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Bayesian Bounds for Parameter Estimation and Nonlinear Filtering/Tracking

Cover Image Copyright Year: 2007
Author(s): Harry L. Van Trees; Kristine L. Bell
Publisher: Wiley-IEEE Press
Content Type : Books & eBooks
Topics: Signal Processing & Analysis
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Abstract

Bayesian Bounds provides a collection of the important papers dealing with the theory and application of Bayesian bounds. The book is essential to both engineers and statisticians whether they are practitioners or theorists. Each part of the book is introduced with the contributions of each selected paper and their interrelationship.

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      Frontmatter

      Page(s): i - 86
      Copyright Year: 2007

      Wiley-IEEE Press eBook Chapters

      The prelims comprise:
      Half Title
      IEEE Press Board Page
      Title
      Copyright
      Dedication
      Contents
      Preface
      Introduction View full abstract»

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      Bayesian CramrRao Bounds

      Page(s): 87
      Copyright Year: 2007

      Wiley-IEEE Press eBook Chapters

      Bayesian Bounds provides a collection of the important papers dealing with the theory and application of Bayesian bounds. The book is essential to both engineers and statisticians whether they are practitioners or theorists. Each part of the book is introduced with the contributions of each selected paper and their interrelationship. View full abstract»

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      Excerpts from Part I of Detection, Estimation, and Modulation Theory

      Page(s): 89 - 109
      Copyright Year: 2007

      Wiley-IEEE Press eBook Chapters

      Bayesian Bounds provides a collection of the important papers dealing with the theory and application of Bayesian bounds. The book is essential to both engineers and statisticians whether they are practitioners or theorists. Each part of the book is introduced with the contributions of each selected paper and their interrelationship. View full abstract»

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      A generalization of the FchetCramr inequality to the case of Bayes estimation

      Page(s): 110
      Copyright Year: 2007

      Wiley-IEEE Press eBook Chapters

      Bayesian Bounds provides a collection of the important papers dealing with the theory and application of Bayesian bounds. The book is essential to both engineers and statisticians whether they are practitioners or theorists. Each part of the book is introduced with the contributions of each selected paper and their interrelationship. View full abstract»

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      Global Bayesian Bounds

      Page(s): 111 - 112
      Copyright Year: 2007

      Wiley-IEEE Press eBook Chapters

      Bayesian Bounds provides a collection of the important papers dealing with the theory and application of Bayesian bounds. The book is essential to both engineers and statisticians whether they are practitioners or theorists. Each part of the book is introduced with the contributions of each selected paper and their interrelationship. View full abstract»

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      Some Classes of Global CramrRao Bounds

      Page(s): 113 - 130
      Copyright Year: 2007

      Wiley-IEEE Press eBook Chapters

      This chapter contains sections titled:
      Introduction
      Notation
      Comparison among different multidimensional bounds
      Generalized bounds
      Some further generalizations This chapter contains sections titled:
      References View full abstract»

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      Excerpts from Part 1 of Detection, Estimation, and Modulation Theory

      Page(s): 131 - 143
      Copyright Year: 2007

      Wiley-IEEE Press eBook Chapters

      This chapter contains sections titled:
      Nonlinear Estimation View full abstract»

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      SingleTone Parameter Estimation from DiscreteTime Observations

      Page(s): 144 - 151
      Copyright Year: 2007

      Wiley-IEEE Press eBook Chapters

      Estimation of the parameters of a single-frequency complex tone from a finite number of noisy discrete-time observations is discussed. The appropriate Cramér-Rao bounds and maximum-likelihood (ML) estimation algorithms are derived. Some properties of the ML estimators are proved. The relationship of ML estimation to the discrete Fourier transform is exploited to obtain practical algorithms. The threshold effect of one algorithm is analyzed and compared to simulation results. Other simulation results verify other aspects of the analysis. View full abstract»

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      Barankin Bounds on Parameter Estimation

      Page(s): 152 - 159
      Copyright Year: 2007

      Wiley-IEEE Press eBook Chapters

      The Schwarz inequality is used to derive the Barankin lowerbounds on the covariance matrix of unbiased estimates of a vector parameter. The bound is applied to communications and radar problems in which the unknown parameter is embedded in a signal of known form and observed in the presence of additive white Gaussian noise. Within this context it is shown that the Barankin bound reducesto the Cramér-Rao bound when the signal-to-noise ratio (SNR) is large. However, as the SNR is reduced beyond a critical value, the Barankln bound deviates radically from the Cramér-Rao bound, exhibiting the so-called threshold effect. The bounds were applied to the linear FM waveform, and within the resulting class of bounds it waspossible to selectone that led to a closed-form expression for the lower bound on the variance of an unbiased range estimate. This expression clearly demonstrates the threshold behavior one must expect when using a nonlinear modulation system. Tighter bounds were easily obtained, but these had to be evaluated numerically. The sidelobe structure of the linear FM compressed pulse leads to a significant increase in the variance of the estimate. For a practical linear FM pulse of 1-s duration and 40-MHz bandwidth, the radar must operate at an SNR greater than 10 dB if meaningful unbiased range estimates are to be obtained. View full abstract»

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      A Modified CramrRao Bound and its Applications

      Page(s): 160 - 162
      Copyright Year: 2007

      Wiley-IEEE Press eBook Chapters

      A new way of treatfng extraneous or nuisance panmeters in applying the Cramér-Rao bound is presented. The modified method produces substantially tighter bounds in two important applications. In particular, new bounds are obtained for the variance of estimates anival times for the Rayleigh channel. View full abstract»

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      A Lower Bound on the MeanSquare Error in Random Parameter Estimation

      Page(s): 163 - 165
      Copyright Year: 2007

      Wiley-IEEE Press eBook Chapters

      A new lower bound on mean-square error in parameter estimation is presented. The bound is tighter than the Cramér-Rao and Bobrovsky-Zakai lower bounds. It requires no bias or regularity assumptions, it is computationally simple, and it can be applied to estimates of vector parameters or functions of the parameters. View full abstract»

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      Lower Bounds on the Mean Square Estimation Error

      Page(s): 166
      Copyright Year: 2007

      Wiley-IEEE Press eBook Chapters

      A general class of lower bounds on the mean square error (mse) in random parameter estimation is formulated. These bounds are generated using functions of the parameter and the data that are orthogonal to the data. A particular choice in the class yields a new lower bound which is superior to both the Cramer-Rao and Bobrovsky-Zakai lower bounds. View full abstract»

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      A General Class of Lower Bounds in Parameter Estimation

      Page(s): 167 - 170
      Copyright Year: 2007

      Wiley-IEEE Press eBook Chapters

      A general class of Bayesian lower bounds on moments of the error in parameter estimation is formulated, and it is shown that the Cramer-Rao, the Bhattacharyya, the Bobrovsky-Zakai, and the Weiss-Weinstein lower bounds are special cases in the class. The bounds can be applied to the estimation of vector parameters and any given function of the parameters. View full abstract»

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      A Bound on MeanSquareEstimate Error

      Page(s): 171 - 175
      Copyright Year: 2007

      Wiley-IEEE Press eBook Chapters

      A lower bound on mean-square-estimate error is derived as an instance of the covariance inequality by concatenating the generating matrices for the Bhattacharyya and Barankin bounds; it represents a generalization of tbe Bhattacharyya, Barankin, Cramér-Rao, Hammersley-Chapman-Robbins, Kiefer, and McAulay-Hofstetter bounds in that all of these bounds may be derived as special cases. The bound is applicable to biased estimates of functions of a multidimensional parameter. Termed the Hybrid Bhattacharyya-Barankin bound, it may be written as the sum or the mth-order Bhattacharyya bound and a nonnegative term similar in form to the rth-order Hammersley-Chapman-Robbins bound. It is intended for use when small-error bounds, such as the Cramér-Rao bound, may not be tight; unlike many large-error bounds, it provides at smooth transition between the small-error and large-error regions. As an example application, bounds are placed on the variance of unbiased position estimates derived from passive array measurements. Here, the hybrid Bhattacharyya-Barankin bound enters the large-error region at a larger SNR and, in the large-error region, is noticeably greater than either the Hammersley-Chapman-Robbins or the Bhattacharyya bounds. View full abstract»

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      The Bayesian Abel Bound on the Mean Square Error

      Page(s): 176 - 179
      Copyright Year: 2007

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      This paper deals with lower bound on the Mean Square Error (MSE). In the Bayesian framework, we present a new bound which is derived from a constrained optimization problem. This bound is found to be tighter than the Bayesian Bhattacharyya bound, the Reuven-Messer bound, the Bobrovsky-Zakaï bound, and the Bayesian Cramér-Rao bound. View full abstract»

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      Some Lower Bounds on Signal Parameter Estimation

      Page(s): 180 - 185
      Copyright Year: 2007

      Wiley-IEEE Press eBook Chapters

      New bounds are presented for the maximum accuracy with which parameters of signals imbedded in white noise can be estimated. The bounds are derived by comparing the estimation problem with related optimal detection problems. They are, with few exceptions, independent of the bias and include explicitly the dependence on the a priori interval. The new results are compared with previously known results. View full abstract»

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      Performance Limitations and Error Calculations for Parameter Estimation

      Page(s): 186 - 194
      Copyright Year: 2007

      Wiley-IEEE Press eBook Chapters

      Error calculations cannot be carried out precisely when parameters are estimated which affect the observation nonlinearly. This paper summarizes the available approaches to studying performance and compares the resulting answers for a specific case. It is shown that the familiar. Cramér-Rao lower bound on rms error yields an accurate answer only for large signal-to-noise ratios (SNR). For low SNR, lower bounds on rms error obtained by Ziv and Zakai give easily calculated and fairly tight answers. Rate distortion theory gives a lower bound on the error achievable with any system. The Barankin lower bound does not appear to give useful information as a computational tool. A technique for approximating the error can be used errectively for a large class of systems. With numerical integration, an upper bound obtained by Seidman gives a fairly tight answer. Recant work by Ziv gives bounds on the bias of estimators but, in general, these appear to be rather weak. Tighter results are obtained for maximum-likelihood estimators with certain symmetry conditions. Applying these techniques makes it possible to locate the threshold level to within a few decibels of channel signal-to-noise ratio. Further, these calculations can be easily carried out for any system. View full abstract»

