Noise Probability Density Function Research Articles

A method for evaluation of the probability density function of white noise filtered non-Gaussian stochastic process

Passing non-Gaussian white noises through a pre-designed linear filter can generate non-Gaussian stochastic processes with desired probabilistic and spectral properties. This linear filtering methodology has been extensively employed in non-Gaussian signal processing and simulation. At present, the investigation of probability properties for the output non-Gaussian processes is primarily concentrated on the higher-order statistics, spectra, and correlation functions. However, evaluating the probability density function (PDF) of the output process remains a challenging task. In order to address this issue, this short communication commences with the theoretical derivation of the PDF formula for the white noise filtered non-Gaussian stochastic process. The derived PDF is the result of a convolution of multiple PDFs, each obtained by applying a scaling transformation to the input white noise PDF. Subsequently, a discretized numerical calculation method is proposed to overcome the analytical difficulties associated with the computation of multiple convolutions. Finally, the accuracy and efficiency of the proposed method are demonstrated through several numerical examples.

Signal detection for FTN system under unknown noise model with imperfect CSI

The problem of signal detection for faster-than-Nyquist (FTN) signaling system with imperfect channel state information (CSI) is studied. Also, the non-Gaussian noise in real applications is considered. A general model, Gaussian mixture noise (GMN) model, is applied to describe the unknown probability density function (PDF) of the non-Gaussian noise. After that, a framework over variational inference is constructed to simplify the problem through finding an auxiliary PDF with factorization form. Based on the factor graph over the signal, joint response and the parameters of GMN, an iterative solution is obtained. Simulations demonstrate the effectiveness of the proposed approach.

A robust direction-of-arrival estimation method for impulsive noise environments

The performance of conventional direction-of-arrival (DOA) estimation methods degrades greatly when there are few snapshots and the noise model is mismatched, especially in impulsive noise environments. To solve this problem, this paper proposes a robust DOA estimation method, which is robust to the probability density function of impulsive noise. First, by exploiting the low-rank decomposition of the residual fitting error matrix and the characteristics of impulsive noise, three conditions of a good cost function in impulsive noise environments are proposed, and a unified cost function framework is established. Based on this, a robust cost function is designed to suppress the contribution of samples with impulsive noise to the cost function. Then, a bi-iterative complex fixed-point algorithm (BI-CFPA) is developed for the nonconvex optimization problem of signal subspace estimation based on the proposed robust cost function. Theoretical analyses indicate that the BI-CFPA has excellent numerical stability and low computational complexity. Finally, with the estimated signal subspace, the multiple signal classification technique is employed to obtain DOA estimates. Simulation results show that the proposed DOA estimator performs better than existing methods in three typical impulsive noise environments, especially under fairly strong impulsive noise.

Beyond Gaussian Noise: A Generalized Approach to Likelihood Analysis with Non-Gaussian Noise

Likelihood analysis is typically limited to normally distributed noise due to the difficulty of determining the probability density function of complex, high-dimensional, non-Gaussian, and anisotropic noise. This is a major limitation for precision measurements in many domains of science, including astrophysics, for example, for the analysis of the cosmic microwave background, gravitational waves, gravitational lensing, and exoplanets. This work presents Score-based LIkelihood Characterization, a framework that resolves this issue by building a data-driven noise model using a set of noise realizations from observations. We show that the approach produces unbiased and precise likelihoods even in the presence of highly non-Gaussian correlated and spatially varying noise. We use diffusion generative models to estimate the gradient of the probability density of noise with respect to data elements. In combination with the Jacobian of the physical model of the signal, we use Langevin sampling to produce independent samples from the unbiased likelihood. We demonstrate the effectiveness of the method using real data from the Hubble Space Telescope and James Webb Space Telescope.

