realizable lower bounds for time delay estimation 2 threshold phenomena in time delay estimation of narrowband signals the mean square error mse plotted as a function of the signal to noise ratio snr exhibits threshold phenomena the thresholds divide the domain of snr values into several disjoint segments for very low snr values the observations are dominated by noise and are essentially useless for delay estimation here the mse is determined largely by the available a priori information for intermediate snr values time delay estimates are possible but are subject to ambiguities resulting from the oscillatory nature of the signal sample correlation for very high snr values these ambiguities are resolvable and the cramer rao bound crb yields a realistic bound for the attainable performance a previous paper by zeira and schultheiss see ieee trans signal processing vol 41 p 3102 3113 nov 1993 used the barankin 1949 bound to obtain a realizable lower bound for the intermediate snr region but left open the question where the transitions from low to medium and medium to high snr operation occur since the attainable mse in the three regions can be very different the location of the snr thresholds is of considerable practical interest the present paper addresses this issue it obtains threshold values for the single and two echo problems with emphasis on the effect of echo separation in the two echo case it concludes that the location of the thresholds is only weakly dependent on echo separation even though the attainable mse in both the intermediate and high snr regions varies drastically as the echo separation changes

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

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

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

estimation of input pulse locations from the response of an all pole transfer system using tapered rank reduction when ordinary rank reduction is used to remove the characteristics of the transfer system and to estimate the multipulse input time series the nonsignificant singular values are sharply cut off by low order truncation if the system q quality factor is high and then a large ripple occurs around each pulse location in order to avoid these difficulties a method is proposed in which the multipulse time series is estimated by a rank reduction using a tapering window in order to suppress the ripple due to the low order sharp truncation and then by applying the pole estimation method to the inverse fourier transform of the resultant time series thus the pulse locations are accurately estimated by using the pulse locations as the initial estimates the maximum likelihood estimates of the pulse locations are obtained from simulation experiments these principles are confirmed

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

the ricci flow on manifolds with boundary we study the short time existence and regularity of solutions to a boundary value problem for the ricci deturck equation on a manifold with boundary using this we prove the short time existence and uniqueness of the ricci flow prescribing the mean curvature and conformal class of the boundary with arbitrary initial data finally we establish that under suitable control of the boundary data the flow exists as long as the ambient curvature and the second fundamental form of the boundary remain bounded

robust linear channel estimation methods for per subcarrier transmit antenna selection the linear minimum mean squared error mse channel estimator for systems employing per subcarrier transmit antenna selection is developed frequency domain correlations after the selection process are shown to be approximated well using a simple function which makes near optimal channel estimation practically possible however the resultant estimators are not robust to errors in the assumed model in terms of the antenna to subcarrier assignment used at transmission as we motivate both analytically and via simulations this is an issue when the channels for conveying this information to the receiver are severely constrained we present windowed channel estimations and new robust estimation algorithms either individually or in combination as solutions to counter this sensitivity the robust estimators are developed based on the principles of 1 minimising the worst case mse over the set of possible models and 2 minimising the expected mse over the set of possible models the latter estimator is preferred due to the lower implementation complexity and better mse performance we also prove theoretical asymptotic in signal to noise ratio performances of the two estimators used with the proposed correlation model simulation results illustrate the performance gains and improved robustness offered by the developed estimators

near optimal range and depth estimation using a vertical array in a correlated multipath environment this paper proposes a near optimal procedure to localize a single stationary source in a two path underwater acoustic environment the investigation is for an m element vertical array with omnidirectional sensors the range and depth estimators are developed using a linear least squares technique when a set of auto and cross correlators is used for time difference of arrival tdoa estimates a weighting matrix is derived to achieve the approximate maximum likelihood ml performance of the weighted least squares range and depth estimators the expressions for error variances and covariances of the range and depth estimates are derived with a small error analysis technique it is verified analytically that the error covariance matrix of the weighted least squares solutions reaches the cramer rao lower bound in the small error region the correlation of the range and depth estimation errors is investigated results show that the range and depth estimation errors are highly correlated in a multipath environment the accuracy properties of the proposed multipath localization procedure are analyzed using different array configurations the results show that the performances of the range and depth estimators are significantly improved if the linear dependent tdoa estimates are included for localizing and that the unweighted range and depth estimators using the entire set of tdoas are approximately optimal for most of the applications the theoretical development of error variance and covariance expressions of the range and depth estimates which incorporates the correlation in the tdoa estimates is corroborated with monte carlo simulations

