Gaussian Noise Source Research Articles

Impact of multiplexing noise on multilayer networks of bistable maps

We explore numerically the complex dynamics of multilayer networks (consisting of three and one hundred layers) of cubic maps in the presence of noise-modulated interlayer coupling (multiplexing noise). The coupling strength is defined by independent discrete-time sources of color Gaussian noise. Uncoupled layers can demonstrate different complex structures, such as double-well chimeras, coherent and spatially incoherent regimes. Regions of partial synchronization of these structures are identified in the presence of multiplexing noise. We elucidate how synchronization of a three-layer network depends on the initially observed structures in the layers and construct synchronization regions in the plane of multiplexing noise parameters “noise spectrum width – noise intensity”. It is shown that in a hundred-layer network, clusters of synchronized layers can be formed at certain optimal values of multiplexing noise parameters. The performed numerical studies confidently indicate that the spatial dynamics of multilayer networks can be controlled by varying multiplexing noise parameters.

Most Probable Dynamics of the Single-Species with Allee Effect under Jump-Diffusion Noise

We explore the most probable phase portrait (MPPP) of a stochastic single-species model incorporating the Allee effect by utilizing the nonlocal Fokker–Planck equation (FPE). This stochastic model incorporates both non-Gaussian and Gaussian noise sources. It has three fixed points in the deterministic case. One is the unstable state, which lies between the two stable equilibria. Our primary focus is on elucidating the transition pathways from extinction to the upper stable state in this single-species model, particularly under the influence of jump-diffusion noise. This helps us to study the biological behavior of species. The identification of the most probable path relies on solving the nonlocal FPE tailored to the population dynamics of the single-species model. This enables us to pinpoint the corresponding maximum possible stable equilibrium state. Additionally, we derive the Onsager–Machlup function for the stochastic model and employ it to determine the corresponding most probable paths. Numerical simulations manifest three key insights: (i) when non-Gaussian noise is present in the system, the peak of the stationary density function aligns with the most probable stable equilibrium state; (ii) if the initial value rises from extinction to the upper stable state, then the most probable trajectory converges towards the maximally probable equilibrium state, situated approximately between 9 and 10; and (iii) the most probable paths exhibit a rapid ascent towards the stable state, then maintain a sustained near-constant level, gradually approaching the upper stable equilibrium as time goes on. These numerical findings pave the way for further experimental investigations aiming to deepen our comprehension of dynamical systems within the context of biological modeling.

Fourier-DeepONet: Fourier-enhanced deep operator networks for full waveform inversion with improved accuracy, generalizability, and robustness

Full waveform inversion (FWI) infers the subsurface structure information from seismic waveform data by solving a non-convex optimization problem. Data-driven FWI has been increasingly studied with various neural network architectures to improve accuracy and computational efficiency. Nevertheless, the applicability of pre-trained neural networks is severely restricted by potential discrepancies between the source function used in the field survey and the one utilized during training. Here, we develop a Fourier-enhanced deep operator network (Fourier-DeepONet) for FWI with the generalization of seismic sources, including the frequencies and locations of sources. Specifically, we employ the Fourier neural operator as the decoder of DeepONet, and we utilize source parameters as one input of Fourier-DeepONet, facilitating the resolution of FWI with variable sources. To test Fourier-DeepONet, we develop three new and realistic FWI benchmark datasets (FWI-F, FWI-L, and FWI-FL) with varying source frequencies, locations, or both. Our experiments demonstrate that compared with existing data-driven FWI methods, Fourier-DeepONet obtains more accurate predictions of subsurface structures in a wide range of source parameters. Moreover, the proposed Fourier-DeepONet exhibits superior robustness when handling data with Gaussian noise or missing traces and sources with Gaussian noise, paving the way for more reliable and accurate subsurface imaging across diverse real conditions.

