fundamental limitations in passive time delay estimation part i narrow band systems 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 snr 1 signal observations from the receiver outputs are completely dominated by noise thus essentially useless for the delay estimation as a result the attainable mean square error bar epsilon 2 is bounded only by the a priori parameter domain if snr 1 2 the modified zzlb coincides with the barankin bound in this regime differential delay observations are subject to ambiguities if snr snr 3 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 snr 2 and ends at snr 3 this is the threshold phenomenon in time delay estimation the various deflection points snr i 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

an accurate algebraic solution for moving source location using tdoa and fdoa measurements this paper proposes an algebraic solution for the position and velocity of a moving source using the time differences of arrival tdoas and frequency differences of arrival fdoas of a signal received at a number of receivers the method employs several weighted least squares minimizations only and does not require initial solution guesses to obtain a location estimate it does not have the initialization and local convergence problem as in the conventional linear iterative method the estimated accuracy of the source position and velocity is shown to achieve the crame spl acute r rao lower bound for gaussian tdoa and fdoa noise at moderate noise level before the thresholding effect occurs simulations are included to examine the algorithm s performance and compare it with the taylor series iterative method

on the accuracy and resolvability of vector parameter estimates in this paper we address the problem of fundamental limitations on resolution in deterministic parameters estimation we introduce a definition of resolvability based on probability and incorporating a requirement for accuracy unlike most existing definitions indeed in many application the key problem is to obtain distributions of estimates that are not only distinguishable but also accurate and compliant with a required precision we exemplify the proposed definition with estimators that produce normal estimates as in the conditional model for which the gaussianity and efficiency of maximum likelihood estimators mles in the asymptotic region of operation in terms of signal to noise ratio and or in large number of snapshots is well established even for a single snapshot in order to measure the convergence in distribution we derive a simple test allowing to check whether the conditional mles operate in the asymptotic region of operation last we discuss the resolution of two complex exponentials with closely spaced frequencies and compare the results obtained with the ones provided by the various statistical resolution limit released in the open literature

an asymptotically unbiased estimator for bearings only and doppler bearing target motion analysis bearings only bo and doppler bearing db target motion analysis tma attempt to obtain a target trajectory based on bearings and on doppler and bearing measurements respectively from an observer to the target the bo tma and db tma problems are nontrivial because the measurement equations are nonlinearly related to the target location parameters the pseudolinear formulation provides a linear estimator solution but the resulting location estimate is biased the instrumental variable method and the numerical maximum likelihood approach can eliminate the bias their convergence behavior however is not easy to control this paper proposes an asymptotically unbiased estimator of the tracking problem the proposed method applies least squares minimization on the pseudolinear equations with a quadratic constraint on the unknown parameters the resulting estimator is shown to be solving the generalized eigenvalue problem the proposed solution does not require initial guesses and does not have convergence problems sequential forms of the proposed algorithms for both bo tma and db tma are derived the sequential algorithms improve the estimation accuracy as a new measurement arrives and do not require generalized eigenvalue decomposition for solution update the proposed estimator achieves the cramer rao lower bound crlb asymptotically for gaussian noise before the thresholding effect occurs

a hybrid lower bound for parameter estimation of signals with multiple change points change point estimation has received much attention in the literature as it plays a significant role in several signal processing applications however the study of the optimal estimation performance in such context is a difficult task since the unknown parameter vector of interest may contain both continuous and discrete parameters namely the parameters associated with the noise distribution and the change point locations in this paper we handle this by deriving a lower bound on the mean square error of these continuous and discrete parameters specifically we propose a hybrid cramer rao weiss weinstein bound and derive its associated closed form expressions numerical simulations assess the tightness of the proposed bound in the case of gaussian and poisson observations

fast estimators of time delay and doppler stretch based on discrete time methods estimation of the time delay and the doppler stretch of a signal is required in several signal processing applications this paper is focused on the joint fine estimation of these two parameters by a fast interpolation of the estimated ambiguity function four discrete time methods viz multirate piecewise scaling linear scaling and indirect estimators based on an orthogonal model are introduced their mean square error is mathematically derived for random signals corrupted by additive random noises the obtained expressions have been evaluated for some typical parameter sets in the reference case of gaussian signals and noises the numerical results compared with the accuracy of a continuous time estimator show the near efficiency of the multirate estimator for a wide range of snrs in fact the adjustable multirate estimator can operate under near optimal conditions unlike the approximate i e piecewise and linear scaling and the indirect methods based on constant sampling rates

boundedness of automorphic l functions in vertical strips the purpose of this paper is to prove the boundedness in vertical strips of finite width for all the l functions that appear in constant terms of eisenstein series under a certain natural assumption on local normalized intertwining operators as a corollary we prove the boundedness for a number of important l functions among them the symmetic cube and triple product l functions attached to cusp forms on gl 2 as well as several rankin selberg product l functions where the local assumption was alredy proved to be valid in each case under the same assumption the finiteness of poles on all of c for all of these l functions is also proved a necessary step in the proof of boundedness in vertical strips

data processing inequalities based on a certain structured class of information measures with application to estimation theory we study data processing inequalities that are derived from a certain class of generalized information measures where a series of convex functions and multiplicative likelihood ratios is nested alternately while these information measures can be viewed as a special case of the most general zakai ziv generalized information measure this special nested structure calls for attention and motivates our study specifically a certain choice of the convex functions leads to an information measure that extends the notion of the bhattacharyya distance or the chernoff divergence while the ordinary bhattacharyya distance is based on the weighted geometric mean of two replicas of the channel s conditional distribution the more general information measure allows an arbitrary number of such replicas we apply the data processing inequality induced by this information measure to a detailed study of lower bounds of parameter estimation under additive white gaussian noise awgn and show that in certain cases tighter bounds can be obtained by using more than two replicas while the resulting lower bound may not compete favorably with the best bounds available for the ordinary awgn channel the advantage of the new lower bound relative to the other bounds becomes significant in the presence of channel uncertainty like unknown fading this different behavior in the presence of channel uncertainty is explained by the convexity property of the information measure