cram acute text e r rao bound for constrained parameter estimation using lehmann unbiasedness the constrained cram acute text e r rao bound ccrb is a lower bound on the mean squared error mse of estimators that satisfy some unbiasedness conditions although the ccrb unbiasedness conditions are satisfied asymptotically by the constrained maximum likelihood cml estimator in the non asymptotic region these conditions are usually too strict and the commonly used estimators such as the cml estimator do not satisfy them therefore the ccrb may not be a lower bound on the mse matrix of such estimators in this paper we propose a new definition for unbiasedness under constraints denoted by c unbiasedness which is based on using lehmann unbiasedness with a weighted mse wmse risk and taking into account the parametric constraints in addition to c unbiasedness a cram acute text e r rao type bound on the wmse of c unbiased estimators denoted as lehmann unbiased ccrb lu ccrb is derived this bound is a scalar bound that depends on the chosen weighted combination of estimation errors it is shown that c unbiasedness is less restrictive than the ccrb unbiasedness conditions thus the set of estimators that satisfy the ccrb unbiasedness conditions is a subset of the set of c unbiased estimators and the proposed lu ccrb may be an informative lower bound in cases where the corresponding ccrb is not in the simulations we examine linear and nonlinear estimation problems under nonlinear constraints in which the cml estimator is shown to be c unbiased and the lu ccrb is an informative lower bound on the wmse while the corresponding ccrb on the wmse is not a lower bound and is not informative in the non asymptotic region

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

on the equality of quasi arithmetic and lagrangian means abstract the paper deals with the equality problem of quasi arithmetic and lagrangian means which is to determine all pairs of continuous strictly monotone functions i r such that for all x y i 1 x y 2 1 1 y x x y t d t holds the main result of the paper shows that there are 6 different cases of equality

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

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

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

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

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

high resolution multipath time delay estimation for broad band random signals the resolution limits to multipath time delay estimation for broadband random signals are examined first appropriate cramer rao lower bounds crlbs are derived to establish baseline performance for unbiased estimation the bounds are then compared to computer simulation results for the two path case a maximum likelihood estimator is implemented while for the three path case the modified forward backward linear prediction algorithm developed for high resolution frequency estimation is used it is shown that both of these techniques can achieve performance close to the appropriate crlb in particular reliable multipath estimation can be achieved even when the multipath time delays are much closer than could be resolved by conventional autocorrelation processing a criterion is developed for predicting when the bound is no longer reachable it is shown that even relatively poor a priori knowledge of the attenuation can significantly help the estimate of time delay for small delays

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

on the effect of nuisance parameters on the threshold snr value of the barankin bound the performance of nonlinear estimators of signal parameters in noise appears to exhibit a threshold phenomenon below a critical value of signal to noise ratio snr the performance of these estimators deviates significantly from the cramer rao bound in a large error ambiguity prone snr region the barankin bound has been proved to be an advantageous tool to assess the attainable performance and the threshold value obviously when the estimation problem involves additional unknown nuisance parameters the mean square error mse of the estimator does not decrease however the impact of these nuisance parameters on the threshold value is not clear in this correspondence we discuss the influence of unknown nuisance parameters on the threshold value the analysis is done for the common problem concerning estimating parameters of a gaussian process we confine our scope to a simplified problem concerning only two estimated parameters however we explain how this simplified analysis can be used to handle a more complicated problem comprising multiple nuisance parameters we derive a sufficient condition applied to the structure of the data covariance matrix when the condition is satisfied then the need to estimate additional nuisance parameters does not change the threshold value using the proposed condition we prove that the threshold snr in passive source localization does not increase when the spectral parameters of the source are unknown whereas the presence of another source at an unknown bearing may change the threshold snr