Noise influence on recurrent neural network with nonlinear neurons

The purpose of this study is to establish the features of noise propagation and accumulation in a recurrent neural network using a simplified echo network as an example. In this work, we studied the influence of activation function of artificial neurons and the connection matrices between them. Methods. We have considered white Gaussian noise sources. We used additive, multiplicative and mixed noise depending on how the noise is introduced into artificial neurons. The noise impact was estimated using the dispersion (variance) of the output signal. Results. It is shown that the activation function plays a significant role in noise accumulation. Two nonlinear activation functions have been considered: the hyperbolic tangent and the sigmoid function with range form 0 to 1. It is shown that some types of noise are suppressed in the case of the second function. As a result of considering the influence of coupling matrices, it was found that diagonal coupling matrices with a large blurring coefficient lead to less noise accumulation in the echo network reservoir with an increase in the reservoir memory influence. Conclusion. It is shown that activation functions of the form of sigmoid with range from 0 to 1 are suitable for suppressing multiplicative and mixed noise. The accumulation of noise in the reservoir was considered for three types of coupling matrices inside the reservoir: a uniform matrix, a band matrix with a small blurring coefficient, and a band matrix with a large blurring coefficient. It has been found that the band matrix echo networks with a high blurring coefficient accumulates the least noise. This holds for both additive and multiplicative noise.

Improving MDM–WDM optical network performance via optimized power allocation using Gaussian noise model

The few-mode fiber nonlinearity has been well modeled by an additive Gaussian Noise (GN) source. The available GN models in Mode Division Multiplexing–Wavelength Division Multiplexing (MDM–WDM) systems are incoherent and neglect the dispersion slope. The incoherent GN model assumes that the Nonlinear Interference (NLI) noise created at each span is summed up in power incoherently at the receiver and leads to underestimating the NLI noise power. In the first part of this paper, a coherent GN model is derived by taking into account the dispersion slope and verified through Split-Step Fourier Method (SSFM) simulations. In the second part of this paper, the total capacity maximization, as well as the minimum Signal to Noise Ratio (SNR) margin maximization problems are presented and solved based on the obtained GN model. MDM–WDM multi-node nonlinear network under different scenarios including Equal/Non-Equal Required SNR (ERS/NERS), and Flat/Non-Flat Amplifier Gain (FG/NFG) are considered in the numerical simulations. Considering 200 different sets of randomized traffic demands (random channel-mode distributions) the cumulative distribution function of minimum SNR margin improvement in optimized power allocation compared with the best flat power allocation is in 99% more than 1 dB, 1.2 dB, 2.4 dB, and 2.8 dB, for ERS-FG, ERS-NFG, NERS-FG, and NERS-NFG scenarios, respectively.

Transition and basin stability in a stochastic tumor growth model with immunization

The phenomenon of noise-induced transition driven by the correlated Gaussian and non-Gaussian colored noises is investigated by the first escape probability (FEP) and the mean first exit time (MFET). To derive the Markovian approximation of the original tumor growth model and obtain the analytical expressions of the FEP and MFET, we reduce the non-Gaussian colored noise and then expand the unified colored noise approximation (UCNA). Additionally, the stochastic basin of attraction (SBA), a recent geometric concept based on the FEP and MFET, is introduced to provide further insight into the effects of noisy fluctuations on the basin stability of a certain domain. A higher FEP or shorter MFET in the high tumor population region B facilitates the transition from B to the low tumor population region Bc, which indicates the weaker stability of domain B. Our main results demonstrate that (i) the transitions from B to Bc can be induced by both the Gaussian and non-Gaussian noise sources; (ii) the stronger noise intensity, especially the non-Gaussian noise intensity, with a larger deviation parameter and immune coefficient improves the FEP, shortens the MFET, and hence benefits the transitions. However, the enlargement of the correlation between noises strengthens the basin stability of domain B and impedes the transitions; (iii) the size of SBA expands due to the larger cross-correlated intensity. In contrast, the enhancements of noise intensities with a larger departure degree reduce the size of SBA, which weakens the basin stability and is less in favor of tumor treatment. Furthermore, the Monte Carlo simulations of the original system are employed to verify the feasibility and accuracy of the analytical predictions.

Kovacs Memory Effect with an Optically Levitated Nanoparticle.

The understanding of the dynamics of nonequilibrium cooling and heating processes at the nanoscale is still an open problem. These processes can follow surprising relaxation paths due to, e.g., memory effects, which significantly alter the expected equilibration routes. The Kovacs effect can take place when a thermalization process is suddenly interrupted by a change of the bath temperature, leading to a nonmonotonic evolution of the energy of the system. Here, we demonstrate that the Kovacs effect can be observed in the thermalization of the center of mass motion of a levitated nanoparticle. The temperature is controlled during the experiment through an external source of white Gaussian noise that mimics an effective thermal bath at a temperature that can be changed faster than any relaxation time of the system. We describe our experiments in terms of the dynamics of a Brownian particle in a harmonic trap without any fitting parameter, suggesting that the Kovacs effect can appear in a large variety of systems.

