analysis of subspace fitting and ml techniques for parameter estimation from sensor array data it is shown that the multidimensional signal subspace method termed weighted subspace fitting wsf is asymptotically efficient this results in a novel compact matrix expression for the cramer rao bound crb on the estimation error variance the asymptotic analysis of the maximum likelihood ml and wsf methods is extended to deterministic emitter signals the asymptotic properties of the estimates for this case are shown to be identical to the gaussian emitter signal case i e independent of the actual signal waveforms conclusions concerning the modeling aspect of the sensor array problem are drawn

on a nonparametric detection method for array signal processing in correlated noise fields the correct derivation of maximization of the likelihood function is presented and the correct form of information theoretic criteria itc for the determination of the number of signals in an unknown correlated noise field is shown the possible pitfalls of using itc under the circumstances are briefly considered

eigenvalues of the laplacian in two dimensions the eigenvalue problem for the laplace operator in two dimensions is classical in mathematics and physics nevertheless computational methods for estimating the eigenvalues are still of much current interest particularly in applications to acoustic and electromagnetic waveguides although our primary interest is with the computational methods there are a number of theoretical results on the behavior of the eigenvalues and eigenfunctions that are useful for understanding the methods and in addition are of interest in themselves these results are discussed first and then the various computational methods that have been used to estimate the eigenvalues are reviewed with particular emphasis on methods that give error bounds some of the more powerful techniques available are illustrated by applying them to a model problem

direction finding using noise covariance modeling we consider the problem of direction finding in the presence of colored noise whose covariance matrix is unknown we show that the ambient noise covariance matrix can be modeled by a sum of hermitian matrices known up to a multiplicative scalar using this model we estimate jointly the directions of arrival of the signals and the noise model parameters we show that under certain conditions it is possible to obtain unbiased and efficient estimates of the signal direction the cramer rao bound is used as the principal analysis tool computer simulations using the maximum likelihood estimator provide a validation of the analytical results

multiple emitter location and signal parameter estimation processing the signals received on an array of sensors for the location of the emitter is of great enough interest to have been treated under many special case assumptions the general problem considers sensors with arbitrary locations and arbitrary directional characteristics gain phase polarization in a noise interference environment of arbitrary covariance matrix this report is concerned first with the multiple emitter aspect of this problem and second with the generality of solution a description is given of the multiple signal classification music algorithm which provides asymptotically unbiased estimates of 1 number of incident wavefronts present 2 directions of arrival doa or emitter locations 3 strengths and cross correlations among the incident waveforms 4 noise interference strength examples and comparisons with methods based on maximum likelihood ml and maximum entropy me as well as conventional beamforming are included an example of its use as a multiple frequency estimator operating on time series is included

doa estimation and detection in colored noise using additional noise only data in a typical array processing scenario noise acting on the array can not be assumed spatially white it is in many cases necessary to use quiet periods when only noise is received to estimate the noise covariance if estimation of the signal parameters and noise covariance is performed jointly performance can be improved this is especially true when stationarity considerations limit the amount of available valid noise only data an asymptotically valid approximative maximum likelihood method aml for the estimation problem is derived in this work the resulting criterion is when concentrated with respect to the signal parameters relatively simple in numerical experiments aml shows promising small sample performance compared to earlier methods the criterion function is also well suited for numerical optimization the new criterion function allows for the development of a novel mode like non iterative estimation procedure if the array belongs to the important class of uniform linear arrays the resulting procedure retains the asymptotic properties of maximum likelihood and numerical simulations indicate superior threshold performance when compared to an optimally weighted subspace fitting wsf formulation of mode for the detection problem no method has been presented that takes the unknown noise covariance into account here a well known detection scheme for wsf is extended to work in this scenario as well the derivations of this scheme further stress the importance of using the correct weighting in wsf when the noise covariance is unknown it is also shown that the minimum value of the criterion function associated with aml can be used for the detection purpose numerical experiments indicate very promising performance for the aml detection scheme

discrete time pontryagin maximum principle under state action frequency constraints we establish a pontryagin maximum principle for discrete time optimal control problems under the following three types of constraints first constraints on the states pointwise in time second constraints on the control actions pointwise in time and third constraints on the frequency spectrum of the optimal control trajectories while the first two types of constraints are already included in the existing versions of the pontryagin maximum principle it turns out that the third type of constraints cannot be recast in any of the standard forms of the existing results for the original control system we include several special cases fine tuned to control affine nonlinear and linear system models

