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

nonlinear semidefinite programming sensitivity convergence and an application in passive reduced order modeling we consider the solution of nonlinear programs with nonlinear semidefiniteness constraints the need for an efficient exploitation of the cone of positive semidefinite matrices makes the solution of such nonlinear semidefinite programs more complicated than the solution of standard nonlinear programs this paper studies a sequential semidefinite programming ssp method which is a generalization of the well known sequential quadratic programming method for standard nonlinear programs we present a sensitivity result for nonlinear semidefinite programs and then based on this result we give a self contained proof of local quadratic convergence of the ssp method we also describe a class of nonlinear semidefinite programs that arise in passive reduced order modeling and we report results of some numerical experiments with the ssp method applied to problems in that class

a global algorithm for nonlinear semidefinite programming in this paper we propose a global algorithm for solving nonlinear semidefinite programming problems this algorithm inspired by the classic sqp sequentially quadratic programming method modifies the s sdp sequentially semidefinite programming local method by using a nondifferentiable merit function combined with a line search strategy

bias reduction for an explicit solution of source localization using tdoa this paper proposes two methods to reduce the bias of the well known algebraic explicit solution chan and ho a simple and efficient estimator for hyperbolic location ieee trans signal process vol 42 pp 1905 1915 aug 1994 for source localization using tdoa bias of a source location estimate is significant when the measurement noise is large and the geolocation geometry is poor bias also dominates performance when multiple times of independent measurements are available such as in uwb localization or in target tracking the paper starts by deriving the bias of the source location estimate from chan and ho the bias is found to be considerably larger than that of the maximum likelihood estimator two methods called biassub and biasred are developed to reduce the bias the biassub method subtracts the expected bias from the solution of chan and ho s work where the expected bias is approximated by the theoretical bias using the estimated source location and noisy data measurements the biasred method augments the equation error formulation and imposes a constraint to improve the source location estimate the biassub method requires the exact knowledge of the noise covariance matrix and biasred only needs the structure of it analysis shows that both methods reduce the bias considerably and achieve the crlb performance for distant source when the noise is gaussian and small the biassub method can nearly eliminate the bias and the biasred method is able to lower the bias to the same level as the maximum likelihood estimator the biasred method is extended for tdoa and fdoa positioning simulations corroborate the performance of the proposed methods

an improved path following method for mixed h _ 2 h _ infty controller design among the few methods available to solve bilinear matrix inequalities bmis occurring in control design the path following method published some years ago appears to be one of the best ones as far as linearization methods are concerned however few details are given in the literature about its implementation and limits in this technical note this method is applied to the design of mixed h 2 h infin controllers with full details of the algorithm and some improvements over the one which has been published a few years ago the results obtained with the numerical example given in that same publication as well as with some other examples are compared with those given by other methods including a direct bmi solving program

a fast and effective multidimensional scaling approach for node localization in wireless sensor networks given a set of pairwise distance estimates between nodes it is often of interest to generate a map of node locations this is an old nonlinear estimation problem that has recently drawn interest in the signal processing community due to the emergence of wireless sensor networks sensor maps are useful for estimating the spatial distribution of measured phenomena and for routing purposes we propose a two stage algorithm that combines algebraic initialization and gradient descent in particular we borrow an algebraic solution known as fastmap from the database literature adapt it to the sensor network context and motivate the placement of anchor pivot nodes on the edges of the network when all nodes can estimate their distance from the anchors the overall algorithm offers very competitive performance at low complexity quadratic in the number of nodes

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

solution and analysis of tdoa localization of a near or distant source in closed form point positioning and direction of arrival doa localization require separate estimators depending on whether the source is near or distant from the sensors the use of modified polar representation mpr for the source location enables the integration of the two estimators together and eliminates the knowledge needed if the source is in the near field or far field the previous work on mpr only provides an iterative implementation of the maximum likelihood estimator mle initialized by a coarse semidefinite relaxation sdr solution this paper proposes a formulation for time difference of arrival tdoa localization of a 2 d or 3 d source in mpr that would lead to a closed form solution through the minimization of a quadratic function with a quadratic constraint two techniques the successive unconstrained minimization sum and the generalized trust region subproblem gtrs are applied to solve the optimization detailed analysis in the first order for mean square and in the second order for estimation bias is performed for both methods the theoretical results illustrate that the closed form solution from each of the methods provides the cramer rao lower bound performance for gaussian noise the gtrs solution has better accuracy than the sum solution when the source signal is arriving at an azimuth or elevation angle close to zero 90 circ or 180 circ or when the measurement noise level is large they yield comparable performance with the mle albeit having less complexity and avoiding possible divergence behavior they outperform the closed form solutions from the literature in angle estimation and have smaller bias

