Zero-mean Gaussian Noise Research Articles

Joint Detection of Bivariate Gaussian Noise Sequences

A method is proposed for joint detection of independent, identically-distributed (IID) zero-mean bivariate Gaussian noise sequences. Received sequences are used to estimate ratio statistics that span a three-dimensional basis, and deviations from an asymptotic form are used to generate the final detection statistic. Sensitivities of the measure to statistical deviations are shown. The technique is applied to radiometric detection and is shown to achieve performance close to ideal energy detection, but without requirements for noise estimation.

Derivation of the Bias of the Normalized Sample Covariance Matrix in a Heterogeneous Noise With Application to Low Rank STAP Filter

In a previous work, we have developed a low-rank (LR) spatio-temporal adaptive processing (STAP) filter when the disturbance is modeled as the sum of a low-rank spherically invariant random vector (SIRV) clutter and a zero-mean white Gaussian noise. This LR-STAP filter is built from the normalized sample covariance matrix (NSCM) and exhibits good robustness properties to secondary data contamination by target components. In this correspondence, we derive the bias of the NSCM with this noise model. We show that the eigenvectors estimated from the NSCM are unbiased. The new expressions of the expectation of NSCM eigenvalues are also given. From these results, we also show that the estimate of the clutter subspace projector based on the NSCM used in our LR-STAP is a consistent estimate of the true one. Results on numerical data validates the theoretical approach.

IMPROVED ANTENNA ARRAY ADAPTIVE BEAMFORMING WITH LOW SIDE LOBE LEVEL USING A NOVEL ADAPTIVE INVASIVE WEED OPTIMIZATION METHOD

An improved adaptive beamforming technique of antenna arrays is introduced. The technique is implemented by using a novel Invasive Weed Optimization (IWO) variant called Adaptive Dispersion Invasive Weed Optimization (ADIWO) where the seeds produced by a weed are dispersed in the search space with standard deviation specifled by the fltness value of the weed. The adaptive seed dispersion makes the ADIWO converge faster than the conventional IWO. This behavior is verifled by applying both the ADIWO and the conventional IWO on well-known test functions. The ADIWO method is utilized here as an adaptive beamformer that makes a uniform linear antenna array steer the main lobe towards the direction of arrival (DoA) of a desired signal, form nulls towards the respective DoA of several interference signals and achieve low side lobe level (SLL). The proposed ADIWO based beamformer is compared to a Particle Swarm Optimization (PSO) based beamformer and a well known beamforming method called Minimum Variance Distortionless Response (MVDR). Several cases have been studied with difierent number of interference signals and difierent power level of additive zero-mean Gaussian noise. The results show that the ADIWO provides su-cient steering ability regarding the main lobe and the nulls, works faster than the PSO and achieves better SLL than the PSO and MVDR.

Deterministic regression methods for unbiased estimation of time-varying autoregressive parameters from noisy observations

A great deal of interest has been paid to autoregressive parameter estimation in the noise-free case or when the observation data are disturbed by random noise. Tracking time-varying autoregressive (TVAR) parameters has been also discussed, but few papers deal with this issue when there is an additive zero-mean white Gaussian measurement noise. In this paper, one considers deterministic regression methods (or evolutive methods) where the TVAR parameters are assumed to be weighted combinations of basis functions. However, the additive white measurement noise leads to a weight-estimation bias when standard least squares methods are used. Therefore, we propose two alternative blind off-line methods that allow both the variance of the additive noise and the weights to be estimated. The first one is based on the errors-in-variable issue whereas the second consists in viewing the estimation issue as a generalized eigenvalue problem. A comparative study with other existing methods confirms the effectiveness of the proposed methods.

