on the relationship between robust minimum variance beamformers with probabilistic and worst case distortionless response constraints an interesting relationship between the probability constrained and worst case optimization based robust minimum variance mv beamformers has been discovered it is shown that both in the cases of circularly symmetric gaussian and worst case distributions of the steering vector mismatch the probability constrained robust mv beamforming problem can be tightly approximated as a convex second order cone programming socp problem the latter problem is mathematically equivalent to that resulting from the deterministic worst case approach and therefore probability constrained beamformers can be interpreted and implemented using their deterministic worst case counterparts however an important advantage of the developed probability constrained mv beamformers with respect to their standard worst case counterparts is that the former approaches enable to explicitly quantify the parameters of the uncertainty region in terms of the beamformer outage probability

besov type spaces of variable smoothness on rough domains abstract the paper puts forward new besov spaces of variable smoothness b p q 0 g t k and b p q r l t k on rough domains a domain g is either a bounded lipschitz domain in r n or the epigraph of a lipschitz function a domain is an e domain these spaces are shown to be the traces of the spaces b p q 0 r n t k and b p q r l r n t k on domains g and respectively the extension operator ext 1 b p q 0 g t k b p q 0 r n t k is linear the operator ext 2 b p q r l t k b p q r l r n t k is nonlinear as a corollary an exact description of the traces of 2 microlocal besov type spaces and weighted besov type spaces on rough domains is obtained

detection of signals by information theoretic criteria a new approach is presented to the problem of detecting the number of signals in a multichannel time series based on the application of the information theoretic criteria for model selection introduced by akaike aic and by schwartz and rissanen mdl unlike the conventional hypothesis testing based approach the new approach does not requite any subjective threshold settings the number of signals is obtained merely by minimizing the aic or the mdl criteria simulation results that illustrate the performance of the new method for the detection of the number of signals received by a sensor array are presented

doubly constrained robust capon beamformer the standard capon beamformer scb is known to have better resolution and much better interference rejection capability than the standard data independent beamformer when the array steering vector is accurately known however the major problem of the scb is that it lacks robustness in the presence of array steering vector errors in this paper we will first provide a complete analysis of a norm constrained capon beamforming nccb approach which uses a norm constraint on the weight vector to improve the robustness against array steering vector errors and noise our analysis of nccb is thorough and sheds more light on the choice of the norm constraint than what was commonly known we also provide a natural extension of the scb which has been obtained via covariance matrix fitting to the case of uncertain steering vectors by enforcing a double constraint on the array steering vector viz a constant norm constraint and a spherical uncertainty set constraint which we refer to as the doubly constrained robust capon beamformer dcrcb nccb and dcrcb can both be efficiently computed at a comparable cost with that of the scb performance comparisons of nccb dcrcb and several other adaptive beamformers via a number of numerical examples are also presented

on the rank of extreme matrices in semidefinite programs and the multiplicity of optimal eigenvalues we derive some basic results on the geometry of semidefinite programming sdp and eigenvalue optimization i e the minimization of the sum of the k largest eigenvalues of a smooth matrix valued function we provide upper bounds on the rank of extreme matrices in sdps and the first theoretically solid explanation of a phenomenon of intrinsic interest in eigenvalue optimization in the spectrum of an optimal matrix the kth and k 1 st largest eigenvalues tend to be equal and frequently have multiplicity greater than two this clustering is intuitively plausible and has been observed as early as 1975 when the matrix valued function is affine we prove that clustering must occur at extreme points of the set of optimal solutions if the number of variables is sufficiently large we also give a lower bound on the multiplicity of the critical eigenvalue these results generalize to the case of a general matrix valued function under appropriate conditions

a competitive mean squared error approach to beamforming we treat the problem of beamforming for signal estimation where the goal is to estimate a signal amplitude from a set of array observations conventional beamforming methods typically aim at maximizing the signal to interference plus noise ratio sinr however this does not guarantee a small mean squared error mse so that on average the resulting signal estimate can be far from the true signal here we consider strategies that attempt to minimize the mse between the estimated and unknown signal waveforms the methods we suggest all maximize the sinr but at the same time are designed to have good mse performance since the mse depends on the signal power which is unknown we develop competitive beamforming approaches that minimize a robust mse measure two design strategies are proposed minimax mse and minimax regret we demonstrate through numerical examples that the suggested minimax beamformers can outperform several existing standard and robust methods over a wide range of signal to noise ratio snr values finally we apply our techniques to subband beamforming and illustrate their advantage in estimating a wideband signal

