explicit expressions of the nth gradient of 1 r explicit expressions of the nth gradient tensor and partial derivatives of the function 1 r are given in the absolute and in the local bases associated respectively with the cartesian and the spherical coordinates of space they may be used in various multipolar developments of r 1 interaction energies

space time transmit code and receive filter design for colocated mimo radar this paper deals with the design of multiple input multiple output radar space time transmit code sttc and space time receive filter strf to enhance moving targets detection in the presence of signal dependent interferences an iterative procedure whose convergence is analytically proved is devised to maximize the signal to interference plus noise ratio sinr accounting for both a similarity constraint and a constant modulus requirement on the probing waveform each iteration of the algorithm involves the solution of hidden convex problems specifically both a convex problem whose solution is provided in closed form and a set of fractional programming problems that can be globally solved in polynomial time via the dinkelback s procedure are settled the computational complexity is linear with the number of iterations and polynomial with the sizes of the sttc and the strf in particular the proposed technique provides a monotonic sinr improvement without limitations on the size of the similarity constraint and ensures convergence to a stationary point filling these important gaps in the open literature besides the reported results highlight that the new devised procedure outperforms both in the optimized sinr value and the computational complexity than the available counterparts

distance between populations using mixed continuous and categorical variables summary a general form of the location model is considered for mixed continuous and categorical variables observed in a number of different populations and some special cases of practical interest are cited the distance between any two populations is derived for each of these models estimation of parameters in these distance measures is discussed and the methods are illustrated by application to previously published data

fast unit modulus least squares with applications in beamforming unit modulus least squares uls problems arise in many applications including phase only beamforming sensor network localization synchronization phase retrieval and radar code design uls formulations can always be recast as unit modulus quadratic programs to which semidefinite relaxation sdr can be applied and is often the state of the art approach however sdr lifts the problem dimension i e the number of variables from n to n 2 which drastically increases the memory burden and computational cost when the problem size is already large e g when designing phase only beamformer weights for massive multiple input multiple output systems this paper focuses on scalable first order algorithms for the uls problem and some of its variants it advocates using simple gradient projection gp as a starting point for solving the uls problem establishes global convergence of gp to a karush kuhn tucker point for this np hard problem and bounds its iteration complexity then it proposes uls extensions tailored to reflect practical beamformer design objectives bringing in and exploiting new degrees of freedom to improve the beampattern designs simple variants of gp are proposed to deal with these extended uls problems simulations are used to showcase the effectiveness of the proposed algorithms in both the plain uls problem and in the context of phase only beamforming

regularization pseudofunction and hadamard finite part abstract first we discuss and correlate the various types of regularizations available in the literature for the singular function h x x k where k is an integer and h x is the heaviside function then we present the corresponding regularization for the function r k where r is the radial distance in r n thereafter we express the recently discovered distributional derivatives of this function in terms of pseudofunctional language commonly used in the coulomb gravitational and interparticle potentials where the function 1 r plays a fundamental role

on the design of constant modulus probing signals for mimo radar probing signal waveforms play a central role in the signal processing performance of a multiple input multiple output mimo radar in practice for a given desired beam pattern we need to design a probing signal waveform whose beam pattern closely matches the desired one and whose autocorrelation and cross correlation sidelobes are kept low the latter properties are important to mitigate undesirable interference caused by multiple targets or scatterers in this correspondence we present an efficient optimization method to design a constant modulus probing signal which can synthesize a desired beam pattern while maximally suppressing both the autocorrelation and cross correlation sidelobes at between given spacial angles we formulate this problem as an unconstrained minimization of a fourth order trigonometric polynomial and propose an efficient quasi newton iterative algorithm to solve it besides we provide an analysis of the local minima of the fourth order trigonometric polynomial and prove that any local minima is a 1 2 approximation of its global optimal solution numerical examples show that the proposed approach compares favorably with the existing approach

efficient implementation of quasi maximum likelihood detection based on semidefinite relaxation in this paper we develop two quasi maximum likelihood ml channel detectors for multiuser detection semidefinite relaxation sdr detector and phase shift keying psk detector these detectors can deliver near ml bit error rate ber performance with a polynomial worst case complexity the sdr detector for binary phase shift keying bpsk constellation is based on a convex sdr whereas the psk detector for m psk constellations is based on a nonconvex low rank sdr the sdr detector is implemented using a dual scaling interior point method while the psk detector is based on a coordinate descent strategy on a feasible region homotopy we use dynamic dimension reduction and warm start techniques to achieve signal to noise ratio snr sensitive improvements for both detectors numerical simulations of ber performance and running time indicate the effectiveness of the two quasi ml detectors when compared to the conventional sphere decoder and its variants

unconstrained synthesis of covariance matrix for mimo radar transmit beampattern multiple input multiple output mimo radars have many advantages over their phased array counterparts improved spatial resolution better parametric identifiably and greater flexibility to design the transmit beampattern the design of the transmit beampatterns generally requires the waveforms to have arbitrary auto and cross correlation properties the correlation covariance matrix r of the waveforms must be positive semidefinite therefore synthesis of a desired beampattern is usually a constrained optimization problem in this paper to simplify the constrained optimization problem two algorithms are proposed to synthesize the waveform covariance matrix for the desired beampattern in the first proposed algorithm the elements of a square root matrix of the covariance matrix r are parameterized using the coordinates of a hypersphere that implicitly fulfil the constraints this yields an iterative algorithm whose convergence speed can be increased significantly by providing good initial values in the second algorithm the constraints and redundant information in the covariance matrix r are exploited to find a closed form solution the drawback of the second algorithm is that it may yield a pseudocovariance matrix pseudo cm that is not guaranteed to be positive semidefinite however the pseudo cm can be easily converted into a covariance matrix using eigenvalue decomposition and or shrinkage methods moreover a pseudo cm can also be used to provide good initial values for the first algorithm to enable faster convergence

