Case Of Additive Noise Research Articles

Approximate maximum likelihood estimators for array processing in multiplicative noise environments

We consider the problem of localizing a source by means of a sensor array when the received signal is corrupted by multiplicative noise. This scenario is encountered, for example, in communications, owing to the presence of local scatterers in the vicinity of the mobile or due to wavefronts that propagate through random inhomogeneous media. Since the exact maximum likelihood (ML) estimator is computationally intensive, two approximate solutions are proposed, originating from the analysis of the high and low signal to-noise ratio (SNR) cases, respectively. First, starting with the no additive noise case, a very simple approximate ML (AML/sub 1/) estimator is derived. The performance of the AML/sub 1/ estimator in the presence of additive noise is studied, and a theoretical expression for its asymptotic variance is derived. Its performance is shown to be close to the Cramer-Rao bound (CRB) for moderate to high SNR. Next, the low SNR case is considered, and the corresponding AML/sub 2/ solution is derived. It is shown that the approximate ML criterion can be concentrated with respect to both the multiplicative and additive noise powers, leaving out a two-dimensional (2-D) minimization problem instead of a four-dimensional (4-D) problem required by the exact ML. Numerical results illustrate the performance of the estimators and confirm the validity of the theoretical analysis.

Statistics of the RSS estimation algorithm for Gaussian measurement noise

The large and small sample properties of the reduced sufficient statistics (RSS) estimator of Kulhavy (1990, 1992) are derived for the nonlinear additive white Gaussian noise measurement model. The RSS algorithm recursively propagates a set of sufficient statistics for a mixture density that approximates the true posterior density of a parameter vector. The joint probability density function for the weighting coefficients of the mixture density is derived for the case of additive white Gaussian noise. Through integration of this density, the estimator bias and mean-squared error are determined. The results are applied to a scalar phase estimation problem in which the sample-averaged statistics are compared with those derived from numerical integration of the density function. The asymptotic bias and variance of the RSS estimator are also derived and compared with simulation results.

Melnikov Processes and Noise-Induced Exits from a Well

For a wide class of near-integrable systems with additive or multiplicative noise the mean zero upcrossing rate for the stochastic system's Melnikov process τu-1, provides an upper bound for the system's mean exit rate, τe-1. Comparisons between τu-1 and τe-1 show that in the particular case of additive white noise this upper bound is weak. For systems excited by processes with tail-limited distributions, the stochastic Melnikov approach yields a simple criterion guaranteeing the nonoccurrence of chaos. This is illustrated for the case of excitation by square-wave, coin-toss dichotomous noise. Finally, we briefly review applications of the stochastic Melnikov approach to a study of the behavior of wind-induced fluctuating currents over a corrugated ocean floor; the snap-trough of buckled columns with continuous mass distribution and distributed random loading; and open-loop control of stochastically excited multistable systems.

Statistical properties of phase-shift algorithms

Statistical properties of phase-shift algorithms are investigated for the case of additive Gaussian intensity noise. Based on a bivariate normal distribution, a generally valid probability-density function for the random phase error is derived. This new description of the random phase error shows properties that cannot be obtained through Gaussian error propagation. The assumption of a normally distributed phase error is compared with the derived probability-density function. For small signal-to-noise ratios the assumption of a normally distributed phase error is not valid. Additionally, it is shown that some advanced systematic-error-compensating algorithms have a disadvantageous effect on the random phase error.

Statistical properties of single-mode oscillations in photorefractive ring resonators

We investigate the statistical properties of the optical field that arises from two-wave mixing in a single-mode unidirectional photorefractive ring resonator. Two cases are considered: additive noise produced by generation-recombination fluctuations in the photorefractive material's ionized donor concentration and multiplicative noise produced by fluctuations in the cavity detuning and losses. Like the homogeneously broadened two-level laser near threshold, the photorefractive resonator has Gaussian and Rayleigh-like intensity distributions in the additive and multiplicative noise cases, respectively. The equations describing the linewidths associated with the intensity fluctuations for the two cases and those describing the linewidth and frequency shift associated with the amplitude fluctuation in the multiplicative noise case also have the same form as the laser equations. The descriptions of the linewidth and frequency shift associated with the amplitude fluctuation in the additive noise case are, in general, different from those of the laser.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Multichannel detection for correlated non-Gaussian random processes based on innovations