a robust adaptive beamformer based on worst case semi definite programming in this correspondence a novel robust adaptive beamformer is proposed based on the worst case semi definite programming sdp a recent paper has reported that a beamformer robust against large steering direction error can be constructed by using linear constraints on magnitude response in sdp formulation in practice however array system also suffers from many other array imperfections other than steering direction error in order to make the adaptive beamformer robust against all kinds of array imperfections the worst case optimization technique is proposed to reformulate the beamformer by minimizing the array output power with respect to the worst case array imperfections the resultant beamformer has the mathematical form of a regularized sdp problem and possesses superior robustness against arbitrary array imperfections although the formulation of robust beamformer uses weighting matrix with the help of spectral factorization approach the weighting vector can be obtained so that the beamformer can be used for both signal power and waveform estimation simple implementation flexible performance control as well as significant signal to interference plus noise ratio sinr enhancement support the practicability of the proposed method

source enumeration via mdl criterion based on linear shrinkage estimation of noise subspace covariance matrix numerous methodologies have been investigated for source enumeration in sample starving environments for those having their root in the framework of random matrix theory the involved distribution of the sample eigenvalues is required instead of relying on the eigenvalue distribution this work devises a linear shrinkage based minimum description length ls mdl criterion by utilizing the identity covariance matrix structure of noise subspace components with linear shrinkage and gaussian assumption of the observations an accurate estimator for the covariance matrix of the noise subspace components is derived the eigenvalues obtained from the estimator turn out to be a linear function of the corresponding sample eigenvalues enabling the ls mdl criterion to accurately detect the source number without incurring significantly additional computational load furthermore the strong consistency of the ls mdl criterion for m n and m n c 0 is proved where m and n are the antenna number and snapshot number respectively simulation results are included for illustrating the effectiveness of the proposed criterion

robust adaptive beamforming based on steering vector estimation with as little as possible prior information a general notion of robustness for robust adaptive beamforming rab problem and a unified principle for minimum variance distortionless response mvdr rab techniques design are formulated this principle is to use standard mvdr beamformer in tandem with an estimate of the desired signal steering vector found based on some imprecise prior information differences between various mvdr rab techniques occur only because of the differences in the assumed prior information and the corresponding signal steering vector estimation techniques a new mvdr rab technique which uses as little as possible and easy to obtain imprecise prior information is developed the objective for estimating the steering vector is the maximization of the beamformer output power while the constraints are the normalization condition and the requirement that the estimate does not converge to any of the interference steering vectors and their linear combinations the prior information used is only the imprecise knowledge of the antenna array geometry and angular sector in which the actual steering vector lies mathematically the proposed mvdr rab is expressed as the well known non convex quadratically constrained quadratic programming problem with two constraints which can be efficiently and exactly solved some new results for the corresponding optimization problem such as a new algebraic way of finding the rank one solution from the general rank solution of the relaxed problem and the condition under which the solution of the relaxed problem is guaranteed to be rank one are derived our simulation results demonstrate the superiority of the proposed method over other previously developed rab techniques

traces of weighted sobolev spaces old and new abstract we give short simple proofs of uspenski s results characterizing besov spaces as trace spaces of weighted sobolev spaces we generalize uspenski s results and prove the optimality of these generalizations we next show how classical results on the functional calculus in the besov spaces can be obtained as straightforward consequences of the theory of weighted sobolev spaces

on various approaches to besov type spaces of variable smoothness abstract the paper is concerned with besov spaces of variable smoothness b p q 0 r n t k in which the norms are defined in terms of convolutions with smooth functions a relation is found between the spaces b p q 0 r n t k and the spaces b p q r l r n t k which were introduced earlier by the author as an application an equivalent description of the trace space of the weighted sobolev space with muckenhoupt weight is given

performance of dmi and eigenspace based beamformers the performance of the direct matrix inversion dmi method for antenna arrays of arbitrary geometry is analyzed by asymptotic statistical techniques the effects of eigenspace disturbance caused by finite samples on the output interference and noise powers are examined under the unit gain constraint in the direction of the desired signal the results show that the performance of the dmi method is degraded mostly by the disturbed noise subspace that suggests the use of an eigenspace based beamformer in which the weight vector is computed by using the signal plus interference subspace component of the sample correlation matrix convergence properties of the eigenspace based beamformer are evaluated for the cases in which the source number is known and in which it is overestimated theoretical analyses validated by computer simulations indicate that the eigenspace based beamformer has faster convergence rate than the dmi method