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      Improved Lower Bounds on Signal Parameter Estimation

      Page(s): 195 - 198
      Copyright Year: 2007

      Wiley-IEEE Press eBook Chapters

      An improved technique for bounding the mean-square error of signal parameter estimates is presented. The resulting bounds are independent of the bias and stronger than previously known bounds. View full abstract»

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      Bounds on Error in Signal Parameter Estimation

      Page(s): 199 - 200
      Copyright Year: 2007

      Wiley-IEEE Press eBook Chapters

      In this correspondence, the problem of lower bounds on mean-square error in parameter estimation is considered. Lower bounds on mean-square error can be used, for instance, to bound the performances, namely the attainable output signal-to-noise ratio, of pulse modulation transmission systems, such as pulse-position modulation (PPM) or pulse-frequency modulation (PFM). The tightest lower bounds to mean-square error previously known are the Ziv-Zakai bounds; the analysis carried out in this paper, which is based on an inequality first obtained by Kotel'nikov, leads to lower bounds tighter than previously known bounds. View full abstract»

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      Correction to "Bounds on Error in Signal Parameter Estimation"

      Page(s): 201
      Copyright Year: 2007

      Wiley-IEEE Press eBook Chapters

      This chapter contains sections titled:
      References View full abstract»

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      Improved Bounds on the Local MeanSquare Error and the Bias of Parameter Estimators

      Page(s): 202 - 203
      Copyright Year: 2007

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      An improved lower bound on the local mean-square error and upper and lower bounds on the bias which are tighter than previously known bounds are derived. View full abstract»

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      Relations Between BeliniTartara, ChazanZakaiZiv, and WaxZiv Lower Bounds

      Page(s): 204 - 205
      Copyright Year: 2007

      Wiley-IEEE Press eBook Chapters

      A class of lower bounds on the mean square error in parameter estimation is presented, based on the Belini-Tartara lower bound. The Wax-Ziv lower bound is shown to be a special case in the class. These bounds often are significantly tighter than the Chazan-Zakai-Ziv lower bound when the parameter to be estimated is subject to ambiguity and threshold effects. View full abstract»

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      Extended ZivZakai Lower Bound for Vector Parameter Estimation

      Page(s): 206 - 219
      Copyright Year: 2007

      Wiley-IEEE Press eBook Chapters

      The Bayesian Ziv-Zakai bound on the mean square error (MSE) in estimating a uniformly distributed continuous random variable is extended for arbitrarily distributed continuous random vectors and for distortion functions other than MSE. The extended bound is evaluated for some representative problems in time-delay and bearing estimation. The resulting bounds have simple closed-form expressions, and closely predict the simulated performance of the maximum-likelihood estimator in all regions of operation. View full abstract»

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      Explicit ZivZakai Lower Bound for Bearing Estimation

      Page(s): 220 - 234
      Copyright Year: 2007

      Wiley-IEEE Press eBook Chapters

      The extended Ziv-Zakai bound for vector parameters is used to develop a lower bound on the mean square error in estimating the 2-D bearing of a narrowband planewave signal using planar arrays of arbitrary geometry. The bound has a simple closed-form expression that is a function of the signal wavelength, the signal-to-noise ratio (SNR), the number of data snapshots, the number of sensors in the array, and the array configuration. Analysis of the bound suggests that there are several regions of operation, and expressions for the thresholds separating the regions are provided. In the asymptotic region where the number of snapshots and/or SNR are large, estimation errors are small, and the bound approaches the inverse Fisher information. This is the same as the asymptotic performance predicted by the local Cramér-Rao bound for each value of bearing. In the a priori performance region where the number of snapshots or SNR is small, estimation errors are distributed throughout the a priori parameter space and the bound approaches the a priori covariance. In the transition region, both small and large errors occur, and the bound varies smoothly between the two extremes. Simulations of the maximum likelihood estimator (MLE) demonstrate that the bound closely predicts the performance of the MLE in all regions. View full abstract»

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      A Global Lower Bound on Parameter Estimation Error with Periodic Distortion Functions

      Page(s): 235 - 240
      Copyright Year: 2007

      Wiley-IEEE Press eBook Chapters

      We present a global Ziv-Zakai-type lower bound on the mean square error for estimation of signal parameter vectors, where some components of the distortion function may be periodic. Periodic distortion functions arise naturally in the context of direction of arrival or phase estimation problems. The bound is applied to an image registration problem, and compared to the performance of the maximum-likelihood estimator (MLE). View full abstract»

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      Excerpts from Part III of Detection, Estimation, and Modulation Theory

      Page(s): 241 - 273
      Copyright Year: 2007

      Wiley-IEEE Press eBook Chapters

      This chapter contains sections titled:
      Receiver Derivation and Signal Design
      Performance of the Optimum Estimator View full abstract»

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      Threshold Region Performance of Maximum Likelihood Direction of Arrival Estimators

      Page(s): 274 - 288
      Copyright Year: 2007

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      This paper presents a performance analysis of the maximum likelihood (ML) estimator for finding the directions of arrival (DOAs) with a sensor array. The asymptotic properties of this estimator are well known. In this paper, the performance under conditions of low signal-to-noise ratio (SNR) and a small number of array snapshots is investigated. It is well known that the ML estimator exhibits a threshold effect, i.e., a rapid deterioration of estimation accuracy below a certain SNR or number of snapshots. This effect is caused by outliers and is not captured by standard techniques such as the Cramér-Rao bound and asymptotic analysis. In this paper, approximations to the mean square estimation error and probability of outlier are derived that can be used to predict the threshold region performance of the ML estimator with high accuracy. Both the deterministic ML and stochastic ML estimators are treated for the single-source and multisource estimation problems. These approximations alleviate the need for time-consuming computer simulations when evaluating the threshold region performance. For the special case of a single stochastic source signal and a single snapshot, it is shown that the ML estimator is not statistically efficient as SNR due to the effect of outliers. View full abstract»

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      Capon Algorithm MeanSquared Error Threshold SNR Prediction and Probability of Resolution

      Page(s): 289 - 305
      Copyright Year: 2007

      Wiley-IEEE Press eBook Chapters

      Below a specific threshold signal-to-noise ratio (SNR), the mean-squared error (MSE) performance of signal parameter estimates derived from the Capon algorithm degrades swiftly. Prediction of this threshold SNR point is of practical significance for robust system design and analysis. The exact pairwise error probabilities for the Capon (and Bartlett) algorithm, derived herein, are given by simple finite sums involving no numerical integration, include finite sample effects, and hold for an arbitrary colored data covariance. Via an adaptation of an interval error based method, these error probabilities, along with the local error MSE predictions of Vaidyanathan and Buckley, facilitate accurate prediction of the Capon threshold region MSE performance for an arbitrary number of well separated sources, circumventing the need for numerous Monte Carlo simulations. A large sample closed-form approximation for the Capon threshold SNR is provided for uniform linear arrays. A new, exact, two-point measure of the probability of resolution for the Capon algorithm, that includes the deleterious effects of signal model mismatch, is a serendipitous byproduct of this analysis that predicts the SNRs required for closely spaced sources to be mutually resolvable by the Capon algorithm. Last, a general strategy is provided for obtaining accurate MSE predictions that account for signal model mismatch. View full abstract»

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      MeanSquared Error and Threshold SNR Prediction of MaximumLikelihood Signal Parameter Estimation With Estimated Colored Noise Covariances

      Page(s): 306 - 324
      Copyright Year: 2007

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      An interval error-based method (MIE) of predicting mean squared error (MSE) performance of maximum-likelihood estimators (MLEs) is extended to the case of signal parameter estimation requiring intermediate estimation of an unknown colored noise covariance matrix; an intermediate step central to adaptive array detection and parameter estimation. The successful application of MIE requires good approximations of two quantities: 1) interval error probabilities and 2) asymptotic (SNR local MSE performance of the MLE. Exact general expressions for the pairwise error probabilities that include the effects of signal model mismatch are derived herein, that in conjunction with the Union Bound provide accurate prediction of the required interval error probabilities. The Cramér-Ran Bound (CRB) often provides adequate prediction of the asymptotic local MSE performance of MLE. The signal parameters, however, are decoupled from the colored noise parameters in the Fisher Information Matrix for the deterministic signal model, rendering the CRB incapable of reflecting loss due to colored noise covariance estimation. A new modification of the CRB involving a complex central beta random variable different from, but analogous to the Reed, Mallett, and Brennan beta loss factor provides a working solution to this problem, facilitating MSE prediction well into the threshold region with remarkable accuracy. View full abstract»

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      Threshold Region Determination of ML Estimation in Known Phase DataAided Frequency Synchronization

      Page(s): 325 - 328
      Copyright Year: 2007

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      This paper studies the performance of the frequency maximum likelihood (ML) etimator of a single tone in Gaussian noise. Two mean-square error (MSE) approximations are first applied and then compared in the case of a known phase offset. The first is proposed by Van Trees [1] and corresponds to the method of interval errors (MIE). The second MSE approximation is proposed by Rife and Boorstyn in [4]. The problem is formulated in the data-aided frequency synchronization frame. View full abstract»

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      Bounds on the Bayes and Minimax Risk for Signal Parameter Estimation

      Page(s): 329 - 337
      Copyright Year: 2007

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      In estimating the parameter from a parametrized signal problem (with 0 L) observed through Gaussian white noise, four useful and computable lower bounds for the Bayes risk were developed. For problems with different L and different signal to noise ratios, some bounds are superior to the others. The lower bound obtained from taking the maximum of the four, serves not only as a good lower bound for the Bayes risk but also as a good lower bound tor the minimax risks. Threshold behavior of the Bayes risk is also evident as shown in our lower bound. View full abstract»

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      The Probability of a Subspace Swap in the SVD

      Page(s): 338 - 344
      Copyright Year: 2007

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      In this paper we extend the work of Tufts, Kot, and Vaccaro (TKV) to improve the analytical characterization of threshold breakdown in SVD methods. Our results sharpen the TKV results by lower bounding the probability of a subspace swap in the SVD. Our key theoretical result is the characteristic function for a random variable whose probability of exceeding zero bounds the probability of a threshold breakdown. View full abstract»