an asymptotic analysis of the fourier laplace transforms of certain oscillatory functions abstract we study the family of fourier laplace transforms f z f p 0 t exp i t i z t d t im z 0 for 1 and c where hadamard finite part is used to regularize the integral when re 1 we prove that each f has analytic continuation to the whole complex plane and determine its asymptotics along any line through the origin we also apply our ideas to show that some of these functions provide concrete extremal examples for the wiener ikehara theorem and a quantified version of the ingham karamata theorem supplying new simple and constructive proofs of optimality results for these complex tauberian theorems

mimo radar transmit beampattern matching design in this paper a novel transmit beampattern matching design method is proposed for a multiple input multiple output mimo radar system whereas previous approaches first solve the correlation matrix r and then use r to design the transmit signal s the proposed one step method obtains the transmit signal s directly through solving the waveform matrix p since mimo radar transmit beampattern matching design is an inverse problem of the transmit beampattern in this paper we first discuss some important properties of the transmit beampattern and standardize the beampattern matching design problem we then theoretically explain the value of the signal waveform matrix p followed by an unconstrained optimization problem to solve p numerical examples demonstrate the advantages of the proposed one step method for the mimo radar transmit beampattern matching design problem including uniform and non uniform linear arrays

two step semi optimal branch and bound algorithm for feature selection in mixed variable discrimination abstract the paper presents a two step procedure for the selection of the most discriminative discrete as well as continuous features in the location model for mixed variable discriminant analysis the multivariate discriminatory measure t 2 is used as a criterion for subset choice the first step consists in finding stepwisely the most discriminative discrete features the second step enables one to select the subset of continuous variables for the previously chosen discrete feature set the two steps may be reversed when the number of discrete variables is already fixed the selection of continuous features is made efficiently by the branch and bound algorithm

design of par constrained sequences for mimo channel estimation via majorization minimization communication systems and radars widely employ sequences of low peak to average power ratio par or unimodulus to meet the hardware constraints and maximize the power efficiency numerous works have probed the unimodular sequence design especially attempting to obtain good correlation properties regarding channel estimation however sequences of such properties do not necessarily qualify for the mission and tailored unimodular sequences for the specific criterion concerned are more desirable when we have access to the prior knowledge of the channel impulse response in this paper we formulate the problem of unimodular sequence design by optimizing minimum mean square error and conditional mutual information respectively the problems turn out nonconvex and we develop efficient algorithms based on the majorization minimization framework with convergence guaranteed more general we also examine optimal sequence design with low par constraints numerical examples demonstrate the improved results of mean square error signal to noise ratio and conditional mutual information by using our proposed training sequences with the efficiency of the derived algorithms illustrated

quadratic maximization and semidefinite relaxation in this paper we study a class of quadratic maximization problems and their semidefinite programming sdp relaxation for a special subclass of the problems we show that the sdp relaxation provides an exact optimal solution another subclass which is hard guarantees that the sdp relaxation yields an approximate solution with a worst case performance ratio of 0 87856 this is a generalization of the well known result of goemans and williamson for the maximum cut problem finally we discuss extensions of these results in the presence of a certain type of sign restrictions

hermitian completely positive matrices abstract in this paper we introduce the hermitian completely positive hcp matrix which has a hermitian completely positive hcp decomposition with all real and imaginary parts of the decomposition vectors being nonnegative some properties of the hermitian completely positive matrix are given a semidefinite algorithm is also proposed for checking whether a hermitian matrix is hcp or not if a matrix is not hcp a certificate for it can be obtained if it is an hcp decomposition can be obtained

a coordinate descent framework to design low psl isl sequences this paper is focused on the design of phase sequences with good aperiodic autocorrelation properties in terms of peak sidelobe level and integrated sidelobe level the problem is formulated as a biobjective pareto optimization forcing either a continuous or a discrete phase constraint at the design stage an iterative procedure based on the coordinate descent method is introduced to deal with the resulting optimization problems that are nonconvex and np hard in general each iteration of the devised method requires the solution of a nonconvex min max problem it is handled either through a novel bisection or an fft based method respectively for the continuous and the discrete phase constraint additionally a heuristic approach to initialize the procedures employing the l p norm minimization technique is proposed simulation results illustrate that the proposed methodologies can outperform some counterparts providing sequences with good autocorrelation features especially in the discrete phase binary case

designing unimodular codes via quadratic optimization the np hard problem of optimizing a quadratic form over the unimodular vector set arises in radar code design scenarios as well as other active sensing and communication applications to tackle this problem which we call unimodular quadratic program uqp several computational approaches are devised and studied power method like iterations are introduced for local optimization of uqp furthermore a monotonically error bound improving technique merit is proposed to obtain the global optimum or a local optimum of uqp with good sub optimality guarantees the provided sub optimality guarantees are case dependent and may outperform the 4 approximation guarantee of semi definite relaxation several numerical examples are presented to illustrate the performance of the proposed method the examples show that for several cases including rank deficient matrices the proposed methods can solve uqps efficiently in the sense of sub optimality guarantee and computational time

extensions of the poisson summation formula the classical poisson summation formula 1 1 and the corresponding distributional formula 1 2 have found extensive applications in various scientific fields however they are not universally valid for instance if x is a smooth function the left hand side of 1 1 is generally divergent even when both sides of 1 1 converge absolutely they may do so to different numbers indeed in example 3 we are faced with the embarrassing situation where the series on the left hand side of 1 1 converges for re s 1 while that on the right hand side converges only for re s 0 our aim is to extend formulas 1 1 and 1 2 with the help of some new results in distributional theory for instance the evaluation of the distribution with zero mean as given by 3 1 at a test function x yields the relation k x dx both the series and the integral in this expression are generally divergent the concept of the cesaro limit is then used to obtain the finite difference of these two terms thereafter we extend the analysis to higher dimensions various innovative examples are presented to illustrate these concepts