a d c optimization algorithm for solving the trust region subproblem this paper is devoted to difference of convex functions d c optimization d c duality local and global optimality conditions in d c programming the d c algorithm dca and its application to solving the trust region problem the dca is an iterative method that is quite different from well known related algorithms thanks to the particular structure of the trust region problem the dca is very simple requiring only matrix vector products and in practice converges to the global solution the inexpensive implicitly restarted lanczos method of sorensen is used to check the optimality of solutions provided by the dca when a nonglobal solution is found a simple numerical procedure is introduced both to find a feasible point having a smaller objective value and to restart the dca at this point it is shown that in the nonconvex case the dca converges to the global solution of the trust region problem using only matrix vector products and requiring at most 2m 2 restarts where m is the number of distinct negative eigenvalues of the coefficient matrix that defines the problem numerical simulations establish the robustness and efficiency of the dca compared to standard related methods especially for large scale problems

a generalized subspace approach for mobile positioning with time of arrival measurements the problem of locating mobile terminals has received considerable attention particularly in the field of wireless communications in this correspondence a simple subspace based algorithm for mobile positioning with the use of time of arrival measurements deduced from signals received at three or more reference base stations is derived and analyzed it is shown that the proposed approach is a generalization of the mobile localization method based on multidimensional similarity analysis computer simulations are included to contrast the estimator performance with cramer rao lower bound

first and second order analysis of nonlinear semidefinite programs in this paper we study nonlinear semidefinite programming problems convexity duality and first order optimality conditions for such problems are presented a second order analysis is also given second order necessary and sufficient optimality conditions are derived finally sensitivity analysis of such programs is discussed

two numerical methods for optimizing matrix stability consider the affine matrix family a x a0 m 1 xkak mapping a design vector x r m into the space of n n real matrices we are interested in the question of how to choose x to optimize the stability of the matrix a x a typical motivation is that one wishes to control the stability of the dynamical system a x z a classic example is stabilization by output feedback we take two approaches let x denote the spectral abscissa the largest real part of the eigenvalues of a matrix x as is well known this quantity bounds the asymptotic decay rate of the trajectories of the associated dynamical system z xz our first approach to optimizing stability is to directly minimize the function a x the spectral abscissa x is a continuous but non smooth in fact non lipschitz function of the matrix argument x and finding a global minimizer of a x is hard we introduce a novel random gradient bundle method for approximating local minimizers motivated by recent work on nonsmooth analysis of the function x our second approach is to minimize a related function a x where is a robustness parameter in 0 1 one motivation for the definition of the robust spectral abscissa x is that it bounds transient peaks as well as asymptotic decay of trajectories of xz the function x is lipschitz but non convex for 0 1

stabilization via nonsmooth nonconvex optimization nonsmooth variational analysis and related computational methods are powerful tools that can be effectively applied to identify local minimizers of nonconvex optimization problems arising in fixed order controller design we support this claim by applying nonsmooth analysis and methods to a challenging belgian chocolate stabilization problem posed in 1994 find a stable minimum phase rational controller that stabilizes a specified second order plant although easily stated this particular problem remained unsolved until 2002 when a solution was found using an eleventh order controller our computational methods find a stabilizing third order controller without difficulty suggesting explicit formulas for the controller and for the closed loop system which has only one pole with multiplicity 5 furthermore our analytical techniques prove that this controller is locally optimal in the sense that there is no nearby controller with the same order for which the closed loop system has all its poles further left in the complex plane although the focus of the paper is stabilization once a stabilizing controller is obtained the same computational techniques can be used to optimize various measures of the closed loop system including its complex stability radius or hinfin performance