Optimal Noise-Boosted Estimator Design Via Adaptive Stochastic Resonance

In signal estimation, an optimal estimator is frequently unachievable because its closed form may not be analytically tractable or is too complex to implement. Alternatively, one can turn to suboptimal yet easily implementable estimators for practical signal estimation tasks. In this paper, an optimal noise-boosted estimator is designed and the adaptive stochastic resonance method is implemented to simultaneously exploit the beneficial role of the injected noise as well as the learning ability of the estimator parameter. Aiming to effectively improve the estimation performance, we use the kernel function method to find an approximate solution for the probability density function (PDF) of the optimal injected noise. During this process, the noise PDF and the estimator parameter establish a finite-dimensional non-convex optimization space for maximizing the estimation performance, which is adaptively searched by the sequential quadratic programming (SQP) algorithm at each iteration. Two representative estimation problems are explored. The obtained results demonstrate that this adaptive stochastic resonance method can improve the performance of the suboptimal estimators and bring it very close to that of the optimal estimator.

GSM Filter: A Learning Model of Measurement Noise Distributions for Bayesian Filtering

In this paper, a new universal Bayesian filter with a Gaussian scale mixture (GSM) model, called the Gaussian scale mixture (GSM) filter, for adaptively learning any measurement noise distribution of a state space system is presented. It is demonstrated that the discrete measurement noise distribution of the system takes on a Gaussian sum form when described by a GSM model. It implies that, as the Gaussian sum model does, the GSM model can also be used to approximate any measurement distribution as closely as desired, whatever the probability density function (pdf) of the measurement noise is. If its covariance is known as a priori or can be estimated, it is found that the scale parameter and matrix of the GSM model at each time step can be achieved adaptively to make the following Bayesian inference much simpler. Further analyses show that the GSM model's auxiliary random variable and the system's state posterior at each time step can be inferred by a variational Bayesian (VB) approach adaptively. At each time step, the posterior distribution of system states shown is a Gaussian whatever the measurement noise pdf of the system is. As a result, the Bayesian filtering for nonlinear systems with any measurement noise pdf can be done by our GSM filter, almost as simple as the general Gaussian filters for filtering a nonlinear Gaussian system. Both simulations and experiments are conducted to verify the effectiveness and correctness of the analytical results. The results on the performances of our algorithms fully outperforming those of the existing state of the art Bayesian filters are given.

NEAR–OPTIMUM DETECTION OF SIGNALS IN UNDERWATER ACOUSTIC NOISE USING LOCALLY OPTIMAL DETECTOR IN TIGERS RIVER

Signal detection has been considered important in under-water signal processing and digital communications, and depending upon noise statistics’ knowledge, near-optimum signal detections in the under-water acoustic noises (UWANs) may be realized in a more effective manner. The theory of the normal (i.e. Gaussian) noise permits using matched filter (MF) detectors; for that reason, a locally optimal (LO) detector has been designed in the present work for improving the probability of the detection ( based on knowing the probability density function (PDF) of noise. Under-water noise that has been utilized for the validation represents the real data that had been gathered from the sea with the use of the broad-band hydrophones at Abo Dali district -Kazem Al Ali Village-Tigris Beaches-Baghdad-Iraq. The LO detector performance is compared after that to the conventional matched filter detector and those have been assessed based on their values. For time-varying signals, the probability of false alarms has been identified as 0.010, and of 90%, energy-to-noise ratios (ENRs) of LO are more efficient compared to the ones of Matched Filter by 4.1dB and for the signals with a fixed frequency, LO is more efficient compared to matched filter by 4.7dB.

Robust Frequency Estimation Under Additive Symmetric α-Stable Gaussian Mixture Noise

Here the estimating problem of a single sinusoidal signal in the additive symmetric α-stable Gaussian (ASαSG) noise is investigated. The ASαSG noise here is expressed as the additive of a Gaussian noise and a symmetric α-stable distributed variable. As the probability density function (PDF) of the ASαSG is complicated, traditional estimators cannot provide optimum estimates. Based on the Metropolis-Hastings (M-H) sampling scheme, a robust frequency estimator is proposed for ASαSG noise. Moreover, to accelerate the convergence rate of the developed algorithm, a new criterion of reconstructing the proposal covariance is derived, whose main idea is updating the proposal variance using several previous samples drawn in each iteration. The approximation PDF of the ASαSG noise, which is referred to the weighted sum of a Voigt function and a Gaussian PDF, is also employed to reduce the computational complexity. The computer simulations show that the performance of our method is better than the maximum likelihood and the -norm estimators.