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      Hybrid Bayesian Bounds

      Page(s): 345
      Copyright Year: 2007

      Wiley-IEEE Press eBook Chapters

      Bayesian Bounds provides a collection of the important papers dealing with the theory and application of Bayesian bounds. The book is essential to both engineers and statisticians whether they are practitioners or theorists. Each part of the book is introduced with the contributions of each selected paper and their interrelationship. View full abstract»

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      Array Shape Calibration Using Sources in Unknown LocationsPart I: FarField Sources

      Page(s): 347 - 360
      Copyright Year: 2007

      Wiley-IEEE Press eBook Chapters

      This paper deals with source localization using a two-dimensional array of sensors whose locations are not known precisely. If only a single source is observed, uncertainties in sensor location increase errors in source bearing and range by and range by an amount which is independent of signal-to-noise ratio and which can easily dominate overall localization accuracy. Major performance gains could therefore result from successful calibration of array geometry. The paper derives Cramer-Rao bounds on calibration and source location accuracies achievable with far-field sources whose bearings are not initially known. The sources are assumed to radiate Gaussian noise and to be spectrally disjoint of each other. When the location of one sensor and the direction to a second sensor is known, three noncollinear sources are sufficient to calibrate sensor positions with errors which decrease to zero as calibrating source strength or time-bandwidth products tend to infinity. The sole exception to this statement is a nominally linear array for which such calibration is not possible. When one sensor location is known but no directional reference is available, three non-collinear sources can determine array shape, but there remains a residual error in angular orientation which is irremovable by the calibration procedure. When no sensor locations are known a priori, one adds to the residual error in rotation a translational component. In the far field, the latter should be unimportant. In addition to the asymptotic results, Cramer-Rao bounds are computed for finite signal-to- noise ratios and observation times. One finds that calibration permits significant reductions in localization errors for parameter values well within the practical range. View full abstract»

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      A BarankinType Lower Bound on the Estimation Error of a Hybrid Parameter Vector

      Page(s): 360 - 370
      Copyright Year: 2007

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      The Barankin bound is a realizable lower bound on the mean-square error (mse) of any unbiased estimator of a (nonrandom) parameter vector. In this correspondence we present a Barankin-type bound which is useful in problems where there is a prior knowledge on some of the parameters to be estimated. That is, the parameter vector is a hybrid vector in the sense that some of its entries are deterministic while other are random variables. We present a simple expression for a positive-definite matrix which provides bounds on the covariance of any unbiased estimator of the nonrandom parameters and an estimator of the random parameters, simultaneously. We show that the Barankin bound for deterministic parameters estimation and the Bobrovsky-Zakai bound for random parameters estimation are special cases of our proposed bound. View full abstract»

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      Efficient computation of the Bayesian CramerRao Bound on Estimating Parameters of Markov Models

      Page(s): 371 - 374
      Copyright Year: 2007

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      This paper presents a novel method for calculating the Hybrid Cramer-Rao lower bound (HCRLB) when the statistical model for the data has a Markovian nature. The method applies to both the non-linear/non-Gaussian as well as linear/Gaussian model. The approach solves the required expectation over unknown random parameters by several one-dimensional integrals computed recursively, thus simplifying a computationally-intensive multi-dimensional integration. The method is applied to the problem of refractivity estimation using radar clutter from the sea surface, where the backscatter cross section is assumed to be a Markov process in range. The HCRLB is evaluated and compared to the performance of the corresponding maximum a-posteriori estimator. Simulation results indicate that the HCRLB provides a tight lower bound in this application. View full abstract»

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      Further Results on CramrRao Bounds for Parameter Estimation in LongCode DS/CDMA Systems

      Page(s): 375 - 380
      Copyright Year: 2007

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      In a recent paper, closed-form formulas for the Cramér-Rao bounds (CRBs) on the error variance of any unbiased estimator of the amplitudes, timing offsets, phase offsets, and directions-of-arrival (DoAs) of multiplexed asynchronous direct-sequence code division multiple access (DS/CDMA)signals adopting aperiodic spreading codes have been derived. These formulas depend on the active users' spreading codes and pilot information symbols. In this paper, we obtain formulas for the modified CRB under the assumption that the spreading codes can be modeled as random nuisance parameters. Moreover, we also consider the case that the timing offsets are modeled as random variates and provide formulas for the hybrid CRB, i.e., the CRB corresponding to the situation that some of the parameters to be estimated are random variates rather than unknown deterministic quantities. Numerical examples showing the impact of the training length, signal-to-noise ratio (SNR), and the number of active users on the CRB are also reported and commented upon. View full abstract»

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      QuasiFluidMechanicsBased QuasiBayesian CramrRao Bounds for Deformed TowedArray Direction Finding

      Page(s): 381 - 392
      Copyright Year: 2007

      Wiley-IEEE Press eBook Chapters

      New quasi-Bayesian (hybrid) Cramér-Rao bound (CRB) expressions are herein derived for far-field deep-sea direction-of-arrival (DOA) estimation with a nominally linear towed-array that 1) is deformed by spatio-temporally correlated oceanic currents, which have been previously overlooked in the towed-array shape-deformation statistical analysis literature, 2) is deformed by temporally correlated motion of the towing vessel, which is modeled only as temporally uncorrelated in prior literature, and 3) sutTers gain-uncertainties and phase-uncertainties in its constituent hydrophones. This paper attempts to bridge an existing literature gap in deformed towed-array DOA-estimation performance analysis, by simultaneously a) incorporating several essential fluid-mechanics considerations to produce a shape-deformation statistical model physically more realistic than those previously used for DOA performance analysis and b) rigorously derive a mathematical analysis to characterize quantitatively and qualitatively the DOA stimation's statistical performance. The derived CRB expressions are parameterized in terms of the towed-array's physically measurable nonidealities for the single-source case. The new hybrid-CRB expressions herein derived are numerically more stable than those in the current literature. View full abstract»

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      Constrained CramrRao Bounds

      Page(s): 393
      Copyright Year: 2007

      Wiley-IEEE Press eBook Chapters

      Bayesian Bounds provides a collection of the important papers dealing with the theory and application of Bayesian bounds. The book is essential to both engineers and statisticians whether they are practitioners or theorists. Each part of the book is introduced with the contributions of each selected paper and their interrelationship. View full abstract»

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      Lower Bounds for Parametric Estimation with Constraints

      Page(s): 395 - 411
      Copyright Year: 2007

      Wiley-IEEE Press eBook Chapters

      A Chapman-Robbins form of the Barankin bound is used to derive a multiparameter Cramï¿¿r-Rao (CR) type lower bound on estimator error covariance when the parameter n is constrained to lie in a subset of the parameter space. A simple form for the constrained CR bound is obtained when the constraint set C can be expressed as a smooth functional inequality constraint, C = {: g }, We show that the constrained CR bound is identical to the unconstrained CR bound at the regular points of n , i.e. where no equality constraints are active. On the other hand, at those points C where pure equality constraints are active the full-rank Fisher information matrix in the unconstrained CR bound must be replaced by a rank-reduced Fisher information matrix obtained as a projection of the full-rank Fisher matrix onto the tangent hyperplane of the constraint set at . A necessary and sufficient condition involving the forms of the constraint and the likelihood function is given for the bound to be achievable, and examples for which the bound is achieved are presented. In addition to providing a useful generalization of the CR bound, our results permit analysis of the gain in achievable mse performance due to the imposition of particular constraints on the parameter space without the need for a global reparameterization. For the purpose of illustration, we apply the constrained bound to problems involving linear constraints and quadratic constraints. Specific examples considered include: linear constraints for Gaussian linear models, object support constraints in image reconstruction, signal subspace constraints in sensor array processing, and average power constraints in spectral estimation and signal extraction. View full abstract»

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      A Simple Derivation of the Constrained Multiple Parameter CramerRao Bound

      Page(s): 412 - 414
      Copyright Year: 2007

      Wiley-IEEE Press eBook Chapters

      Recently Gorman and Hero obtained a remarkable extension to tbe classical multiple parameter Cramer-Rao (CR) lower bound tbat accounts for deterministic, nonlinear equality constraints on the parameters. The virtue of Gorman and Hero's new result is that the constrained CR bound on all of the parameters is obtained by a subtracting an easily computed nonnegative definite correction matrix from the unconstrained CR bound matrix. This correspondence presents a new, simple derivation of the constrained CR bound and a new necessary condition for an estimator to satisfy the constrained CR bound with equality. View full abstract»

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      On the CramrRan Bound Under Parametric Constraints

      Page(s): 415 - 417
      Copyright Year: 2007

      Wiley-IEEE Press eBook Chapters

      This letter presents a simple expression for the Cramér-Rao bound (CRB) for parametric estimation under differentiable, deterministic constraints on the parameters. In contrast to previous works, the constrained CRB presented here does not require that the Fisher information matrix (FIM) for the unconstrained problem be of full rank. This is a useful extension because, for several signal processing problems (such as blind channel identification), the unconstrained problem is unidentifiable. Our expression for the constrained CRB depends only on the unconstrained FIM and a basis of the nullspace of the constraint's gradient matrix. We show that our constrained CRB formula reduces to the known expression when the FIM for the unconstrained problem is nonsingular. A necessary and sufficient condition for the existence of the constrained CRB is also derived. View full abstract»

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      Regularity and Strict Identifiability in MIMO Systems

      Page(s): 418 - 429
      Copyright Year: 2007

      Wiley-IEEE Press eBook Chapters

      We study finite impulse response (FIR) multi-input multi-output (MIMO) systems with additive noise, treating the finite- length sources and channel coefficients as deterministic unknowns, considering both regularity and identifiability. In blind estimation, the ambiguity set is large, admitting linear combinations of the sources. We show that the Fisher information matrix (FIM) is always rank deficient by at least the number of sources squared and develop necessary and sufficient conditions for the FIM to achieve its minimum nullity. Tight bounds are given on the required source data lengths to achieve minimum nullity of the FIM. We consider combinations of constraints that lead to regularity (i.e., to a full-rank FIM and, thus, a meaningful Cramér-Rao bound). Exploiting the null space ofthe FIM, we show how parameters must be specified to obtain a full-rank FIM, with implications for training sequence design in multisource systems. Together with constrained Cramér-Rao bounds (CRBs), this approach provides practical techniques for obtaining appropriate MIMO CRBs for many cases. Necessary and sufficient conditions are also developed for strict identifiability (ID). The conditions for strict ID are shown to be nearly equivalent to those for the FIM nullity to be minimized. View full abstract»