Performance Predictions for Parameter Estimators That Minimize Cost-Functions Using Wirtinger Calculus With Application to CM Blind Equalization

We present calculations of the first-order bias and first-order variance of parameters which are estimated batch-wise from the global minimum of a cost-function of complex-valued signals embedded in zero-mean Gaussian noise. The derivation involves the calculation of the multidimensional Taylor series of the non-analytic cost-function up to third order using the elegant Wirtinger calculus. Whereas closed-form expressions for the variance can be obtained straightforwardly from a second-order Taylor series, and have been presented in various other contexts, an exact expression for the bias cannot be derived, in general. In this paper, we propose approximate expressions for the first-order bias and confirm them in a comparison of results from extensive analytical calculations with results from Monte Carlo (MC) simulations for the statistical efficiency of a batch-processing blind equalizer using the constant-modulus (CM) criterion. We study the equalization of independent and identically distributed (i.i.d.) random symbols and obtain asymptotic (for large batch size) expressions for the averages of the bias and variance over zero-mean random (real-valued) signals of binary phase shift keying (BPSK), and (complex-valued) signals of M-ary PSK modulation (M >; 2) . Finally, we compare the statistical efficiency of the CM estimator with the one of the maximum likelihood (ML) blind estimation of the path parameters and equalized symbols with CM constraint.

Transition points in the capacity-achieving distribution for the peak-power limited AWGN and free-space optical intensity channels

The capacity-achieving input distribution for many channels like the additive white Gaussian noise (AWGN) channel and the free-space optical intensity (FSOI) channel under the peak-power constraint is discrete with a finite number of mass points. The number of mass points is itself a variable, and figuring it out is a part of the optimization problem. We wish to understand the behavior of the optimal input distribution at the transition points where the number of mass points changes. To this end, we give a new set of necessary and sufficient conditions at the transition points, which offer new insights into the transition and make the computation of the optimal distribution easier. For the real AWGN channel case, we show that for the zero-mean unit-variance Gaussian noise, the peak amplitude A of 1.671 and 2.786 mark the points where the binary and ternary signaling, respectively, are no longer optimal. For the FSOI channel, we give transition points where binary gives way to ternary, and in some cases where ternary gives way to quaternary, in the presence of the peak-power constraint and with or without the average-power constraint.

Robust Scheme for Detection of an Expanding Moving Object Using a Facet-Based Model in Infrared Imaging

In detection and tracking of a small or large moving object in infrared (IR) imaging systems, it is necessary to perform analysis of the object in real time. The authors proposed the facet-based detection scheme for a small moving object with zero-mean Gaussian noise in previous research. However, it is difficult to detect larger moving objects using the facet-based model because the kernel size in the facet-based model is 5×5 pixels. In this article, the authors propose a robust detection scheme using the facet-based model in IR for larger moving objects. A new condition for the object is proposed for the robust facet-based detection of a larger object with zero-mean Gaussian noise. In the proposed algorithm, first, we extract a mean of image intensity from the center of the facet in the region of interest (ROI) of the first frame. Second, we apply the facet-based model to the same positioned pixel in a subsequent frame. The pixels are detected from the maximum extreme condition. The pixels are detected from the maximum extreme condition. The experimental results show that the proposed algorithm is efficient and robust.

Hyperplane-Based Vector Quantization for Distributed Estimation in Wireless Sensor Networks

This paper considers distributed estimation of a vector parameter in the presence of zero-mean additive multivariate Gaussian noise in wireless sensor networks. Due to stringent power and bandwidth constraints, vector quantization is performed at each sensor to convert its local noisy vector observation into one bit of information, which is then forwarded to a fusion center where a final estimate of the vector parameter is obtained. Within such a context, this paper focuses on a class of hyperplane-based vector quantizers which linearly convert the observation vector into a scalar by using a compression vector and then carry out a scalar quantization. It is shown that the key of the vector quantization design is to find a compression vector for each sensor. Under the framework of the Cramer-Rao bound (CRB) analysis, the compression vector design problem is formulated as an optimization problem that minimizes the trace of the CRB matrix. Such an optimization problem is extensively studied. In particular, an efficient iterative algorithm is developed for the general case, along with optimal and near-optimal solutions for some specific but important noise scenarios. Performance analysis and simulation results are carried out to illustrate the effectiveness of the proposed scheme.