Forward-looking ground penetrating radar image reconstruction using fully connected neural network

One way to identify targets buried in the ground is to use forward-looking ground penetration radars. The two components of speed and accuracy are very important in reconstructing the images obtained from this method. Distinguishing the target from the Earth clutter is one of the main challenges of this type of imaging. This article proposes a novel fast algorithm that reconstructs scattered images created by forward-looking ground-penetrating radar. The linear back-projection algorithm is used for fast image reconstruction. Because our problem is nonlinear ill-posed inverse, errors and distortions are compensated by a deep fully connected neural network. The difference between received and reconstructed signal is the input of the neural network, and output is the error between reconstructed and actual coefficients of the Interested region. Our assumption is the region of interest has a few electromagnetic scatterers. Because of types of soils and non-stationary situations, it is difficult to estimate the probability density function of noise plus clutter. Therefore, we modified order statistic constant false alarm rate to detect shallowly buried scatterers. The simulation results show that the algorithm improves signal-to-noise and clutter ratio in output and detects objects with a constant probability of false alarm in a fast way.

The Residual ISI for Which the Convolutional Noise Probability Density Function Associated with the Blind Adaptive Deconvolution Problem Turns Approximately Gaussian.

In a blind adaptive deconvolution problem, the convolutional noise observed at the output of the deconvolution process, in addition to the required source signal, is—according to the literature—assumed to be a Gaussian process when the deconvolution process (the blind adaptive equalizer) is deep in its convergence state. Namely, when the convolutional noise sequence or, equivalently, the residual inter-symbol interference (ISI) is considered small. Up to now, no closed-form approximated expression is given for the residual ISI, where the Gaussian model can be used to describe the convolutional noise probability density function (pdf). In this paper, we use the Maximum Entropy density technique, Lagrange’s Integral method, and quasi-moment truncation technique to obtain an approximated closed-form equation for the residual ISI where the Gaussian model can be used to approximately describe the convolutional noise pdf. We will show, based on this approximated closed-form equation for the residual ISI, that the Gaussian model can be used to approximately describe the convolutional noise pdf just before the equalizer has converged, even at a residual ISI level where the “eye diagram” is still very closed, namely, where the residual ISI can not be considered as small.

Randomized Machine Learning and Forecasting of Nonlinear Dynamic Models Applied to SIR Epidemiological Model

We propose an approach to estimation of the parameters of non-linear dynamic models using the concept of Randomized Machine Learning (RML), based on the transition from deterministic models to random ones (with random parameters), followed by estimation of the probability distributions of parameters and noises on real data. The main feature of this method is its efficiency in conditions of a small amount of real data. The paper considers models formulated in terms of ordinary differential equations, which are converted to a discrete form for setting and solving the problem of entropy optimization. The application of the proposed approach is demonstrated on the problem of predicting the total number of infected COVID-19 using adynamic SIR epidemiological model. To do this, we construct a randomized SIR model (R-SIR) with one parameter, the entropy-optimal estimate of which is realized by its probability density function, as well as the probability density functions of the measurement noise at the points where training is performed. Next, the technique of randomized prediction with noise filtering is applied, based on the generation of the corresponding distributions and the construction of an ensemble of predictive trajectories with the calculation of the trajectory averaged over the ensemble. The paper implements a computational experiment using real operational data on the infection cases in the form of a comparative study with a well-known method for estimating model parameters based on the least squares method. The results obtained in the experiment demonstrate a significant decrease in the mean absolute percentage error (MAPE) with respect to real observations in the forecast interval, which shows the efficiency of the proposed method and its effectiveness in problems of the type considered in the work.

The five-point non-coherent model of a distributed radar object providing the independent control of angle noise probability characteristics along two orthogonal coordinate axes

The two-dimensional five-point non-coherent model replacing a distributed radar target is explored in this work. Four fixed model points are set in corners of the square but the fifth movable point lies inside of this square. Model points are supplied by normal uncorrelated random processes. The possibilities of the five-point non-coherent model of a distributed radar object for independent control of the producing angle noise parameters along two orthogonal coordinate axes are explored. The disadvantage of this model is noted - the connection of parameters values of angle noise probability density function for two coordinate axes. The expression describing this connection is specified. Expressions determining the boundaries of the allowable coordinate values of the fifth movable point of the five-point non-coherent model, within which the model provides the set parameters of the angle noise probability density function, are defined. The arrived results are validated by program simulations.

