White Gaussian Noise Model Research Articles

Lower bounds for the trade-off between bias and mean absolute deviation

In nonparametric statistics, rate-optimal estimators typically balance bias and stochastic error. The recent work on overparametrization raises the question whether rate-optimal estimators exist that do not obey this trade-off. In this work we consider pointwise estimation in the Gaussian white noise model with regression function f in a class of β-Hölder smooth functions. Let ’worst-case’ refer to the supremum over all functions f in the Hölder class. It is shown that any estimator with worst-case bias ≲n−β/(2β+1)≕ψn must necessarily also have a worst-case mean absolute deviation that is lower bounded by ≳ψn. To derive the result, we establish abstract inequalities relating the change of expectation for two probability measures to the mean absolute deviation.

Multi-task learning with self-learning weight for image denoising

BackgroundImage denoising technology removes noise from the corrupted image by utilizing different features between image and noise. Convolutional neural network (CNN)-based algorithms have been the concern of the recent progress on diverse image restoration problems and become an efficient solution in image denoising.ObjectiveAlthough a quite number of existing CNN-based image denoising methods perform well on the simplified additive white Gaussian noise (AWGN) model, their performance often degrades severely on the real-world noisy images which are corrupted by more complicated noise.MethodsIn this paper, we utilized the multi-task learning (MTL) framework to integrate multiple loss functions for collaborative training of CNN. This approach aims to improve the denoising performance of CNNs on real-world images with non-Gaussian noise. Simultaneously, to automatically optimize the weights of individual sub-tasks within the MTL framework, we incorporated a self-learning weight layer into the CNN.ResultsExtensive experiments demonstrate that our approach effectively enhances the denoising performance of CNN-based image denoising algorithms on real-world images. It reduces excessive image smoothing, improves quantitative metrics, and enhances visual quality in the restored images.ConclusionOur method shows the effectiveness of the improved performance of denoising CNNS for real-world image denoising processing.

Multi-objective seismic optimization and evaluation of core-damper-frame tall buildings considering SSI effect

To control the seismic response of the core-frame (CF) structure and attenuate the stress level of the core tube, a novel core-damper-frame (CDF) system is proposed by decoupling the perimeter frame and the central core with flexible damper connectors. In this way, the large inertia force of the perimeter frame cannot be fully transmitted to the core, and the stress of the core is mitigated. In the meantime, the damper connectors would absorb a considerable amount of seismic energy, protecting the perimeter frame as the relative displacement between the two subsystems (perimeter frame and core tube) occurs. By incorporating the soil-structure-interaction (SSI) and the filtered Gaussian white-noise excitation model, the "excitation-soil-core-damper-frame" augmented system is established. Combing the augmented system and the genetic algorithm, a stochastic optimization procedure of the CDF considering the SSI effect is proposed. The optimization aims to minimize a series of performance objectives including displacement, acceleration, shear force, and overturning moment of the CDF system. The conflicting relationships between the multiple objectives are discussed; the influence of the SSI effect and the excitation parameters are evaluated. From an illustrative 40-story CDF structural example, the system shows great superiority in controlling the higher mode vibrations, and could significantly reduce the peak core acceleration (by 54.44 %), frame acceleration (by 56.48 %), core drift (by 28.83 %), frame drift (by 39.28 %), core shear (by 83.67 %), and frame moment (by 22.41 %), as compared to its conventional CF counterpart. In special, over 50 % of the input seismic energy is dissipated by the damper connectors in the CDF system.

Adding realistic noise models to synthetic ground‐penetrating radar data

ABSTRACTCost‐effective computing capabilities have paved the road for the use of numerical modelling to develop advanced methods and applications of ground‐penetrating radar (GPR). Realistic synthetic data and the corresponding modelling techniques, respectively, should consider all subsurface and above‐ground aspects that influence GPR wave propagation and the characteristics of recorded signals. Critical aspects that can be realized in modern GPR modelling tools include heterogeneous and frequency‐dependent material properties, complex structures and interface geometries as well as three‐dimensional antenna models, including the interaction between the antenna and the subsurface. However, realistic noise related to the electronic components of a GPR system or ambient electromagnetic noise is often not considered, or simplified by assuming a white Gaussian noise model which is added to the modelled data. We present an approach to include realistic noise scenarios as typically observed in GPR field data into the flow of modelling synthetic GPR data. In our approach, we extract the noise from recorded GPR traces and add it to the modelled GPR data via a convolution‐based process. We illustrate our methodology using a modelling exercise, where we contaminate a synthetic two‐dimensional GPR dataset with frequency‐dependent noise recorded in an urban environment. Comparing our noise‐contaminated synthetic data with field data recorded in a similar environment illustrates that our method allows the generation of synthetic GPR with realistic noise characteristics and further highlights the limitations of assuming pure white Gaussian noise models.