distributional solutions of the hypergeometric differential equation abstract we present the distributional solutions to the hypergeometric differential equation these solutions are obtained in the form of infinite series of the dirac delta functions and its derivatives we employ these solutions to observe their interesting features furthermore the form of these solutions is the same as the ones required for the weight distributions for a certain class of orthogonal polynomials

combining propensity score matching with additional adjustments for prognostic covariates abstract propensity score matching refers to a class of multivariate methods used in comparative studies to construct treated and matched control samples that have similar distributions on many covariates this matching is the observational study analog of randomization in ideal experiments but is far less complete as it can only balance the distribution of observed covariates whereas randomization balances the distribution of all covariates both observed and unobserved an important feature of propensity score matching is that it can be easily combined with model based regression adjustments or with matching on a subset of special prognostic covariates or combinations of prognostic covariates that have been identified as being especially predictive of the outcome variables we extend earlier results by developing approximations for the distributions of covariates in matched samples created with linear propensity score methods for the practically important situation where matching uses both the estimat

design of phase codes for radar performance optimization with a similarity constraint this paper deals with the design of coded waveforms which optimize radar performances in the presence of colored gaussian disturbance we focus on the class of phase coded pulse trains and determine the radar code which approximately maximizes the detection performance under a similarity constraint with a prefixed radar code this is tantamount to forcing a similarity between the ambiguity functions of the devised waveform and of the pulse train encoded with the prefixed sequence we consider the cases of both continuous and finite phase alphabet and formulate the code design in terms of a nonconvex np hard quadratic optimization problem in order to approximate the optimal solutions we propose techniques with polynomial computational complexity based on the method of semidefinite program sdp relaxation and randomization moreover we also derive approximation bounds yielding a ldquomeasure of goodnessrdquo of the devised algorithms at the analysis stage we assess the performance of the new encoding techniques both in terms of detection performance and ambiguity function under different choices for the similarity parameter we also show that the new algorithms achieve an accurate approximation of the optimal solution with a modest number of randomizations

mimo radar waveform design with constant modulus and similarity constraints we consider the problem of waveform design for multiple input multiple output mimo radar in the presence of signal dependent interference embedded in white gaussian disturbance we present two sequential optimization procedures to maximize the signal to interference plus noise ratio sinr accounting for a constant modulus constraint as well as a similarity constraint involving a known radar waveform with some desired properties e g in terms of pulse compression and ambiguity the presented sequential optimization algorithms based on a relaxation method yield solutions with good accuracy their computational complexity is linear in the number of iterations and trials in the randomized procedure and polynomial in the receive filter length finally we evaluate the proposed techniques by considering their sinr performance beam pattern as well as pulse compression property via numerical simulations

on asymptotic expansions of twisted products the series development of the quantum mechanical twisted product is studied the series is shown to make sense as a moment asymptotic expansion of the integral formula for the twisted product either pointwise or in the distributional sense depending on the nature of the factors a condition is given that ensures convergence and is stronger than previously known results possible applications are examined

asymptotic expansion of a general integral for note see the image of page 2655 for this formatted text b in r ast a real function h x and also complex pseudo functions f x and k x u obeying weak assumptions an asymptotic expansion of the general integral i lambda fp int 0 b f x k h x lambda x dx is proposed with respect to the real and large parameter lambda where the notation fp designates an integration in the finite part sense of hadamard

asymptotic taylor expansion for fourier hyperfunctions a necessary and sufficient condition is proved that the asymptotic taylor expansion for a fourier hyperfunction converges in the space of fourier hyperfunctions

mixtures of continuous and categorical variables in discriminant analysis a hypothesis testing approach summary in the classical case of discrimination between two multivariate normal populations an allocation rule based on a hypothesis testing approach john s z statistic has long been in competition with the rule based on simple parameter estimation and substitution in the bayes procedure anderson s w statistic recent evidence suggests that the z statistic may be preferable in certain circumstances the purpose of this paper is to extend this approach to mixed discrete and continuous variables accordingly an allocation rule based on the location model is derived by use of a hypothesis testing argument a method for the estimation of error rates with this rule is outlined some examples are given and the new rule is compared with its direct competitors some of its possible merits are discussed briefly let v be a p component random vector which has probability densities f1 v and t2 v in two distinct populations 7r1 and 7r2 respectively and suppose that it is required to use v in discriminating between the two populations more particularly suppose that it is necessary to decide from which of the two populations an observed value v of v has come this is the classical problem of discriminant analysis which has been described by many authors a standard treatment is given by anderson 1958 chapter 6 anderson shows that if the probability densities f1 v and t2 v are completely specified the optimal decision rule is given by the bayes procedure which is to allocate v to w1 or 7r2 according to fl v lf2 v s k ae cut off point k is determined by the prior probabilities of an observation coming from each population and by the costs incurred when an individual is misclassified from each population in practice the densities t1 v and f2 v are rarely known and the only information about 7r1 and 7r2 which is usually available comes from two training sets v ll v21 v l known to come from 7r1 and v l2 v22 v m2 known to come from 7r2 in such circumstances estimation is necessary various methods have been proposed each involving a different level of assumptions at one extreme no assumptions at all are made about the functional form of the densities ti v i 1 2 and these densities are estimated from the training sets by some nonparametric method a comprehensive bibliography of such methods is provided by wertz and schneider 1979 at the other extreme exact functional forms which include unknown parameters are assumed for the densities a classical approach to the problem involves estimation of the unknown parameters from the training sets replacing the unknown parameters by these estimates in fl v lt2 v leads

successive qcqp refinement for mimo radar waveform design under practical constraints the authors address the problem of designing a waveform for multiple input multiple output mimo radar under the important practical constraints of constant modulus and waveform similarity incorporating these constraints in an analytically tractable manner is a longstanding open challenge this is due to the fact that the optimization problem that results from signal to interference plus noise ratio sinr maximization subject to these constraints is a hard non convex problem the authors develop a new analytical approach that involves solving a sequence of convex quadratically constrained quadratic programing qcqp problems which they prove converges to a sub optimal solution because an improvement in sinr results via solving each problem in the sequence they call the method successive qcqp refinement sqr furthermore the proposed sqr method can be easily extended to incorporate emerging requirements of spectral coexistence as shown briefly in this paper the authors evaluate sqr against other candidate techniques with respect to sinr performance beam pattern and pulse compression properties in a variety of scenarios results show that sqr outperforms state of the art methods that also employ constant modulus and or similarity constraints while being computationally less burdensome