Spectrum sensing for cognitive radio network with multiple receive antennas under impulsive noise environments

Spectrum sensing with multiple receive antennas is addressed in the cognitive radio network under impulsive noise environments. Based on order statistics, we propose a non-linear combining scheme to cope with the heavy-tail characteristics of the probability density function of impulsive noise. Through computer simulations, it is shown that the proposed scheme exhibits better detection performance than the conventional schemes in impulsive noise environments with Rayleigh fading.

QoS Analysis and Signal Characteristics for Short-Range Visible-Light Communications

Visible-light communications (VLCs) are energy-efficient communication means and a popular alternative to the radio-frequency technology. VLCs are referred to as free-space optical communication systems which can be built upon the existing lighting infrastructure. Due to the cost-effectiveness concern, light emitting diodes (LEDs) are adopted as the visible-light sources. In this paper, we demonstrate a VLC prototype using white LEDs and pulse position modulation (PPM), which is aimed for short-range communications. Based on the acquired VLC signal waveforms, characteristics related to electronic components, LEDs, and photo diodes (PDs) are studied thoroughly. The non-square shape of received pulses is modeled by new template functions involving rising and setting time intervals. A new PPM demodulator is devised thereupon. Meanwhile, the probability density function of noise is derived using the Edgeworth expansion. As all of the aforementioned aspects are taken into account, the VLC quality-of-service (QoS) in terms of package error rate and throughput is studied in this work. Experiments are taken to justify the theoretical QoS metric and evaluate our proposed new PPM demodulator.

Optimizing Coherence Suppression in a Laser Broadened by Phase Modulation With Noise

Phase noise can be modulated onto the output of a laser with an electro-optic phase modulator (EOM) to create a highly incoherent broadened source with low intensity noise. This technique leaves a small but finite fraction of the coherent carrier power that can be highly detrimental in applications requiring incoherent light. This article shows that the carrier suppression in a laser broadened by this technique can be calculated for an arbitrary noise probability density function. The carrier suppression can be varied experimentally by adjusting the noise voltage standard deviation Gv <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">σ</sub> and saturation voltage V <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">sat</sub> of the amplifier that amplifies the noise source that drives the EOM. Simulations show that suppressions better than -30 dB are attainable for reasonable tolerances in Gv <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">σ</sub> , for example an EOM with a V <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">π</sub> of 4.7 V, a V <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">sat</sub> of 6.3 V, and Gv <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">σ</sub> /V <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">π</sub> = 0.73 ± 0.03. Even greater tolerances can be achieved with a higher V <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">sat</sub> (21 V), lower V <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">π</sub> (3.5 V), and Gv <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">σ</sub> /V <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">π</sub> ≈ 3.2 ± 2.3. This model aids in selecting these parameters such that the carrier suppression is resilient to fluctuations in these voltages due to temperature variations and aging of the components. The model predictions are validated by testing a fiber optic gyroscope interrogated with a broadened laser for various Gv <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">σ</sub> and V <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">sat</sub> and inferring the carrier suppression from its measured noise. A -44-dB carrier suppression was observed for a nonlinear amplifier with Gv <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">σ</sub> = 3.43 V, V <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">sat</sub> = 6.3 V, and an EOM with V <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">π</sub> = 4.7 V, in agreement with predictions.

One-Bit Sine-Fit

This article introduces a novel procedure for the estimation of the frequency, amplitude, and phase of a sinusoidal signal after one-bit quantization. The estimator requires knowledge of the quantizer threshold but does not require information about the noise probability density function (PDF), possibly affecting measurement results. The threshold must be different from zero for estimating the amplitude but not necessarily for estimating the frequency or the phase. The algorithm requires an irrational value of the ratio between the sinusoidal frequency and the sampling rate. The procedure allows also the estimation of the noise variance and the cumulative distribution function (CDF). Theoretical properties are analyzed, and experimental results are provided to validate the estimation procedure.