Enhancement the Time Variation of Channel Gain by Realize Doppler Spectrum Using Filtered White Gaussian Noise (FWGN) Model

Enhancement the Time Variation of Channel Gain by Realize Doppler Spectrum Using Filtered White Gaussian Noise (FWGN) Model

Channel Estimation for Indoor Massive MIMO Visible Light Communication With Deep Residual Convolutional Blind Denoising Network

Massive multiple-input multiple-output (m-MIMO) visible light communication (VLC) has been considered as one of the promising components in the optical wireless communication system. However, the envisioned benefits may be limited due to the high computational complexity to estimate accurate channel state information (CSI). Besides, several propagation paths in the practical communication channels induces significant difficulties. In this paper, a deep residual convolutional blind denoising network (ResCBDNet) has been proposed to predict more realistic channels in indoor m-MIMO VLC communication system. Unlike the other convolutional networks which may overfit on the simplified additive white gaussian noise model, the ResCBDNet exploits noise level estimation subnetwork to improve the generalization ability to real noise as well as interactively reduce the noise in the channel matrix by adjusting the noise level map. More specifically, we have investigated to optimize the ResCBDNet over a large range of SNRs. During the simulation, the sparse channel matrix has been treated as a two dimensional natural image. Simulation results validate that the proposed network is very promising in practical channel estimation for the indoor m-MIMO VLC communication system and outperforms the state-of-the-art channel estimators in terms of normalized mean square error and peak signal-to-noise ratio.

Minimax detection of localized signals in statistical inverse problems

Abstract We investigate minimax testing for detecting local signals or linear combinations of such signals when only indirect data are available. Naturally, in the presence of noise, signals that are too small cannot be reliably detected. In a Gaussian white noise model, we discuss upper and lower bounds for the minimal size of the signal such that testing with small error probabilities is possible. In certain situations we are able to characterize the asymptotic minimax detection boundary. Our results are applied to inverse problems such as numerical differentiation, deconvolution and the inversion of the Radon transform.

A Unified, SNR-Aware SC-LDPC Code Design With Applications to Magnetic Recording

Spatially coupled (SC)-low-density parity-check (LDPC) codes are known to have outstanding error-correction performance and low decoding latency, which make them an excellent choice for high-density magnetic recording (MR) technologies. Whereas previous works on LDPC and SC-LDPC codes mostly take either an asymptotic or a finite-length design approach, we propose a unified framework for jointly optimizing the codes’ thresholds and cycle counts to address both regimes. We focus on circulant-based (CB) SC-LDPC code family as a representative, high-performance exemplar of structured SC-LDPC codes. The framework is based on efficient traversal and pruning of the code search space, building on the fact that the performance of a CB SC-LDPC code depends on some characteristics of the code’s partitioning matrix, which by itself is much smaller than the code’s full parity-check matrix. We then propose an algorithm that traverses all non-equivalent partitioning matrices and outputs a list of codes, each offering an attractive point on the trade-off between asymptotic and finite-length performance. Our simulations show that our framework results in SC-LDPC codes that outperform the state-of-the-art constructions, over both additive white Gaussian noise (AWGN) and partial response (PR) channel models, and that it offers the flexibility to choose low-signal-to-noise ratio (SNR), high-SNR, or in- between SNR region considering system requirements, e.g., that of the MR device.