an invariance of geometric mean with respect to lagrangian means abstract the invariance of the geometric mean g with respect to the lagrangian mean type mapping l f l g i e the equation g l f l g g is considered we show that the functions f and g must be of high class regularity this fact allows to reduce the problem to a differential equation and determine the second derivatives of the generators f and g

a competitive minimax approach to robust estimation of random parameters we consider the problem of estimating in the presence of model uncertainties a random vector x that is observed through a linear transformation h and corrupted by additive noise we first assume that both the covariance matrix of x and the transformation h are not completely specified and develop the linear estimator that minimizes the worst case mean squared error mse across all possible covariance matrices and transformations h in the region of uncertainty although the minimax approach has enjoyed widespread use in the design of robust methods we show that its performance is often unsatisfactory to improve the performance over the minimax mse estimator we develop a competitive minimax approach for the case where h is known but the covariance of x is subject to uncertainties and seek the linear estimator that minimizes the worst case regret namely the worst case difference between the mse attainable using a linear estimator ignorant of the signal covariance and the optimal mse attained using a linear estimator that knows the signal covariance the linear minimax regret estimator is shown to be equal to a minimum mse mmse estimator corresponding to a certain choice of signal covariance that depends explicitly on the uncertainty region we demonstrate through examples that the minimax regret approach can improve the performance over both the minimax mse approach and a plug in approach in which the estimator is chosen to be equal to the mmse estimator with an estimated covariance matrix replacing the true unknown covariance we then show that although the optimal minimax regret estimator in the case in which the signal and noise are jointly gaussian is nonlinear we often do not lose much by restricting attention to linear estimators

on the cramer rao bound for time delay and doppler estimation corresp using a theorem due to whittle simple derivations of the cramer rao lower bound are presented for some delay estimation problems related to a single source multiple sources and multipath the problem of doppler estimation is briefly discussed

a new time delay estimation in multipath this paper addresses a new approach to time delay estimation based upon the autocorrelation estimator ae the primary aim of this paper is to estimate time delays in a multipath environment in absence of prior knowledge of the channel the maximum likelihood estimator mle and ae are two computational tools that are used to determine the parameters of a multipath channel mle requires some priori knowledge of the source signal and the channel ae can be a blind estimator but it is more suitable for a simple propagation model one extra path under the multipath assumption we prove that if the observation sequence is zero padded the performance of mle exceeds that of ae however at the price of higher computational efforts the general autocorrelator estimator gae based on autocorrelation of the received signal is introduced the gae is formulated as a blind estimator and the pertinent cramer rao lower bounds crlb are derived we also develop an algorithm to estimate the parameters of a multipath environment based on the new generalization the performance of this algorithm is examined for different signal noise scenarios our results show that the time delays are estimated accurately based on the proposed algorithm

cyclic barankin type bounds for non bayesian periodic parameter estimation in many practical periodic parameter estimation problems the appropriate performance criteria are periodic in the parameter space the existing mean square error mse lower bounds such as cramer rao bound crb and barankin type bounds do not provide valid lower bounds in such problems in this paper cyclic versions of the crb and the barankin type bounds hammersley chapman robbins and mcaulay seidman are derived for non bayesian parameter estimation the proposed bounds are lower bounds on the mean cyclic error mce of any cyclic unbiased estimator where the cyclic unbiasedness is defined by using lehmann unbiasedness these mce lower bounds can be readily obtained from existing mse lower bounds and thus can be easily calculated the cyclic barankin type bounds and the performance of the maximum likelihood ml estimator are compared in terms of mce in von mises distributed measurements problem and for frequency and amplitude estimation with gaussian noise in these problems the ml estimator is found to be cyclic unbiased