a new class of bayesian cyclic bounds for periodic parameter estimation many practical signal processing applications involve estimation of parameters with periodic nature such as phase frequency and direction of arrival estimation the commonly used mean squared error mse risk does not take periodicity into account and thus is inappropriate for periodic parameter estimation in which one is interested in the modulo t error rather than the plain error value as a result mse lower bounds are not valid for periodic estimation in addition conventional bayesian mse lower bounds such as the bayesian cramer rao bound bcrb and the bobrovsky zakai bound bzb require restrictive regularity conditions and usually do not exist in periodic settings an alternative risk which is commonly used in periodic parameter estimation problems is the mean cyclic error mce in this paper we establish a new class of bayesian lower bounds on the mce of any estimator the new class includes cyclic versions of the bcrb and the bzb that have less restrictive regularity conditions than those of the conventional bcrb and bzb respectively the tightest bound in the proposed class is derived and its tightness is discussed in addition the proposed class is extended to mixed vector parameter estimation with both periodic and nonperiodic parameters the new cyclic lower bounds are compared with the mce performance of the minimum mce and maximum a posteriori probability estimators for von mises parameter estimation and for frequency estimation

a general class of lower bounds in parameter estimation 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 the extension of these bounds to multiple parameter is discussed

delay and doppler estimation by time space partition of the array data an instrumentationally attractive technique for estimating the doppler time compression and its time derivatives is proposed and analyzed in this paper the basic scheme consists of m 1 generalized cross correlators gcc s in parallel where m is the number of sensors in the receiving array each of which is implemented successively to generate a sequence of differential delay measurements the resulting spatial temporal differential delay data are then compressed into the estimates of the doppler and its derivatives using a least squares ls filter it is shown that for signal to noise ratio snr conditions above the threshold level the indicated two step estimation procedure is very nearly optimal when operating near the threshold point however the proposed scheme is only distinctly suboptimal

time domain analysis of cross correlation for time delay estimation with an autocorrelated signal the estimation accuracy of the time difference of arrival tdoa of an exponentially autocorrelated signal at two sensors in white noise is analyzed the estimate is obtained by cross correlating samples taken as short term integrals of the noisy signals from the two sensors this technique avoids ambiguities in the cross correlation and it is shown that the best sampling rate is double the nyquist rate for which the cramer rao lower bound crlb is met in practice

on lower bounds for nonstandard deterministic estimation we consider deterministic parameter estimation and the situation where the probability density function p d f parameterized by unknown deterministic parameters results from the marginalization of a joint p d f depending on random variables as well unfortunately in the general case this marginalization is mathematically intractable which prevents from using the known standard deterministic lower bounds lbs on the mean squared error mse actually the general case can be tackled by embedding the initial observation space in a hybrid one where any standard lb can be transformed into a modified one fitted to nonstandard deterministic estimation at the expense of tightness however furthermore these modified lbs mlbs appears to include the submatrix of hybrid lbs which is an lb for the deterministic parameters moreover since in the nonstandard estimation maximum likelihood estimators mles can be no longer derived suboptimal nonstandard mles nsmles are proposed as being a substitute we show that any standard lb on the mse of mles has a nonstandard version lower bounding the mse of nsmles we provide an analysis of the relative performance of the nsmles as well as a comparison with the mlbs for a large class of estimation problems last the general approach introduced is exemplified among other things with a new look at the well known gaussian complex observation models

threshold region performance of maximum likelihood direction of arrival estimators 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 crame spl acute r 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 spl rarr spl infin due to the effect of outliers

time delay estimation of signals with uncertain spectra the performance of generalized correlation gc methods for time delay estimation when a priori knowledge of the received source spectrum is limited is improved the treatment is restricted to finite time observation of a single low pass source where the uncertainty in its autospectrum is characterized by a single unknown parameter the constrained processor cp selects an optimum postcorrelator filter from a finite set of alternatives that are designed to span the expected range of source spectra an approach proposed previously for this situation is the lms adaptive filtering method for time delay estimation lmstde for purposes of comparison both the theoretical and simulation performance of the lmstde method are examined simulation experiments indicate that lmstde offers performance comparable to conventional gc methods without a priori knowledge of the source spectrum the cp in the same setting however can yield significantly lower variance time delay estimates than either the lmstde or conventional gc methods with performance approaching the cramer rao lower bound

a fresh look at the bayesian bounds of the weiss weinstein family minimal bounds on the mean square error mse are generally used in order to predict the best achievable performance of an estimator for a given observation model in this paper we are interested in the bayesian bound of the weiss weinstein family among this family we have bayesian cramer rao bound the bobrovsky mayerwolf zakai bound the bayesian bhattacharyya bound the bobrovsky zakai bound the reuven messer bound and the weiss weinstein bound we present a unification of all these minimal bounds based on a rewriting of the minimum mean square error estimator mmsee and on a constrained optimization problem with this approach we obtain a useful theoretical framework to derive new bayesian bounds for that purpose we propose two bounds first we propose a generalization of the bayesian bhattacharyya bound extending the works of bobrovsky mayer wolf and zakai second we propose a bound based on the bayesian bhattacharyya bound and on the reuven messer bound representing a generalization of these bounds the proposed bound is the bayesian extension of the deterministic abel bound and is found to be tighter than the bayesian bhattacharyya bound the reuven messer bound the bobrovsky zakai bound and the bayesian cramer rao bound we propose some closed form expressions of these bounds for a general gaussian observation model with parameterized mean in order to illustrate our results we present simulation results in the context of a spectral analysis problem