esprit estimation of signal parameters via rotational invariance techniques an approach to the general problem of signal parameter estimation is described the algorithm differs from its predecessor in that a total least squares rather than a standard least squares criterion is used although discussed in the context of direction of arrival estimation esprit can be applied to a wide variety of problems including accurate detection and estimation of sinusoids in noise it exploits an underlying rotational invariance among signal subspaces induced by an array of sensors with a translational invariance structure the technique when applicable manifests significant performance and computational advantages over previous algorithms such as mem capon s mlm and music

estimation of the directions of arrival of signals in unknown correlated noise i the map approach and its implementation the authors propose a method of direction of arrival doa estimation of signals in the presence of noise whose covariance matrix is unknown and arbitrary other than being positive definite they examine the projection of the data onto the noise subspace the conditional probability density function pdf of the projected data given the signal parameters and the unknown projected noise covariance matrix is first formed the a posteriori pdf of the signal parameters alone is then obtained by assigning a noninformative a priori pdf to the unknown noise covariance matrix and integrating out this quantity a simple criterion for the maximum a posteriori map estimate of the doas of the signals is established some properties of this criterion are discussed and an efficient numerical algorithm for the implementation of this criterion is developed the advantage of this method is that the noise covariance matrix does not have to be known nor must it be estimated

on unique localization of multiple sources by passive sensor arrays necessary and sufficient conditions are derived for the unique localization of narrowband sources having the same known center frequency by passive sensor arrays the conditions specify the maximum number of sources that can be uniquely localized by a general array that satisfies some mild geometric constraints the conditions are expressed in terms of the number of sensors and the rank of the correlation matrix of the sources two different conditions are presented the first guarantees uniqueness for every batch of sampled data the second which is weaker guarantees uniqueness for almost every batch of sampled data with the exception of a set of batches of measure zero it is shown that a condition that is slightly weaker than the second one is also necessary

information theoretic criteria for the determination of the number of signals in spatially correlated noise the problem of determining the number of signals in high resolution array processing when the noise is spatially correlated having an unknown covariance matrix is examined by considering a model in which two sensor arrays are well separated such that their noise outputs are uncorrelated the authors develop a likelihood function whose maximum can be expressed in a very simple form involving the canonical correlation coefficients this likelihood function and a choice of penalty functions constitute a number of new information theoretic criteria suitable for the determination of the number of signals in an unknown correlated noise environment furthermore it is demonstrated that the new criteria are applicable in the case when only one sensor array is available

maximum likelihood estimation for array processing in colored noise direction of arrival estimation of multiple sources using a uniform linear array in noise with unknown covariance is considered the noise is modeled as a spatial autoregressive process with unknown parameters both stochastic and deterministic signal models are considered for the random signal case an approximate maximum likelihood estimator of the signal and noise parameters is derived it requires numerical maximization of a compressed likelihood function over the unknown arrival angles analytical expressions for the mles of the signal covariance and the ar parameters are given similar results for the case of deterministic signals are also presented

the rayleigh ritz method still competitive abstract an extended rayleigh ritz method for computing two sided eigenvalue bounds of the one dimensional schrodinger equation is presented the method is based on the b spline approximation over the truncated domain for the low lying eigenvalues and problems with rapidly increasing spectrum the corresponding lower bounds are much more accurate than the known temple bounds numerical results for problems exhibiting different kinds of the eigenfunction behaviour are presented

two sided eigenvalue bounds for the spherically symmetric states of the schro dinger equation the eigenvalues of the radial schr6dinger equation are calculated very accurately by obtaining exact upper and lower bounds by truncating the usual unbounded domain 0 cx of the system to a finite interval of the form 0 g two auxiliary eigenvalue problems are defined it is then proved that the eigenvalues of the resulting confined systems provide upper and lower bounds converging monotonically to the true eigenvalues required moreover each auxiliary eigenvalue problem gives rise to an orthonormal set involving bessel functions the matrix representation of the hamiltonian is therefore derived by expanding the wave function into a fourier bessel series numerical results for single and doublewell polynomial oscillators as well as gaussian type non polynomial potentials illustrate that the eigenvalues can be calculated to an arbitrary accuracy whenever the boundary parameter g is in the neighborhood of some critical value denoted by ecr 1998 elsevier science b v all rights reserved