a general and formal slender body theory in the non lifting case in this paper an alternative and integral method is proposed in order to build a formal slender body theory valid up to high orders with respect to the slenderness ratio epsilon and for a non lifting body which is not necessarily of circular cross section the method consists of asymptotically expanding and inverting the fredholm integral equation of the second kind bearing on the unknown source density which may be spread on the boundary of the body for such a treatment the concept of integration in the finite part sense of hadamard is powerful the source density is then given up to order o 3 and the pressure coefficient is provided on the body up to order o 4 log throughout this paper special attention is also paid to the main considered case of the axially symmetric slender body

tensor decompositions and applications this survey provides an overview of higher order tensor decompositions their applications and available software a tensor is a multidimensional or n way array decompositions of higher order tensors i e n way arrays with n geq 3 have applications in psycho metrics chemometrics signal processing numerical linear algebra computer vision numerical analysis data mining neuroscience graph analysis and elsewhere two particular tensor decompositions can be considered to be higher order extensions of the matrix singular value decomposition candecomp parafac cp decomposes a tensor as a sum of rank one tensors and the tucker decomposition is a higher order form of principal component analysis there are many other tensor decompositions including indscal parafac2 candelinc dedicom and paratuck2 as well as nonnegative variants of all of the above the n way toolbox tensor toolbox and multilinear engine are examples of software packages for working with tensors

generalized function solutions of differential and functional differential equations recently there has been considerable interest in problems concerning the existence of solutions to differential and functional differential equations fde in various spaces of generalized functions many important areas in theoretical and mathematical physics theory of partial differential equations quantum electrodynamics operational calculus and functional analysis use the methods of the distributions theory yet for ordinary differential equations ode and especially fde research in this direction is insufficiently developed and remains restricted to isolated results for some secondorder equations or special higher order systems we note in particular papers l 6 which contain references to previous work it is weil known that normal linear homogeneous systems of ode with infinitely smooth coefficients have no generalized function solutions other than the classical solutions in contrast to this case for equations with singularities in the coefficients new solutions in generalized functions may appear and some classical solutions may disappear the number m is called the order of the distribution

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

inhomogeneous polynomial optimization over a convex set an approximation approach in this paper we consider computational methods for optimizing a multivariate inhomogeneous polynomial function over a general convex set the focus is on the design and analysis of polynomial time approximation algorithms the methods are able to deal with optimization models with polynomial objective functions in any fixed degrees in particular we first study the problem of maximizing an inhomogeneous polynomial over the euclidean ball a polynomial time approximation algorithm is proposed for this problem with an assured relative worst case performance ratio which is dependent only on the dimensions of the model the method and approximation ratio are then generalized to optimize an inhomogeneous polynomial over the intersection of a finite number of co centered ellipsoids furthermore the constraint set is extended to a general convex compact set specifically we propose a polynomial time approximation algorithm with a relative worst case performance ratio for polynomial optimization over some convex compact sets e g a polytope finally numerical results are reported revealing good practical performance of the proposed algorithms for solving some randomly generated instances

selection of variables in mixed variable discriminant analysis a manova log linear formulation of the location model for mixed variable discriminant analysis is considered a strategy for the selection of variables and terms in such a model is based on akaike s criterion to overcome problems caused by noncomparable submodels some modifications to akaike s criterion are proposed an example of discrimination between bad and good melons illustrates the selection procedure

a distributional theory for asymptotic expansions the purpose of this chapter is to present the distributional theory of asymptotic expansions for functions of one variable this chapter and the next where the multidimensional expansions are studied are the central part of the book

applications of second order cone programming in a second order cone program socp a linear function is minimized over the intersection of an affine set and the product of second order quadratic cones socps are nonlinear convex problems that include linear and convex quadratic programs as special cases but are less general than semidefinite programs sdps several efficient primaldual interior point methods for socp have been developed in the last few years after reviewing the basic theory of socps we describe general families of problems that tan be recast as socps these include robust linear programming and robust leastsquares problems problems involving sums or maxima of norms or with convex hyperbolic constraints we discuss a variety of engineering applications such as filter design antenna array weight design truss design and grasping forte optimization in robotics we describe an efficient primaldual interior point method for solving socps which shares many of the features of primaldual interior point methods for linear program

the asymptotic expansion of certain multi dimensional generalized functions abstract we apply regularization of divergent integrals in the derivation of the asymptotic expansion of certain multi dimensional generalized functions we further present several illustrations to demonstrate that the asymptotic development of generalized functions provides a lucid formulation of many concepts in asymptotic analysis such as the expansion of oscillatory integrals and the expansion of certain series

maximum block improvement and polynomial optimization in this paper we propose an efficient method for solving the spherically constrained homogeneous polynomial optimization problem the new approach has the following three main ingredients first we establish a block coordinate descent type search method for nonlinear optimization with the novelty being that we accept only a block update that achieves the maximum improvement hence the name of our new search method maximum block improvement mbi convergence of the sequence produced by the mbi method to a stationary point is proved second we establish that maximizing a homogeneous polynomial over a sphere is equivalent to its tensor relaxation problem thus we can maximize a homogeneous polynomial function over a sphere by its tensor relaxation via the mbi approach third we propose a scheme to reach a kkt point of the polynomial optimization provided that a stationary solution for the relaxed tensor problem is available numerical experiments have shown that our new method works very efficiently for a majority of the test instances that we have experimented with the method finds the global optimal solution at a low computational cost