a novel adaptive beamformer based on semidefinite programming sdp with magnitude response constraints a novel robust adaptive beamformer formulated as a semidefinite programming sdp problem is proposed in this paper with new constraints on the magnitude response the beamwidth and response ripple of the robust response region can be well controlled moreover only a small part of these inequality constraints on the magnitude response are active during optimization so that few degrees of freedom dofs of the adaptive beamformer are consumed consequently the resultant beamformer has significant improvement on signal to interference plus noise ratio sinr an important problem in the proposed beamformer is how to generate the array weight vector from the optimal semidefinite matrix in this paper a method utilizing the extended spectral factorization method is proposed to solve this problem simple implementation flexible performance control as well as significant sinr enhancement support the practicability of the proposed method

robust minimum variance beamforming this paper introduces an extension of minimum variance beamforming that explicitly takes into account variation or uncertainty in the array response sources of this uncertainty include imprecise knowledge of the angle of arrival and uncertainty in the array manifold in our method uncertainty in the array manifold is explicitly modeled via an ellipsoid that gives the possible values of the array for a particular look direction we choose weights that minimize the total weighted power output of the array subject to the constraint that the gain should exceed unity for all array responses in this ellipsoid the robust weight selection process can be cast as a second order cone program that can be solved efficiently using lagrange multiplier techniques if the ellipsoid reduces to a single point the method coincides with capon s method we describe in detail several methods that can be used to derive an appropriate uncertainty ellipsoid for the array response we form separate uncertainty ellipsoids for each component in the signal path e g antenna electronics and then determine an aggregate uncertainty ellipsoid from these we give new results for modeling the element wise products of ellipsoids we demonstrate the robust beamforming and the ellipsoidal modeling methods with several numerical examples

an interior point method for semidefinite programming we propose a new interior point based method to minimize a linear function of a matrix variable subject to linear equality and inequality constraints over the set of positive semidefinite matrices

doubly constrained robust capon beamformer with ellipsoidal uncertainty sets the doubly constrained robust dcr capon beamformer with a spherical uncertainty set was introduced and studied by stoica and wang here we consider the generalized dcr problem gdcr in which the uncertainty set is an ellipsoid rather than a sphere although as noted previously by stoica and wang this problem is nonconvex and appears to be intractable we show that it can be solved efficiently in fact we prove that the gdcr beamformer can be obtained as a solution to a convex optimization problem to this end we first derive a strong duality result for nonconvex quadratic programs with two quadratic constraints over the complex domain specializing the results to our context leads to a semidefinite programming formulation of the gdcr beamformer

theory and application of covariance matrix tapers for robust adaptive beamforming we unify several seemingly disparate approaches to robust adaptive beamforming through the introduction of the concept of a covariance matrix taper cmt this is accomplished by recognizing that an important class of adapted pattern modification techniques are realized by the application of a conformal matrix taper to the original sample covariance matrix from the schur product theorem for positive semi definite matrices and kolmogorov s existence theorem we further establish that cmts are in fact the solution to a minimum variance optimum beamformer associated with an auxiliary stochastic process that is related to the original by a hadamard schur product this allows us to gain deeper insight into the design of both existing pattern modification techniques and new cmts that can for example simultaneously address several different design constraints such as pattern distortion due to insufficient sample support and weights mismatch due to nonstationary interference a new two dimensional 2 d cmt for space time adaptive radar applications designed to provide more robust clutter cancellation is also introduced since the cmt approach only involves a single matrix haddamard product it is also inherently low complexity the practical utility of the cmt approach is illustrated through its application to both spatial and spatio temporal adaptive beamforming examples

reduced rank mdl method for source enumeration in high resolution array processing this paper proposes a reduced rank minimum description length mdl method to enumerate the incident waves impinging on a uniform linear array ula first a new observation data and a reference signal are formed from sensor data by means of the shift invariance property of the ula a cross correlation between them is calculated which is able to capture signal information and efficiently suppress additive noise second the normalized cross correlation is used as initial information for a recursion procedure to quickly partition the observation data into two orthogonal components in a signal subspace and a reduced rank noise subspace the components in the noise subspace are employed to calculate the total code length that is required to encode the observation data finally the model with the shortest code length namely the minimum description length is chosen as the best model unlike the traditional mdl methods this method partitions the observation data into the cleaner signal and noise subspace components by means of the recursion procedure avoiding the estimation of a covariance matrix and its eigendecomposition thus the method has the advantage of computational simplicity its performance is demonstrated via numerical results

