller with an active tuned mass damper (ATMD) is used for reducing the along-wind response of a tall building. ATMD consists of a second mass with optimum parameters for tuning frequency and damping ratio of the tuned mass damper (TMD), under the stationary random load, was used. A fluctuating along-wind load, acting on a tall building, was treated as a stationary Gaussian white noise and was simulated numerically, in the time domain, using the along-wind load spectra proposed by G. Solari in 1993. Using this simulated wind load, it was possible to calculate the along-wind responses of a tall building (with and without the ATMD), using an LQG controller. Comparing the RMS (root mean square) response revealed that the numerically simulated along-wind responses, without ATMD, are a good approximation to the closed form response, and that the reduced responses with ATMD and LQG controller were estimated by varying the values of control design parameters.

Active Control of Along-Wind Response of a Tall Building with AMD Using LQR Controller

Most of modern tall buildings using lighter construction materials are more flexible so could be excessive wind-induced vibrations resulting in occupant discomfort and structural unsafety. The optimal control technique for reducing along-wind vibration of a tall building based on the linear quadratic regulator (LQR) is presented in this work. Actively controlled reduced along-wind vibration response is obtained from the tall building installed in an active mass damper (AMD) with a LQR controller. Fluctuating along-wind load is generated using numerical simulation method, which can formulate a stationary Gaussian white noise process. Simulating wind load in the time domain using known spectra data of fluctuating along-wind load is particularly useful for estimation of windinduced vibration which is more or less narrow banded process such as a along-wind response of a tall building. In this work, fluctuating along-wind load acting on a tall building treated as a stationary Gaussian white noise process is simulated numerically using the along-wind load spectra proposed by G. Solari in1992. And using this simulated along-wind load estimated the reduced along-wind vibration response of a tall building installed in an AMD with a LQR controller.

LQR Control of Along-Wind Response of a Tall Building

Modern tall buildings using high strength and lighter construction materials are more flexible so could be excessive wind-induced vibrations resulting in occupant discomfort and structural unsafety. Recently, many studies have been advanced in using actuator force as an active control force based on the linear quadratic optimum control theory .It needs to predict the wind-induced response and the optimum control force to reduce the excessive wind-induced vibration. It takes a lot of time and cost to do wind tunnel test needed it, so numerical simulation approach instead of that is recommended sometimes. Simulating wind load in the time domain using known spectra data of fluctuating wind load is particularly useful for some prediction of windinduced vibration which is more or less narrow banded process such as a along-wind response of a tall building. In this study, fluctuating along-wind load acting on a tall building treated as a stationary Gaussian white noise process was simulated numerically in the time domain using the along-wind load spectra proposed by G.Solari in1992. And using this simulated along-wind load estimated the reduced along-wind vibration response of a tall building which has an active tuned mass damper (ATMD) installed based on the linear quadratic regulator (LQR) control method.

Perception of Thai distinctive vowel length in noise

A forced choice identification perception experiment using 150 monosyllabic rhyming-word stimulus pairs (with identical consonants and tone) in four conditions of white Gaussian noise was conducted to explore vowel confusions in Thai, a language with nine monophthongs and length (short-long) contrast for all vowels (e.g., /i/-/i:/ and /o/-/o:/). Each stimulus containing speech and noise portions is equal in length. Perceptual results of 18 vowels from 36 Thai listeners at a noise level (SNR) of -24 dB, where the percent intelligibility is the most interpretable, showed that stimuli with short vowels are more accurately perceived than those with long vowels (93.46 vs. 85.64%) with /o:/ and /e:/ as the most confusable. Interestingly, asymmetrical confusions are observed with very few short vowels being misperceived as long vowels, but a larger number of long vowels misperceived as short. Consistent with previous studies of perception of English vowels in white noise [e.g., Benki (2003)], the findings confirm perceptual robustness of vowel height (correlating with F1) over vowel front/backness (correlating with F2). Lastly, an analysis for listeners’ misidentified responses shows that the listeners generally favor short over long vowels.

Suppression of strong random noise in seismic data by using time-frequency peak filtering

Time-frequency peak filtering (TFPF) is highly efficient in suppressing random noise in seismic data. Although the hypothesis of stationary Gaussian white noise cannot be fulfilled in practical seismic data, TFPF can effectively suppress white and colored random noise with different intensities, as can be theoretically demonstrated by detecting such noise in synthetic seismic data. However, a “zero-drift” effect is observed in the filtered signal and is independent of the average power and variance of the random noise, but related to its mean value. Furthermore, we consider the situation where the local linearization of the seismic data cannot be satisfied absolutely and study the “distortion” characteristics of the filtered signal using TFPF on a triangular wave. We found that over-compensation is possible in the frequency band for the triangular wave. In addition, it is nonsymmetrical and has a relationship to the time-varying curvature of the seismic wavelet. The results also present an improved approach for TFPF.

Parametric control of structural responses using an optimal passive tuned mass damper under stationary Gaussian white noise excitations

In this study, the structural control strategy utilizing a passive tuned mass damper (TMD) system as a seismic damping device is outlined, highlighting the parametric optimization approach for displacement and acceleration control. The theory of stationary random processes and complex frequency response functions are explained and adopted. For the vibration control of an undamped structure, the optimal parameters of a TMD, such as the optimal tuning frequency and optimal damping ratio, to stationary Gaussian white noise acceleration are investigated by using a parametric optimization procedure. For damped structures, a numerical searching technique is used to obtain the optimal parameters of the TMD, and then the explicit formulae for these optimal parameters are derived through a sequence of curve-fitting schemes. Using these specified optimal parameters, several different controlled responses are examined, and then the displacement and acceleration based control effectiveness indices of the TMD are examined from the view point of RMS values. From the viewpoint of the RMS values of displacement and acceleration, the optimal TMDs adopted in this study shows clear performance improvements for the simplified model examined, and this means that the effective optimization of the TMD has a good potential as a customized target response-based structural strategy.