asymptotic design of quantizers for decentralized mmse estimation conceptual and practical encoding decoding aimed at accurately reproducing remotely collected observations has been heavily investigated since the pioneering works by shannon about source coding however when the goal is not to reproduce the observables but making inference about an embedded parameter and the scenario consists of many unconnected remote nodes the landscape is less certain we consider a multiterminal system designed for efficiently estimating a random parameter according to the minimum mean square error mmse criterion the analysis is limited to scalar quantizers followed by a joint entropy encoder and it is performed in the high resolution regime where the problem can be more easily mathematically tackled focus is made on the peculiarities deriving from the estimation task as opposed to that of reconstruction as well as on the multiterminal as opposite to centralized character of the inference the general form of the optimal nonuniform quantizer is derived and examples are given

optimistic and pessimistic approximations to variance of time delay estimators a general theorem is presented that can be used to find an optimistic or if preferred pessimistic expression for the variance of a time delay estimator the theorem concerns the relative locations of the maxima of two functions which are contaminated with identical additive noise the utility of the method developed is demonstrated by applying the approach to the generalized cross correlator to obtain both an optimistic expression and a pessimistic expression for the variance of the time delay

lower bounds on worst case probability of large error for two channel time delay estimation a lower bound on the worst case probability of large error for two channel time delay estimation using random signals is developed the bound is based on the minimum probability of error m ary hypothesis test using a maximum likelihood estimator mle the bound can only be evaluated approximately however it can be determined arbitrarily well via simulation if the mle can be instrumented the bound is evaluated in detail for uniform lowpass signal and noise spectra and large bandwidth time product it is then compared with another lower bound based on a binary hypothesis test with computer simulation results for an approximate mle instrumentation and with an upper bound established via computer simulation for a correlator for a uniform lowpass spectrum the m ary bound is shown to be very nearly a greatest lower bound over the full range of input signal to noise ratio further the correlator is shown to be very nearly an optimal instrumentation in the sense of reaching the minimum probability of large error

semicontinuity problems in the calculus of variations soit f r 3 r une fonction deux fois continument differentiable et pour u w 1 1 a b f u a b a b f x u x u x dx si f r n r m r nm r satisfait 1 f est mesurable en x continue en cs et 2 0 f x s a x c s p p ou p 1 a est une fonction localement sommable non negative et c une constante non negative alors la fonctionnelle f u a b est sequentiellement faiblement semi continue inferieurement sur w 1 p r m si et seulement si f x s est quasi convexe pour chaque s et presque chaque x

delay estimation using narrow band processes array processing of narrow band gaussian signals is studied with emphasis on delay estimation the barankin bound is used to examine the effect of ambiguity on mean square measurement error when the bound is plotted as a function of signal to noise ratio one observes a distinct threshold above the critical signal to noise ratio the lower bound on mean square error is given by the cramer rao inequality which is approached by the barankin inequality under these conditions below the threshold the barankin bound can exceed the cramer rao bound by large factors the relative magnitude of the bounds in that region depends critically on the ratio of signal center frequency to signal bandwidth

capon algorithm mean squared error threshold snr prediction and probability of resolution 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

statistics of the mle and approximate upper and lower bounds part i application to toa estimation in nonlinear deterministic parameter estimation the maximum likelihood estimator mle is unable to attain the cramer rao lower bound at low and medium signal to noise ratios snrs due the threshold and ambiguity phenomena in order to evaluate the achieved mean squared error mse at those snr levels we propose new mse approximations msea and an approximate upper bound by using the method of interval estimation mie the mean and the distribution of the mle are approximated as well the mie consists in splitting the a priori domain of the unknown parameter into intervals and computing the statistics of the estimator in each interval also we derive an approximate lower bound alb based on the taylor series expansion of noise and an alb family by employing the binary detection principle the accuracy of the proposed mseas and the tightness of the derived approximate bounds are validated by considering the example of time of arrival estimation

a minimum discrepancy estimator in parameter estimation in statistical estimation theory a satisfactory estimator should be able to embody a large portion of the available information which may be known a priori or provided by the data hence the loss of information is minimum when this estimator is employed in previous work an estimator criterion based on the discrepancy between the estimator s error covariance and its information lower bound was proposed conceptually this criterion is a measure of the loss of information carried by a parameter estimator based on the bayesian approach a minimum discrepancy estimator mde was derived under the linearity assumption it was however pointed out that the minimal information loss could not be guaranteed by the linear version moreover some good asymptotic properties were not obtainable therefore in this paper the existence and uniqueness conditions of the general mde are studied under certain regularity conditions the mde can be obtained by solving a fredholm equation of the second kind furthermore it is shown to be consistent and asymptotically efficient as a result the mde is ensured to have the minimum loss of information in finite samples and no loss of information when sample size tends to infinity examples indicate that if the prior information is vague the mde is superior to the minimum variance estimator mve in terms of information loss if the prior distribution is suitably chosen the mde is superior to the maximum likelihood estimator mle on the basis of deficiency

relations between belini tartara chazan zakai ziv and wax ziv lower bounds a class of lower bounds on the mean square error in parameter estimation is presented that are based on the belini tartara lower bound 1974 the wax ziv lower bound 1977 is shown to be a special case in the class these bounds often are significantly tighter than the chazan zakai ziv lower bound 1975 when the parameter to be estimated is subject to ambiguity and threshold effects

