parameter dependent lmis in robust analysis characterization of homogeneous polynomially parameter dependent solutions via lmi relaxations this note investigates the robust stability of uncertain linear time invariant systems in polytopic domains by means of parameter dependent linear matrix inequality pd lmi conditions exploiting some algebraic properties provided by the uncertainty representation a systematic procedure to construct a family of finite dimensional lmi relaxations is provided the robust stability is assessed by means of the existence of a lyapunov function more specifically a homogeneous polynomially parameter dependent lyapunov hppdl function of arbitrary degree for a given degree if an hppdl solution exists a sequence of relaxations based on real algebraic properties provides sufficient lmi conditions of increasing precision and constant number of decision variables for the existence of an hppdl function which tend to the necessity alternatively if an hppdl solution of degree exists a sequence of relaxations which increases the number of variables and the number of lmis will provide an hppdl solution of larger degree the method proposed can be applied to determine homogeneous parameter dependent matrix solutions to a wide variety of pd lmis by transforming the infinite dimensional lmi problem described in terms of uncertain parameters belonging to the unit simplex in a sequence of finite dimensional lmi conditions which converges to the necessary conditions for the existence of a homogeneous polynomially parameter dependent solution of arbitrary degree illustrative examples show the efficacy of the proposed conditions when compared with other methods from the literature

reduced order generalized h_ infty filtering for linear discrete time systems with application to channel equalization this paper investigates the problem of reduced order generalized h infty filtering for linear discrete time systems generalized h infty filtering covers standard h infty filtering as a special case and can make use of the frequency domain characteristics of practical signals moreover reduced order filters are more attractive than full order ones regarding implementability by virtue of the generalized bounded real lemma a new necessary and sufficient condition is proposed for analyzing the generalized h infty performance of the filtering error system to compute reduced order generalized h infty filters a necessary and sufficient condition is then derived in terms of matrix inequalities to solve which iterative linear matrix inequality algorithms are further developed in addition the proposed filter design method is applied to cope with the channel equalization problem examples are finally presented for illustrating the proposed method

h _ infty filtering for discrete time state delayed systems with finite frequency specifications this article researches the problem of finite frequency ff h filtering for linear discrete time state delayed systems the disturbance is assumed to reside in low middle high frequency ranges to reduce the conservatism of the results delay partitioning idea is used to derive a new ff bounded real lemma brl by applying the generalized kalman yakubovich popov lemma two equivalent approaches to the proof of the proposed ff brl are given respectively starting from transfer function and lyapunov krasovskii functional a new ff h filter design method is proposed in terms of solving a set of linear matrix inequalities finally a numerical example clearly demonstrates the merits and effectiveness of the proposed method

mixture densities maximum likelihood and the em algorithm the problem of estimating the parameters which determine a mixture density has been the subject of a large diverse body of literature spanning nearly ninety years during the last two decades the

robust spl hscr sub spl infin filtering for uncertain discrete time state delayed systems this paper addresses the problem of robust spl hscr sub spl infin filtering for linear discrete time systems subject to parameter uncertainties in the system state space model and with multiple time delays in the state variables the uncertain parameters are supposed to belong to a given convex bounded polyhedral domain a methodology is developed to design a stable linear filter that assures asymptotic stability and a prescribed spl hscr sub spl infin performance for the filtering error irrespective of the uncertainty and the time delays the proposed design is given in terms of linear matrix inequalities which has the advantage in that it can be implemented numerically very efficiently

an lmi condition for the robust stability of uncertain continuous time linear systems a new sufficient condition for the robust stability of continuous time uncertain linear systems with convex bounded uncertainties is proposed in this note the results are based on linear matrix inequalities lmis formulated at the vertices of the uncertainty polytope which provide a parameter dependent lyapunov function that assures the stability of any matrix inside the uncertainty domain with the aid of numerical procedures based on unidimensional search and the lmis feasibility tests a simple and constructive way to compute robust stability domains can be established