on binary probing signals and instrumental variables receivers for radar the so called merit factor approach mfa to radar binary sequence design has led to several theoretical contributions in fairly diverse research areas including information theory computer science combinatorial optimization and analytical number theory however the mfa which basically aims at minimizing the clutter effect on radar performance implicitly assumes the use of a least squares ls receiver that is optimal only when there is no clutter this problem can be eliminated by using a more general optimal instrumental variables iv receiver in lieu of the ls receiver the iv receiver can reject clutter more efficiently than the ls receiver additionally the binary sequence design problem associated with the iv approach has an interesting form

tractable transmit mimo beampattern design under a constant modulus constraint multiple input multiple output mimo radar systems allow each antenna element to transmit a different waveform this waveform diversity can be exploited to enhance the beampattern design in particular effective management of radar radiation power in directions of interest we address the problem of designing a beampattern for mimo radar which in turn is determined by the transmit waveform while unconstrained design is straightforward a key open challenge is enforcing the constant modulus constraint on the radar waveform it is well known that the problem of minimizing deviation of the designed beampattern from an idealized one subject to the constant modulus constraint constitutes a hard nonconvex problem existing methods that address constant modulus invariably lead to a stiff tradeoff between analytical tractability achieved by relaxations and approximations and realistic design that exactly achieves constant modulus but is computationally burdensome a new approach is proposed in our paper which involves solving a sequence of convex equality constrained quadratic programs each of which has a closed form solution and such that constant modulus is achieved at convergence we further prove that the converged solution satisfies the karush kuhn tucker optimality conditions of the aforementioned hard nonconvex problem we evaluate the proposed successive closed forms scf algorithm against the state of the art mimo beampattern design techniques in both narrowband and wideband setups and show that the scf breaks the tradeoff between desirable performance and the associated computation cost

relative entropy based waveform design for mimo radar detection in the presence of clutter and interference this paper considers the waveform design problem for multiple input multiple output mimo radars in order to improve the detection performance of the systems we assume that target echoes are embedded in signal dependent clutter as well as colored interference considering the optimal neyman pearson detector obtaining waveforms which maximize the detection probability for a fixed value of the probability of false alarm is intractable therefore we employ relative entropy associated with the detection problem as the figure of merit for the waveform design we devise an iterative method based on minorization maximization mm technique to tackle the nonconvex design problem this method also includes a novel trick for replacing a nonconvex constraint set associated with an equivalent form of the design problem with a convex one iteratively the proposed method increases the design metric monotonically and is guaranteed to converge we extend the devised method for using in design with peak to average power ratio par and similarity constraints the method can be applied to both statistical and colocated mimo radars several numerical examples are included to demonstrate the effectiveness of the proposed method

multiple discriminant analysis in the presence of mixed continuous and categorical data abstract it is shown that for the two group case classification by minimum distance is equivalent to classification from the maximum likelihood discriminant rule based on the location model for mixed data this enables straightforward extension to be made to classification in the g 2 group case with such data an example is given in which the performance of the proposed rule is compared with the performance of the traditional normal based rule

on mimo detection under non gaussian target scattering in this paper we consider a multiple input multiple output mimo detection problem with m widely spaced transmit antennas and l widely spaced receive antennas and we study the problem of designing the signal waveforms transmitted by each source node under non gaussian target scattering and temporally correlated gaussian clutter two figures of merit are investigated for space time code stc optimization under a semidefinite rank constraint 1 the lower chernoff bound lcb to the detection probability for fixed probability of false alarm and 2 the mutual information mi between the observations available at the receive nodes and the channel response generated by a point like target assumed present tout court both receive and transmit power constraints are discussed if the scattering distribution possesses some suitably defined properties of unitary invariance see both mi optimal and lcb optimal stcs have a simple canonical structure the same set of clutter dependant temporal codewords are employed at the transmit nodes the only difference among the many solutions being the amount of power radiated by each antenna such a spatial power allocation critically depends upon the adopted figure of merit the specified power constraint and the underlying scattering model sufficient conditions to determine the optimal power allocation for all design criteria are provided asymptotic power distributions are also derived in the limit of vanishingly small and increasingly large signal to clutter ratios proving that assuming gaussian scattering at the design stage is a robust choice a case study of relevant practical interest is examined in depth so as to compare the proposed design criteria and to assess the impact of signal non gaussianity on the system performances

new algorithms for designing unimodular sequences with good correlation properties unimodular i e constant modulus sequences with good autocorrelation properties are useful in several areas including communications and radar the integrated sidelobe level isl of the correlation function is often used to express the goodness of the correlation properties of a given sequence in this paper we present several cyclic algorithms for the local minimization of isl related metrics these cyclic algorithms can be initialized with a good existing sequence such as a golomb sequence a frank sequence or even a pseudo random sequence to illustrate the performance of the proposed algorithms we present a number of examples including the design of sequences that have virtually zero autocorrelation sidelobes in a specified lag interval and of long sequences that could hardly be handled by means of other algorithms previously suggested in the literature

design of optimized radar codes with a peak to average power ratio constraint this paper considers the problem of radar waveform design in the presence of colored gaussian disturbance under a peak to average power ratio par and an energy constraint first of all we focus on the selection of the radar signal optimizing the signal to noise ratio snr in correspondence of a given expected target doppler frequency algorithm 1 then through a max min approach we make robust the technique with respect to the received doppler algorithm 2 namely we optimize the worst case snr under the same constraints as in the previous problem since algorithms 1 and 2 do not impose any condition on the waveform phase we also devise their phase quantized versions algorithms 3 and 4 respectively which force the waveform phase to lie within a finite alphabet all the problems are formulated in terms of nonconvex quadratic optimization programs with either a finite or an infinite number of quadratic constraints we prove that these problems are np hard and hence introduce design techniques relying on semidefinite programming sdp relaxation and randomization as well as on the theory of trigonometric polynomials providing high quality suboptimal solutions with a polynomial time computational complexity finally we analyze the performance of the new waveform design algorithms in terms of detection performance and robustness with respect to doppler shifts