The paper addresses the problem of multichannel signal detection in additive correlated non-Gaussian noise using the innovations approach. Although this problem has been addressed extensively for the case of additive Gaussian noise, the corresponding problem for the non-Gaussian case has received limited attention. This is due to the fact that there is no unique specification for the joint probability density function (PDF) of N correlated non-Gaussian random variables. The authors overcome this problem by using the theory of spherically invariant random processes (SIRPs) and derive the innovations-based detector. It is found that the optimal estimators for obtaining the innovations, processes are linear and that the resulting detector is canonical for the class of PDFs arising from SIRPs. The authors also present a performance analysis of the innovations-based detector for the case of a K-distributed SIRP.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Asymptotically robust detection of known signals in nonadditive noise

Robust detection of known weak signals of unknown amplitude is considered for a class of combined additive and nonadditive noise models and an asymptotically large set of independent observations. This class includes observation models that may have combinations of multiplicative and signal-dependent noise terms. Sufficient conditions are given for robust detection schemes for cases where the general form of the observation model is known but the additive noise distribution is known only to be a member of a general convex uncertainty class. Robust schemes satisfying these conditions are found for some example cases where additive signals and noise have been processed by memoryless nonlinearities. Some interesting example cases of combined multiplicative and signal-dependent noise are shown to use redescending nonlinearities, instead of the limiter nonlinearities typically found for similar additive noise cases. >

Pulsating laser oscillations depend on extremely-small-amplitude noise.

Pulsating laser oscillations consist of short and intense pulses separated by long regimes with nearly zero intensities. These oscillations appear if the dimensionless relaxation rate of the inversion of population \ensuremath{\gamma} is a small quantity (class-B lasers). During the low-intensity regimes, the laser is particularly sensitive to small-amplitude noise. We investigate the effect of noise by studying the limit-cycle oscillations of the laser with a saturable absorber. Specifically, we analyze how the size of the limit cycle is modified as the amplitude of the noise d is progressively increased from zero. We show that if \ensuremath{\gamma}\ensuremath{\ll}1 and d&gt;${\mathit{d}}_{\mathit{c}}$=O(exp(-1/\ensuremath{\gamma})), the size of the limit cycle decreases by an O(1) quantity and depends on d. The mathematical problem is a singular perturbation problem that involves two small parameters (d and \ensuremath{\gamma}). We first show that the slow evolution of the limit-cycle solution can be studied as a slow passage through a steady bifurcation point. We then analyze the case of a constant imperfection (small injected signal) and the case of additive noise (small-amplitude Gaussian white noise). Both cases lead to the same conclusions.

Maximum likelihood estimation of object location in diffraction tomography

The problem is formulated within the context of diffraction tomography, where the complex phase of the diffracted wavefield is modeled using the Rytov approximation and the measurements consist of noisy renditions of this complex phase at a single frequency. The log likelihood function is computed for the case of additive zero mean Gaussian white noise and shown to be expressible in the form of the filtered backpropagation algorithm of diffraction tomography. In this form however, the filter function is no longer the rho filter appropriate to least square reconstruction but is now the generalized projection (propagation) of the object (centered at the origin) onto the line(s) parallel to the measurement line(s), but passing through the origin. This result allows the estimation problem to be solved via a diffraction tomographic imaging procedure where the noisy data is filtered and backpropagated in a first step, and the point of maximum value of the resulting image is then the maximum likelihood (ML) estimate of the object's location. The authors include a calculation of the Cramer-Rao bound for the estimation error and a computer simulation study illustrating the estimation procedure. >