maximum likelihood approach to strain imaging using ultrasound a maximum likelihood ml strategy for strain estimation is presented as a framework for designing and evaluating bioelasticity imaging systems concepts from continuum mechanics signal analysis and acoustic scattering are combined to develop a mathematical model of the ultrasonic waveforms used to form strain images the model includes three dimensional 3 d object motion described by affine transformations rayleigh scattering from random media and 3 d system response functions the likelihood function for these waveforms is derived to express the fisher information matrix and variance bounds for displacement and strain estimation the ml estimator is a generalized cross correlator for pre and post compression echo waveforms that is realized by waveform warping and filtering prior to cross correlation and peak detection experiments involving soft tissuelike media show the ml estimator approaches the cramer rao error bound for small scaling deformations at 5 mhz and 1 2 compression the predicted

composite bound on the attainable mean square error in passive time delay estimation from ambiguity prone signals corresp the location of a radiating source can be determined by measuring the relative time delay of its signal wavefront to several spatially separated receivers a new technique is presented to investigate the performance of time delay measurement schemes based on a modified version of the ziv zakai lower bound this technique is shown to yield a tight bound on the attainable mean square measurement error for any prespecified signal to noise ratio conditions

everything you always wanted to know about training guidelines derived using the affine precoding framework and the crb in this paper affine precoding is used to investigate the tradeoffs that exist while using the transmitter resources on training versus information symbols the channel input is a training vector superimposed on a linearly precoded vector of symbols a block fading frequency selective multi input multi output mimo channel is considered to highlight the tradeoffs between training and data symbols the fisher information matrix fim is derived under two circumstances the random parameter vector to be estimated contains 1 only fading channel coefficients and 2 unknown data symbols as well as the channel coefficients while strategy 1 corresponds to the receiver structure in which the channel is estimated initially and the channel measurement is utilized to retrieve the data symbols strategy 2 corresponds to the structure in which channel and symbol estimations are performed jointly the interesting outcome of the study in this paper is that minimizing the channel cramer rao bound crb for strategies 1 and 2 under a total average transmit power constraint leads to different affine precoder design guidelines

a class of weiss weinstein bounds and its relationship with the bobrovsky mayer wolf zakai bounds a fairly general class of bayesian large error lower bounds of the weiss weinstein family essentially free from regularity conditions on the probability density functions support and for which a limiting form yields a generalized bayesian cramer rao bound bcrb is introduced in a large number of cases the generalized bcrb appears to be the bobrovsky mayer wolf zakai bound bmzb interestingly enough a regularized form of the bobrovsky zakai bound bzb applicable when the support of the prior is a constrained parameter set is obtained modified weiss weinstein bound and bzb which limiting form is the bmzb are proposed in expectation of an increased tightness in the threshold region some of the proposed results are exemplified with a reference problem in signal processing the gaussian observation model with parameterized mean and uniform prior

bayesian bounds for matched field parameter estimation 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 crame spl acute r 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

extended ziv zakai lower bound for vector parameter estimation 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

analytic sequential weiss weinstein bounds in this paper we explore a sequential bayesian bound for state space models focusing on hybrid continuous and discrete random states we provide an analytic recursion for the sequential weiss weinstein sww bound for linear state space models with solutions for gaussian uniform and exponential distributions as derived as well as for a combination of these we compare the sww bound for discretized states with the corresponding bound for the continuous states the sww bound is contrasted with the sequential cramer rao bound for gaussian distributions practical issues of sww bounds are discussed and numerical simulation results provide insights into their behavior

robust adapative kalman filtering for systems with unknown step inputs and non gaussian measurement errors target tracking with kalman filters is hampered by target maneuvering and unknown process and measurement noises we show that moving data windows may be used to analyze state and measurement error sequences determining robust estimates of bias and covariance for steps in the system forcing functions and non gaussian measurement errors the robust estimators yield improvements over linear bias and covariance estimators extensive simulations compare conventional linear adaptive and robust adaptive average step responses of a first order system filter quantities examined are state estimate state error process and measurement covariance estimates kalman gain and input step estimate

time delay estimation in unknown gaussian spatially correlated noise a novel class of methods that estimate the difference in arrival time between signals corrupted by spatially correlated gaussian noise sources of unknown cross correlation is presented the methods are based on the idea of comparing the similarities between the two sensor measurements in higher order spectrum domains bispectrum rather than in the cross correlation domain it is demonstrated that estimation techniques based on higher order cumulants suppress the effect of correlated gaussian noise sources and therefore exhibit improved performance over generalized cross correlation methods results are reported for different types of signals lengths of data records and signal to noise ratios

a lower bound on the mean square error in random parameter estimation corresp a new lower bound on mean square error in parameter estimation is presented the bound is tighter than the cramer 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

general classes of performance lower bounds for parameter estimation part i non bayesian bounds for unbiased estimators in this paper a new class of lower bounds on the mean square error mse of unbiased estimators of deterministic parameters is proposed derivation of the proposed class is performed by projecting each entry of the vector of estimation error on a hilbert subspace of l2 this hilbert subspace contains linear transformations of elements in the domain of an integral transform of the likelihood ratio function the integral transform generalizes the traditional derivative and sampling operators which are applied on the likelihood ratio function for computation of performance lower bounds such as cramer rao bhattacharyya and mcaulay seidman bounds it is shown that some well known lower bounds on the mse of unbiased estimators can be derived from this class by modifying the kernel of the integral transform a new lower bound is derived from the proposed class using the kernel of the fourier transform in comparison with other existing bounds the proposed bound is computationally manageable and provides better prediction of the threshold region of the maximum likelihood estimator in the problem of single tone estimation