pre asymptotic expansions we introduce the concepts of pre asymptotic schemes and pre asymptotic expansions to study the divergent series that formally are solutions of various types of equations

code design to optimize radar detection performance under accuracy and similarity constraints this paper deals with the design of coded waveforms which optimize radar performances in the presence of colored gaussian disturbance we focus on the class of linearly coded pulse trains and determine the optimum radar code according to the following criterion maximization of the detection performance under a control on the region of achievable doppler estimation accuracies and imposing a similarity constraint with a prefixed radar code this last constraint is tantamount to requiring a similarity between the ambiguity functions of the devised waveform and of the pulse train encoded with the prefixed sequence the resulting optimization problem is nonconvex and quadratic in order to solve it we propose a technique with polynomial computational complexity based on the relaxation of the original problem into a semidefinite program thus the best code is determined through a rank one decomposition of an optimal solution of the relaxed problem at the analysis stage we assess the performance of the new encoding technique in terms of detection performance region of achievable doppler estimation accuracies and ambiguity function

exactness of semidefinite relaxations for nonlinear optimization problems with underlying graph structure this work is concerned with finding a global optimization technique for a broad class of nonlinear optimization problems including quadratic and polynomial optimization problems the main objective of this paper is to investigate how the hidden structure of a given real complex valued optimization problem makes it easy to solve to this end three conic relaxations are proposed necessary and sufficient conditions are derived for the exactness of each of these relaxations and it is shown that these conditions are satisfied if the optimization problem is highly structured more precisely the structure of the optimization problem is mapped into a generalized weighted graph where each edge is associated with a weight set extracted from the coefficients of the optimization problem in the real valued case it is shown that the relaxations are all exact if each weight set is sign definite and in addition a condition is satisfied for each cycle of the graph it is also proved that if some of these conditi

mimo radar waveform design in colored noise based on information theory in this paper we consider multiple input multiple output mimo radar waveform design in colored noise two information theoretic measures are used as criterions for optimal waveform design under transmitted power constraint the first one is by maximizing the mutual information between target impulse response and target echoes the second one is by maximizing the relative entropy between two hypotheses in the first hypothesis we assume the target is not present in the echoes while in the second hypothesis we assume the target exists in the echoes we derive optimal solutions for both cases interestingly both optimal solutions require that transmitted waveform should match with the target and noise however the optimal solutions of the two problems lead to different power allocation strategies

optimal weighted average power similar tests for the covariance structure in the linear regression model this paper suggests a procedure for the construction of optimal weighted average power similar tests for the error covariance matrix of a gaussian linear regression model when the alternative model belongs to the exponential family the paper uses a saddlepoint approximation to construct simple test statistics for a large class of problems and overcomes the computational burden of evaluating the complicated integrals arising in the derivation of optimal weighted average power tests extensions to panel data models are considered applications are given to tests for error autocorrelation in the linear regression model and in a panel data framework

complex quadratic optimization and semidefinite programming in this paper we study the approximation algorithms for a class of discrete quadratic optimization problems in the hermitian complex form a special case of the problem that we study corresponds to the max 3 cut model used in a recent paper of goemans and williamson j comput system sci 68 2004 pp 442 470 we first develop a closed form formula to compute the probability of a complex valued normally distributed bivariate random vector to be in a given angular region this formula allows us to compute the expected value of a randomized with a specific rounding rule solution based on the optimal solution of the complex semidefinite programming relaxation problem in particular we present an m 2 1 cos frac 2 pi m 8 pi approximation algorithm and then study the limit of that model in which the problem remains np hard we show that if the objective is to maximize a positive semidefinite hermitian form then the randomization rounding procedure guarantees a worst case performance ratio of pi 4 approx 0 7854 which is better than the ratio of 2 pi approx 0 6366 for its counterpart in the real case due to nesterov furthermore if the objective matrix is real valued positive semidefinite with nonpositive off diagonal elements then the performance ratio improves to 0 9349

code design for radar stap via optimization theory in this paper we deal with the problem of constrained code optimization for radar space time adaptive processing stap in the presence of colored gaussian disturbance at the design stage we devise a code design algorithm complying with the following optimality criterion maximization of the detection performance under a control on the regions of achievable values for the temporal and spatial doppler estimation accuracy and on the degree of similarity with a pre fixed radar code the resulting quadratic optimization problem is solved resorting to a convex relaxation that belongs to the semidefinite program sdp class an optimal solution of the initial problem is then constructed through a suitable rank one decomposition of an optimal solution of the relaxed one at the analysis stage we assess the performance of the new algorithm both on simulated data and on the standard challenging the knowledge aided sensor signal processing and expert reasoning kassper datacube

waveform design for radar stap in signal dependent interference waveform design is a pivotal component of the fully adaptive radar construct in this paper we consider waveform design for radar space time adaptive processing stap accounting for the waveform dependence of the clutter correlation matrix due to this dependence in general the joint problem of receiver filter optimization and radar waveform design becomes an intractable nonconvex optimization problem nevertheless it is however shown to be individually convex either in the filter or in the waveform variables we derive constrained versions of a the alternating minimization algorithm b proximal alternating minimization and c the constant modulus alternating minimization which at each step iteratively optimizes either the stap filter or the waveform independently a fast and slow time model permits waveform design in radar stap but the primary bottleneck is the computational complexity of the algorithms

taylor expansion for generalized functions a necessary and sufficient condition is given that the asymptotic taylor expansion for a distribution for an ultradistribution is the taylor series convergent in the space of distributions of ultradistributions