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      Covariance, Subspace, and Intrinsic CramrRao Bounds

      Page(s): 430 - 450
      Copyright Year: 2007

      Wiley-IEEE Press eBook Chapters

      Cramï¿¿r-Rao bounds on estimation accuracy are established for estimation problems on arbitrary manifolds in which no set of intrinsic coordinates exists. The frequently encountered examples of estimating either an unknown subspace or a covariance matrix are examined in detail. The set of subspaces, called the Grassmann manifold, and the set of covariance (positive-definite Hermitian) matrices have no fixed coordinate system associated with them and do not possess a vector space structure, both of which are required for deriving classical Cramï¿¿r-Rao bounds. Intrinsic versions of the Cramï¿¿r-Rao bound on manifolds utilizing an arbitrary affine connection with arbitrary geodesics are derived for both biased and unbiased estimators. In the example of covariance matrix estimation, closed-form expressions for both the intrinsic and flat bounds are derived and compared with the root-mean-square error (RMSE) of the sample covariance matrix (SCM) estimator for varying sample support K. The accuracy bound on unbiased covariance matrix estimators is shown to be about (10/log 10)n/K 1/2 dB, where n is the matrix order. Remarkably, it is shown that from an intrinsic perspective, the SCM is a biased and inefficient estimator and that the bias term reveals the dependency of estimation accuracy on sample support observed in theory and practice. The RMSE of the standard method of estimating subspaces using the singular value decomposition (SVD)is compared with the intrinsic subspace Cramï¿¿r-Rao bound derived in closed form by varying both the signal-to-noise ratio (SNR) of the unknown p-dimensional subspace and the sample support. In the simplest case, the Cramï¿¿r-Rao bound on subspace estimation accuracy is shown to be about (p(n - p)1/2 K-1/2SN-1/2 rad for p-dimensional subspaces. It is seen that the SVD-based method yields accuracies very close to the Cramï¿¿r-Rao bound, esta blishing that the principal invariant subspace of a random sample provides an excellent estimator of an unknown subspace. The analysis approach developed is directly applicable to many other estimation problems on manifolds encountered in signal processing and elsewhere, such as estimating rotation matrices in computer vision and estimating subspace basis vectors in blind source separation. View full abstract»

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      Exploring Estimator BiasVariance Tradeoffs Using the Uniform CR Bound

      Page(s): 451 - 466
      Copyright Year: 2007

      Wiley-IEEE Press eBook Chapters

      We introduce a plane, which we call the delta-sigma plane, that is indexed by tbe norm of the estimator bias gradient and the variance of the estimator. The norm of the bias gradient is related to the maximum variation in the estimator bias function over a neighborhood of parameter space. Using a uniform Cramér-Rao (CR) bound on estimator variance, a delta-sigma tradeoff curve is specified that defines an unachievable region of the delta-sigma plane for a specifiedstatistical model. In order to place an estimator on this plane for comparison with the delta-sigma tradeoff curve, the estimator variance, bias gradient, and bias gradient norm must be evaluated. We present a simple and accurate method for experimentally determining the bias gradient norm based on applying a bootstrap estimator to a sample mean constructed from the gradient of the log-likelihood. We demonstrate the methods developed in this paper for linear Gaussian and nonlinear Poisson inverse problems. View full abstract»

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      Performance Bounds for Estimating Vector Systems

      Page(s): 467 - 479
      Copyright Year: 2007

      Wiley-IEEE Press eBook Chapters

      We propose a unified framework for the analysis of estimators of geometrical vector quantities and vector systems through a collection of performance measures. Unlike standard performance indicators, these measures have intuitive geometrical and physical interpretations, are independent of the coordinate reference frame, and are applicable to arbitrary parameterizations of the unknown vector or system of vectors. For each measure, we derive both finite-sample and asymptotic lower bounds that hold for large classes of estimators and serve as benchmarks for the assessment of estimation algorithms. Like the performance measures themselves, these bounds are independent of the reference coordinate frame, and we discuss their use as system design criteria. View full abstract»

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      Properties of Quadratic Covariance Bounds

      Page(s): 480 - 484
      Copyright Year: 2007

      Wiley-IEEE Press eBook Chapters

      In this paper we investigate the properties of quadratic covariance bounds for parametric estimators. The Cramér-Rao, Bhattacharyya, and Barankin bounds have this quadratic structure and the properties of these bounds are uniquely determined by their respective score functions. We enumerate some characteristics of score functions which generate tight bounds. We also introduce projection operator and integral/kernel representations for this class of quadratic covariance bounds. These representations are useful as analysis and synthesis tools. In the last section of this paper we address the issue of efficiency for this class of bounds. View full abstract»

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      Applications: Static Parameters

      Page(s): 485
      Copyright Year: 2007

      Wiley-IEEE Press eBook Chapters

      Bayesian Bounds provides a collection of the important papers dealing with the theory and application of Bayesian bounds. The book is essential to both engineers and statisticians whether they are practitioners or theorists. Each part of the book is introduced with the contributions of each selected paper and their interrelationship. View full abstract»

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      Fundamental Limitations in Passive Time Delay EstimationPart I: NarrowBand Systems

      Page(s): 487 - 501
      Copyright Year: 2007

      Wiley-IEEE Press eBook Chapters

      Time delay estimation of a noise-like random signal observed at two or more spatially separated receivers is a problem of considerable practical interest in passive radar/sonar applications. A new method is presented to analyze the mean-square error performance of delay estimation schemes based on a modified (improved) version of the Ziv-Zakai lower bound (ZZLB). This technique is shown to yield the tightest results on the attainable system performance for a wide range of signal-to-noise ratio (SNR) conditions. For delay estimation using narrow-band (ambiguity-prone) signals, the fundamental result of this study is illustrated in Fig. 3. The entire domain of SNR is divided into several disjoint segments indicating several distinct modes of operation. If the available SNR does not exceed SNR1, signal observations from the receiver outputs are completely dominated by noise thus essentially useless for the delay estimation. As a resuit, the attainable mean-square error -2 is bounded only by the a priori parameter domain. If SNR1 > SNR > SNR2, the modified ZZLB coincides. with the Barankin bound. In this regime differential delay observations are subject to ambiguities. If SNR < SNR3 the modified ZZLB coincides with the Cramer-Rao lower bound indicating that the ambiguity in the differential delay estimation can essentially be resolved. The transition from the ambiguity-dominated mode of operation to the ambiguity-free mode of operation starts at SNR2 and ends at SNR3. This is the threshold phenomenon in time delay estimation. The various deflection points SNRi and the various segments of the bound (Fig. 3) are given as functions of such important system parameters as time-bandwidth product (WT), signal bandwidth to center frequency ratio (W/0) and the number of half wavelengths of the signal center frequency contained in the spacing between receivers. With this information the composite bound illustrated in Fig. 3 provides the most complete characterization of the attainable system performance under any prespecified SNR conditions. View full abstract»

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      Lower Bound on the Achievable DSP Performance for Localizing StepLike Continuous Signals in Noise

      Page(s): 502 - 508
      Copyright Year: 2007

      Wiley-IEEE Press eBook Chapters

      Estimating the time of arrival (TOA) of step-like signals (e.g., a rectangular pulse), which are, theoretically, of infinite bandwidth, is essential for many applications. In modern signal processing, the TOA estimator is implemented by digital signal processing (DSP) techniques. Existing tools for studying the TOA estimation performance do not take into consideration the estimation error caused by the finite sampling rate of the system. In this paper, we present a new Cramér-Rao type lower bound that is used to evaluate the achievable performance of TOA estimation in a given processing sampling rate. We use it to refer to the important question of what processing sampling rate to use when localizing a step-like signal. We show that for a given signal-to-noise ratio (SNR), there exists a certain sampling rate threshold beyond which performance does not improve by increasing the sampling rate, and we show how to find it. View full abstract»

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      A Survey of Time Delay Estimation Performance Bounds

      Page(s): 509 - 515
      Copyright Year: 2007

      Wiley-IEEE Press eBook Chapters

      In this paper we review performance bounds, as well as some current trends, in time delay estimation (TOE). Research over several decades reveals that a few key parameters determine TOE performance. The most basic are the signal-to-noise ratio (SNR), and the signal time-bandwidth (TB) product; larger values for each are desirable. The Cramér-Rao bound (CRR) reveals asymptotic maximum-likelihood estimation (MLE) behavior with respect to TB and SNR. At moderate to lower SNR, TOEs generally break down as ambiguities arise due to increased noise and the cross-correlation of the signal, causing the TOE to deviate (often quite sharply) away from the CRB. Because it is a local bound, the eRB does not indicate the threshold behavior, and Ziv-Zakai and other bounds have been developed to handle this. When TO is measured between multiple sensors, the coherence between them can fundamentally limit the result, an effect that occurs in acoustics due to the turbulent atmosphere. We discuss modifications to the classical bounds that accommodate the coherence loss, and reveal a threshold coherence phenomenon. When communications and other signals are utilized for TOE, they may have significant nuisance parameters, including carrier uncertainty, unknown symbols, as well as effects due to an unknown channel. Recent TDE performance limits reveal the effect of these parameters for various signal models, including the impact of diversity channels on TDE. View full abstract»

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      Bayesian Bounds for MatchedField Parameter Estimation