Data-driven probabilistic quantification and assessment of the prediction error model in damage detection applications

Advances in computer hardware and sensor technologies have led to a surge in the use of data-driven modeling and machine learning for structural engineering applications, with Structural Health Monitoring (SHM) being one of them. Despite considerable interest, it remains a research topic due to the difficulty in accurately quantifying aleatoric and epistemic uncertainty in SHM systems. Sources of uncertainty are related to operational and environmental variability, as well as measurement noise and the model prediction error associated with the data used to train damage identification algorithms. In this work, the authors aim to explicitly quantify the statistical structure of model prediction error and assess its influence on the detection performance of strain-based SHM architectures under the existence of aleatoric variability. A structural beam, subjected to probabilistic static loading is used as the reference structure and strain measurements as the damage-sensitive features. Model prediction error is quantified explicitly using robust statistical tools through available laboratory observations and synthetic (Finite Element) data. Monte Carlo simulations enabled the forward propagation of uncertainty to the feature space to generate training data for three binary detectors (Likelihood Ratio Test, Quadratic Discriminant Analysis and Mahalanobis Distance), based on statistical pattern recognition. Detection performance was compared between the explicitly quantified prediction model error and the commonly assumed white Gaussian noise model, showcasing the influence of systematic error (bias) and correlation on the robustness of an SHM system using real-world data.

Shot-noise-limited photon-counting precoding scheme for MIMO ultraviolet communication in atmospheric turbulence.

Photon counting is a promising solution to detecting low-power optical signals for ultraviolet (UV) communications in the forthcoming sixth-generation (6G) network. Different from the conventional additive white Gaussian noise (AWGN) model, the discrete signal-dependent Poisson shot noise poses challenges to the signal processing of photon-counting systems. In this paper, a joint design of precoder and equalizer is proposed for photon-counting multiple-input multiple-output (PhC-MIMO) UV systems. To circumvent the impasse arising from the signal-dependent shot noise, we propose an alternating optimization algorithm based on the minimum mean squared error (MMSE) criterion. The algorithm decomposes the joint design into convex subproblems solved in an alternating manner, and guarantees at least a stationary point solution. Numerical results corroborate that the proposed system exhibits robustness to turbulence fading and offers high throughput while mitigating the adverse effect of background radiation noise. Specifically, the 32 × 8 system can achieve a bit error rate (BER) of 10-5 at the signal energy of -154.0 dBJ per bit under strong Gamma-Gamma turbulence with the scintillation index (S.I.) of 3.

A Novel Approach for Modeling and Digital Generation of RF Signals Distorted by Bandlimited Phase Noise

In radar and communication systems, phase noise is one of the main causes of performance degradation. Phase noise increases the uncertainty in radar measurements and limits the achievable data rates in communication systems. When radio frequency (RF) signals distorted by bandlimited phase noise are modeled, the phase noise power spectral density (PSD) is usually approximated by a scalar root-mean-square (RMS) phase error derived from a simple integration of the PSD. This disregards the close-to-carrier noise excess. In this article, we show that this convention simplification describes the real behavior of phase noise inadequately. In addition, we present a simulation of realistic phase noise behavior. The novel additive colored noise (ACN) model requires a representative phase noise PSD of the phase-locked loop (PLL) signal generator phase noise to be modeled. The developed ACN approach is validated by comparing the measured PSD of different PLL signal generators with the respective simulated RF signals distorted by phase noise. As a simple metric for assessing the quality of the phase noise models, we use the influence of phase noise on carrier frequency estimation. It is shown that the novel approach shows a significant improvement in the agreement between the simulated and measured precision compared to standard additive white Gaussian noise (AWGN) models. The results show that advanced phase noise models, as proposed in this paper, are necessary to adequately model and predict the performance of radar and communication systems.

Detailed Analysis and Modeling of Phase Noise and Systematic Phase Distortions in FMCW Radar Systems