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

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

the risk unbiased cramer rao bound for non bayesian multivariate parameter estimation how accurately can one estimate a deterministic parameter subject to other unknown deterministic model nuisance parameters the most popular answer to this question is given by the cramer rao bound crb the main assumption behind the derivation of the crb is local unbiased estimation of all model parameters the foundations of this paper rely on doubting this assumption generally in multivariate parameter estimation each parameter in its turn can be treated as a single parameter of interest whereas the other model parameters are treated as nuisance as their misknowledge interferes with the estimation of the parameter of interest this approach is utilized in this paper to provide a fresh look at deterministic parameter estimation a new cramer rao cr type bound is derived without assuming unbiased estimation of the nuisance parameters rather than that we apply lehmann s concept of unbiasedness for a risk that measures the distance between the estimator and the locally best unbiased estimator which assumes perfect knowledge of the nuisance parameters the proposed risk unbiased crb rucrb is proven to be asymptotically attainable by the maximum likelihood estimator while being tighter than the conventional crb furthermore simulations verify the asymptotic achievability of the rucrb by the maximum likelihood estimator for an array processing problem

the mass of an asymptotically flat manifold we show that the mass of an asymptotically flat n manifold is a geometric invariant the proof is based on harmonic coordinates and to develop a suitable existence theory results about elliptic operators with rough coefficients on weighted sobolev spaces are summarised some relations between the mass scalar curvature and harmonic maps are described and the positive mass theorem for n dimensional spin manifolds is proved

least squares algorithms for time of arrival based mobile location localization of mobile phones is of considerable interest in wireless communications in this correspondence two algorithms are developed for accurate mobile location using the time of arrival measurements of the signal from the mobile station received at three or more base stations the first algorithm is an unconstrained least squares ls estimator that has implementation simplicity the second algorithm solves a nonconvex constrained weighted least squares cwls problem for improving estimation accuracy it is shown that the cwls estimator yields better performance than the ls method and achieves both the crame spl acute r rao lower bound and the optimal circular error probability at sufficiently high signal to noise ratio conditions

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

passive range difference estimation in a dispersive medium passive source localization can be achieved by utilizing the range difference rd measurements between the source and several spatially separated sensors when the signal propagation medium is nondispersive the rd is typically obtained from time difference of arrival tdoa between sensor outputs however for a dispersive medium which corresponds to in solid localization the tdoa information is not reliable in this correspondence two rd estimation algorithms for dispersive medium are derived and analyzed assuming that the dispersion curve is available computer simulations are included to contrast the estimation performance of the two methods with the cramer rao lower bound crlb for different conditions

cramer rao bound for location systems in multipath environments the cramer rao lower bound for source localization is studied in the context of multipath stochastic sources multipath propagation and observations in an array of sensors a general expression is derived and then specialized to simpler configurations and related to results previously reported in the literature the special case of a single stochastic source in a multipath environment is treated the relative importance for source localization of the temporal multipath and spatial array baseline structures of the incoming wavefield is assessed it is shown that for an array of k sensors the multipath contribution to the fisher information matrix can be interpreted as the contribution of k independent arrays whose size depends on the number of spatially resolved replicas the degradation due to unknown source spectra is analyzed when the source spectrum is completely arbitrary source location is not possible with a single sensor if a parametric form of the source spectrum is available the multipath structure can be used to locate the acoustic source

a useful form of the abel bound and its application to estimator threshold prediction this correspondence investigates the abel bound in order to predict the estimators mean square error mse threshold effect a tractable and computationally efficient form of this bound is derived this form combines the chapman robbins and the crameacuter rao bounds this bound is applied to a data aided carrier frequency estimation problem for which a closed form expression is provided an indicator of the signal to noise ratio threshold is proposed a comparison with recent results on the barankin bound chapman robbins version shows the superiority of the abel bound version to predict the mse threshold without increasing the computational complexity

closed form formulae for time difference of arrival estimation for a positioning system with l sensors a maximum of l l 1 2 distinct time difference of arrival tdoa measurements which are referred to as the full tdoa set can be obtained in this paper closed form expressions regarding optimum conversion of the full tdoa set to the nonredundant tdoa set which corresponds to l 1 tdoa measurements with respect to a common reference receiver in the case of white signal source and noise are derived the most interesting finding is that optimum conversion can be achieved via the standard least squares estimation procedure furthermore the cramer rao lower bound for tdoa based positioning is produced in closed form which will be useful for optimum sensor array design