a comment on the weiss weinstein bound for constrained parameter sets the weiss weinstein bound wwb provides a lower limit on the mean squared error mse achievable by an estimator of an unknown random parameter in this correspondence it is shown that some previously proposed simplified versions of the bound do not always hold for constrained parameters i e parameters whose distribution has finite support these simplifications can produce results which are no longer lower bounds on the mse sufficient conditions are provided for the reductions to be valid

general classes of performance lower bounds for parameter estimation part ii bayesian bounds in this paper a new class of bayesian lower bounds is proposed derivation of the proposed class is performed via projection of each entry of the vector function to be estimated on a hilbert subspace of l2 this hilbert subspace contains linear transformations of elements in the domain of an integral transform applied on functions used for computation of bounds in the weiss weinstein class the integral transform generalizes the traditional derivative and sampling operators used for computation of existing performance lower bounds such as the bayesian cramer rao bayesian bhattacharyya and weiss weinstein bounds it is shown that some well known bayesian lower bounds can be derived from the proposed class by specific choice of the integral transform kernel a new lower bound is derived from the proposed class using the fourier transform kernel the proposed bound is compared with other existing bounds in terms of signal to noise ratio snr threshold region prediction in the problem of frequency estimation the bound is shown to be computationally manageable and provides better prediction of the snr threshold region exhibited by the maximum a posteriori probability map and minimum mean square error mmse estimators

an automorphic generalization of the hermite minkowski theorem we show that for any integer n there are only finitely many cuspidal algebraic automorphic representations of rm gl n over mathbb q with n varying whose conductor is n and whose weights are in the interval 0 1 23 more generally we define a simple sequence r w w geq 0 such that for any integer w any number field e whose root discriminant is less than r w and any ideal n in the ring of integers of e there are only finitely many cuspidal algebraic automorphic representations of general linear groups over e whose conductor is n and whose weights are in the interval 0 1 w assuming a version of grh we also show that we may replace r w with 8 pi e gamma h w in this statement where gamma is euler s constant and h w the w th harmonic number the proofs are based on some new positivity properties of certain real quadratic forms which occur in the study of the weil explicit formula for rankin selberg l functions both the effectiveness and the optimality of the methods are discussed

a general class of outage error probability lower bounds in bayesian parameter estimation in this paper a new class of lower bounds on the outage error probability in bayesian parameter estimation is proposed the outage error probability is an important criterion in parameter estimation that provides meaningful information even in the presence of large errors and is useful for prediction of the system operation region computation of the minimum outage error probability is usually not tractable and thus lower bounds on this probability can be very useful for performance analysis the proposed class of lower bounds on the outage error probability is derived using holder s inequality several bounds in the proposed class are presented it is shown that the ziv zakai lower bound on the outage error probability can be obtained from a subclass in the proposed class of bounds the proposed class of bounds is utilized to derive a new class of bayesian bounds on the mean square error it is shown that for unimodal posterior probability density functions the tightest lower bound on the probability of outage error in the proposed class attains the minimum probability of outage error the proposed bounds are exemplified in linear gaussian model parameter estimation and time delay estimation

statistics of the mle and approximate upper and lower bounds part ii threshold computation and optimal pulse design for toa estimation threshold and ambiguity phenomena are studied in part i of this paper where approximations for the mean squared error mse of the maximum likelihood estimator are proposed using the method of interval estimation mie and where approximate upper and lower bounds are derived in this part we consider time of arrival estimation and we employ the mie to derive closed form expressions of the begin ambiguity end ambiguity and asymptotic signal to noise ratio snr thresholds with respect to some features of the transmitted signal both baseband and passband pulses are considered we prove that the begin ambiguity threshold depends only on the shape of the envelope of the acr whereas the end ambiguity and asymptotic thresholds only on the shape of the acr we exploit the results on the begin ambiguity and asymptotic thresholds to optimize with respect to the available snr the pulse that achieves the minimum attainable mse the results of this paper are valid for various estimation problems

relaxation results for a class of variational integrals abstract the relaxation problem for functionals of the form u du dx with s z not necessarily continuous with respect to s is studied

joint time scale and tdoa estimation analysis and fast approximation relative motion rm between a signal source and a receiver causes a time scaling of the signal arriving at the receiver when estimating the time difference of arrival tdoa of a signal at two receivers time scaling when not properly accounted for can introduce a bias that could dominate the estimation errors segmentization processing cannot reduce this bias following the derivation of the formulae for bias and mean square errors of tdoa estimation under rm this paper moves on to the joint estimation of tdoa and time scale it proposes an iterative search for the maximization of the cross ambiguity function caf which is also the maximum likelihood function for additive gaussian bandlimited white noise disturbance in addition a quadratic lagrange interpolator is also proposed to obtain the initial parameter values for the iterative search which can increase the chance of converging to the global minimum solution it is necessary to time scale a digital sequence by a noninteger in the maximization process for an n point sequence this operation which first interpolates the samples by sinc functions and then resamples is in the order o n sup 2 noting that the magnitude of the sinc function decreases rapidly from its peak this paper uses a fast approximation fa method that applies only five sinc coefficients for the interpolation reducing the computation to o n simulation results have corroborated that the maximization of the caf does provide estimates that reach the cramer rao lower bound and that the degradation in accuracy is negligible when fa is applied