maximum likelihood array processing in spatially correlated noise fields using parameterized signals this paper deals with the problem of estimating signal parameters using an array of sensors this problem is of interest in a variety of applications such as radar and sonar source localization a vast number of estimation techniques have been proposed in the literature during the past two decades most of these can deliver consistent estimates only if the covariance matrix of the background noise is known in many applications the aforementioned assumption is unrealistic recently a number of contributions have addressed the problem of signal parameter estimation in unknown noise environments based on various assumptions on the noise herein a different approach is taken we assume instead that the signals are partially known the received signals are modeled as linear combinations of certain known basis functions the exact maximum likelihood ml estimator for the problem at hand is derived as well as computationally more attractive approximation the cramer rao lower bound crb on the estimation error variance is also derived and found to coincide with the crb assuming an arbitrary deterministic model and known noise covariance

an a posteriori error estimation of the rayleigh ritz eigenvalues leta b be essentially self adjoint and positive definite differential operators defined inl 2 g using svirskij s construction of the base operator and some results from the analytic perturbation theory of linear operators a formula providing eigenvalue lower bounds of the problemau bu is derived in this formula a rough lower bound of some higher eigenvalue and the residual convergence of the rayleigh ritz eigenfunction approximations are needed some numerical results are presented

maximum likelihood doa estimation and asymptotic cramer rao bounds for additive unknown colored noise this paper is devoted to the maximum likelihood estimation of multiple sources in the presence of unknown noise with the spatial noise covariance modeled as a function of certain unknown parameters e g an autoregressive ar model a direct and systematic way is developed to find the exact maximum likelihood ml estimates of all parameters associated with the direction finding problem including the direction of arrival doa angles spl theta the noise parameters spl alpha the signal covariance spl phi sub s and the noise power spl sigma sup 2 we show that the estimates of the linear part of the parameter set spl phi sub s and spl sigma sup 2 can be separated from the nonlinear parts spl theta and spl alpha thus the estimates of spl phi sub s and spl sigma sup 2 become explicit functions of spl theta and spl alpha this results in a significant reduction in the dimensionality of the nonlinear optimization problem asymptotic analysis is performed on the estimates of spl theta and spl alpha and compact formulas are obtained for the cramer rao bounds crb s finally a newton type algorithm is designed to solve the nonlinear optimization problem and simulations show that the asymptotic crb agrees well with the results from monte carlo trials even for small numbers of snapshots

a performance analysis of subspace based methods in the presence of model errors i the music algorithm application of subspace based algorithms to narrowband direction of arrival doa estimation requires that both the array response in all directions of interest and the spatial covariance of the noise must be known in practice however neither of these quantities is known precisely depending on the degree to which they deviate from their nominal values serious performance degradation can result the performance of the music algorithm is examined for situations in which the noise covariance and array response are perturbed from their assumed values theoretical expressions for the error in the music doa estimates are derived and compared with simulations performed for several representative cases and with the appropriate cramer rao bound an optimally weighted version of music is proposed for a particular class of array errors

maximum likelihood array processing for stochastic coherent sources maximum likelihood ml estimation in array signal processing for the stochastic noncoherent signal case is well documented in the literature we focus on the equally relevant case of stochastic coherent signals explicit large sample realizations are derived for the ml estimates of the noise power and the singular signal covariance matrix the asymptotic properties of the estimates are examined and some numerical examples are provided in addition we show the surprising fact that the ml estimates of the signal parameters obtained by ignoring the information that the sources are coherent coincide in large samples with the ml estimates obtained by exploiting the coherent source information thus the ml signal parameter estimator derived for the noncoherent case or its large sample realizations asymptotically achieves the lowest possible estimation error variance corresponding to the coherent cramer rao bound

music maximum likelihood and cramer rao bound further results and comparisons the problem of determining the direction of arrival of narrowband plane waves using sensor arrays and the related problem of estimating the parameters of superimposed signals from noisy measurements are studied a number of results have been recently presented by the authors on the statistical performance of the multiple signal characterization music and the maximum likelihood ml estimators for the above problems this work extends those results in several directions first it establishes that in the class of weighted music estimators the unweighted music achieves the best performance i e the minimum variance of estimation errors in large samples next it derives the covariance matrix of the ml estimator and presents detailed analytic studies of the statistical efficiency of music and ml estimators these studies include performance comparisons of music and mle with each other as well as with the ultimate performance corresponding to the cramer rao bound finally some numerical examples are given which provide a more quantitative study of performance for the problem of finding two directions with uniform linear sensor arrays