time delay estimation via cross correlation in the presence of large estimation errors the estimate of the difference in time of arrival of a common random signal received at two sensors each of which also receives uncorrelated noise is examined for both small and large estimation errors it is shown that as the post integration signal to noise ratio decreases the correlator exhibits a thresholding effect that is the probability of a large error an anomalous estimate increases rapidly approximate theoretical results for the probability of an anomaly are presented and are verified experimentally the variance of the time delay estimate is examined for both a gated mode in which the time delay corresponding to the correlation peak closest to the true time delay is used as the estimate of time delay and an ungated mode in which the time delay corresponding to the largest peak over the full range of the correlator delay times is used as the estimate the observed variance for both modes is compared with the theoretical variance based on a small error analysis for the gated modes the signal to noise ratio below which the observed variance begins to differ significantly from the small error theory can be reliably predicted from a linearity criterion it is shown however that the expected variance for the ungated mode can depart from the small error theory at a higher signal to noise ratio than for the gated modes thus the variance due to anomalies can be the most important factor in determining the region of applicability of the small error analysis

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

on the capacity of surfaces in manifolds with nonnegative scalar curvature given a surface in an asymptotically flat 3 manifold with nonnegative scalar curvature we derive an upper bound for the capacity of the surface in terms of the area of the surface and the willmore functional of the surface the capacity of a surface is defined to be the energy of the harmonic function which equals 0 on the surface and goes to 1 at even in the special case of 3 this is a new estimate more generally equality holds precisely for a spherically symmetric sphere in a spatial schwarzschild 3 manifold as applications we obtain inequalities relating the capacity of the surface to the hawking mass of the surface and the total mass of the asymptotically flat manifold

scalar curvature deformation and a gluing construction for the einstein constraint equations on a compact manifold the scalar curvature map at generic metrics is a local surjection f m we show that this result may be localized to compact subdomains in an arbitrary riemannian manifold the method is extended to establish the existence of asymptotically flat scalar flat metrics on n n 3 which are spherically symmetric hence schwarzschild at infinity i e outside a compact set such metrics provide cauchy data for the einstein vacuum equations which evolve into nontrivial vacuum spacetimes which are identically schwarzschild near spatial infinity

the barankin bound and threshold behavior in frequency estimation this article presents the barankin 1949 bound as a fundamental statistical tool for the understanding of the threshold effect associated with the estimation of the frequency of a sinusoid in additive white gaussian noise it is shown that the threshold effect takes hold whenever the barankin bound departs significantly from the cramer rao bound in terms of the signal to noise ratio snr and the data length t the quantity snr spl times t ln t is shown to be a good indicator for deciding whether the snr is above threshold or not

uniformly best biased estimators in non bayesian parameter estimation in this paper a new structured approach for obtaining uniformly best non bayesian biased estimators which attain minimum mean square error performance at any point in the parameter space is established we show that if a uniformly best biased ubb estimator exists then it is unique and it can be directly obtained from any locally best biased lbb estimator a necessary and sufficient condition for the existence of a ubb estimator is derived it is shown that if there exists an optimal bias such that this condition is satisfied then it is unique and its closed form expression is obtained the proposed approach is exemplified in two nonlinear estimation problems where uniformly minimum variance unbiased estimators do not exist in the considered examples we show that the ubb estimators outperform the corresponding maximum likelihood estimators in the mse sense