      Page(s): 516 - 527
      Copyright Year: 2007

      Wiley-IEEE Press eBook Chapters

      Matched-field methods concern estimation of source locations and/or ocean environmental parameters by exploiting full wave modeling of acoustic waveguide propagation. Because of the nonlinear parameter-dependence of the signal field, the estimate is subject to ambiguities and the sidelobe contribution often dominates the estimation error below a threshold signal-to-noise ratio (SNR). To study the matched-field performance, three Bayesian lower bounds on mean-square error are developed: the Bayesian Cramér-Rao bound (BCRB), the Weiss-Weinstein bound (WWB), and the Ziv-Zakai bound (ZZB). Particularly, for a multiple-frequency, multiple-snapshot random signal model, a closed-form minimum probability of error associated with the likelihood ratio test is derived, which facilitates error analysis in a wide scope of applications. Analysis and example simulations demonstrate that 1) unlike the local CRB, the BCRB is not achieved by the maximum likelihood estimate (MLE) even at high SNR if the local performance is not uniform across the prior parameter space; 2) the ZZB gives the closest MLE performance prediction at most SNR levels of practical interest; 3) the ZZB can also be used to determine the necessary number of independent snapshots achieving the asymptotic performance of the MLE at a given SNR; 4) incoherent frequency averaging, which is a popular multitone processing approach, reduces the peak sidelobe error but may not improve the overall performance due to the increased ambiguity baseline; and finally, 5) effects of adding additional parameters (e.g.,environmental uncertainty) can be well predicted from the parameter coupling. View full abstract»

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      Barankin Bounds for Source Localization in an Uncertain Ocean Environment

      Page(s): 529 - 539
      Copyright Year: 2007

      Wiley-IEEE Press eBook Chapters

      Ambiguity surfaces for underwater acoustic matched-field processing are prone to having high secondary peaks. This leads to anomalous source localization estimates below a threshold signal-to-noise ratio (SNR) at which the performance rapidly departs from that predicted by the Cramér-Rao lower bound (CRLB). In this paper, Barankln bounds are used to predict the threshold SNR under two different models including known or uncertain shallow-water environments and monochromatic or random narrowband sources. Evaluation of the Barankln bound suggests that although asymptotic localization performance degrades with increasing environmental uncertainty, the threshold SNR is relatively unaffected. View full abstract»

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      A Bayesian Approach to Array Geometry Design

      Page(s): 540 - 544
      Copyright Year: 2007

      Wiley-IEEE Press eBook Chapters

      In this paper, weconsider the design of planar arrays that optimize direction-of-arrival (DOA) estimation performance. We assume that the single-source DOA is a random variable with a known prior probability distribution, and the sensors of the array are constrained to lie in a region with an arbitrary boundary. The Cramér-Rao Bound (CRB) and the Fisher Information Matrix (FIM) for single-source DOA constitute the basis of the optimality criteria. We relate the design criteria to a Bayesian CRB criterion and to array beamwidth; we also derive closed-form expressions for the design criteria when the DOA prior is uniform on a sector of angles. We show that optimal arrays have elements on the constraint boundary, thus providing a reduced dimension iterative solution procedure. Finally, we present example designs. View full abstract»

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      Optimization of Element Positions for direction Finding with Sparse Arrays

      Page(s): 545 - 548
      Copyright Year: 2007

      Wiley-IEEE Press eBook Chapters

      Sparse arrays are attractive for direction-Of-Arrival (DOA) estimation since they can provideaccurate estimates at a low cost. A problem of great interest in this matter is to determinethe element positions that yield the best DOA estimation performance. A major difficulty with this problem is to define a suitable performance measure to optimize. In this paper, a novel criterion is proposed for optimizing element positions. The ambiguity threshold of the Weiss-Weinstein Bound (WWB) is used to optimize the element positionsof a sparse linear array. The array obtainedfrom the optimization is compared withsome other sparse arraystructuresthat have been proposed in the literature. View full abstract»

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      DirectionOfArrival Estimation Using Separated SubarraysThis workwassupported in part by Ericsson Microwave Systems AS

      Page(s): 549 - 553
      Copyright Year: 2007

      Wiley-IEEE Press eBook Chapters

      In this paper we study direction-Of-Arrival (DOA) estimation with a particular class of sparse linear arrays, characterized by two widely separated subarrays. Since a large array aperture is obtained with a small number of elements, this structure can provide very accurate angle estimates at a reasonable cost, but at the expense of near ambiguities. Thepotential and theoretical limitations in DOA estimation with this class of arrays are investigated. We evaluate the performances of DOA estimation algorithms and compare with theoretical bounds, taking the effects of near ambiguities into account. View full abstract»

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      Comparison of performance bounds for DOA estimation

      Page(s): 554 - 557
      Copyright Year: 2007

      Wiley-IEEE Press eBook Chapters

      For a single signal, numerical comparison of the Bayesian Cramer-Rao lower bound, the Chazan-Ziv-Zakai lower bound and the Weiss-Weinstein lower bound on DOA estimation MSE shows that the CZZLB is the tightest lower bound and can accurately predict the threshold SNR, which is a critical system design parameter. The analysis is extended to two signals where the multiple parameter WWLB is applicable but appears to be a weak lower bound. Simulation results show that below the threshold SNR, a maximum likelihood DOA estimation procedure such as the Expectation Maximization(EM) algorithm seeded by the Alternating Projection Maximization (APM) algorithm should be used to provide the best DOA estimation performance. Above the threshold SNR, the use of the statistically efficient MUSIC algorithm is adequate. View full abstract»

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      On Geolocation Accuracy with Prior Information in Nonlineofsight Environment

      Page(s): 558 - 561
      Copyright Year: 2007

      Wiley-IEEE Press eBook Chapters

      Non-line-of-sight (NLOS) geolocation becomes an important issue with the fast development of mobile communications in recent years. Several methods have been proposed to address this problem. However, a comprehensive study on the best geolocation accuracy that these methods may possibly achieve is called for. In [1], [2], we reported a unified analysis of the Cramer-Rao Lower Bound (CRLB) and achievable bounds applicable to NLOS geolocation, assuming no prior information on the mobile station (MS) position or NLOS induced paths is available. In practice, however,we often have some information about these parameters beforehand. In this paper, we derive a lower bound for the geolocation accuracy in the presence of such prior information, and explore its physical interpretation. Some numerical examples are discussed. View full abstract»

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      On the Application of the CramerRao and Detection, Theory Bounds to Mean Square Error of, Symbol Timing Recovery

      Page(s): 562 - 570
      Copyright Year: 2007

      Wiley-IEEE Press eBook Chapters

      The Cramer-Rao bound is often used to lower bound the mean square error of symbol timing estimators. However, the valid application of this bound requires that the estimator be unbiased and that the signaling waveform be sufficiently smooth. In some cases of practical interest, these conditions are not satisfied. Furthermore, the Cramer-Rao bound is known to yield poor results for small signal-to-noise ratios. A modified Cramer-Rao bound is derived that takes account of the fact that the symbol timing estimate is restricted to a finite interval of one symbol duration. A detection theory bound is also applied to the symbol timing problem. Results obtained using the Cramer-Rao and detection theory bounds are compared. It is shown that the detection theory bound is superior to the traditionally used Cramer-Rao bound. In particular, the detection theory bound gives useful results for sharp pulse shapes and shows a dependence of the mean square error on the data sequence. It also yields meaningful results for small signal-to-noise ratios. View full abstract»

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      Barankin Bound for Range and Doppler Estimation Using Orthogonal Signal Transmission

      Page(s): 571 - 576
      Copyright Year: 2007

      Wiley-IEEE Press eBook Chapters

      In this paper, the Barankin bound for performance evaluation of target range and Doppler estimation by an active radar (or sonar) is derived. The Barankin bound is analyzed for two signal cases: pulse train with identical (coherent) signals between pulses, and pulse train with orthogonal coded signals. At high pulse repetition frequencies (PRF's), identical signal transmission results in high sidelobes in the ambiguity function, while orthogonal signal transmissions allows to reduce the sidelobes and the ambiguity level. The Barankin bound is shown to be an efficient tool for system analysis in the presence of ambiguities. It is shown that for the identical signals case, the threshold signal-to-noise ratio (SNR) predicted by the Barankin bound is higher than the orthogonal signals case. The results are accompanied by maximum likelihood (ML) simulations which show that the Barankin bound predicts the threshold SNR with a good accuracy. It is shown that at high SNR's, the Barankin bound, the Cramér-Rao bound and the ML coincide. View full abstract»

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      Barankin Bounds for Target Localization by MIMO Radars

      Page(s): 577 - 580
      Copyright Year: 2007

      Wiley-IEEE Press eBook Chapters

      Multiple-input multiple-output (MIMO) radar/sonar systems transmit signals coded in time and space domains and have several advantages upon conventional systems. In this paper, the Barankin bound on target localization errors is used for analysis of the threshold signal-to-noise ratio (SNR) of MIMO radar/sonar systems. It is shown that the threshold SNR of MIMO systems with spatially orthogonal transmit signals, is significantly lower than with coherent transmit signals. Orthogonal signal transmission results in lower threshold SNR, because of the lower sidelobes in the likelihood function. The lower sidelobes are achieved, since MIMO configuration allows to process the signal in both transmit and receive modes. Simulation results show that the threshold SNR obtained by orthogonal signal transmission is lower by more than 10 dB compared to coherent signal transmission. View full abstract»

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      WeissWeinstein Bound for DataAided Carrier Estimation

      Page(s): 581 - 584
      Copyright Year: 2007

      Wiley-IEEE Press eBook Chapters

      This letter investigates Bayesian bounds on the mean-square error (MSE) applied to a data-aided carrier estimation problem. The presented bounds are derived from a covariance inequality principle: the so-called Weiss and Weinstein family. These bounds are of utmost interest to find the fundamental MSE limits of an estimator, even for critical scenarios (low signal-to-noise ratio and/or low number of observations). In a data-aided carrier estimation problem, a closed-form expression of the Weiss-Weinstein bound (WWB) that is known to be the tightest bound of the Weiss and Weinstein family is given. A comparison with the maximum likelihood estimator and the other bounds of the Weiss and Weinstein family is given. The WWB is shown to be an efficient tool to approximate this estimator's MSE and to predict the well-known threshold effect. View full abstract»

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      Nonlinear Stochastic Dynamic Systems

      Page(s): 585 - 586
      Copyright Year: 2007

      Wiley-IEEE Press eBook Chapters

      Bayesian Bounds provides a collection of the important papers dealing with the theory and application of Bayesian bounds. The book is essential to both engineers and statisticians whether they are practitioners or theorists. Each part of the book is introduced with the contributions of each selected paper and their interrelationship. View full abstract»

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      On the Differential Equations Satisfied by Conditional Probability Densities of Markov Processes, with ApplicationsReceived by the editors January 24, 1964. This research was supported by the United States Air Force, under Contracts No. AF 33(657)8559, and No. AF 49(638)1206.