Spectrum Sensing With Non-Gaussian Noise Over Multi-Path Fading Channels Towards Smart Cities With IoT

As the limited communication spectrum can not meet the demand of the exponential growth of intelligent connected devices in the internet of things(IoT) and typical smart city applications, in this paper, we propose a tractable spectrum sensing method based on Rao detection over non-Gaussian noise, such as generalized Gaussian noise(GGN), Gaussian mixture noise(GMN) and symmetric alpha-stable distribution ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\text{S}\alpha \text{S}$ </tex-math></inline-formula> ) noise, multi-path fading channels environment to alleviate the issue of spectrum scarcity. In this method, there are unknown parameters in the multi-path fading channels. When the probability density function (P.D.F.) of non-Gaussian noise has a closed-form expression, the spectrum sensing method based on Rao detection is used. Otherwise the P.D.F. for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\text{S}\alpha \text{S}$ </tex-math></inline-formula> noise is estimated firstly by using non-parametric kernel estimation method, which addresses the issue that <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\text{S}\alpha \text{S}$ </tex-math></inline-formula> noise has no closed-form P.D.F. expression, and then the performance of spectrum sensing is derived based on the theory of Rao detection in multi-path fading channels over typical smart city applications. Simulation results show that the accuracy of estimated P.D.F. for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\text{S}\alpha \text{S}$ </tex-math></inline-formula> noise and the performance of spectrum sensing under different <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\alpha $ </tex-math></inline-formula> values over indoor, outdoor, and vehicle fading channels environment.

Idejni projekat generatora analognog diskretnog uniformnog šuma

The paper presents the basic idea of the construction and a practical realization of a novel analog discrete uniform noise generator. The source of noise is a carbon resistor, the noise signal is linearly amplified and its value is limited in the finite small range [xg, xg]. The probability density function (PDF) of the carbon resistor thermal noise in that region is square one. By narrowing the symmetric allowable gap (interval) around zero, PDF of the noise approaches a uniform distribution. The factor of deviation from the uniform distribution is precisely defined and its value is practically negligible. In addition, the paper discusses the application of the proposed dither noise, both in the two-bit and in the multi-bit stochastic digital measurement method (SDMM). It is shown that noise is more suitable for application in multi-bit SDMM, because it is less sensitive to deviations from the uniform distribution.

Elements of Randomized Forecasting and Its Application to Daily Electrical Load Prediction in a Regional Power System

A randomized forecasting method based on the generation of ensembles of entropy-optimal forecasting trajectories is developed. The latter are generated by randomized dynamic regression models containing random parameters, measurement noises, and a random input. The probability density functions of random parameters and measurement noises are estimated using real data within the randomized machine learning procedure. The ensembles of forecasting trajectories are generated by the sampling of the entropy-optimal probability distributions. This procedure is used for the randomized prediction of the daily electrical load of a regional power system. A stochastic oscillatory dynamic regression model is designed. One-, two-, and three-day forecasts of the electrical load are constructed, and their errors are analyzed.

Probability density function of ocean noise based on a variational Bayesian Gaussian mixture model.

Extensive ocean noise records have kurtoses markedly different from the Gaussian distribution and therefore exhibit non-Gaussianity, which influences the performance of many sonar signal processing methods. To model the amplitude distribution, this paper studies a Bayesian Gaussian mixture model (BGMM) and its associated learning algorithm, which exploits the variational inference method. The most compelling feature of the BGMM is that it automatically selects a suitable number of effective components and then can approximate a sophisticated distribution in practical applications. The probability density functions (PDFs) of three types of noise in different frequency bands collected in the South China Sea-ambient noise, ship noise, and typhoon noise-are modeled and the goodness of fit is examined by applying the one-sample Kolmogorov-Smirnov test. The results demonstrate that: (i) Ambient noise in the low-frequency band may be slightly non-Gaussian, ship noise in each considered band is apparently non-Gaussian, and typhoons affect the noise in the low-frequency band to make it apparently non-Gaussian, while the noise in the high-frequency band is less affected and appears to be Gaussian. (ii) BGMM has higher goodness of fit than the Gaussian or Gaussian mixture model. (iii) In the non-Gaussian case, despite some components having small mixing coefficients, they are of great significance for describing the PDF.