In coherent continuous-wave (CW) radar systems, such as frequency-modulated CW (FMCW) radar systems, measurement precision is distorted by phase noise and systematic phase errors during radar signal generation. Unfortunately, to date, these phase distortions have typically been modeled based on many simplifications, often leading to overoptimistic predictions of radar performance. For example, phase noise is regularly considered based on additive white Gaussian noise (AWGN) models, ignoring its colored spectral property. Systematic errors are also frequently not properly separated from stochastic distortions, or not precisely measured and modeled. To overcome these issues, an advanced CW radar and frequency synthesizer model is proposed, analyzed, and experimentally verified in this paper. It is shown how systematic phase errors can be measured and how the influence of phase distortions on radar measurement can be accurately predicted. The proposed modeling approach was investigated for range correlation and range precision. The key metrics are the phase noise power spectral density (PSD) for range correlation and the variance of the frequency estimate for range precision. A 24-GHz FMCW radar system was used for the experimental verification. Long-range radar targets with distances of 38 to 880 m were created with an analog fiber optical link. Therefore, a precise and systematic evaluation of all effects was possible. The proposed phase distortion modeling approach realistically represented the range-dependent radar measurement effects. This enables the most precise simulation of and prediction for FMCW radar performance to date.

End-to-End Learning of a Constellation Shape Robust to Channel Condition Uncertainties

Vendor interoperability is one of the desired future characteristics of optical networks. This means that the transmission system needs to support a variety of hardware with different components, leading to system uncertainties throughout the network. For example, uncertainties in signal-to-noise ratio and laser linewidth can negatively affect the quality of transmission within an optical network due to e.g. mis-parametrization of the transceiver signal processing algorithms. In this paper, we propose to geometrically optimize a constellation shape that is robust to uncertainties in the channel conditions by utilizing end-to-end learning. In the optimization step, the channel model includes additive noise and residual phase noise. In the testing step, the channel model consists of laser phase noise, additive noise and blind phase search as the carrier phase recovery algorithm. Two noise models are considered for the additive noise: white Gaussian noise and nonlinear interference noise model for fiber nonlinearities. The latter models the behavior of an optical fiber channel more accurately because it considers the nonlinear effects of the optical fiber. For this model, the uncertainty in the signal-to-noise ratio can be divided between amplifier noise figures and launch power variations. For both noise models, our results indicate that the learned constellations are more robust to the uncertainties in channel conditions compared to a standard constellation scheme such as quadrature amplitude modulation and standard geometric constellation shaping techniques.

Asymptotic equivalence for nonparametric regression with dependent errors: Gauss–Markov processes

For the class of Gauss–Markov processes we study the problem of asymptotic equivalence of the nonparametric regression model with errors given by the increments of the process and the continuous time model, where a whole path of a sum of a deterministic signal and the Gauss–Markov process can be observed. We derive sufficient conditions which imply asymptotic equivalence of the two models. We verify these conditions for the special cases of Sobolev ellipsoids and Hölder classes with smoothness index \(>1/2\) under mild assumptions on the Gauss–Markov process. To give a counterexample, we show that asymptotic equivalence fails to hold for the special case of Brownian bridge. Our findings demonstrate that the well-known asymptotic equivalence of the Gaussian white noise model and the nonparametric regression model with i.i.d. standard normal errors (see Brown and Low (Ann Stat 24:2384–2398, 1996)) can be extended to a setup with general Gauss–Markov noises.

Distributed nonparametric function estimation: Optimal rate of convergence and cost of adaptation

Distributed minimax estimation and distributed adaptive estimation under communication constraints for Gaussian sequence model and white noise model are studied. The minimax rate of convergence for distributed estimation over a given Besov class, which serves as a benchmark for the cost of adaptation, is established. We then quantify the exact communication cost for adaptation and construct an optimally adaptive procedure for distributed estimation over a range of Besov classes. The results demonstrate significant differences between nonparametric function estimation in the distributed setting and the conventional centralized setting. For global estimation, adaptation in general cannot be achieved for free in the distributed setting. The new technical tools to obtain the exact characterization for the cost of adaptation can be of independent interest.

Minimax nonparametric estimation of pure quantum states

In classical statistics, Pinsker’s theorem provides an exact asymptotic minimax bound in nonparametric estimation, improving upon optimal rates of convergence results. We obtain a quantum version of the theorem by establishing asymptotic minimax results for estimation of the displacement vector in a quantum Gaussian white noise model, given by a sequence of shifted vacuum states. Analogous results are then obtained for estimation of a general pure state from an ensemble of identically prepared, independent quantum systems, using the recently established local asymptotic equivalence to the quantum Gaussian white noise model. Optimality holds with respect to the most fundamental distance measure for quantum states, that is, trace norm distance, and in a true quantum sense, allowing for all possible measurements. Adaptive estimators are also obtained for the above cases. As an application, we obtain asymptotic minimax adaptive estimators for Wigner functions of pure states.