Equivocation in nonlinear digital carrier synchronizers

The nonlinear (nondecision-aided) digital phase estimator provides rapid phase estimation of an M-ary phase-shift-keyed modulated carrier. This carrier synchronization technique exhibits an anomaly, dubbed equivocation, which can degrade performance. A statistical description of equivocation, a jump in the reference phase trajectory of approximately=2 pi /M rad in a carrier synchronization technique, for the simple case of negligible frequency offset and additive white Gaussian noise is derived. Equivocation is shown to significantly degrade the bit error probability performance even in this benign channel. Since the algorithm as proposed is unsuitable for general use, a modified nonlinear (nondecision-aided) equivocation-free phase estimator is presented. The results of a simulation of this modified architecture developed to demonstrate the improved performance are presented. >

Influence of random fluctuations on delayed bifurcations: The case of additive white noise.

The influence of additive white noise on the delay of a bifurcation point in the presence of a swept control parameter has been studied by analog experiment and digital simulation. Measurements of the time taken for the second moment [x^2(t)] to reach a given threshold are in excellent agreement with the calculations of Zeghlache, Mandel, and Van den Broeck [Phys, Rev. A 40, 286 (1989)). It is demonstrated, however, that the mean first-passage time (MFPT) for x^2(t) to attain the same threshold, corresponding to the quantity that is usually determined in laser experiments, can be markedly different. It may be either larger or smaller, depending on the conditions under which the measurements are made. A calculation of the MFPT is presented and shown to be in excellent agreement with the experimental measurements and to reduce, in the relevant limit, to the theoretical results previously published by Torrent and San Miguel [Phys. Rev. A 38, 245 (1988)].

Minimax estimation of unknown deterministic signals in colored noise

The estimation of a deterministic signal corrupted by random noise is considered. The strategy is to find a linear noncausal estimator which minimizes the maximum mean square error over an a priori set of signals. This signal set is specified in terms of frequency/energy constraints via the discrete Fourier transform. Exact filter expressions are given for the case of additive white noise. For the case of additive colored noise possessing a continuous power spectral density, a suboptimal filter is derived whose asymptotic performance is optimal. Asymptotic expressions for the minimax estimator error are developed for both cases. The minimax filter is applied to random data and is shown to solve asymptotically a certain worst-case Wiener filter problem. >

Bounds from noisy linear measurements

An estimation problem in which a finite number of linear measurements of an unknown function is available, and in which the only prior information available concerning the unknown function consists of inequality constraints on its magnitude, is ill-posed in that insufficient information is available from which point estimates of the unknown function can be made with any reliability, even with exact measurements. An alternative to point estimation involves the computation of bounds on linear functionals of the unknown function in terms of the measurements. A generalization is described of the bounding technique to problems in which the measurements are inexact. The bounds are defined in terms of a primal optimization problem. A deterministic interpretation of the bounds is given, as well as a probabilistic one for the case of additive Gaussian measurement noise. An unconstrained dual optimization problem is derived that has an interesting data-adaptive filtering interpretation and provides an attractive basis for computation. Several aspects of the primal and dual optimization problems are investigated that have important implications for the reliable computation of the bounds.

The Hopf bifurcation of twodimensional systems under the influence of one external noise source

The behaviour of the Hopf bifurcation under the influence of external noise is investigated by means of a twodimensional model which uses Gaussian white noise as input. The model includes the case of multiplicative and/or additive noise. Applying the Birkhoff transformation the model is transformed to the coordinates normally used to discuss the deterministic Hopf bifurcation. Then the stationary solution of the model is calculated as an expansion for weak noise: The Hopf bifurcation under the influence of noise exhibits a bifurcation interval with width and position depending on the noise power. Moreover, a class of the systems described by the model can perform noise driven bifurcations.

Refined filtering of image noise using local statistics

An effective algorithm for digital image noise filtering is presented in this paper. Mostnoise filtering techniques such as Kalman filter and transform domain methods require extensive image modeling and produce filtered images with considerable contrast loss. The algorithm proposed in this paper is an extension of Lee's local statistics method modified to utilize local gradient information. It does not require image modeling, and it will not smear edges and subtle details. For both the additive and multiplicative noise cases, the local mean and variance are computed from a reduced set of pixels depending on the orientation of the edge. Consequently, noise along the edge is removed, and the sharpness of the edge is enhanced. For practical applications when the noise variance is spatially varying and unknown, an adaptive filtering algorithm is developed. Experiments show its good potential for processing real-life images. Examples on images containing 256×256 pixels are given to substantiate the theoretical development.