      Page(s): 587 - 600
      Copyright Year: 2007

      Wiley-IEEE Press eBook Chapters

      Bayesian Bounds provides a collection of the important papers dealing with the theory and application of Bayesian bounds. The book is essential to both engineers and statisticians whether they are practitioners or theorists. Each part of the book is introduced with the contributions of each selected paper and their interrelationship. View full abstract»

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      A Bayesian Approach to Problems in Stochastic Estimation and Control

      Page(s): 601 - 607
      Copyright Year: 2007

      Wiley-IEEE Press eBook Chapters

      In this paper, a general class of stochastic estimation and control problems is formulated from the Bayesian Decision-Theoretic viewpoint. A discussion as to how these problems can be solved step by step in principle and practice from this approach is presented. As a specific example, the closed form Wiener-Kalman solution for linear estimation in Gaussian noise is derived. The purpose of the paper is to show that the Bayesian approach provides; 1) a general unifying framework within which to pursue further researches in stochastic estimation and control problems, and 2) the necessary computations and difficulties that must be overcome for these problems. An example of a nonlinear, non-Gaussian estimation problem is also solved. View full abstract»

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      On the Estimation of State Variables and Parameters for Noisy Dynamic SystemsReceived November 7, 1963. The work described in this paper is based on a dissertation submitted in partial fulfillment of the requirements for the Sc.D. degree in the Department of Electrical Engineering, Massachusetts Institute of Technology, Cambridge.

      Page(s): 608 - 615
      Copyright Year: 2007

      Wiley-IEEE Press eBook Chapters

      The problem of estimating state variables and parameters is considered for discrete-time systems in the presence of random disturbances and measurement noise. The solution of the linear problem is given and an approximation technique is developed for nonlinear systems. A dynamic programming formulation of the estimation problem is also developed. View full abstract»

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      Bounds on the Accuracy Attainable in the Estimation of Continuous Random Processes

      Page(s): 616 - 623
      Copyright Year: 2007

      Wiley-IEEE Press eBook Chapters

      The problem of estimating a message, a(t), which is a sample function from a continuous Gaussian random process is considered. The message to be estimated may be contained in the transmitted signal in a nonlinear manner. The signal is corrupted by additive noise before observation. The received waveform is available over some observation interval [Ti , Tf ]. We want to estimate a(t) over the same interval. Instead of considering explicit estimation procedures, we find bounds on how well any procedure could do. The principle results are as follows: 1) a lower bound on the mean-square estimation error. This bound is a generalization of bounds derived previously by Cramer, Rao, and Slepian for estimating finite sets of parameters. 2) The bound is evaluated for several practical examples. Possible extension and applications are discussed briefly. View full abstract»

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      Filtering and control performance bounds with implications on asymptotic separation

      Page(s): 624 - 630
      Copyright Year: 2007

      Wiley-IEEE Press eBook Chapters

      A bound is derived on the accuray in causally estimating a Gaussian process from nonlinear observations. Both additive Gaussian noiac and Poisson observations are included. The bound is used to study the control of a stochastic linear dynamical system with nonlinear observations of either type and an average quadratic cost. An asymptotic Separation Theorem is established showing that a linear feedback cootrol law. involving a state estimate. is asymptotically optimum as the accuracy of the state estimate approaches the bound. View full abstract»

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      A Lower Bound on the Estimation Error for Markov Processes

      Page(s): 631 - 634
      Copyright Year: 2007

      Wiley-IEEE Press eBook Chapters

      A lower bound on the minimal mean-square error in estimating nonlinear Markov processes is presented. The. bound holds for causal and uncausal flltering. The derivation is based on the Van Trees' version of the Cramér-Rao inequality. View full abstract»

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      A Lower Bound on the Estimation Error for Certain Diffusion Processes

      Page(s): 635 - 642
      Copyright Year: 2007

      Wiley-IEEE Press eBook Chapters

      A lower bound on the minimal mean-square error in estimating nonlinear diffusion processes is derived. The bound holds for causal and noncausal filtering. View full abstract»

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      Error Bounds for the Nonlinear Filtering of Signals with Small Diffusion Coefficients

      Page(s): 643 - 654
      Copyright Year: 2007

      Wiley-IEEE Press eBook Chapters

      New upper and lower bounds for the nonlinear filtering problem are presented. The lower bounds are especially useful in the region of small diffusion coefficients where previously known bounds are inefficient. The upper and lower bounds are shown to be tight. An example demonstrating the tightness of the bounds is presented. View full abstract»

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      The CramrRao Estimation Error Lower Bound Computation for Deterministic Nonlinear Systems

      Page(s): 655 - 656
      Copyright Year: 2007

      Wiley-IEEE Press eBook Chapters

      For condouous-time nonlinear deterministie system models with diserete nonlinear measuremeuts in additive Gaussian white noise,the extended Kalman filter (EKF) convariance propagation equations linearized about the true unknown trajectory provide the Cramér-Rao lower bound to the estimadon error covariance matrix. A useful application is establishing the optimum filter performance for a given nonlinear estimation problem by developing a simulation of the nonlinear system and an EKF linearized about the true trajectory. View full abstract»

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      Two Lower Bounds on the Covariance for Nonlinear Estimation Problems

      Page(s): 657 - 659
      Copyright Year: 2007

      Wiley-IEEE Press eBook Chapters

      Two convariance lower bounds for nonlinear state estimation problems are presented. These bounds are based upon the Cramer-Rao bound for treating nuisance parameters and they can be applied to filtering, smoothing. and prediction problems. The tightness of these bounds are examined using a nonlinear system where the recursive equation for covariance computation can be obtained. These results are also compared wirh the bound of Bobrovsky and Zakai. View full abstract»

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      Excerpts from Part II of Detection, Estimation, and Modulation Theory

      Page(s): 660 - 678
      Copyright Year: 2007

      Wiley-IEEE Press eBook Chapters

      This chapter contains sections titled:
      Bounds on the Performance of Analog Messagetransmission Systems
      Rate-distortion Bound
      Performance Comparison: Infinite-bandwidth Channel View full abstract»

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      Lower and Upper Bounds on the Optimal Filtering Error of Certain Diffusion Processes

      Page(s): 679 - 685
      Copyright Year: 2007

      Wiley-IEEE Press eBook Chapters

      The optimal nonlinear filtering of certain vector-valued diffusion processes embedded in white noise is considered. We derive upper and lower bounds on the minimal causal mean-square error. The derivation of the lower bound is based on information-theoretic considerations, namely the rate-dlstortlon function (-entropy), The upper bounds are based on linear-filtering arguments. It is demonstrated that for a wide class of high-precision systems, the upper and lower bounds are tight within a factor of 2 or better. View full abstract»

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      Posterior CramrRao Bounds for DiscreteTime Nonlinear Filtering

      Page(s): 686 - 696
      Copyright Year: 2007

      Wiley-IEEE Press eBook Chapters

      A mean-square error lower bound for the discretetime nonlinear filtering problem is derived based on the Van Trees (posterior) version of the Cramér-Rao inequality. This lower bound is applicable to multidimensional nonlinear, possibly non-Gaussian, dynamical systems and is more general than the previous bounds in the literature. The case of singular conditional distribution of the one-step-ahead state vector given the present state is considered. The bound is evaluated for three important examples: the recursive estimation of slowly varying parameters of an autoregressive process, tracking a slowly varying frequency of a single cisoid in noise, and tracking parameters of a sinusoidal frequency with sinusoidal phase modulation. View full abstract»

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      Filtering, predictive, and smoothing CramrRao bounds for discretetime nonlinear dynamic systemsThis is an expanded version of the paper presented at the 14th IFAC World Congress, 59 July 1999, Beijing. This paper was recommended for publication in revised form by Associate Editor Hkan Hjalmarsson under the direction of Editor Torsten Sderstrm.

      Page(s): 697 - 710
      Copyright Year: 2007

      Wiley-IEEE Press eBook Chapters

      Cramér-Rao lower bounds for the discrete-time nonlinear state estimation problem are treated. The Cramér-Rao bound for the mean-square error matrix of a state estimate is particularly important for quality evaluation of nonlinear state estimators as it represents a limit of cognizability of the state. Recursive relations for filtering, predictive, and smoothing Cramér-Rao bounds are derived to establish a unifying framework for several previously published derivation procedures and results. Lower bounds for systems with unknown parameters are newly provided. Computation of filtering, predictive, and smoothing Cramér-Rao bounds, their mutual comparison and utilization for quality evaluation of some nonlinear filters are shown in numerical examples. © 2001 Elsevier Science Ltd. All rights reserved. View full abstract»

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      Recursive WeissWeinstein Lower Bounds for DiscreteTime Nonlinear Filtering

      Page(s): 711 - 716
      Copyright Year: 2007

      Wiley-IEEE Press eBook Chapters

      Being essentially free from regularity conditions, the Weiss-Weinstein lower bound can be applied to a larger class of systems than the well-known Cramér-Rao lower bound. Thus, this bound is of special interest in applications involving hybrid systems, i.e., systems with both continuously and discretely-distributed parameters, which can represent in practice fault-prone systems. However, the requirement to know explicitly the joint distribution of the estimated parameters with all the measurements renders the application of the Weiss-Weinstein lower bound to Markovian dynamic systems impractical. A new algorithm is presented in this paper for the recursive computation of the Weiss-Weinstein lower bound for a wide class of Markovian dynamic systems. The algorithm makes use of the transitional distribution of the Markovian state process, and the distribution of the measurements at each time step conditioned on the appropriate states, both easily obtainable from the system equations. For systems satisfying the Cramér-Rao lower bound regularity conditions, and for a particular choice of its parameters, it is shown that the recursive Weiss-Weinstein lower bound reduces to the recently introduced recursive Cramer-Rao lower bound. Moreover, it is shown that several recently reported lower bounds, derived for systems with fault-prone measurements, are special cases of the proposed recursive Weiss-Weinstein lower bound. View full abstract»