robust spl hscr sub spl infin filtering for continuous time varying uncertain systems with deterministic input signals many dynamical systems involve not only process and measurement noise signals but also parameter uncertainty and known input signals when spl lscr sub 2 or spl hscr sub spl infin filters that were designed based on a nominal model of the system are applied the presence of parameter uncertainty will not only affect the noise attenuation property of the filter but also introduce a bias proportional to the known input signal and the latter may be very appreciable we introduce a finite horizon robust spl hscr sub spl infin filtering method that provides a guaranteed spl hscr sub spl infin bound for the estimation error in the presence of both parameter uncertainty and a known input signal this method is developed by using a game theoretic approach and the results generalize those obtained for cases without parameter uncertainty or without a known input signal it is also demonstrated via an example that the proposed method provides significantly improved signal estimates

on mmse crossing properties and implications in parallel vector gaussian channels the scalar additive gaussian noise channel has the single crossing point property between the minimum mean square error mmse in the estimation of the input given the channel output assuming a gaussian input to the channel and the mmse assuming an arbitrary input this paper extends the result to the parallel vector additive gaussian channel in three phases the channel matrix is the identity matrix and we limit the gaussian input to a vector of gaussian i i d elements the single crossing point property is with respect to the signal to noise ratio as in the scalar case the channel matrix is arbitrary and the gaussian input is limited to an independent gaussian input a single crossing point property is derived for each diagonal element of the mmse matrix the gaussian input is allowed to be an arbitrary gaussian random vector a single crossing point property is derived for each eigenvalue of the difference matrix between the two mmse matrices these three extensions are then translated to new information theoretic properties on the mutual information using the i mmse relationship a fundamental relationship between estimation theory and information theory revealed by guo and coworkers the results of the last phase are also translated to a new property of fisher information finally the applicability of all three extensions on information theoretic problems is demonstrated through a proof of a special case of shannon s vector entropy power inequality a converse proof of the capacity region of the parallel degraded broadcast channel bc under an input per antenna power constraint and under an input covariance constraint and a converse proof of the capacity region of the compound parallel degraded bc under an input covariance constraint

robust h filter design with past output measurements for uncertain discrete time systems this paper is concerned with the design of robust filters for uncertain discrete time linear systems the uncertainty is assumed to be time invariant belonging to a polytope an upper bound to the h norm of the transfer function from the system input to the filtering error is used as performance criterion the novelty is that the resulting robust filter has order greater than the order of the system being filtered by incorporating past output measurements the design conditions are expressed in terms of convex problems constrained by linear matrix inequalities performance is assessed through parameter dependent lyapunov functions the results generalize existing robust filtering procedures without bringing in any additional conservatism simple numerical examples illustrate the procedure

h 2 and h_ infty robust filtering for discrete time linear systems this paper investigates robust filtering design problems in h2 and h infty spaces for discrete time systems subjected to parameter uncertainty which is assumed to belong to a convex bounded polyhedral domain it is shown that by a suitable change of variables both design problems can be converted into convex programming problems written in terms of linear matrix inequalities lmi the results generalize the ones available in the literature to date in several directions first all system matrices can be corrupted by parameter uncertainty and the admissible uncertainty may be structured then assuming the order of the uncertain system is known the optimal guaranteed performance h2 and h infty filters are proven to be of the same order as the order of the system comparisons with robust filters for systems subjected to norm bounded uncertainty are provided in both theoretical and practical settings in particular it is shown that under the same assumptions the results here are generally better as far as the minimization of a guaranteed cost expressed in terms of h2 or h infty norms is considered some numerical examples illustrate the theoretical results

estimating entropy on m bins given fewer than m samples consider a sequence p sub n of discrete probability measures supported on m sub n points and assume that we observe n independent and identically distributed i i d samples from each p sub n we demonstrate the existence of an estimator of the entropy h p sub n which is consistent even if the ratio n m sub n is bounded and as a corollary even if this ratio tends to zero albeit at a sufficiently slow rate

h 2 robust filter design with performance certificate via convex programming in this paper a new approach to h 2 robust filter design is proposed both continuous and discrete time invariant systems subject to polytopic parameter uncertainty are considered after a brief discussion on some of the most expressive methods available for h 2 robust filter design a new one based on a performance certificate calculation is presented the performance certificate is given in terms of the gap produced by the robust filter between lower and upper bounds of a minimax programming problem where the h 2 norm of the estimation error is maximized with respect to the feasible uncertainties and minimized with respect to all linear rational and causal filters the calculations are performed through convex programming methods developed to deal with linear matrix inequality lmi many examples borrowed from the literature to date are solved and it is shown that the proposed method outperforms all other designs