maximally robust 2 d channel estimation for ofdm systems two dimensional minimum mean square error mmse channel estimation for orthogonal frequency division multiplexing ofdm systems is known to perform better than the least squares as well as the conventional 1 d mmse estimation owing to its ability of exploiting both the time and the frequency correlations among the channel frequency response cfr coefficients its superior performance comes however at the price of increased requirements the knowledge of observation noise power and that of the channel frequency as well as time correlation functions in practical transmission scenarios where channel correlation functions are not known or cannot be easily estimated it is desirable to have an estimator that is robust to mismatches between the assumed and the actual channel correlation function while such a robust estimator for the case of an infinite number of observations is well known for various uncertainty classes not much attention has been paid to the practical case of a finite number of observations we derive in this paper the maximally robust mr 2 d channel estimator for the case of a finite number of pilot observations under some realistic constraints on the uncertainty class to which the 2 d channel correlation sequence belongs we demonstrate that the correlation sequence associated with the mr estimator can be obtained by a simple semi definite optimization procedure and is interestingly different from the well known heuristic proposals simulation results establish the superiority of the proposed mr estimator over commonly employed heuristic robust estimator by as much as 3 db in terms of the worst case estimation mse and around 1 db in terms of the average ber performance under different practical transmission scenarios of interest

fundamentals of statistical signal processing estimation theory minimum variance unbiased estimation cramer rao lower bound linear models general minimum variance unbiased estimation best linear unbiased estimators maximum likelihood estimation least squares method of moments the bayesian philosophy general bayesian estimators linear bayesian estimators kalman filters summary of estimators extension for complex data and parameters

on the performance limits of data aided synchronization this paper addresses data aided da synchronization in which the reference parameter acquisition is aided by a training sequence known to the receiver the cramer rao lower bound crb for the da timing and or carrier phase recovery is presented for da parameter estimation the crb typically varies with the training sequence this indicates that different training sequences offer fundamentally different performance in the literature the widely cited closed form crb for timing and carrier phase recovery was derived under the assumption that the training sequence is independent and identically distributed i i d and sufficiently long we derive a closed form crb for timing and carrier phase recovery with respect to an arbitrary training sequence and pulse shaping function for the over and under sampling cases it turns out that the crb is a weighted summation of the aperiodic correlation of the training sequence and the weighting factor is determined by the pulse shaping filter therefore this paper reveals the fundamental link between a training sequence and its corresponding performance limit

on existence of static metric extensions in general relativity motivated by problems related to quasi local mass in general relativity we study the static metric extension conjecture proposed by r bartnik 4 we show that for any metric on b1 that is close enough to the euclidean metric and has reflection invariant boundary data there always exists an asymptotically flat and scalar flat static metric extension in m 3 b1 such that it satisfies bartnik s geometric boundary condition 4 on b1

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

an invariance of geometric mean with respect to generalized quasi arithmetic means in this paper we study the invariance of the geometric mean with respect to some generalized quasi arithmetic means namely we present some results concerning the functional equation 1 01 tx 1 t y d t 1 01 tx 1 t y d t xy x y i where i is an open real interval i r are continuous strictly monotonic unknown functions and are borel probability measures on the interval 0 1

reformulation of pisarenko harmonic decomposition method for single tone frequency estimation based on the linear prediction lp property of sinusoidal signals a closed form unbiased frequency estimator for a real sinusoid in white noise is proposed the frequency estimator which is derived by minimizing a constrained least squares cost function can be considered as a reformulation of the well known pisarenko harmonic decomposer phd online computation of the frequency estimate can be achieved in a very simple manner and its variance is derived computer simulations are included to corroborate the theoretical development and to contrast the estimator performance with the phd maximum likelihood and lp based methods as well as crame spl acute r rao lower bound

a bound on mean square estimate error 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 the bhattacharyya barankin cramer 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 of 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 cramer rao bound may not be tight unlike many large error bounds it provides a 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

aggregation functions means this two part state of the art overview on aggregation theory summarizes the essential information concerning aggregation issues an overview of aggregation properties is given including the basic classification on aggregation functions in this first part the stress is put on means i e averaging aggregation functions both with fixed arity n ary means and with multiple arities extended means