analysis of wide band cross correlation for time delay estimation this correspondence deals with the estimation of the time delay of a wide band signal based on cross correlation of samples of the received noisy signals at two locations the accuracy of the time delay estimate is expressed explicitly in terms of the sampling period resolution cell the signal to noise ratio snr and the number of samples in the data window simulations that confirm the validity of the estimation accuracy expressions for snr as low as 6 db for a 128 sample window are shown

application of the weiss weinstein bound to a two dimensional antenna array the weiss weinstein ww bound is applied to an 8 8 element uniform antenna array for a single source impinging on the array the ww bound for this problem is evaluated and compared to simulations of the music algorithm and classical beamforming the simulations show that for high signal to noise ratio snr the ww bound is met by both algorithms however the threshold for music and classical beamforming occurs at a higher snr than the bound predicts this suggests the possibility of other spectrum estimation algorithms with much lower thresholds than those currently used today

a unified approach to closed loop time delay estimation systems two generalized configurations for a closed loop delay estimation system cldes are discussed the indirect correlation ic configuration is compared to the conventional cldes the direct correlation dc configuration it is shown that the baseband equivalents of the two configurations differ only by the input noise however the noise power in the ic configuration is far less than that of the dc configuration using the known procedure for calculation of the variance of the time delay estimator it is shown that the two configurations result in the same variance however this procedure is known to be incorrect since for the direct correlation configuration which is known to have an intrinsic noise that persists even when the input signals are ideal the variance is nonzero for zero input signal to noise ratio snr thus another procedure is suggested for variance calculation that overcomes this problem it is shown using this expression that the variance of the time delay estimator is smaller for the ic configuration provided the input snr is not less than 0 db

a new generalized cross correlator a new generalized cross correlator gcc for the passive time delay estimation problem is presented the interpretation of this gcc is that of estimating the cross correlation function by cross correlating the least mean square estimates of the signal component in each of the observed waveforms the implementation is simply a gcc with the weighting filter equal to the magnitude coherency squared numerical evaluation of the performance of this processor and a robust version indicate that they compare favorably to some of the well known gcc procedures

a lower bound on the bayesian mse based on the optimal bias function a lower bound on the minimum mean squared error mse in a bayesian estimation problem is proposed in this paper this bound utilizes a well known connection to the deterministic estimation setting using the prior distribution the bias function which minimizes the cramer rao bound can be determined resulting in a lower bound on the bayesian mse the bound is developed for the general case of a vector parameter with an arbitrary probability distribution and is shown to be asymptotically tight in both the high and low signal to noise ratio snr regimes a numerical study demonstrates several cases in which the proposed technique is both simpler to compute and tighter than alternative methods

explicit ziv zakai lower bound for bearing estimation 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 cramer 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

mean squared error and threshold snr prediction of maximum likelihood signal parameter estimation with estimated colored noise covariances 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 snrrarrinfin 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 crameacuter rao 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

time domain adaptive delay estimation in this correspondence a new adaptive delay estimator using only two lms filters is presented compared with other algorithms a significant reduction in computational complexity is obtained because only time domain processing is required in addition a new frequency domain adaptive delay estimator has been introduced and the link between both methods is clarified comparison with algorithms described in literature turns out favorably

weiss weinstein lower bounds for markovian systems part 1 theory being essentially free from regularity conditions the weiss weinstein estimation error lower bound can be applied to a larger class of systems than the well known crameacuter 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 makes the application of the weiss weinstein lower bound to markovian dynamic systems cumbersome a sequential algorithm for the computation of the crameacuter rao lower bound for such systems has been recently reported in the literature along with the marginal state distribution 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 a similar technique is employed herein to develop sequential weiss weinstein lower bounds for a class of markovian dynamic systems in particular it is shown that in systems satisfying the crameacuter rao lower bound regularity conditions the sequential weiss weinstein lower bound derived herein reduces for a judicious choice of its parameters to the sequential crameacuter rao lower bound

a bound on mean square estimation error with background parameter mismatch in typical parameter estimation problems the signal observation is a function of the parameter set to be estimated as well as some background environmental system parameters assumed known the assumed background could differ from the true one leading to biased estimates even at high signal to noise ratio snr to analyze this background mismatch problem a ziv zakai type lower bound on the mean square error mse is developed based on the mismatched likelihood ratio test mlrt at high snr the bound incorporates the increase in mse due to estimation bias at low snr it includes the threshold effect due to estimation ambiguity the kernel of the bound s evaluation is the error probability associated with the mlrt a closed form expression for this error probability is derived under a random signal model typical of the bearing estimation passive source localization problem the mismatch is then analyzed in terms of the related ambiguity functions examples of bearing estimation with system array shape mismatch demonstrate that the developed bound describes the simulations of the maximum likelihood estimate well including the sidelobe introduced threshold behavior and the bias at high snr

necessary conditions for a maximum likelihood estimate to become asymptotically unbiased and attain the cramer rao lower bound ii range and depth localization of a sound source in an ocean waveguide analytic expressions for the first order bias and second order covariance of a maximum likelihood estimate mle are applied to the problem of localizing an acoustic source in range and depth in a shallow water waveguide with a vertical hydrophone array these expressions are then used to determine necessary conditions on sample size or equivalently signal to noise ratio snr for the localization mle to become asymptotically unbiased and attain minimum variance as expressed by the cramer rao lower bound crlb these analytic expressions can be applied in a similar fashion to any ocean acoustic inverse problem involving random data both deterministic and completely randomized signals embedded in independent and additive waveguide noise are investigated as the energy ratio of received signal to additive noise sanr descends to the lower operational range of a typical passive localization system source range and depth estimates exhibit significant biases and have variances that can exceed the crlb by orders of magnitude the spatial structure of the bias suggests that acoustic range and depth estimates tend to converge around particular range and depth cells for moderate sanr values