optimal utilization of signal free samples for array processing in unknown colored noise fields many algorithms for direction of arrival doa estimation require the noise covariance matrix to be known or to possess a known structure in many cases the noise covariance is in fact estimated from separate measurements this paper addresses the combined effects of finite sample sizes both in the estimated noise covariance matrix and in the data with signals present it is assumed that a batch of signal free samples is available in addition to the signal containing samples no assumption is made on the structure of the noise covariance in this paper the asymptotic covariance of the weighted subspace fitting wsf algorithm is derived for the case in which the data are whitened using an estimated noise covariance the expression obtained suggests an optimal weighting that improves performance compared to the standard choice in addition a new method based on covariance matching is proposed both methods are asymptotically statistically efficient the cramer rao lower bound crb on the covariance of the estimate for the data model is also derived monte carlo simulations show promising small sample performance for the two new methods and confirm the asymptotic results

bounds for maximum likelihood regular and non regular doa estimation in k distributed noise we consider the problem of estimating the direction of arrival of a signal embedded in k distributed noise when secondary data which contains noise only are assumed to be available based upon a recent formula of the fisher information matrix fim for complex elliptically distributed data we provide a simple expression of the fim with the two data sets framework in the specific case of k distributed noise we show that under certain conditions the fim for the deterministic part of the model can be unbounded while the fim for the covariance part of the model is always bounded in the general case of elliptical distributions we provide a sufficient condition for unboundedness of the fim accurate approximations of the fim for k distributed noise are also derived when it is bounded additionally the maximum likelihood estimator of the signal doa and an approximated version are derived assuming known covariance matrix the latter is then estimated from secondary data using a conventional regularization technique when the fim is unbounded an analysis of the estimators reveals a rate of convergence much faster than the usual t 1 simulations illustrate the different behaviors of the estimators depending on the fim being bounded or not

technical communique a simple proof of the pontryagin maximum principle on manifolds applying the tubular neighborhood theorem we give a simple proof of the pontryagin maximum principle on a smooth manifold the idea is as follows given a control system on a manifold m we embed it into some r n and extend the control system to r n then we apply the pontryagin maximum principle on r n to the extended system and project the consequence to m

direction of arrival estimation using a sparse representation of array covariance vectors a new direction of arrival doa estimation method is proposed based on a novel data model using the concept of a sparse representation of array covariance vectors sracv in which doa estimation is achieved by jointly finding the sparsest coefficients of the array covariance vectors in an overcomplete basis the proposed method not only has high resolution and the capability of estimating coherent signals based on an arbitrary array but also gives an explicit error suppression criterion that makes it statistically robust even in low signal to noise ratio snr cases simulation experiments are conducted to validate the effectiveness of the proposed method the performance is compared with several existing doa estimation methods and the cramer rao lower bound crlb

maximum likelihood direction of arrival estimation in the presence of unknown nonuniform noise we consider the problem of estimating directions of arrival doas of multiple sources observed on the background of nonuniform white noise with an arbitrary diagonal covariance matrix a new deterministic maximum likelihood ml doa estimator is derived its implementation is based on an iterative procedure which includes a stepwise concentration of the log likelihood ll function with respect to the signal and noise nuisance parameters and requires only a few iterations to converge new closed form expressions for the deterministic and stochastic direction estimation cramer rao bounds crbs are derived for the considered nonuniform model our expressions can be viewed as an extension of the well known results by stoica and nehorai 1989 1990 and weiss and friedlander 1993 to a more general noise model than the commonly used uniform one in addition these expressions extend the results obtained by matveyev et al see circuits syst signal process vol 18 p 479 87 1999 to the multiple source case comparisons with the above mentioned earlier results help to discover several interesting properties of doa estimation in the nonuniform noise case to compare the estimation performance of the proposed ml technique with the results of our crb analysis and with the performance of conventional uniform ml simulation results are presented additionally we test our technique using experimental seismic array data our simulations and experimental results both validate essential performance improvements achieved by means of the approach proposed

a simple proof of the discrete time geometric pontryagin maximum principle on smooth manifolds we establish a geometric pontryagin maximum principle for discrete time optimal control problems on finite dimensional smooth manifolds under the following three types of constraints a constraints on the states pointwise in time b constraints on the control actions pointwise in time c constraints on the frequency spectrum of the optimal control trajectories our proof follows in spirit the path to establish geometric versions of the pontryagin maximum principle on smooth manifolds indicated in chang 2011 in the context of continuous time optimal control

