Modulo-2piphase sequence estimation (Corresp.)

The probabilistic evolution of random walk on the circle is studied, and the results are used to derive a maximum {em a posteriori} probability (MAP) sequence estimator for phase. The sequence estimator is a Viterbi tracker for tracking phase on a finite-dimensional grid in[-pi,pi). The algorithm is shown to provide a convenient method for obtaining fixed-lag phase estimates. Performance characteristics are presented and compared with several published nonlinear filtering algorithms.     

An adaptive digital filter for improving the signal-to-noise ratio of a signal with additive noise (Corresp.)

An adaptive digital filtering scheme is developed to deal with the problem of improving the signal-to-noise ratio of a signal corrupted by noise when only general {em a priori} assumptions regarding the signal and the noise are possible. Specifically, the noise is assumed to be white zero-mean and uncorrelated; while the signal is considered to be band-limited, possibly with slowly varying spectrum. The proposed adaptive digital filtering scheme is based upon a class of variable wave digital filters. Adaptation of the digital filter multipliers is accomplished through the use of an identification procedure based on an adaptive spectral estimation method.     

A generalized likelihood ratio formula: Arbitrary noise statistics for doubly composite hypotheses (Corresp.)

A relationship between the likelihood ratio and a generalized causal conditional mean estimator is presented for the doubly composite hypotheses problem. The observation statistics are arbitrary and need not be Gaussian. The relationship parallelsthe well-known relationship for the Gaussian noise and composite signal hypothesis case, and the likelihood ratio can still be viewed as an estimator-correlator operation.     

Estimation of the relative delay of two similar signals of unknown phases in white Gaussian noise

In this paper we consider the estimation of the delay of a signal of unknown phase obscured by white noise and by another signal of unknown phase and known delay. We also consider the situation where both delays are unknown. Use is made of Shannon's sampling analysis^{1}as developed by Woodward.^{2}We conclude that the relative carrier phase of the two signals must be first estimated in order to maximize the {em a posteriori} probability of the estimate, but that knowledge of the absolute phase is only of secondary value.     

Detection and estimation of frequency-random signals (Corresp.)

A random frequency-varying signal is approximated by a finite-parameter model. A maximum {em a posteriori} estimate of the parameter set leads to an estimation criterion, which when maximized is also the decision statistic obtained with an estimator-correlator detector. An efficient implementation of the frequency-parameter estimate (which corresponds to maximizing the criterion) is presented. The resulting estimate is compared to previous techniques found in the literature. Finally, performance bounds for the resulting detector are evaluated and compared to the asymptotically optimum low-energy coherence detector.     

Convergence of the maximum a posteriori path estimator in hiddenMarkov models

In a hidden Markov model (HMM) the underlying finite-state Markov chain cannot be observed directly but only by an additional process. We are interested in estimating the unknown path of the Markov chain. The most widely used estimator is the maximum a posteriori path estimator (MAP path estimator). It can be calculated effectively by the Viterbi (1967) algorithm as is, e.g., frequently done in the field of coding theory, correction of intersymbol interference, and speech recognition. We investigate (component-wise) convergence of the MAP path estimator. Convergence is shown under the condition of unbounded likelihood ratios. This condition is satisfied in the important case of HMMs with additive white Gaussian noise. We also prove convergence, if the Markov chain has two states. The so-called Viterbi paths are an important tool for obtaining these results     

Structural characterization of locally optimum detectors in terms of locally optimum estimators and correlators

Explicit formulas for locally (SNRrightarrow 0)optimum (MMSE) signal estimators (smoother, filter, and predictor) for discrete-time observations of a random signal in additive random noise are derived and used to characterize the locally optimum (likelihood ratio) signal detector for on-off signaling. The characterizations are canonical (distribution-free) detector structures involving estimator-correlators. These structural characterizations provide new interpretations of known detectors for various special cases. If the one-step signal predictor is recursive and the noise is white (possibly non-Gaussian or nonstationary), there is a canonical structure that admits recursive computation. The primary motivation for these structural characterizations is to render the estimator-correlator design philosophy applicable for the purpose of simplifying implementations and enhancing adaptability. Unlike the known esfimator-correlator structural characterizations for continuous-time globally optimum detectors, the new characterizations apply for non-Gaussian as well as Gaussian noise, and the estimators are explicit rather than implicit.     