Trapping enhanced by noise in nonhyperbolic and hyperbolic chaotic scattering

The noise-enhanced trapping is a surprising phenomenon that has already been studied in chaotic scattering problems where the noise affects the physical variables but not the parameters of the system. Following this research, in this work we provide strong numerical evidence to show that an additional mechanism that enhances the trapping arises when the noise influences the energy of the system. For this purpose, we have included a source of Gaussian white noise in the HȨnon-Heiles system, which is a paradigmatic example of open Hamiltonian system. For a particular value of the noise intensity, some trajectories decrease their energy due to the stochastic fluctuations. This drop in energy allows the particles to spend very long transients in the scattering region, increasing their average escape times. This result, together with the previously studied mechanisms, points out the generality of the noise-enhanced trapping in chaotic scattering problems.

Lévy noise-induced transition and stochastic resonance in a tumor growth model

The stationary probability density and stochastic resonance phenomenon of a tumor growth model under the excitation of Lévy noise and Gaussian white noise are investigated in this paper. The fourth-order Runge-Kutta method and the Janick-Weron algorithm are used to simulate the stationary probability density. Meanwhile, the signal-to-noise ratio(SNR) is studied as a function of Lévy noise intensity and Gaussian white noise intensity by numerical simulation respectively. The results indicate that: (i)both Lévy and Gaussian noise sources give rise to noise-induced transitions for the system, with a peculiarity that smaller stability indexα and Lévy noise intensity D enhance the likelihood of tumor cell death; (ii)both noise parameters and system parameters can induce the occurrence of stochastic resonance. However, the effect of Gaussian white noise on SNR is different from that of Lévy noise.

Observation of stochastic resonance for weak periodic magnetic field signal using a chaotic system

Stochastic resonance, found in many natural and engineered bistable systems, refers to a physical phenomenon wherein the response of nonlinear dynamical systems to weak input signal is enhanced by the presence of noise, tuned to the optimal level. However, the observation of this phenomenon for weak subthreshold periodic magnetic field signals in chaotic bistable systems has not been reported. In this paper we present the experimental observation of stochastic resonance in a Chua circuit. The experiment was performed using an internal Gaussian noise source, and an external driving subthreshold sinusoidal magnetic field below 1mT, with a frequency range in the order of kHz. The circuit was operating near the switching threshold between single-scroll and double-scroll chaotic regimes. The spectral characteristics of interest (the output signal power at forcing signal frequency and the signal-to-noise ratio) pass through a maximum with varying noise intensity. The residence times probability distribution quantifying the synchronization between the forcing and the response has a multi-peaked structure for the optimal noise intensity. Our experiments show the possibility of stochastic resonance for weak subthreshold magnetic field signals in chaotic systems, with parts of the attractor located in different regions of the phase space and with internal noise.

Frequency Domain Maximum Likelihood Identification With Gaussian Input–Output Uncertainty

This letter deals with the identification of dynamical systems corrupted by additive and independent identically distributed Gaussian noise sources when the noise-free-input is an arbitrary signal. We review two stochastic models: 1) the errors-in-variables (EIV) and 2) the random variable setting. We derive a maximum likelihood (ML) estimator for the latter and compare with a previously proposed EIV ML method. The problem is formulated in the frequency domain and it is assumed that the ratio of the noise variances is known. We discuss the similarity between the two approaches and prove that one can be seen as the regularized counterpart of the other, advocating the suggested estimator when only few data samples are available.

Practical Prediction of CFO-Made OFDM Symbol Distortion

The Carrier Frequency Offset (CFO) is considered to be a major drawback of the Orthogonal Frequency-Division Multiplexing (OFDM) signal. So, in many practical situations, specifically with LTE-Advanced downlink introducing carrier aggregation, estimation of the CFO-induced OFDM symbol phase deviation, is of interest. However, this demands complex test equipment, such as e.g. a Vector Signal Analyzer (VSA), which might not be always and everywhere available. Therefore, we applied the link abstraction principle on the Bit Error Rate (BER) that is considered to be determined just by the CFO-caused phase deviation, i.e. as if the channel was noiseless and time-dispersion-free (so that evident bit errors occur just due to the actual CFO). Furthermore, as the CFO-caused squared phase deviation is linear with the instantaneous (per-OFDM-symbol) Peak-to-Average Power Ratio (PAPR), which is related to the Error Vector Magnitude (EVM) and so with BER, we develop a simple model for analytical BER-based estimating of CFO. In this sense, we considered the easy-to-measure BER degradation as resulting just from the according additive white Gaussian noise (AWGN) source, which abstracts the CFO distortion. The proposed analytical model is validated by according Monte-Carlo simulations.