the pre image problem in kernel methods in this paper we address the problem of finding the pre image of a feature vector in the feature space induced by a kernel this is of central importance in some kernel applications such as on using kernel principal component analysis pca for image denoising unlike the traditional method in which relies on nonlinear optimization our proposed method directly finds the location of the pre image based on distance constraints in the feature space it is noniterative involves only linear algebra and does not suffer from numerical instability or local minimum problems evaluations on performing kernel pca and kernel clustering on the usps data set show much improved performance

global attractor for a class of nonlinear lattices abstract we consider a class of nonlinear lattices with nonlinear damping 0 1 u n t 1 p p u n t u n t h u n t g n u n t f n where n z t r is a real positive constant p is any positive integer and is the discrete one dimensional laplace operator under suitable conditions on h and g we prove the existence of a global attractor for the continuous semigroup associated with 0 1 our proofs are based on a difference inequality due to m nakao m nakao global attractors for nonlinear wave equations with nonlinear dissipative terms j differential equations 227 2006 204 229

differential delay doppler ml estimation with unknown signals previously published analyses on the maximum likelihood estimation of joint frequency and time offsets between two noisy versions of a common signal have assumed a gaussian random signal with a known power spectrum in many applications the common signal is not gaussian and there may be no prior knowledge of its detailed structure instead estimation of a hypothesized common signal can be construed as another element in the estimation process the analysis is straightforward and shows that the complex ambiguity function caf still represents the ml approach when the noise is gaussian and spectrally flat additional interpretation shows that use of interference rejection filtering followed by a caf is an attractive suboptimum approach in an environment of narrowband interferers

combining convex concave decompositions and linearization approaches for solving bmis with application to static output feedback a novel optimization method is proposed to minimize a convex function subject to bilinear matrix inequality bmi constraints the key idea is to decompose the bilinear mapping as a difference between two positive semidefinite convex mappings at each iteration of the algorithm the concave part is linearized leading to a convex subproblem applications to various output feedback controller synthesis problems are presented in these applications the subproblem in each iteration step can be turned into a convex optimization problem with linear matrix inequality lmi constraints the performance of the algorithm has been benchmarked on the data from the compleib library

solving semidefinite quadratic linear programs using sdpt3 this paper discusses computational experiments with linear optimization problems involving semidefinite quadratic and linear cone constraints sqlps many test problems of this type are solved using a new release of sdpt3 a matlab implementation of infeasible primal dual path following algorithms the software developed by the authors uses mehrotra type predictor corrector variants of interior point methods and two types of search directions the hkm and nt directions a discussion of implementation details is provided and computational results on problems from the sdplib and dimacs challenge collections are reported

on bending invariant signatures for surfaces isometric surfaces share the same geometric structure also known as the first fundamental form for example all possible bendings of a given surface that includes all length preserving deformations without tearing or stretching the surface are considered to be isometric we present a method to construct a bending invariant signature for such surfaces this invariant representation is an embedding of the geometric structure of the surface in a small dimensional euclidean space in which geodesic distances are approximated by euclidean ones the bending invariant representation is constructed by first measuring the intergeodesic distances between uniformly distributed points on the surface next a multidimensional scaling technique is applied to extract coordinates in a finite dimensional euclidean space in which geodesic distances are replaced by euclidean ones applying this transform to various surfaces with similar geodesic structures first fundamental form maps them into similar signature surfaces we thereby translate the problem of matching nonrigid objects in various postures into a simpler problem of matching rigid objects as an example we show a simple surface classification method that uses our bending invariant signatures

a newton like method for solving rank constrained linear matrix inequalities this paper presents a newton like algorithm for solving systems of rank constrained linear matrix inequalities though local quadratic convergence of the algorithm is not a priori guaranteed or observed in all cases numerical experiments including application to an output feedback stabilization problem show the effectiveness of the algorithm

a novel weighted multidimensional scaling analysis for time of arrival based mobile location recently multidimensional scaling mds algorithms have been verified to be robust for the mobile localization however they do not achieve the cramer rao lower bound crlb even though the measurement noise is small in this paper a novel weighted mds method is proposed for time of arrival based mobile location it can achieve the crlb at moderate noise level before the threshold effect occurs in addition unlike the classical mds methods depending on the eigendecomposition of the scalar product matrix the proposed method is based on the matrix itself directly which in turn decreases the computational complexity