Statistical Destriping of Pushbroom-Type Images Based on an Affine Detector Response

Remote sensing pushbroom-type imaging systems acquire entire columns of an image with a single detector. As a consequence, the miss-calibration of the detectors produces stripes on the image. In this context, this paper introduces a new self-calibration destriping method based on an affine response model for the detectors, called Statistical Affine Destriping (SAD). In contrast, some previous contributions were limited to a purely linear model, while many others only considered an additive structured noise model. It is based on the maximum a posteriori estimation of the gain and offset parameters attached to each detector given the observed image. Simple statistical prior assumptions are adopted: respectively, a Gaussian white noise model for the gains and offsets, and a first-order, homogeneous Markov model for the observed scene. Based on a simplification of the posterior likelihood, we propose a very efficient optimization scheme based on a constrained Majorize-Minimize principle, allowing us to process large dimension images. Moreover, simple empirical rules are given to tune the hyperparameters of the destriping method for high-resolution PLEIADES-type images. Compared to the performance of a destriping method limited to gain correction, we observe that the new version provides reliable results in a wider range of situations. We also extend the method in two directions. On the one hand, we consider that some detectors may be atypical, with very high or very low gains or offsets. On the other hand, we extend the method to multispectral image destriping.

Efficient estimation methods for non-Gaussian regression models in continuous time

In this paper, we develop an efficient nonparametric estimation theory for continuous time regression models with non-Gaussian Levy noises in the case when the unknown functions belong to Sobolev ellipses. Using the Pinsker’s approach, we provide a sharp lower bound for the normalized asymptotic mean square accuracy. However, the main result obtained by Pinsker for the Gaussian white noise model is not correct without additional conditions for the ellipse coefficients. We find such constructive sufficient conditions under which we develop efficient estimation methods. We show that the obtained conditions hold for the ellipse coefficients of an exponential form. For exponential coefficients, the sharp lower bound is calculated in explicit form. Finally, we apply this result to signals number detection problems in multi-pass connection channels and we obtain an almost parametric convergence rate that is natural for this case, which significantly improves the rate with respect to power-form coefficients.

Signal-Dependent Noise Estimation for a Real-Camera Model via Weight and Shape Constraints

Most computer vision algorithms require parameter adjustment according to the image noise level. Conventionally, the additive white Gaussian noise (AWGN) model is widely used in most noise estimation algorithms; however, this assumption does not hold in the real world where the noise from cameras is more complex, and it is more appropriate to assume the signal-dependent noise (SDN) model. In this paper, we focus on the SDN model while considering the nonlinear radiometric calibration inside an actual camera, and propose an algorithm to efficiently estimate the noise level function (NLF), which is defined as the noise standard deviation with respect to image intensity. First, the input image is divided into overlapping patches, and noise samples are estimated in the linear transform domain. The confidence levels of the noise samples and the prior of the camera response function are then employed as constraints for the recovery of the NLF. Finally, the noise samples and constraints are represented in a convex optimization problem. The experimental results using both real and synthetic noisy images demonstrate the superiority of the proposed method. In addition, the estimated NLFs are incorporated into two well-known denoising schemes, non-local means and BM3D, and shows significant improvements in denoising SDN-polluted images.

Characterization of a resonant capacitively coupled wireless power transfer system for communication purposes at 6 MHz

This work characterizes a wireless power transmission (WPT) system operating at the 6 MHz frequency band for wireless information transfer purposes. To perform the characterization, the channel is modelled using its 2-port network scattering parameters. Then, a long string of symbols encoded with on–off keying and modified Miller encoding is passed through the channel model at various data rates. The original data is recovered from the channel output signal using a software-defined receiver. Bit error rate curves and eye diagrams are created for each encoding scheme and data rate. To verify the simulated results, the aforementioned data strings are transferred through the actual WPT system and output signals are post-processed in the digital domain with the same software-defined receiver. For comparison purposes, the performance of the WPT channel is compared to that of a conventional wireless communication channel with a flat additive white Gaussian noise model. The simulation and measurement results demonstrate that the provided WPT system can be used for data transfer with an acceptable error rate.