The Estimated Signal Parameter Detector: Incorporating Signal Parameter Statistics Into the Signal Processor

Acoustic propagation through a time-varying and spatially varying environment can produce variations in the received signal over multiple observations, possibly degrading receiver performance. This paper presents a signal processing structure that utilizes knowledge of received signal statistics to recoup lost performance. Recent research has shown that received signal parameter statistics can be calculated using Monte Carlo simulation and knowledge of ocean environment properties and processes. The processor possesses an estimator-correlator structure, and is referred to in this paper as the estimated signal parameter detector (ESPD). To demonstrate ESPD performance, the derivation is implemented to distinguish between monotone sinusoids with Gaussian-distributed amplitudes with identical means but different variances, embedded in zero-mean white Gaussian noise. In general, the amplitude distributions can possess any form and the noise distribution must belong to a general class of probability density functions (pdfs). The present assumptions allow for analytical results, and performance of the ESPD is seen to depend upon the signal-to-noise ratio (SNR) as well as the difference between the amplitude variances. Larger SNR and greater difference in amplitude variance result in better receiver performance, eventually leading to an asymptotic performance bound prediction.

Optimum estimate of the position of zero-mean Gaussian pulse noise on images

Algorithms have been developed for estimating the spatial position of pulse noise with a Gaussian distribution centered and truncated at zero of the brightness on images. The optimality criteria of the algorithms are the unconditional pass probability for a fixed unconditional false-alarm probability and vice versa. Results are presented of numerical studies of the algorithms, which show higher quality by comparison with procedures known earlier.

A Simplified Design Model for Random Process Variability

A method for modeling the effects of random process variation through measured transistor current is introduced. The methodology culminates by modeling random current variation at a given operating point above threshold as a zero-mean Additive White Gaussian Noise (AWGN) current source with a standard deviation dependent on nominal operating current and design variables, the transistor sizes, W and L, and the transistor operating points, V <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">gs</sub> and V <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ds</sub> . The model has a simple posynomial form and is accurate compared to measured data within an RMS error of 5.4% for narrow-, wide-, short-, and long-channel transistors. The model's simplified form bridges the gap between existing statistical methods and circuit design. The efficacy of the model in the circuit design space will also be presented. In the analog domain, we will show that the contributions of this model can be used as a replacement for Monte Carlo methods for calculating circuit voltage and current variances. In the digital space, we will investigate the calculation of timing delay distributions. Results calculated via an alpha-power (α <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</sub> ) law model for average current show that scaling the supply voltage by γ results in an approximately 1/γ scaling in the path-delay standard deviation for a 65-nm process.

Stabilised weighted linear prediction

Weighted linear prediction (WLP) is a method to compute all-pole models of speech by applying temporal weighting of the square of the residual signal. By using short-time energy (STE) as a weighting function, this algorithm was originally proposed as an improved linear predictive (LP) method based on emphasising those samples that fit the underlying speech production model well. The original formulation of WLP, however, did not guarantee stability of all-pole models. Therefore, the current work revisits the concept of WLP by introducing a modified short-time energy function leading always to stable all-pole models. This new method, stabilised weighted linear prediction (SWLP), is shown to yield all-pole models whose general performance can be adjusted by properly choosing the length of the STE window, a parameter denoted by M. The study compares the performances of SWLP, minimum variance distortionless response (MVDR), and conventional LP in spectral modelling of speech corrupted by additive noise. The comparisons were performed by computing, for each method, the logarithmic spectral differences between the all-pole spectra extracted from clean and noisy speech in different segmental signal-to-noise ratio (SNR) categories. The results showed that the proposed SWLP algorithm was the most robust method against zero-mean Gaussian noise and the robustness was largest for SWLP with a small M-value. These findings were corroborated by a small listening test in which the majority of the listeners assessed the quality of impulse-train-excited SWLP filters, extracted from noisy speech, to be perceptually closer to original clean speech than the corresponding all-pole responses computed by MVDR. Finally, SWLP was compared to other short-time spectral estimation methods (FFT, LP, MVDR) in isolated word recognition experiments. Recognition accuracy obtained by SWLP, in comparison to other short-time spectral estimation methods, improved already at moderate segmental SNR values for sounds corrupted by zero-mean Gaussian noise. For realistic factory noise of low pass characteristics, the SWLP method improved the recognition results at segmental SNR levels below 0 dB.