some relaxation problems in optimal control theory usually the space u r is a space lp q endowed with its weak topology the space y g is a sobolev space w y q endowed with its weak topology or with the l q topology in the one dimensional case and condition ii is guaranteed by some boundedness or growth hypotheses on j and a when condition i is not fulfilled in general the existence of a solution of 1 1 may fail nevertheless it may be interesting to study the asymptotic 212 0022 247x 87 3 00

a pessimistic approximation for the fisher information measure the problem of determining the intrinsic quality of a signal processing system with respect to the inference of an unknown deterministic parameter theta is considered while the fisher information measure f theta forms a classical tool for such a problem direct computation of the information measure can become difficult in various situations for the estimation theoretic performance analysis of nonlinear measurement systems the form of the likelihood function can make the calculation of the information measure f theta challenging in situations where no closed form expression of the statistical system model is available the analytical derivation of f theta is not possible at all based on the cauchy schwarz inequality we derive an alternative information measure s theta it provides a lower bound on the fisher information f theta and has the property of being evaluated with the mean the variance the skewness and the kurtosis of the system model at hand these entities usually exhibit good mathematical tractability or can be determined at low complexity by real world measurements in a calibrated setup with various examples we show that s theta provides a good conservative approximation for f theta and outline different estimation theoretic problems where the presented information bound turns out to be useful

bayesian estimation in the presence of deterministic nuisance parameters part i performance bounds how accurately can one estimate a random parameter subject to unknown deterministic nuisance parameters the hybrid cramer rao bound hcrb provides an answer to this question for a restricted class of estimators the hcrb is the most popular performance bound on the mean square error mse for random parameter estimation problems which involve deterministic parameters the hcrb is useful when one is interested in both the random and the deterministic parameters and in the coupling between their estimation errors this bound refers to locally weak sense unbiased estimators with respect to w r t the deterministic parameters however if these parameters are nuisance it is unnecessary to restrict their estimation as unbiased this paper is the first of a two part study of bayesian parameter estimation in the presence of deterministic nuisance parameters it begins with a study on order relations between existing cramer rao cr type bounds of mean unbiased bayesian estimators then a new cr type bound is developed with no assumption of unbiasedness on the nuisance parameters alternatively lehmann s concept of unbiasedness is employed rather than conventional mean unbiasedness it is imposed on a risk that measures the distance between the estimator and the minimum mse mmse estimator which assumes perfect knowledge of the nuisance parameters in the succeeding paper asymptotic performances of some bayesian estimators with maximum likelihood based estimates for the nuisance parameters are investigated the proposed risk unbiased bound rub is proved to be asymptotically achieved by the mmse estimator with maximum likelihood estimates for the nuisance parameters while the existing cr type bounds are not necessarily achievable

a new barankin bound approximation for the prediction of the threshold region performance of maximum likelihood estimators it is well known that the ml estimator exhibits a threshold effect i e a rapid deterioration of estimation accuracy below a certain signal to noise ratio snr or number of snapshots this effect is caused by outliers and is not captured by standard tools such as the cramer rao bound crb the search of the snr threshold value where the crb becomes unreliable for prediction of maximum likelihood estimator variance can be achieved with the help of the barankin bound bb as proposed by many authors the major drawback of the bb in comparison with the crb is the absence of a general analytical formula which compels one to resort to a discrete form usually the mcaulay seidman bound msb requesting the search of an optimum over a set of test points in this paper we propose a new practical bb discrete form that provides for a given set of test points an improved snr threshold prediction in comparison with existing approximations msb abel bound mcaulay hofstetter bound at the expense of the computational complexity increased by a factor les p 1 3 where p is the number of unknown parameters we have derived its expression for the general gaussian observation model to be used in place of existing approximations

nonparametric narrowband wavefront estimation from wavefront gradients the wavefield sampled by a two dimensional 2 d planar array can he characterized by a delayed signal this correspondence investigates the problem of the nonparametric estimation of the delay function or wavefront when the waveform is narrowband the wavefront is estimated by integrating the local estimates of the differential time of delay tod of the waveform between neighboring sensors the ambiguity of the tod estimator for the narrowband waveform can be conveniently limited by the a priori search interval for the cross correlation method the reconstruction of the wavefronts is formulated as a least squares problem that is solved iteratively errors of the solution are quantified and the contribution caused by noise is shown to be negligible

weiss weinstein lower bounds for markovian systems part 2 applications to fault tolerant filtering 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 crameacuter 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

tissue elasticity reconstruction using linear perturbation method a new method to reconstruct the elastic modulus of soft tissue subjected to an external static compression is presented in this approach the newton raphson method is used to vary a finite element fe model of the elasticity equations to fit in a least squared sense a set of axial tissue displacement fields estimated using a correlation technique applied to ultrasound signals the ill conditioning of the hessian matrix is eliminated using the tikhonov regularization technique this regularization provides a compromise between fidelity to the observed data and a priori information of the solution using an echographic image formation model it is shown that the method converges within a few iterations 8 10 and that strain images artifacts which are common in elastography are significantly reduced after the resolution of the inverse problem

poisson models and mean squared error for correlator estimator of time delay a method for modeling large errors in correlation based time delay estimation is developed in terms of level crossing probabilities the level crossing interpretation for peak ambiguity leads directly to an exact expression for the probability of large error involving the hazard function associated with the level crossing process two models for the distribution of the error over the level crossing time yield approximations to the mean square error mse that involve the low order