constant modulus waveform design for mimo radar transmit beampattern a multiple input multiple output radar has great flexibility to design the transmit beampattern via selecting the probing waveform the idea of current transmit beampattern design is to approximate the disired transimit beampattern and minimize the cross correlation sidelobes in this paper under the constant modulus constraint two algorithms are proposed to design the probing waveform directly in the first algorithm the optimization criterion is minimizing the squared error between the designed beampattern and the given beampattern since the objective function is a nonconvex fourth order polynomial and the constant modulus constraint can be regarded as many nonconvex quadratic equality constraints an efficient alternating direction method of multipliers admm algorithm whose convergence speed is very fast is proposed to solve it in the second algorithm the criterion is minimizing the absolute error between the designed beampattern and the given beampattern this nonconvex problem can be formulated as l 1 norm problem which can be solved through a double admm algorithm finally we assess the performance of the two proposed algorithms via numerical results

ideal point discriminant analysis a new method of multiple discriminant analysis was developed that allows a mixture of continuous and discrete predictors the method can be justified under a wide class of distributional assumptions on the predictor variables the method can also handle three different sampling situations conditional joint and separate in this method both subjects cases or any other sampling units and criterion groups are represented as points in a multidimensional euclidean space the probability of a particular subject belonging to a particular criterion group is stated as a decreasing function of the distance between the corresponding points a maximum likelihood estimation procedure was developed and implemented in the form of a fortran program detailed analyses of two real data sets were reported to demonstrate various advantages of the proposed method these advantages mostly derive from model evaluation capabilities based on the akaike information criterion aic

the cesaro behaviour of distributions a theory of summability for generalized functions of one real variable is presented this theory generalizes the classical theory of summability of series and integrals the relationship between the average behaviour of generalized functions according to this summability theory and the distributional behaviour is studied several applications are given

discrimillant analysis with discrete and continuous variables a paradigmatic methodologic approach does not exist for the problem of discriminant analysis with both discrete and continuous explanatory variables procedures that have been employed in this situation include fisher s linear discriminant function the quadratic discriminant function the linear discriminant furlction with higher order terms discriminant functions with logistic regression estimates of the coefficients and the location model average optimal error rates for these procedures are reported for the case of one normal variable and two binary variables these approaches are also compared for the prediction of two year survival following recovery from myocardial infarction employing resubstitution jackknife and independent sample estimates of the error rates recommendations for applications are presented

cognitive radar based sequence design via sinr maximization the ambiguity function plays an important role in radar systems in fact many radar design problems can be interpreted from the perspective of persuing desired ambiguity functions to adapt to various application scenes in this paper we consider designing a radar sequence subject to a peak to average power ratio par constraint to maximize the signal to interference plus noise ratio which can also be interpreted as designing a sequence with a desired ambiguity function from an optimization point of view this is equivalent to optimizing a complex quartic function with the par constraint an efficient algorithm based on the general majorization minimization mm method is developed to solve this problem with guaranteed convergence to a stationary point under some mild conditions in addition the unit modulus constraint as a special case is considered and another algorithm is proposed which is the combination of the general mm and the coordinate descent method numerical experiments show that the proposed algorithms can shape a desired ambiguity function based on the prior information and the performance is much better compared with the existing methods

discriminant analysis algorithm based on a distance function and on a bayesian decision we propose a new algorithm for the allocation of an individual to one of several possible groups or populations the algorithm enables us to define a finite partition over the sample space based on distance function this partition is used jointly with the application of a standard bayesian decision rule to allocate individuals to the populations the algorithm also provides a measure of the allocation confidence for each individual in a similar manner to that of logistic regression the error rates for classification are also computed using the leave one out method results are compared with those obtained with other discriminant analysis techniques previously reported fisher s linear discriminant function the quadratic discriminant function logistic discrimination and others

asymptotic expansions of a class of integrals the purpose of this paper is to study the expansion with respect to the large and real parameter of the integral i fp b 0 f x k x x d x when weak assumptions are required for the pseudofunctions f x and k x u and the symbol fp means the integration in the finite part sense of hadamard

phase recovery maxcut and complex semidefinite programming phase retrieval seeks to recover a signal x in mathbb c p x c p from the amplitude a x a x of linear measurements ax in mathbb c n a x c n we cast the phase retrieval problem as a non convex quadratic program over a complex phase vector and formulate a tractable relaxation called phasecut similar to the classical maxcut semidefinite program we solve this problem using a provably convergent block coordinate descent algorithm whose structure is similar to that of the original greedy algorithm in gerchberg and saxton optik 35 237 246 1972 where each iteration is a matrix vector product numerical results show the performance of this approach over three different phase retrieval problems in comparison with greedy phase retrieval algorithms and matrix completion formulations

semidefinite relaxation bounds for indefinite homogeneous quadratic optimization this paper studies the relationship between the optimal value of a homogeneous quadratic optimization problem and its semidefinite programming sdp relaxation we consider two quadratic optimization models 1 min x c x mid x a k x ge 1 k 0 1 ldots m x in mathbb f n and 2 max x c x mid x a k x le 1 k 0 1 ldots m x in mathbb f n where mathbb f is either the real field mathbb r or the complex field mathbb c and a k c are symmetric matrices for the minimization model 1 we prove that if the matrix c and all but one of the a k s are positive semidefinite then the ratio between the optimal value of 1 and its sdp relaxation is upper bounded by o m 2 when mathbb f mathbb r and by o m when mathbb f mathbb c moreover when two or more of the a k s are indefinite this ratio can be arbitrarily large for the maximization model 2 we show that if c and at most one of the a k s are indefinite while other a k s are positive semidefinite then the ratio between the optimal value of 2 and its sdp relaxation is bounded from below by o 1 log m for both the real and the complex case this result improves the bound based on the so called approximate s lemma of ben tal nemirovski and roos siam j optim 13 2002 pp 535 560 when two or more of the a k s in 2 are indefinite we derive a general bound in terms of the problem data and the sdp solution for both optimization models we present examples to show that the derived approximation bounds are essentially tight