An Analysis of Deterministic Chaos as an Entropy Source for Random Number Generators.

This paper presents an analytical study on the use of deterministic chaos as an entropy source for the generation of random numbers. The chaotic signal generated by a phase-locked loop (PLL) device is investigated using numerical simulations. Depending on the system parameters, the chaos originating from the PLL device can be either bounded or unbounded in the phase direction. Bounded and unbounded chaos differs in terms of the flatness of the power spectrum associated with the chaotic signal. Random bits are generated by regular sampling of the signal from bounded and unbounded chaos. A white Gaussian noise source is also sampled regularly to generate random bits. By varying the sampling frequency, and based on the autocorrelation and the approximate entropy analysis of the resulting bit sequences, a comparison is made between bounded chaos, unbounded chaos and Gaussian white noise as an entropy source for random number generators.

Identification of suitable embedding dimensions and lags for time series generated by chaotic, finite-dimensional systems

In the field of nonlinear dynamics, many methods have been proposed to tackle the issue of optimally setting embedding dimension and lag in order to analyze sampled scalar signals. However, intrinsic statistical uncertainties due to the finiteness of input sequences severely hinder a general solution to the problem. A more achievable approach consists of assessing sets of dimension and lag pairs that are equivalently suitable to embed a time series. Here we present a method to identify these sets of embedding pairs, under the hypothesis that the time series of interest is generated by a chaotic, finite-dimensional dynamical system. We first introduce a ``distance gauge transformation'' based on the analytical forms of correlation integrals corresponding to a Gaussian white noise source. We show that in this new distance gauge, correlation integrals generated by chaotic, finite-dimensional systems are characterized by distinctive features, whose absence is a marker of the unsuitability of the underlying embedding choice. By means of a new estimator of the correlation dimension that relies on the new distance gauge, sets of suitable embedding pairs are finally identified by looking at the uniformity of the estimation. The method is completely automatic and was successfully tested on both synthetic and experimental time series. It also provides a tool to estimate the redundance and irrelevance timescales of the system that underlie the time series as well as a lower constraint to the sampling rate. The method is suitable for applications in research fields where a chaotic behavior has to be identified, such as neuroscience, geophysics, and economics.

Gaussian Source Detection and Spatial Spectral Estimation Using a Coprime Sensor Array With the Min Processor

A coprime sensor array (CSA) interleaves two undersampled uniform linear arrays with coprime undersampling factors and has recently found broad applications in signal detection and estimation. CSAs commonly use the product processor by multiplying the scanned responses of two colinear subarrays to estimate the spatial power spectral density (PSD) of the received signal. This paper proposes a new CSA processor, the CSAmin processor, which chooses the minimum between the two CSA subarray periodograms at each bearing to estimate the spatial PSD. The proposed CSAmin processor resolves the CSA subarray spatial aliasing equally well as the product processor. For an extended aperture CSA, the CSAmin reduces the peak sidelobe height and total sidelobe area over the product processor for the same CSA geometry. Moreover, unlike the PSD estimate from the product processor, the PSD estimate from the CSAmin is guaranteed to be positive semidefinite. This paper derives the probability density function, the complementary cumulative distribution function (CCDF, or tail distribution), and the first two moments of the CSAmin PSD estimator in closed form for Gaussian sources in white Gaussian noise. Numerical simulations verify the derived CSAmin statistics and demonstrate that the CSAmin improves the performance over the product processor in detecting narrowband Gaussian sources in the presence of loud interferers and noise. The CSAmin spatial PSD estimate achieves lower variance than the product processor estimate, and keeps the PSD estimate unbiased for white Gaussian processes and asymptotically unbiased for nonwhite Gaussian processes.