lower bounds on parameter modulation estimation under bandwidth constraints the problem of modulating the value of a parameter onto a band limited signal to be transmitted over a continuous time additive white gaussian noise awgn channel and then estimating this parameter at the receiver is considered the performance is measured by the mean power alpha error mp alpha text e which is defined as the worst case alpha th order moment of the absolute estimation error the optimal exponential decay rate of the mp alpha text e as a function of the transmission time is investigated two upper converse bounds on the mp alpha text e exponent are derived on the basis of known bounds for the awgn channel of inputs with unlimited bandwidth the bounds are computed for typical values of the error moment and the signal to noise ratio snr and the snr asymptotics of the different bounds are analyzed the new bounds are compared with known converse and achievability bounds which were derived from channel coding considerations

fundamental limitations on the resolution of deterministic signals we exploit results from detection theory for deriving fundamental limitations on resolution the results are general and are not based on any specific resolution technique and therefore hold for any method and for any resolution success rate we show that for signals with additive white gaussian noise the resolution limit is approximated by a simple expression related to the signal to noise ratio the signal waveform and the resolution success rate as an example we discuss the resolution of two complex exponentials with closely spaced frequencies the theoretical result is compared with the empirical performance of model order selection methods such as the akaike information criterion and the minimum description length

realizable lower bounds for time delay estimation the accuracy of time delay estimates obtainable in active localization systems is studied focusing on the effect of ambiguities in the time delay estimates such ambiguities occur when the transmitted signal has small relative bandwidth then for signal to noise ratios below a certain threshold the commonly used cramer rao lower bound is not realizable the study concentrates on the region of intermediate snr values where the cramer rao bound is no longer achievable but useful information on time delays can still be obtained from the measurement realizable bounds for the single and two echo cases are obtained by deriving a new form of the barankin 1949 bound for active time delay estimation this form maintains the realizability property of the most general form but is of reasonable complexity new bounds are derived for the multiple echo case examples are presented to illustrate the dependence of the bound on parameters such as snr relative bandwidth and echo separation

necessary conditions for a maximum likelihood estimate to become asymptotically unbiased and attain the cramer rao lower bound part i general approach with an application to time delay and doppler shift estimation analytic expressions for the first order bias and second order covariance of a general maximum likelihood estimate mle are presented these expressions are used to determine general analytic conditions on sample size or signal to noise ratio snr that are necessary for a mle to become asymptotically unbiased and attain minimum variance as expressed by the cramer rao lower bound crlb the expressions are then evaluated for multivariate gaussian data the results can be used to determine asymptotic biases variances and conditions for estimator optimality in a wide range of inverse problems encountered in ocean acoustics and many other disciplines the results are then applied to rigorously determine conditions on snr necessary for the mle to become unbiased and attain minimum variance in the classical active sonar and radar time delay and doppler shift estimation problems the time delay mle is the time lag at the peak value of a matched filter output it is shown that the matched filter estimate a

sensitivity analysis for relaxed optimal control problems with final state constraints in this article we compute a second order expansion of the value function of a family of relaxed optimal control problems with final state constraints parameterized by a perturbation variable the sensitivity analysis is performed for controls that we call r strong solutions they are optimal solutions with respect to the set of feasible controls with a uniform norm smaller than a given r and having an associated trajectory in a small neighborhood for the uniform norm in this framework relaxation enables us to consider a wide class of perturbations and therefore to derive sharp estimates of the value function

a barankin type lower bound on the estimation error of a hybrid parameter vector the barankin 1949 bound is a realizable lower bound on the mean square error mse of any unbiased estimator of a nonrandom parameter vector 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 1976 bound for random parameters estimation are special cases of our proposed bound

exact maximum likelihood time delay estimation for short observation intervals an exact solution is presented to the problem of maximum likelihood time delay estimation for a gaussian source signal observed at two different locations in the presence of additive spatially uncorrelated gaussian white noise the solution is valid for arbitrarily small observation intervals that is the assumption t tau sub c mod d mod made in the derivation of the conventional asymptotic maximum likelihood aml time delay estimator where tau sub c is the correlation time of the various random processes involved and d is the differential time delay is relaxed the resulting exact maximum likelihood eml instrumentation is shown to consist of a finite time delay and sum beamformer followed by a quadratic postprocessor based on the eigenvalues and eigenfunctions of a one dimensional integral equation with nonconstant weight the solution of this integral equation is obtained for the case of stationary signals with rational power spectral densities finally the performance of the eml and aml estimators is compared by means of computer simulations

large and small error performance limits for multipath time delay estimation the cramer rao lower bound crlb and a lower bound on the probability of large errors for multipath time delay estimation for a single resolvable multipath are developed these theoretical results are compared to the performance of the maximum likelihood estimator mle and an autocorrelator via computer simulation for small errors and large snr the mle reaches the crlb the variance of the time delay estimate for the autocorrelator is much larger than the crlb if the reflection coefficient for the delayed path is near unity for small errors and small snr the mle reduces to an autocorrelator hence their performance is identical for large errors and bandwidth time products greater than 500 the error probability for the autocorrelator is within 1 db in input snr of the lower bound on large error probability thus the autocorrelator is nearly an optimal instrumentation from the point of view of large errors at least for large bandwidth time products and the flat low pass spectrum considered here

on mmse estimation a linear model under gaussian mixture statistics in a bayesian linear model suppose observation y hx n stems from independent inputs x and n which are gaussian mixture gm distributed with known matrix h the minimum mean square error mmse estimator for x has analytical form however its performance measure the mmse itself has no such closed form because existing bayesian mmse bounds prove to have limited practical value under these settings we instead seek analytical bounds for the mmse both upper and lower this paper provides such bounds and relates them to the signal to noise ratio snr