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      Tighter Alternatives to the CramerRao Lower Bound for DiscreteTime Filtering

      Page(s): 717 - 722
      Copyright Year: 2007

      Wiley-IEEE Press eBook Chapters

      The Cramér-Rao Lower Bound establishes a fundamental perfonnance baseline for gauging parameter estimation accuracy in tracking and data fusion. However, it is known to be a weak lower bound for some problems. This paper presents a set of tighter alternatives: the Bhattacharyya,Bobrovsky-Zakai and Weiss-Weinstein lower bounds. General mathematical expressions are obtained for these bounds and their calculation is described. Then the bounds are applied to a nonBnear/nonGaussian estimation problem. It is found that the alternative bounds are tighter than the Cramér-Rao bound, but they are still somewhat conservative. View full abstract»

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      A Tutorial on Particle Filters for Online Nonlinear/NonGaussian Bayesian Tracking

      Page(s): 723 - 737
      Copyright Year: 2007

      Wiley-IEEE Press eBook Chapters

      Increasingly, for many application areas, it is becoming important to include elements of nonlinearity and non-Gaussianity in order to model accurately the underlying dynamics of a physical system. Moreover, it is typically crucial to process data on-line as it arrives, both from the point of view of storage costs as well as for rapid adaptation to changing signal characteristics. In this paper, we review both optimal and suboptimal Bayesian algorithms for nonlinear/non-Gaussian tracking problems, with a focus on particle filters. Particle filters are sequential Monte Carlo methods based on point mass (or particle) representations of probability densities, which can be applied to any state-space model and which generalize the traditional Kalman filtering methods. Several variants of the particle filter such as SIR, ASIR, and RPF are introduced within a generic framework of the sequential importance sampling (SIS) algorithm. These are discussed and compared with the standard EKF through an illustrative example. View full abstract»

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      Applications: Nonlinear Dynamic Systems

      Page(s): 739
      Copyright Year: 2007

      Wiley-IEEE Press eBook Chapters

      Bayesian Bounds provides a collection of the important papers dealing with the theory and application of Bayesian bounds. The book is essential to both engineers and statisticians whether they are practitioners or theorists. Each part of the book is introduced with the contributions of each selected paper and their interrelationship. View full abstract»

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      Utilization of Modified Polar Coordinates for BearingsOnly Tracking

      Page(s): 741 - 752
      Copyright Year: 2007

      Wiley-IEEE Press eBook Chapters

      Previous studies have shown that the Cartesian coordinate extended Kalman filter exhibits unstable behavior characteristics when utilized for bearings-only target motion analysis (TMA). In contrast. fonnulating the TMA estimation problem in modified polar (MP) coordinates leads to an extended Kalman filter which is both stable and asymptotically unbiased. Exact state equations for the MP filter are derived without imposing any restrictions on own-ship motion; thus, prediction accuracy inherent in the traditional Cartesian formulation is completely preserved. In addition, these equations reveal that MP coordinates are well-suited for bearings-only TMA because they automatically decouple observable and unobservable components of the estimated state vector.Such decoupling is shown to prevent covariance matrix ill-condttioning, which is the primary cause of filter instability. Further investigation also confirms that the MP state estimates are asymptotically unbiased. Realistic simulation data are presented to support these findings and to compare algorithm performance with respect to the Cramer-Rao lower bound (ideal) as well as the Cartesian and pseudolinear filters. View full abstract»

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      A Performance Bound for Mamoeuvring Target Tracking Using BestFitting Gaussian Distributions

      Page(s): 753 - 760
      Copyright Year: 2007

      Wiley-IEEE Press eBook Chapters

      In this paper, we consider the problem of calculating the Posterior Cramér-Rao Lower Bound (PCRLB) in the case of tracking a manoeuvring target. In a recent article [1] the anthors calculated the PCRLB conditional on the manoeuvre sequence and then determined the bound as a weighted average, giving an unconditional PCRLB (referred to herein as the Enumer-PCRLB). However, we argue that this approach can produce an optimistic lower bound because the sequence of manoeuvres is implicitly assumed known. Indeed, in simulations we show that in tracking a target that can switch between a nearly constant-velocity (NCV) model and a coordinated turn (CT) model, the Enumer-PCRLB can be lower than the PCRLB in the case of tracking a target whose motion is governed purely by tbe NCV model. Motivated by this, in this paper we develop a general approach to calculating the manoeuvring target PCRLB based on utilizing best-fitting Gaussian distributions. The basis of the technique is, at each stage, to approximate the multi-modal prior target probability density function using a best-fitting Gaussian distribution. We present a recursive formula for calculating the mean and covarlanee of this Gaussian distribution, and demonstrate bow the covariance increases as a result of the potential manoeuvres. We are then able to calculate the PCRLB using a standard Riccati-like recursion. Returning to our previous example, we show that this best-fitting Gaussian approach gives a bound that shows the correct qualitative behavior, namely that the bound is greater when the target can manoeuvre. Moreover, for simulated scenarios taken from (1], we show that the best-fitting Gaussian PCRLB is both greater than the existing bound (the Enumer-PCRLB) and more consistent with the performance of the variable structure interacting multiple model (VS-IMM) tracker utilized therein. View full abstract»

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      Optimal Observer Maneuver for BearingsOnly Tracking

      Page(s): 761 - 772
      Copyright Year: 2007

      Wiley-IEEE Press eBook Chapters

      In bearings-only tracking, observer maneuver is critical to ensure observability and to obtain an accurate target localization. Here, optimal control theory is applied to the determination of the course of a constant speed observer that minimizes an accuracy criterion deduced from the Fisher information matrix (FIM). Necessary conditions for optimal maneuver (Euler equations) are established and resolved, partly by analytical means and partly by an iterative numerical procedure. Examples of optimal observer maneuvers are presented and discussed. View full abstract»

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      Posterior CramrRao Bound for Tracking Target Bearing

      Page(s): 773 - 778
      Copyright Year: 2007

      Wiley-IEEE Press eBook Chapters

      The posterior Cramér-Rao bound on the mean square error in tracking the bearing, bearing rate, and power level of a narrowband source is developed. The formulation uses a linear process model with additive noise and a general nonlinear measurement model, where the measurements are the sensor array data. The joint Bayesian Cramer-Rao bound on the state variables over the entire observation interval is formulated and a recursive bound on the state variables as a function of time is derived based on the nonlinear filtering bound developed by Tichavsky et al (1998) and analyzed by Ristic et al (2004). The bound is shown to have the same form as when the measurements are bearing and power estimates with variance equal to the deterministic Cramér-Rao bound for a single data snapshot. The bound is compared against simulated performance of the maximum a posteriori penalty function (MAP-PF) tracking algorithm developed in Zamich et al (2001). View full abstract»

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      Matrix CRLB Scaling Due to Measurements of Uncertain Origin

      Page(s): 779 - 789
      Copyright Year: 2007

      Wiley-IEEE Press eBook Chapters

      In many estimation situations, measurements are of uncertain origin. This is best exemplified by the target-tracking situation in which at each scan (of a radar, sonar, or electro-optical sensor), a number of measurements are obtained, and it is not known which, if any, of these is target originated. The source of extraneous measurements can be false alarms-especially in low-SNR situations that force the detector at the end of the signal processing chain to operate with a reduced threshold-or spurious targets. In several earlier papers, the surprising observation was made that the Cramér-Rao lower bound (CRLB) for the estimation of a fixed parameter vector (e.g., initial position and velocity) that characterizes the target motion, for the special case of multidimensional measurements in the presence of additive white Gaussian noise, is simply a multiple of that for the case with no uncertainty. That is, there is a scalar information-reduction factor; this is particularly useful as it allows comparison in terms of a scalar. In this paper, we explore this result to determine how wide the class of such problems is. It turns out to include many non-Gaussian situations. Simulations corroborate the analysis. View full abstract»

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      PCRLB for Tracking in Cluttered Enironments: Measurement Sequence Conditioning Approach

      Page(s): 790 - 813
      Copyright Year: 2007

      Wiley-IEEE Press eBook Chapters

      We consider the problem of calculating the posterior Cramï¿¿r-Rao lower bound (PCRLB) for tracking in cluttered domains in which there can be both missed detections and false alarms. We present a novel formulation of the PCRLB in which we initially determine a bound conditional on the sequence of measurements available. We then create an unconditional bound as a weighted average of these conditional PCRLBs. This new bound is proven to be less optimistic than the standard formulation of the PCRLB for cluttered environments recently developed [1, 2] and will therefore better predict optimal estimator performance. At each stage, the conditional PCRLB must evaluate the effect of the uncertain measurements, and we extend previous work [2] to show that the measurement origin uncertainty manifests itself as a single information reduction factor (IRF) that is dependent on the number of measurements available. We also present some useful approximations when the false alarm rate is low. Simulations then consider the problems of 1) determining the CRLB for the point of impact of a ballistic missile, and 2) determining the PCRLB for tracking a nearly constant-velocity (NCV) target in a high clutter environment. In each case, we compare the new bound with the standard approach, and as expected the new CRLB/PCRLB can be seen to be less optimistic. Moreover, in case 1) we compare the new CRLB with a heuristic bound specially constructed for this problem, and a maximum likelihood estimator (MLE). The new bound both compares favorably with the heuristic bound, and shows close agreement with the performance of the MLE. The new bound is therefore an accurate predictor of filter performance in this case. In example 2) we demonstrate some interesting features of the new theory. Of particular interest we determine both precisely when the new bound will be significantly greater than the sta ndard bound and when the two bounds will be virtually identical. This is useful in determining when the new approach, with its greater computational burden, should be preferred to the established approach. We conclude that the novel PCRLB formulation introduced herein represents an exciting development in the determination of RMSE performance bounds in cluttered environments. View full abstract»

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      Dynamic CramrRao Bound for Target Tracking in Clutter