Estimation of the directions of arrival of signals in unknowncorrelated noise. I. The MAP approach and its implementation

The authors propose a method of direction of arrival (DOA) estimation of signals in the presence of noise whose covariance matrix is unknown and arbitrary, other than being positive definite. They examine the projection of the data onto the noise subspace. The conditional probability density function (PDF) of the projected data given the signal parameters and the unknown projected noise covariance matrix is first formed. The a posteriori PDF of the signal parameters alone is then obtained by assigning a noninformative a priori PDF to the unknown noise covariance matrix and integrating out this quantity. A simple criterion for the maximum a posteriori (MAP) estimate of the DOAs of the signals is established. Some properties of this criterion are discussed, and an efficient numerical algorithm for the implementation of this criterion is developed. The advantage of this method is that the noise covariance matrix does not have to be known, nor must it be estimated     

Gaussian arbitrarily varying channels

The {em arbitrarily varying channel} (AVC) can be interpreted as a model of a channel jammed by an intelligent and unpredictable adversary. We investigate the asymptotic reliability of optimal random block codes on Gaussian arbitrarily varying channels (GAVC's). A GAVC is a discrete-time memoryless Gaussian channel with input power constraintP_{T}and noise powerN_{e}, which is further corrupted by an additive "jamming signal." The statistics of this signal are unknown and may be arbitrary, except that they are subject to a power constraintP_{J}. We distinguish between two types of power constraints: {em peak} and {em average.} For peak constraints on the input power and the jamming power we show that the GAVC has a random coding capacity. For the remaining cases in which either the transmitter or the jammer or both are subject to average power constraints, no capacities exist and onlylambda-capacities are found. The asymptotic error probability suffered by optimal random codes in these cases is determined. Our results suggest that if the jammer is subject only to an average power constraint, reliable communication is impossible at any positive code rate.     

Estimation of Constrained Parameters With Guaranteed MSE Improvement

We address the problem of estimating an unknown parameter vector x in a linear model y=Cx+v subject to the a priori information that the true parameter vector x belongs to a known convex polytope X. The proposed estimator has the parametrized structure of the maximum a posteriori probability (MAP) estimator with prior Gaussian distribution, whose mean and covariance parameters are suitably designed via a linear matrix inequality approach so as to guarantee, for any xisinX, an improvement of the mean-squared error (MSE) matrix over the least-squares (LS) estimator. It is shown that this approach outperforms existing "superefficient" estimators for constrained parameters based on different parametrized structures and/or shapes of the parameter membership region X     

Direction estimation in partially unknown noise fields

The problem of direction of arrival estimation in the presence of colored noise with unknown covariance is considered. The unknown noise covariance is assumed to obey a linear parametric model. Using this model, the maximum likelihood directions parameter estimate is derived, and a large sample approximation is formed. It is shown that a priori information on the source signal correlation structure is easily incorporated into this approximate ML (AML) estimator. Furthermore, a closed form expression of the Cramer-Rao bound on the direction parameter is provided. A perturbation analysis with respect to a small error in the assumed noise model is carried out, and an expression of the asymptotic bias due to the model mismatch is given. Computer simulations and an application of the proposed technique to a full-scale passive sonar experiment is provided to illustrate the results     

The estimator-correlator for discrete-time problems

A class of discrete-time detection problems is formulated and a general recursive formula for the likelihood ratio is obtained. The result is analogous to the general likelihood-ratio formula or "estimator-correlator" receiver which has been derived in continuous time for a number of detection problems. As in the continuous-time versions, a conditional-mean estimate again plays a central role in the receiver structure.     

Trellis quantization with MAP detection for noisy channels

Maximum a posteriori (MAP) detection is applied to trellis quantizers operating over additive white Gaussian noise (AWGN) channels. The use of the MAP method instead of maximum likelihood is motivated by the fact that the source coder output probabilities, conditioned on the previous outputs (i.e. the state), are not equal. Simulation results indicate that by using MAP detection instead of maximum likelihood, gains as high as 0.57 dB and 2.2 dB can be achieved in terms of signal-to-quantization noise ratio (SQNR) for Gauss-Markov source and speech samples, respectively     

A maximum a posteriori probability expectation maximization algorithm for image reconstruction in emission tomography

The expectation maximization method for maximum likelihood image reconstruction in emission tomography, based on the Poisson distribution of the statistically independent components of the image and measurement vectors, is extended to a maximum aposteriori image reconstruction using a multivariate Gaussian a priori probability distribution of the image vector. The approach is equivalent to a penalized maximum likelihood estimation with a special choice of the penalty function. The expectation maximization method is applied to find the a posteriori probability maximizer. A simple iterative formula is derived for a penalty function that is a weighted sum of the squared deviations of image vector components from their a priori mean values. The method is demonstrated to be superior to pure likelihood maximization, in that the penalty function prevents the occurrence of irregular high amplitude patterns in the image with a large number of iterations (the so-called "checkerboard effect" or "noise artifact").     