sensor array processing based on subspace fitting algorithms for estimating unknown signal parameters from the measured output of a sensor array are considered in connection with the subspace fitting problem the methods considered are the deterministic maximum likelihood method ml esprit and a recently proposed multidimensional signal subspace method these methods are formulated in a subspace fitting based framework which provides insight into their algebraic and asymptotic relations it is shown that by introducing a specific weighting matrix the multidimensional signal subspace method can achieve the same asymptotic properties as the ml method the asymptotic distribution of the estimation error is derived for a general subspace weighting and the weighting that provides minimum variance estimates is identified the resulting optimal technique is termed the weighted subspace fitting wsf method numerical examples indicate that the asymptotic variance of the wsf estimates coincides with the cramer rao bound the performance improvement compared to the other techniques is found to be most prominent for highly correlated signals

a performance analysis of subspace based methods in the presence of model error ii multidimensional algorithms for pt i see ibid vol 40 no 7 p 1758 74 1992 in pt i the performance of the music algorithms for narrowband direction of arrival doa estimation when the array manifold and noise covariance are not correctly modeled was investigated this analysis is extended to multidimensional subspace based algorithms including deterministic or conditional maximum likelihood md music weighted subspace fitting wsf mode and esprit a general expression for the variance of the doa estimates that can be applied to any of the above algorithms and to any of a wide variety of scenarios is presented optimally weighted subspace fitting algorithms are presented for special cases involving random unstructured errors of the array manifold and noise covariance it is shown that one dimensional music outperforms all of the above multidimensional algorithms for random angle independent array perturbations

performance study of conditional and unconditional direction of arrival estimation a numerical and analytical study of conditional and unconditional direction of arrival doa estimation is presented explicit expressions for the unconditional cramer rao bounds on the doa estimation accuracy and the covariance matrix of the conditional maximum likelihood method are given it is shown that many doa estimation methods have the same asymptotic statistical properties under conditional and unconditional models the situation of two narrowband plane signals impinging on a uniformly spaced linear array is discussed

maximum likelihood methods for direction of arrival estimation five methods of direction of arrival doa estimation which can be derived from the maximum likelihood ml principle are considered the ml method mlm results from the application of the ml principle to the statistics of the observed raw data the standard multiple signal classification music procedure called music 1 is obtained as a brute force approximation of the mlm an improved music procedure named music 2 is obtained by applying the ml principle to the statistics of certain linear combinations of the sample noise space eigenvectors a procedure which compromises between the good performance of the mlm and the computational simplicity of music is a method of direction estimation mode 1 which is derived as a large sample realization of the mlm a fifth method called mode 2 is obtained by using the ml principle on the statistics of certain linear combinations of the sample eigenvectors mode 2 is computationally less demanding than the mlm it is of the same complexity as mode 1 and statistically more efficient a numerical comparison of these five doa estimation methods is presented it confirms the analytic results on their theoretical performance levels

direction estimation in partially unknown noise fields the problem of direction of arrival estimation in the presence of colored noise with unknown covariance is considered the unknown noise covariance is assumed to obey a linear parametric model using this model the maximum likelihood directions parameter estimate is derived and a large sample approximation is formed it is shown that a priori information on the source signal correlation structure is easily incorporated into this approximate ml aml estimator furthermore a closed form expression of the cramer rao bound on the direction parameter is provided a perturbation analysis with respect to a small error in the assumed noise model is carried out and an expression of the asymptotic bias due to the model mismatch is given computer simulations and an application of the proposed technique to a full scale passive sonar experiment is provided to illustrate the results

bracketing of cutoff frequencies of uniform waveguides with eccentric circular geometry by the rayleigh ritz method abstract this paper deals with an extension of the rayleigh ritz method for bracketing eigenvalues of self adjoint differential operators the proposed extension uses a simple formula which gives lower bounds of eigenvalues requiring only the rayleigh ritz eigenfunction approximations and a rough lower bound of some higher eigenvalue the combination of this extended rayleigh ritz method with the widely used exact conformal mapping technique enables one to obtain eigenvalue bounds of the helmholtz equation defined on the regions with complicated boundary shape numerical results concerning the cutoff frequencies of the eccentric annular eccentric lunar and eccentric inverted lunar waveguide are presented