gradient of mutual information in linear vector gaussian channels this paper considers a general linear vector gaussian channel with arbitrary signaling and pursues two closely related goals i closed form expressions for the gradient of the mutual information with respect to arbitrary parameters of the system and ii fundamental connections between information theory and estimation theory generalizing the fundamental relationship recently unveiled by guo shamai and verdu spl acute we show that the gradient of the mutual information with respect to the channel matrix is equal to the product of the channel matrix and the error covariance matrix of the best estimate of the input given the output gradients and derivatives with respect to other parameters are then found via the differentiation chain rule

undersmoothed kernel entropy estimators we develop a ldquoplug inrdquo kernel estimator for the differential entropy that is consistent even if the kernel width tends to zero as quickly as 1 n where n is the number of independent and identically distributed i i d samples thus accurate density estimates are not required for accurate kernel entropy estimates in fact it is a good idea when estimating entropy to sacrifice some accuracy in the quality of the corresponding density estimate

extended h 2 and h norm characterizations and controller parametrizations for discrete time systems this paper presents new synthesis procedures for discrete time linear systems it is based on a recently developed stability condition which contains as particular cases both the celebrated lyapunov theorem for precisely known systems and the quadratic stability condition for systems with uncertain parameters these new synthesis conditions have some nice properties a they can be expressed in terms of lmi linear matrix inequalities and b the optimization variables associated with the controller parameters are independent of the symmetric matrix that defines a quadratic lyapunov function used to test stability this second feature is important for several reasons first structural constraints as those appearing in the decentralized and static output feedback control design can be addressed less conservatively second parameter dependent lyapunov function can be considered with a very positive impact on the design of robust h 2 and h x control problems third the design of controller with mixed

optimal linear filtering under parameter uncertainty this paper addresses the problem of designing a guaranteed minimum error variance robust filter for convex bounded parameter uncertainty in the state output and input matrices the design procedure is valid for linear filters that are obtained from the minimization of an upper bound of the error variance holding for all admissible parameter uncertainty the results provided generalize the ones available in the literature to date in several directions first all system matrices can be corrupted by parameter uncertainty and the admissible uncertainty may be structured assuming the order of the uncertain system is known the optimal robust linear filter is proved to be of the same order as the order of the system in the present case of convex bounded parameter uncertainty the basic numerical design tools are linear matrix inequality lmi solvers instead of the riccati equation solvers used for the design of robust filters available in the literature to date the paper that contains the most important and very recent results on robust filtering is used for comparison purposes in particular it is shown that under the same assumptions our results are generally better as far as the minimization of a guaranteed error variance is considered some numerical examples illustrate the theoretical results

bridging mixture model estimation and information bounds using i mmse we derive bounds on mutual information for arbitrary estimation problems in additive noise modeled using gaussian mixtures previous work exploiting the i minimum mean squared error mmse formula to formulate a bridge between bounds on the mmse for gaussian mixture model estimation problems and bounds on the mutual information are generalized to allow arbitrary noise modeling a novel upper bound on estimation information is also developed for the general estimation case in addition limits are analyzed to develop bounds on arbitrary entropy asymptotic behavior of all bounds and bound errors with some results bridged back to the mmse domain

mmse dimension if n is standard gaussian the minimum mean square error mmse of estimating a random variable x based on snr x n vanishes at least as fast as 1 snr as snr we define the mmse dimension of x as the limit as snr of the product of snr and the mmse mmse dimension is also shown to be the asymptotic ratio of nonlinear mmse to linear mmse for discrete absolutely continuous or mixed distribution we show that mmse dimension equals renyi s information dimension however for a class of self similar singular x e g cantor dis tribution we show that the product of snr and mmse oscillates around information dimension periodically in snr db we also show that these results extend considerably beyond gaussian noise under various technical conditions