Evaluation of Stochastic Differential Equation Approximation of Ion Channel Gating Models

Fox and Lu derived an algorithm based on stochastic differential equations for approximating the kinetics of ion channel gating that is simpler and faster than "exact" algorithms for simulating Markov process models of channel gating. However, the approximation may not be sufficiently accurate to predict statistics of action potential generation in some cases. The objective of this study was to develop a framework for analyzing the inaccuracies and determining their origin. Simulations of a patch of membrane with voltage-gated sodium and potassium channels were performed using an exact algorithm for the kinetics of channel gating and the approximate algorithm of Fox & Lu. The Fox & Lu algorithm assumes that channel gating particle dynamics have a stochastic term that is uncorrelated, zero-mean Gaussian noise, whereas the results of this study demonstrate that in many cases the stochastic term in the Fox & Lu algorithm should be correlated and non-Gaussian noise with a non-zero mean. The results indicate that: (i) the source of the inaccuracy is that the Fox & Lu algorithm does not adequately describe the combined behavior of the multiple activation particles in each sodium and potassium channel, and (ii) the accuracy does not improve with increasing numbers of channels.

Image Sequence Denoising via Sparse and Redundant Representations

In this paper, we consider denoising of image sequences that are corrupted by zero-mean additive white Gaussian noise. Relative to single image denoising techniques, denoising of sequences aims to also utilize the temporal dimension. This assists in getting both faster algorithms and better output quality. This paper focuses on utilizing sparse and redundant representations for image sequence denoising, extending the work reported in. In the single image setting, the K-SVD algorithm is used to train a sparsifying dictionary for the corrupted image. This paper generalizes the above algorithm by offering several extensions: i) the atoms used are 3-D; ii) the dictionary is propagated from one frame to the next, reducing the number of required iterations; and iii) averaging is done on patches in both spatial and temporal neighboring locations. These modifications lead to substantial benefits in complexity and denoising performance, compared to simply running the single image algorithm sequentially. The algorithm's performance is experimentally compared to several state-of-the-art algorithms, demonstrating comparable or favorable results.

Tracking Analysis of Complex Adaptive IIR Notch Filter for a Linear Chirp Signal

This paper analyzes frequency tracking characteristics of a complex-coefficient adaptive infinite impulse response (IIR) notch filter with a simplified gradient-based algorithm. The input signal to the complex notch filter is a complex linear chirp embedded in a complex zero-mean white Gaussian noise. The analysis starts with derivation of a first-order real-coefficient difference equation with respect to steady-state instantaneous frequency tracking error. Closed-form expression for frequency tracking mean square error (MSE) is then derived from the difference equation. Lastly, closed-form expressions for optimum notch bandwidth coefficient and step size constant that minimize the frequency tracking MSE are derived. Computer simulations are presented to validate the analysis.

A Spatiotemporal Framework for Estimating Trial-to-Trial Amplitude Variation in Event-Related MEG/EEG

A spatiotemporal framework for estimating trial-to-trial variability in evoked response (ER) data is presented. Spatial and temporal bases capture the aspects of the response that are consistent across trials, while the basis expansion coefficients represent the variable components of the response. We focus on the simplest case of constant spatiotemporal response shape and varying amplitude across trials. Two different constraints on the amplitude evolution are employed to effectively integrate the individual responses and improve robustness at low SNR. The linear dynamical system response constraint estimates the current trial amplitude as an unknown constant scaling of the estimate in the previous trial plus zero-mean Gaussian noise with unknown variance. The independent response constraint estimates response amplitudes across trials as independent Gaussian random variables having unknown mean and variance. We develop a generalized expectation-maximization algorithm to obtain the maximum-likelihood (ML) estimates of the signal waveform, noise covariance matrix, and unknown constraint parameters. ML source localization is achieved by scanning the likelihood over different sets of spatial bases. We demonstrate the variability estimation and source localization effectiveness of the proposed algorithms using both real and simulated ER data.

Detection of Edges in Spectral Data III—Refinement of the Concentration Method

Edge detection from Fourier spectral data is important in many applications including image processing and the post-processing of solutions to numerical partial differential equations. The concentration method, introduced by Gelb and Tadmor in 1999, locates jump discontinuities in piecewise smooth functions from their Fourier spectral data. However, as is true for all global techniques, the method yields strong oscillations near the jump discontinuities, which makes it difficult to distinguish true discontinuities from artificial oscillations. This paper introduces refinements to the concentration method to reduce the oscillations. These refinements also improve the results in noisy environments. One technique adds filtering to the concentration method. Another uses convolution to determine the strongest correlations between the waveform produced by the concentration method and the one produced by the jump function approximation of an indicator function. A zero crossing based concentration factor, which creates a more localized formulation of the jump function approximation, is also introduced. Finally, the effects of zero-mean white Gaussian noise on the refined concentration method are analyzed. The investigation confirms that by applying the refined techniques, the variance of the concentration method is significantly reduced in the presence of noise.