      Page(s): 814 - 827
      Copyright Year: 2007

      Wiley-IEEE Press eBook Chapters

      Recently, there have been several new results for an old topic, the Cramér-Rao lower bound (CRLB). Specifically, it has been shown that for a wide class of parameter estimation problems (e.g. for objects with deterministic dynamics) the matrix CRLB, with both measurement origin uncertainty (i.e., in the presence of false alarms or random clutter) and measurement noise, is simply that without measurement origin uncertainty times a scalar information reduction factor (IRF). Conversely, there has arisen a neat expression for the CRLB for state estimation of a stochastic dynamic nonlinear system (i.e., objects with a stochastic motion); but this is only valid without measurement origin uncertainty. The present paper can be considered a marriage of the two topics: the clever Riccati-like form from the latter is preserved, but it includes the IRF from the former. The effects of plant and observation dynamics on the CRLB are explored. Further, the CRLB is compared via simulation to two common target tracking algorithms, the probabilistic data association filter (PDAF) and the multi-frame (N-D) assignment algorithm. View full abstract»

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      CramrRao lower bound for tracking multiple targets

      Page(s): 828 - 833
      Copyright Year: 2007

      Wiley-IEEE Press eBook Chapters

      The derivation and computation of the theoretical Cramér-Rao lower bounds for multiple target tracking has traditionally been considered to be a notoriously difficult problem. The authors present a simple and exact solution based on the assumption that raw sensor data (before thresholding) are available. The multi-target tracking problem can then be formulated as recursive Bayesian track-before-detect estimation. The advantage of this formulation is that it is identical to nonlinear filtering, for which the exact posterior Cramér-Rao bound is already known. The paper presents several numerical examples in support of the theoretical findings. View full abstract»

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      Posterior CramerRao Bounds for MultiTarget Tracking

      Page(s): 834 - 846
      Copyright Year: 2007

      Wiley-IEEE Press eBook Chapters

      This study is concerned with multi-target tracking (MTT). The Cramér-Rao lower bound (CRB) is the basic tool for investigating estimation performance. Though basically defined for estimation of deterministic parameters, it has been extended to stochastic ones in a Bayesian setting. In the target tracking area, we have thus to deal with the estimation of the whole trajectory, itself described by a Markovian model. This leads up to the recursive formulation of the posterior CRB (PCRB). The aim of the work presented here is to extend this calculation of the PCRB to MTT under various assumptions. View full abstract»

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      Bounds on Performance for Multiple Target Tracking

      Page(s): 847 - 849
      Copyright Year: 2007

      Wiley-IEEE Press eBook Chapters

      Bounds on performance are derived for tracking in a dense multiple tarlet emironment. The best possible performance ean be estimated without implementing the optimal algorithm; this is analogous to the Cramér-Rao bound for nonlinear estimadon. The effects of multipie targets, clutter, false alarms, unresolved measurements, missed detections, and multiple sensors can be evaluated using these bounds. View full abstract»

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      Pointmass filter and CramerRao bound for TerrainAided Navigation

      Page(s): 850 - 855
      Copyright Year: 2007

      Wiley-IEEE Press eBook Chapters

      The nonlinear estimation problem in navigation using terrain height variations is studied. The optimal Bayesian solution to the problem is derived. The implementation is grid based, calculating the probability of a set of points on an adaptively dense mesh. The Cramer-Rao bound is derived. Monte Carlo simulations over a commercial map shows that the algorithm, after convergence, reaches the Cramer-Rao lower bound View full abstract»

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      Bayesian CramrRao Bounds For Multistatic Radar

      Page(s): 856 - 859
      Copyright Year: 2007

      Wiley-IEEE Press eBook Chapters

      The Bayesian Cramér-Rao bound (BCRB) on the mean square error in tracking the position and velocity of a moving target in a multistatic radar system is formulated and a recursive bound on the state variables as a function of time is derived based on the nonlinear filtering bound developed by Tichavsky et al (1998). The result is an error bound ellipse in the xy-plane that evolves as the target moves along its trajectory. The recursive BCRB provides an efficient technique to analyze various system design trade-offs including the effect of transmitter-receiver geometry, the contribution of each transmitter to tracking accuracy, the effect of angular estimation accuracy. View full abstract»

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      WeissWeinstein Lower Bounds for Markovian Systems. Part 2: Applications to FaultTolerant Filtering

      Page(s): 860 - 871
      Copyright Year: 2007

      Wiley-IEEE Press eBook Chapters

      Characterized by sudden structural changes, fault-prone systems are modeled using the framework of systems with switching parameters or hybrid systems. Since a closed-form mean-square optimal filtering algorithm for this class of systems does not exist, it is of particular interest to derive a lower bound on the state estimation error covariance. The well known Cramér-Rao bound is not applicable to fault-prone systems because of the discrete distribution of the fault indicators, which violates the regularity conditions associated with this bound. On the other hand, the Weiss-Weinstein lower bound is essentially free from regularity conditions. Moreover, a sequential version of the Weiss-Weinstein bound, suitable for Markovian dynamic systems, is presented by the authors in a companion paper. In the present paper, this sequential version is applied to several classes of fault-prone dynamic systems. The resulting bounds can be used to examine fault detectability and identifiability in these systems. Moreover, it is shown that several recently reported lower bounds for fault-prone systems are special cases of, or closely related to, the sequential version of the Weiss-Weinstein lower bound. View full abstract»

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      Combined cramrRao/WeissWeinstein Bound for Tracking Target Bearing

      Page(s): 872 - 876
      Copyright Year: 2007

      Wiley-IEEE Press eBook Chapters

      A recursive Bayesian Cramér-Rao/Weiss-Weinstein bound for the discrete-time nonlinear filtering problem for the special case when the state-space model consists of a linear process model and a general (nonlinear) measurement model is developed. This type of model arises in many applications including target tracking. It is often the case that the recursive Bayesian Cramér-Rao bound (BCRB) developed by Tichavsky et al is a good predictor of meansquare error performance for some of the state vector parameters, but is a weak bound for other components. The recursive Weiss-Weinstein bound (WWB) developed by Rapoport & Oshman and Reece & Nicholson offers a potentially higher bound but is generally more difficult to derive and implement, and its evaluation involves choosing "test points" in the parameter space. It becomes equal to the BRCB in the limiting case when the test points equal zero. A bound which combines the BCRB with the WWB using non-zero test points only for a subset of the state-vector components can provide as tight a bound as the WWB while keeping the complexity manageable. We first derive the recursive bound for the linear process/nonlinear measurement model, then apply the bound to the problem of tracking the bearing and bearing rate of a narrowband source using observations from a sparse linear array. The bound is compared to the recursive BCRB and to simulated tracking performance. The BCRWWB provides a tighter bound than the BCRB for the bearing tracking error, which is subject to ambiguities. View full abstract»

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      Statistical Literature

      Page(s): 877
      Copyright Year: 2007

      Wiley-IEEE Press eBook Chapters

      Bayesian Bounds provides a collection of the important papers dealing with the theory and application of Bayesian bounds. The book is essential to both engineers and statisticians whether they are practitioners or theorists. Each part of the book is introduced with the contributions of each selected paper and their interrelationship. View full abstract»

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      Applications of the van Trees inequality: A Bayesian CramrRao bound

      Page(s): 879 - 899
      Copyright Year: 2007

      Wiley-IEEE Press eBook Chapters

      We use a Bayesian version of the Cramér-Rao lower bound due to van Trees to give an elementary proof that the limiting distribution of any regular estimator cannot have a variance less than the classical information bound, under minimal regularity conditions. We also show how minimax convergence rates can be derived in various non- and semi-parametric problems from the van Trees inequality. Finally we develop multivariate versions of the inequality and give applications. View full abstract»

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      On CramrRao Type Integral Inequalities

      Page(s): 900 - 922
      Copyright Year: 2007

      Wiley-IEEE Press eBook Chapters

      A brief but comprehensive survey of recent work in the atea of cramér-Rao type integral inequalities leading to lower bounds for the risk of estimators in a Bayesian contest is given. View full abstract»

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      An Extension of the CramrRao InequalityResearch sponsored by the National Science Foundation under grant NSF G1858.

      Page(s): 923 - 936
      Copyright Year: 2007

      Wiley-IEEE Press eBook Chapters

      Bayesian Bounds provides a collection of the important papers dealing with the theory and application of Bayesian bounds. The book is essential to both engineers and statisticians whether they are practitioners or theorists. Each part of the book is introduced with the contributions of each selected paper and their interrelationship. View full abstract»

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      CramrRao bounds for Posterior variances

      Page(s): 937 - 942
      Copyright Year: 2007

      Wiley-IEEE Press eBook Chapters

      The paper obtains Cramér-Rao type bounds for posterior variances. Conditions are provided for attainability of these bounds. The present bounds are usually more widely applicable, and are often sharper than the ones reported in Schutzenberger (1957). Also, our results extend immediately to find lower bounds of posterior risks, and subsequently of Bayes risks of parameters of interest. Such results can be contrasted to the findings of Gart (1959) who obtained lower bounds for Bayes risks of estimators from which posterior risks are not easy to retrieve. We have provided also multiparameter generalization of our results. Finally, certain Bhattacharyya type bounds for posterior variances are also obtained. In contrast to the usual technique of differentiation under integration for finding Cramér-Rao bounds, our method of proof involves integration by parts. View full abstract»

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      References

      Page(s): 943 - 950
      Copyright Year: 2007

      Wiley-IEEE Press eBook Chapters

      Bayesian Bounds provides a collection of the important papers dealing with the theory and application of Bayesian bounds. The book is essential to both engineers and statisticians whether they are practitioners or theorists. Each part of the book is introduced with the contributions of each selected paper and their interrelationship. View full abstract»

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      Author Index

      Page(s): 951
      Copyright Year: 2007

      Wiley-IEEE Press eBook Chapters

      Bayesian Bounds provides a collection of the important papers dealing with the theory and application of Bayesian bounds. The book is essential to both engineers and statisticians whether they are practitioners or theorists. Each part of the book is introduced with the contributions of each selected paper and their interrelationship. View full abstract»