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

a supplement to multidimensional scaling framework for mobile location a unified view recently while the classical multidimensional scaling mds algorithm had been claimed to be infeasible mds based algorithms including modified mds and subspace approach have been applied to the range measurements based mobile localization in this correspondence we first prove that the classical mds algorithm is feasible for the mobile localization then we show that all mds based algorithms can be unified under the same framework and they are not optimal according to the gaussian markov theorem based on the unified framework the weighted mds analysis is applied to obtain the optimum performance the performances of both the mds algorithms and their corresponding weighted mds based algorithms in the unified framework are demonstrated through simulations

iterative constrained weighted least squares source localization using tdoa and fdoa measurements this paper investigates the constrained weighted least squares cwls source localization problem by using time difference of arrival and frequency difference of arrival measurements the problem can be formulated as a quadratic programming with two indefinite quadratic equality constraints which is nonconvex and np hard moreover the weighting matrix is coupled with the unknown source position and velocity we propose an iterative cwls method that can efficiently solve this problem it iteratively performs a linearization procedure on the quadratic equality constraints to obtain an approximate programming with linear constraints which can be analytically solved and the weighting matrix is updated in each iteration theoretical analysis reveals that the proposed method if converges can lead to the global optimal solution of the formulated problem that reaches the crlb accuracy under mild assumptions on the measurement noises the monte carlo simulation results indicate that the percentage of convergence within mathbf 20 iterations is more than mathbf 96 and the localization accuracy is significantly improved over the previous methods with less computation time requirement moreover it is found from simulations that the iterative cwls method retains acceptable performance even under the ill conditioned situation when the sensor geometry is not desirable

a novel closed form estimator for 3d tma using heterogeneous sensors this paper considers the problem of three dimensional target motion analysis using a combination of bearing elevation and time difference of arrival tdoa measurements we propose a hybrid closed form solution based on the weighted instrumental variables for this highly nonlinear problem the proposed solution avoids the high computational complexity and convergence problems associated with iterative estimators furthermore the simulation results suggest that the proposed estimator is nearly efficient since its performance is very close to the best achievable performance as predicted by the cramer rao lower bound

an asymptotically efficient estimator for tdoa and fdoa positioning of multiple disjoint sources in the presence of sensor location uncertainties this correspondence considers the problem of locating multiple disjoint sources using time differences of arrival tdoas and frequency differences of arrival fdoas in the presence of sensor position and velocity errors previous work applies to one source only with suboptimum performance for near source or requires joint estimation of the source and sensor locations that could be computation demanding through nonlinearly transforming the measurements and converting them with respect to the inaccurate sensor locations we have developed an algebraic solution to this problem the solution is shown analytically to achieve the cramer rao lower bound crlb performance over small noise region and does not require joint estimation with sensor locations

the strong second order sufficient condition and constraint nondegeneracy in nonlinear semidefinite programming and their implications for a locally optimal solution to the nonlinear semidefinite programming problem under robinson s constraint qualification the following conditions are proved to be equivalent the strong second order sufficient condition and constraint nondegeneracy the nonsingularity of clarke s jacobian of the karush kuhn tucker system the strong regularity of the karush kuhn tucker point and others

multidimensional scaling analysis for passive moving target localization with tdoa and fdoa measurements a new framework for positioning a moving target is introduced by utilizing time differences of arrival tdoa and frequency differences of arrival fdoa measurements collected using an array of passive sensors it exploits the multidimensional scaling mds analysis which has been developed for data analysis in the field such as physics geography and biology particularly we present an accurate and closed form solution for the position and velocity of a moving target unlike most passive target localization methods focusing on minimizing a loss function with respect to the measurement vector the proposed method is based on the optimization of a cost function related to the scalar product matrix in the classical mds framework it is robust to the large measurement noise the bias and variance of the proposed estimator is also derived simulation results show that the proposed estimator achieves better performance than the spherical interpolation si method and the two step weighted least squares wls approach and it attains the cramer rao lower bound at a sufficiently high noise level before the threshold effect occurs moreover for the proposed estimator the threshold effect which is a result of the nonlinear nature of the localization problem occurs apparently later as the measurement noise increases for a near field target