Systematic errors and susceptibility to noise of four methods for calculating anatomical dead space from the CO2 expirogram

Anatomical dead space is usually measured using the Fowler equal area method. Alternative methods include the Hatch, Cumming, and Bowes methods, in which first, second, and third order polynomials, respectively, fitted to an expired CO2 volume vs expired volume curve, intercept the x-axis at the anatomical dead space. This study assessed systematic errors and susceptibility to noise of the Fowler, Hatch, Cumming, and Bowes dead spaces calculated over 40-80% of the CO2 expirogram. Simulated CO2 expirograms with 220 ml anatomical dead space and varying alveolar plateau slopes were generated digitally and zero-mean Gaussian noise added. CO2 expirograms were recorded in 10 anaesthetized human subjects. Anatomical dead space was calculated by the Fowler, Hatch, Cumming, and Bowes methods. The Fowler, Hatch, Cumming, and Bowes methods displayed systematic biases of -1.8%, 13.2%, 2.4%, and -1.3%, respectively, at a normalized simulated alveolar plateau slope of 1.6 litre(-1). At a noise level of 0.0066 vol/vol, the standard deviations of recovered simulated dead spaces were 70.6, 1.8, 2.4, and 3.7 ml, respectively. The Hatch, Cumming, and Bowes methods applied to human expirograms differed significantly from that of Fowler by 13, -4, and -11 ml, respectively. In the human study, the Hatch and Cumming methods yielded the lowest intra-individual dead space variability. The Fowler method shows greatest susceptibility to measurement noise and the Hatch method exhibits the largest systematic error. The Cumming method, which exhibits both low bias and low noise susceptibility, is preferred for estimating anatomical dead space from CO2 expirograms.

Extraction of Ocean Wave Spectra From Simulated Noisy Bistatic High-Frequency Radar Data

An algorithm is developed for the inversion of bistatic high-frequency (HF) radar sea echo to give the nondirectional wave spectrum. The bistatic HF radar second-order cross section of patch scattering, consisting of a combination of four Fredholm-type integral equations, contains a nonlinear product of ocean wave directional spectrum factors. The energy inside the first-order cross section is used to normalize this integrand. The unknown ocean wave spectrum is represented by a truncated Fourier series. The integral equation is then converted to a matrix equation and a singular value decomposition (SVD) method is invoked to pseudoinvert the kernel matrix. The new algorithm is verified with simulated radar Doppler spectrum for varying water depths, wind velocities, and radar operating frequencies. To make the simulation more realistic, zero-mean Gaussian noise from external sources is also taken into account

Bandwidth-constrained distributed estimation for wireless sensor networks-part II: unknown probability density function

Wireless sensor networks (WSNs) deployed to perform surveillance and monitoring tasks have to operate under stringent energy and bandwidth limitations. These motivate well distributed estimation scenarios where sensors quantize and transmit only one, or a few bits per observation, for use in forming parameter estimators of interest. In a companion paper, we developed algorithms and studied interesting tradeoffs that emerge even in the simplest distributed setup of estimating a scalar location parameter in the presence of zero-mean additive white Gaussian noise of known variance. Herein, we derive distributed estimators based on binary observations along with their fundamental error-variance limits for more pragmatic signal models: i) known univariate but generally non-Gaussian noise probability density functions (pdfs); ii) known noise pdfs with a finite number of unknown parameters; iii) completely unknown noise pdfs; and iv) practical generalizations to multivariate and possibly correlated pdfs. Estimators utilizing either independent or colored binary observations are developed and analyzed. Corroborating simulations present comparisons with the clairvoyant sample-mean estimator based on unquantized sensor observations, and include a motivating application entailing distributed parameter estimation where a WSN is used for habitat monitoring