rank constrained separable semidefinite programming with applications to optimal beamforming consider a downlink communication system where multiantenna base stations transmit independent data streams to decentralized single antenna users over a common frequency band the goal of the base stations is to jointly adjust the beamforming vectors to minimize the transmission powers while ensuring the signal to interference noise ratio requirement of each user within the system at the same time it may be necessary to keep the interference generated on other coexisting systems under a certain tolerable level in addition one may want to include general individual shaping constraints on the beamforming vectors this beamforming problem is a separable homogeneous quadratically constrained quadratic program and it is difficult to solve in general in this paper we give conditions under which strong duality holds and propose efficient algorithms for the optimal beamforming problem first we study rank constrained solutions of general separable semidefinite programs sdps and propose rank reduction procedures to achieve a lower rank solution then we show that the sdp relaxation of three classes of optimal beamforming problem always has a rank one solution which can be obtained by invoking the rank reduction procedures

a projection approach for robust adaptive beamforming it is well known that calibration errors can seriously degrade performance in adaptive arrays particularly when the input signal to noise ratio is large the effect is caused by the perturbation of the presumed steering vector from its optimal value although it is not as widely known similar degradation occurs in sampled matrix inversion processing when the covariance matrix is estimated while the desired signal is present in the snapshot data under these conditions performance loss is due to errors in the estimated covariance matrix and occurs even when the steering vector is known exactly in the paper a new method based of modification of the steering vector is proposed to overcome both the problems of perturbation and of sample covariance errors the method involves projection of the presumed steering vector onto the observed signal plus interference subspace an analysis is also presented illustrating that the sample covariance errors can be viewed as a particular type of perturbation error and a simple approximation is derived for the expected beamformer performance as a function of the number of data snapshots both analytical and experimental results are presented that illustrate the advantages of the proposed method

response vector constrained robust lcmv beamforming based on semidefinite programming although linearly constrained minimum variance lcmv beamforming is robust against imprecise target information it usually leads to relatively high sidelobe and distorted mainlobe which would induce a high false alarm probability to circumvent this problem this work devises a novel robust lcmv beamforming approach by utilizing response vector optimization it intends to find the optimal response vector in lieu of the all one response vector in traditional lcmv beamformer the proposed robust beamformer is first formulated as a nonconvex quadratically constrained quadratic programming problem and then transformed into a semidefinite programming problem which can be efficiently and exactly solved the proposed beamformer not only improves the performance in terms of signal to interference plus noise ratio substantially but also possesses low sidelobe and well maintained mainlobe moreover since the response vector is quite small in size the complexity of calculating the optimal response vector is negligible additionally the proposed beamformer is also extended to two dimensional space time adaptive processing simulation results are presented to demonstrate the superiority of the proposed approach

robust adaptive beamformers based on worst case optimization and constraints on magnitude response in this paper novel robust adaptive beamformers are proposed with constraints on array magnitude response with the transformation from the array output power and the magnitude response to linear functions of the autocorrelation sequence of the array weight the optimization of an adaptive beamformer which is often described as a quadratic optimization problem in conventional beamforming methods is then reformulated as a linear programming lp problem unlike conventional robust beamformers the proposed method is able to flexibly control the robust response region with specified beamwidth and response ripple in practice an array has many imperfections besides steering direction error in order to make the adaptive beamformer robust against all kinds of imperfections worst case optimization is exploited to reconstruct the robust beamformer by minimizing array output power with the existence of the worst case array imperfections the robust beamforming can be expressed as a second order cone programming socp problem the resultant beamformer possesses superior robustness against arbitrary array imperfections with the proposed methods a large robust response region and a high signal to interference plus noise ratio sinr enhancement can be achieved readily simple implementation flexible performance control as well as significant sinr enhancement support the practicability of the proposed methods

sensitivity to perturbation of the stationary distribution some refinements abstract the sensitivity of the unique stationary distribution of a finite markov chain which has a state accessible in one step from every other state is examined using ordinary matrix inversion involving an arbitrary vector a result of funderlic and meyer is sharpened the presence of the arbitrary vector allows extension of the result to stochastic matrices with other structures the examples given by funderlic and meyer are reconsidered

traces of weighted sobolev spaces with muckenhoupt weight the case p 1 abstract a complete description of traces on r n of functions from the weighted sobolev space w 1 l r n 1 l n with weight a 1 loc r n 1 is obtained in the case l 1 the proof of the trace theorem is based on a special nonlinear algorithm for constructing a system of tilings of the space r n as the trace of the space w 1 1 r n 1 we have the new function space z k m