Estimating the number of sources in white Gaussian noise: simple eigenvalues based approaches

Estimating the number of sources is a key task in many array signal processing applications. Conventional algorithms such as Akaike's information criterion (AIC) and minimum description length (MDL) suffer from underestimation and overestimation errors. In this study, the authors propose four algorithms to estimate the number of sources in white Gaussian noise. The authors’ proposed algorithms are categorised into two main categories; namely, sample correlation matrix (CorrM) based and correlation coefficient matrix (CoefM) based. Their proposed algorithms are applied on the CorrM and CoefM eigenvalues. They propose to use two decision statistics, which are the moving increment and the moving standard deviation of the estimated eigenvalues as metrics to estimate the number of sources. For their two CorrM based algorithms, the decision statistics are compared to thresholds to decide on the number of sources. They show that the conventional process to estimate the threshold is mathematically tedious with high computational complexity. Alternatively, they define two threshold formulas through linear regression fitting. For their two CoefM based algorithms, they re-define the problem as a simple maximum value search problem. Results show that the proposed algorithms perform on par or better than AIC and MDL as well as recently modified algorithms at medium and high signal-to-noise ratio (SNR) levels and better at low SNR levels and low number of samples, while using a lower complexity criterion function.

Performance improvement of a buck converter using Kalman filtering

This paper presents a predictive current control algorithm for a synchronous buck converter using an extended Kalman filter (EKF) algorithm. The predictive approach avoids the need of current-sensing circuitry and provides insensitivity to Gaussian noise sources at the output of the buck converter, which is the same as the control loop input. The method requires a model for the buck converter, the EKF design, and current loop tuning. All these prerequisites are described in this work along with the implementation of the algorithm in a state of the art microcontroller. Simulation and experimental results show that while maintaining a good step response, the proposed method provides better results than standard methods when Gaussian noise is present at the output voltage.

The Foraging Brain: Evidence of Lévy Dynamics in Brain Networks.

In this research we have analyzed functional magnetic resonance imaging (fMRI) signals of different networks in the brain under resting state condition. To such end, the dynamics of signal variation, have been conceived as a stochastic motion, namely it has been modelled through a generalized Langevin stochastic differential equation, which combines a deterministic drift component with a stochastic component where the Gaussian noise source has been replaced with α-stable noise. The parameters of the deterministic and stochastic parts of the model have been fitted from fluctuating data. Results show that the deterministic part is characterized by a simple, linear decreasing trend, and, most important, the α-stable noise, at varying characteristic index α, is the source of a spectrum of activity modes across the networks, from those originated by classic Gaussian noise (α = 2), to longer tailed behaviors generated by the more general Lévy noise (1 ≤ α < 2). Lévy motion is a specific instance of scale-free behavior, it is a source of anomalous diffusion and it has been related to many aspects of human cognition, such as information foraging through memory retrieval or visual exploration. Finally, some conclusions have been drawn on the functional significance of the dynamics corresponding to different α values.

Evaluation of closure strategies for a periodically-forced Duffing oscillator with slowly modulated frequency subject to Gaussian white noise

The response of a Duffing oscillator subject to a periodic forcing with slowly and stochastically modulated frequency is analyzed numerically. The results of both moment and cumulant-based stochastic reductions are compared to Monte Carlo simulations. It is shown how the explicit characterization of higher-order central moments of the (Gaussian) noise source and the periodic nature of the forcing enable a reliable reduction strategy providing a faithful description of the mean behavior of stochastic solutions. The reduced model is then used to illustrate how a large noise level and fast frequency drift may combine to sustain a strong response that is normally associated to resonance in the noiseless static case.

Adaptive Digital Beamformer Autocalibration Using an Injected Noise Source

This paper presents a wideband closed-loop relative adaptive calibration method for digital beamformers in phased array applications. Using an internally distributed and injected additive white Gaussian noise source, we equalized the channels relative to a selected channel before beamforming. This relative injection method has a size, weight, and power (SWaP) advantage compared with an absolute method where a copy of the illumination waveform is used as the reference signal. Measured data from an existing 22-phase-center elemental digital beamformer array were used to evaluate the relative method against existing absolute methods. The channel response was extracted and the bit error rate was evaluated using a simulation test bench. A parametric conceptual design was also performed to evaluate the relative SWaP between the relative and absolute methods. The results show that the relative method is 17.6 times smaller, uses 1.3 times less power, and is 2.4 times lower in cost than the absolute method. The results also show that the relative calibration method can be as effective as an absolute method, given that the injected waveform has a higher signal-to-noise ratio than the ambient scene.