ambiguity function shaping for cognitive radar via complex quartic optimization in this paper we propose a cognitive approach to design phase only modulated waveforms sharing a desired range doppler response the idea is to minimize the average value of the ambiguity function of the transmitted signal over some range doppler bins which are identified exploiting a plurality of knowledge sources from a technical point of view this is tantamount to optimizing a real and homogeneous complex quartic order polynomial with a constant modulus constraint on each optimization variable after proving some interesting properties of the considered problem we devise a polynomial time waveform optimization procedure based on the maximum block improvement mbi method and the theory of conjugate partial symmetric conjugate super symmetric fourth order tensors at the analysis stage we assess the performance of the proposed technique showing its capability to properly shape the range doppler response of the transmitted waveform

quadratic optimization with similarity constraint for unimodular sequence synthesis this paper considers unimodular sequence synthesis under similarity constraint for both the continuous and discrete phase cases a computationally efficient iterative algorithm for the continuous phase case ia cpc is proposed to sequentially optimize the quadratic objective function the quadratic optimization problem is turned into multiple one dimensional optimization problems with closed form solutions for the discrete phase case we present an iterative block optimization algorithm specifically we partition the design variables into k blocks and then we sequentially optimize each block via exhaustive search while fixing the remaining k 1 blocks finally we evaluate the computational costs and performance gains of the proposed algorithms in comparison with power method like and semidefinite relaxation related techniques

extended matrix cube theorems with applications to µ theory in control in this paper we study semi infinite systems of linear matrix inequalities which are generically np hard for these systems we introduce computationally tractable approximations and derive quantitative guarantees of their quality as applications we discuss the problem of maximizing a hermitian quadratic form over the complex unit cube and the problem of bounding the complex structured singular value with the help of our complex matrix cube theorem we demonstrate that the standard scaling upper bound on m is a tight upper bound on the largest level of structured perturbations of the matrixm for which all perturbed matrices share a common lyapunov certificate for the discrete time stability

computational design of sequences with good correlation properties in this paper we introduce a computational framework based on an iterative twisted approximation itrox and a set of associated algorithms for various sequence design problems the proposed computational framework can be used to obtain sequences or complementary sets of sequences possessing good periodic or aperiodic correlation properties and in an extended form to construct zero or low correlation zone sequences furthermore as constrained e g finite alphabets are of interest in many applications we introduce a modified version of our general framework that can be useful in these cases several applications of itrox are studied and numerical examples focusing on the construction of real valued and binary sequences are provided to illustrate the performance of itrox for each application

approximating quadratic programming with bound and quadratic constraints we consider the problem of approximating the global maximum of a quadratic program qp subject to bound and simple quadratic constraints based on several early results we show that a 4 7 approximate solution can be obtained in polynomial time

synthetic aperture radar autofocus via semidefinite relaxation the autofocus problem in synthetic aperture radar imaging amounts to estimating unknown phase errors caused by unknown platform or target motion at the heart of three state of the art autofocus algorithms namely phase gradient autofocus multichannel autofocus mca and fourier domain multichannel autofocus fmca is the solution of a constant modulus quadratic program cmqp currently these algorithms solve a cmqp by using an eigenvalue relaxation approach we propose an alternative relaxation approach based on semidefinite programming which has recently attracted considerable attention in other signal processing problems experimental results show that our proposed methods provide promising performance improvements for mca and fmca through an increase in computational complexity

approximation bounds for quadratic optimization with homogeneous quadratic constraints we consider the np hard problem of finding a minimum norm vector in n dimensional real or complex euclidean space subject to m concave homogeneous quadratic constraints we show that a semidefinite programming sdp relaxation for this nonconvex quadratically constrained quadratic program qp provides an o m 2 approximation in the real case and an o m approximation in the complex case moreover we show that these bounds are tight up to a constant factor when the hessian of each constraint function is of rank 1 namely outer products of some given so called steering vectors and the phase spread of the entries of these steering vectors are bounded away from pi 2 we establish a certain constant factor approximation depending on the phase spread but independent of m and n for both the sdp relaxation and a convex qp restriction of the original np hard problem finally we consider a related problem of finding a maximum norm vector subject to m convex homogeneous quadratic constraints we show that an sdp relaxation for this nonconvex qp provides an o 1 ln m approximation which is analogous to a result of nemirovski e math program 86 1999 pp 463 473 for the real case

on the complexity of purely complex spl mu computation and related problems in multidimensional systems in this paper the following robust control problems are shown to be np hard given a purely complex uncertainty structure spl delta determine if 1 spl mu sub spl delta m 1 for a given rational matrix m 2 spl par m spl middot spl par sub spl mu 1 for a given rational transfer matrix m s and 3 inf q spl isin h sup spl infin spl par f t q spl par sub spl mu 1 for a given linear fractional transformation f t q with rational coefficients in other words purely complex spl mu computation analysis and synthesis problems are np hard it is also shown that checking stability and computing the h sup spl infin norm of a multidimensional system are np hard problems therefore it is rather unlikely to find nonconservative polynomial time algorithms for solving the problems in complete generality

approximation algorithms for homogeneous polynomial optimization with quadratic constraints in this paper we consider approximation algorithms for optimizing a generic multi variate homogeneous polynomial function subject to homogeneous quadratic constraints such optimization models have wide applications e g in signal processing magnetic resonance imaging mri data training approximation theory and portfolio selection since polynomial functions are non convex the problems under consideration are all np hard in general in this paper we shall focus on polynomial time approximation algorithms in particular we first study optimization of a multi linear tensor function over the cartesian product of spheres we shall propose approximation algorithms for such problem and derive worst case performance ratios which are shown to be dependent only on the dimensions of the model the methods are then extended to optimize a generic multi variate homogeneous polynomial function with spherical constraint likewise approximation algorithms are proposed with provable approximation performance ratios furthermore the constraint set is relaxed to be an intersection of co centered ellipsoids namely we consider maximization of a homogeneous polynomial over the intersection of ellipsoids centered at the origin and propose polynomial time approximation algorithms with provable worst case performance ratios numerical results are reported illustrating the effectiveness of the approximation algorithms studied