Maximum likelihood estimation for array processing in colored noise

Direction of arrival estimation of multiple sources, using a uniform linear array, in noise with unknown covariance is considered. The noise is modeled as a spatial autoregressive process with unknown parameters. Both stochastic and deterministic signal models are considered. For the random signal case, an approximate maximum likelihood estimator of the signal and noise parameters is derived. It requires numerical maximization of a compressed likelihood function over the unknown arrival angles. Analytical expressions for the MLEs of the signal covariance and the AR parameters are given. Similar results for the case of deterministic signals are also presented     

The CFAR adaptive subspace detector is a scale-invariant GLRT

The constant false alarm rate (CFAR) matched subspace detector (CFAR MSD) is the uniformly most-powerful-invariant test and the generalized likelihood ratio test (GLRT) for detecting a target signal in noise whose covariance structure is known but whose level is unknown. Previously, the CFAR adaptive subspace detector (CFAR ASD), or adaptive coherence estimator (ACE), was proposed for detecting a target signal in noise whose covariance structure and level are both unknown and whose covariance structure is estimated with a sample covariance matrix based on training data. We show here that the CFAR ASD is GLRT when the test measurement is not constrained to have the same noise level as the training data, As a consequence, this GLRT is invariant to a more general scaling condition on the test and training data than the well-known GLRT of Kelly (1986)     

Robust matched filters

The problem of designing robust matched filters for situations in which there is uncertainty in the signal structure or noise statistics is considered. Two general aspects of this problem are treated. First, maximin robust designs are characterized for a general Hilbert-space formulation of the matched filtering problem and explicit solutions are given for two intuitively appealing models for uncertainty. These results are seen to generalize earlier results for specific matched filtering problems. The second general aspect treated is the application of the theoretical maximin results to the particular problem of designing filters to combat uncertain nonlinear channel distortion. Channel distortion is modeled by considering an unknown received signal which may differ inL_{2}-norm from the transmitted or nominal signal by no more than a fixed amount. The effect of the channel distortion is seen to be equivalent to that of adding a white noise to the channel whose spectral height depends on the degree of distortion. General expressions are developed for the determination of the robust filter and its performance, and numerical results are presented for the case of baseband detection in Gauss-Markov noise.     

Some results on the stochastic signal parameter estimation problem

The problem of finding maximum-likelihood estimates of the partially or completely unknown autocorrelation function of a zero-mean Gaussian stochastic signal corrupted by additive, white Gaussian noise is analyzed. It is shown that these estimates can be found by maximizing the output of a Wiener estimator-correlator receiver biased by a smoothed version of the output noise-to-signal ratio of the Wiener estimator over the class of admissible autocorrelation functions. For the case where the autocorrelation function is known except for an amplitude scale parameter, an illuminating expression for the Cramer-Rao minimum estimation variance is derived. Detailed study of this expression yields, among other results, an interpretation of the maximum-likelihood estimator as an adaptive processor.     

Asymptotic performance analysis of cyclic detectors

The paper deals with the analytical evaluation of the asymptotic detection and false-alarm probabilities of multicycle and single-cycle detectors operating in additive white Gaussian noise with unknown and nonrandom spectral level, which are based on the cyclostationarity properties of the signal to be intercepted. The receiver operating characteristics are derived by using the asymptotic complex normality and the covariance expression of the sample average estimator of the cyclic covariance in an observed discrete-time series. A numerical example for interception of a binary phase-shift keying signal is considered     

Bayesian motion estimation for temporally recursive noise reduction in X-ray fluoroscopy

This paper develops a Bayesian motion estimation algorithm for motion-compensated temporally recursive filtering of moving low-dose X-ray images (X-ray fluoroscopy). These images often exhibit a very low signal-to-noise ratio. The described motion estimation algorithm is made robust against noise by spatial and temporal regularization. A priori expectations about the spatial and temporal smoothness of the motion vector field are expressed by a generalized Gauss-Markov random field. The advantage of using a generalized Gauss-Markov random field is that, apart from smoothness, it also captures motion edges without requiring an edge detection threshold. The costs of edges are controlled by a single parameter, by means of which the influence of the regularization can be tuned from a median-filter-like behaviour to a linear-filter-like one.