Analog signals which optimally minimize interpulse interference (Corresp.)

THOMAS R. BENEDICT, SENIOR MEMBER, IEEE Pulsed analog communication where periodic identical known pulse shapes carry amplitude information is considered. The Cramér Rao bound in amplitude estimation is derived for the case of additive Gaussian noise and arbitrary interpulse interference. It is observed that maximum-likelihood processing can achieve this bound. Then the class of signals that minimizes this bound is described. The optimum pulses appear to outperform Gaussian pulses by about 3 dB at interesting values of pulse spacing; and by comparison to <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\sin x/x</tex> pulses, they appear to have a potential for reducing interchannel interference by on the order of 3 dB while keeping interpulse interference constant;

Linear filtering in multiplicative noise

Linear filtering of a continuous dynamical system is considered where the noise appears multiplicatively. It is first shown that the solution process is a log normal process. The estimator equation and the variance equation for the filter are then derived, which are shown to be coupled. A successive approximation scheme is proposed whereby at each iteration step the estimator equation and the variance equation are separable and take forms similar to that of the Kalman-Bucy filter for the additive-noise case.

Exponential Fourier densities on S2 and optimal estimation and detection for directional processes

Two classes of probability densities, the exponential Fourier densities and the exponential trigonometric densities, are introduced on the unit sphere, as well as four kinds of displacements. In general, neither class is closed under the operation of taking conditional distributions with respect to any of the displacements. A combined usage of both classes is required to study the estimation and detection models obtained from various combinations of the displacements. The merits and disadvantages of each model are discussed. Recursive formulas for the conditional densities and the likelihood ratios are derived for many of the models. The additive measurement noise case is also considered in detail. An error criterion for direction estimation is presented with respect to which the optimal estimates can be easily computed from the probability distribution. A deficiency of the models and techniques developed in this paper is that random driving terms are disallowed in the signal processes.

Characterization of cyclostationary random signal processes

Many communication and control systems employ signal formats that involve some form of periodic processing operation. Signals produced by samplers, scanners, multiplexors, and modulators are familiar examples. Often these signals are appropriately modeled by random processes that are cyclostationary (CS), i.e., processes with statistical parameters, such as mean and autocorrelation, that fluctuate periodically with time. In this paper we examine two methods for representing nonstationary processes that reveal the special properties possessed by CS processes. These representations are the harmonic series representation (HSR) and the translation series representation (TSR). We show that the HSR is particularly appropriate for characterizing the structural properties of CS processes and that the TSR provides natural models for many types of communication signal formats. The advantages gained by modeling signals as CS processes rather than stationary processes is illustrated by consideration of the optimum filtering problem. We present general solutions for filters that minimize mean-square error for continuous-waveform estimation, and we discuss several specific examples for the particular case of additive noise. These examples demonstrate improvement in performance over that of filter designs based on stationary models for the signal processes.

A class of estimators for optimum adaptive detection

It is shown that in the operation of optimum detection of a signal whose form or descriptive parameters are specified by nonsingular a priori probability distributions, the receiver's structure can be interpreted as an estimator of the unknown signal followed by a detector that treats the estimate as a perfectly known waveform. This result holds, under very broad conditions, for any signal and noise distribution. For some special noise distributions, however, including the Gaussian, the interpretation is found to be obviously not unique. It is shown that the Price-Kailath results on the optimum receiver for the Gaussian channel (identified as an estimator-correlator or adaptive matched filter) correspond to one of the possible interpretations of the detection operation. In addition, it is shown that in this case the resulting estimator is minimum-variance only if the signal also is a realization of a normal process. As an illustrative example of an alternative, equivalent interpretation of the detection operation, a nonminimum-variance estimator is discussed for the case of Rayleigh fading and additive Gaussian noise, and its performance is evaluated.