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

lagrangian mean type mappings for which the arithmetic mean is invariant we determine the class of all pairs of the lagrangian means forming mean type mappings which are invariant with respect to the arithmetic mean

blind minimax estimation we consider the linear regression problem of estimating an unknown deterministic parameter vector based on measurements corrupted by colored gaussian noise we present and analyze blind minimax estimators bmes which consist of a bounded parameter set minimax estimator whose parameter set is itself estimated from measurements thus our approach does not require any prior assumption or knowledge and the proposed estimator can be applied to any linear regression problem we demonstrate analytically that the bmes strictly dominate the least squares ls estimator i e they achieve lower mean squared error mse for any value of the parameter vector both stein s estimator and its positive part correction can be derived within the blind minimax framework furthermore our approach can be readily extended to a wider class of estimation problems than stein s estimator which is defined only for white noise and nontransformed measurements we show through simulations that the bmes generally outperform previous extensions of stein s technique

the structure and performance of estimators for real time estimation of randomly varying time delay structures for time delay estimation are described which are deduced using the maximum likelihood technique the performance of estimators designed for time invariant delays is discussed for the case of signals with a time variable delay a two step algorithm for the estimation of a randomly varying time delay between two stochastic signals is developed in the first step the maximum likelihood estimate is computed over an observation interval small enough to consider the delay to be approximately constant due to the short averaging interval the probability of ambiguous peaks in the estimation function and the variance of the estimate are greatly increased therefore in the second step a decision directed adaptive postfiltering algorithm is presented that effectively suppresses outliers and reduces the variance

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

some cauchy mean type mappings for which the geometric mean is invariant abstract the invariance of the geometric mean g with respect to the cauchy mean type mapping d f g d h k i e the equation g d f g d h k g is considered we give some necessary and necessary and sufficient conditions under assumption that one of the generators of each cauchy means is a power function

on existence and behaviour of asymptotically flat solutions to the stationary einstein equations we show existence and uniqueness of asymptotically flat solutions to the stationary einstein equations ins 3 b r whereb r is a ball of radiousr 0 when a small enough continuous complex function u on s is given regularity and decay estimates imply that these solutions are analytic in the interior ofs and also at infinity when suitably conformally rescaled

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

new results on deterministic cramer rao bounds for real and complex parameters the cramer rao bounds crb is a lower bound of great interest for system analysis and design in the asymptotic region high signal to noise ratio snr and or large number of snapshots as it is simple to calculate and it is usually possible to obtain closed form expressions the first part of the paper is a generalization to complex parameters of the barankin rationale for deriving mse lower bounds that is the minimization of a norm under a set of linear constraints with the norm minimization approach the study of fisher information matrix fim singularity constrained crb and regularity conditions become straightforward corollaries of the derivation the second part provides new results useful for system analysis and design a general reparameterization inequality the equivalence between reparameterization and equality constraints and an explicit relationship between parameters unidentifiability and fim singularity

deterministic performance bounds on the mean square error for near field source localization this correspondence investigates lower bounds on estimator s mean square error applied to the passive near field source localization more precisely we focus on the so called threshold prediction for which these bounds are known to be useful we give closed form expressions of the mcaulay seidman the hammersley chapman robbins the mcaulay hofstetter bounds and also a recently proposed bound the so called todros tabrikian bound for the deterministic observation model i e parameterized mean and the stochastic observation model i e parameterized covariance matrix finally numerical simulations are given to assess the efficiency of these lower bounds to approximate the estimator s mean square error and to predict the threshold effect

lower bound on the localization error in infinite networks with random sensor locations we present novel lower bounds on the mean square error mse of the location estimation of an emitting source via a network where the sensors are deployed randomly the sensor locations are modeled as a homogenous poisson point process in contrast to previous bounds that are a function of the specific locations of all the sensors we present cramer rao bound crb type bounds on the expected mean square error that is we first derive the crb on the mse as a function of the sensors location and then take expectation with respect to the distribution of the sensors location thus these bounds are not a function of a particular sensor configuration but rather of the sensor statistics hence these novel bounds can be evaluated prior to sensor deployment and provide insights into design issues such as the necessary sensor density the effect of the channel model the effect of the signal power and others the derived bounds are simple to evaluate and provide a good prediction of the actual network performance numerical results show that the novel bounds give a good approximation even in other deployment scenarios such as a square or a triangular grid