brief homogeneous lyapunov functions for systems with structured uncertainties the problem addressed in this paper is the construction of homogeneous polynomial lyapunov functions hplfs for linear systems with time varying structured uncertainties a sufficient condition for the existence of an hplf of given degree is formulated in terms of a linear matrix inequalities lmi feasibility problem this condition turns out to be also necessary in some cases depending on the dimension of the system and the degree of the lyapunov function the maximum norm of the parametric uncertainty for which there exists a homogeneous polynomial lyapunov function is computed by solving a generalized eigenvalue problem the construction of such lyapunov functions is efficiently performed by means of popular convex optimization tools for the solution of problems in lmi form comparisons with other classes of lyapunov functions through numerical examples taken from the literature show that hplfs are a powerful tool for robustness analysis

stability of a mixed additive and cubic functional equation in quasi banach spaces abstract in this paper we establish the general solution and investigate the hyers ulam rassias stability of the following functional equation f 2 x y f 2 x y 2 f x y 2 f x y 2 f 2 x 2 f x in the quasi banach spaces

high snr asymptotics of mutual information for discrete constellations with applications to bicm asymptotic expressions of the mutual information between any discrete input and the corresponding output of the scalar additive white gaussian noise channel are presented in the limit as the signal to noise ratio snr tends to infinity asymptotic expressions of the symbol error probability sep and the minimum mean square error mmse achieved by estimating the channel input given the channel output are also developed it is shown that for any input distribution the conditional entropy of the channel input given the output mmse and sep have an asymptotic behavior proportional to the gaussian q function the argument of the q function depends only on the minimum euclidean distance med of the constellation and the snr and the proportionality constants are functions of the med and the probabilities of the pairs of constellation points at med the developed expressions are then generalized to study the high snr behavior of the generalized mutual information gmi for bit interleaved coded modulation bicm by means of these asymptotic expressions the long standing conjecture that gray codes are the binary labelings that maximize the bicm gmi at high snr is proven it is further shown that for any equally spaced constellation whose size is a power of two there always exists an anti gray code giving the lowest bicm gmi at high snr

technical communique establishing tightness in robust h analysis via homogeneous parameter dependent lyapunov functions several methods have been proposed to provide upper bounds to the robust h performance of a linear system with polytopic state space uncertainty a common problem of these methods is the lack of a test for establishing if the found upper bound is tight or not which is essential to determine if the search of the sought performance is finished or should continue this paper provides a necessary and sufficient condition for establishing the tightness of the upper bound provided by homogeneous polynomially parameter dependent quadratic lyapunov functions hpd qlfs which recently have been shown to be a powerful tool for computing these upper bounds this condition is based on the existence of particular vectors in suitable linear spaces techniques for finding these vectors are also provided

estimation of entropy and mutual information we present some new results on the nonparametric estimation of entropy and mutual information first we use an exact local expansion of the entropy function to prove almost sure consistency and central limit theorems for three of the most commonly used discretized information estimators the setup is related to grenander s method of sieves and places no assumptions on the underlying probability measure generating the data second we prove a converse to these consistency theorems demonstrating that a misapplication of the most common estimation techniques leads to an arbitrarily poor estimate of the true information even given unlimited data this inconsistency theorem leads to an analytical approximation of the bias valid in surprisingly small sample regimes and more accurate than the usual 1 n formula of miller and madow over a large region of parameter space the two most practical implications of these results are negative 1 information estimates in a certain data regime are likely contaminated by bias even if bias corrected estimators are used and 2 confidence intervals calculated by standard techniques drastically underestimate the error of the most common estimation methods finally we note a very useful connection between the bias of entropy estimators and a certain polynomial approximation problem by casting bias calculation problems in this approximation theory framework we obtain the best possible generalization of known asymptotic bias results more interesting this framework leads to an estimator with some nice properties the estimator comes equipped with rigorous bounds on the maximum error over all possible underlying probability distributions and this maximum error turns out to be surprisingly small we demonstrate the application of this new estimator on both real and simulated data

on mmse estimation a linear model under gaussian mixture statistics in a bayesian linear model suppose observation y hx n stems from independent inputs x and n which are gaussian mixture gm distributed with known matrix h the minimum mean square error mmse estimator for x has analytical form however its performance measure the mmse itself has no such closed form because existing bayesian mmse bounds prove to have limited practical value under these settings we instead seek analytical bounds for the mmse both upper and lower this paper provides such bounds and relates them to the signal to noise ratio snr