a polynomial approach to robust estimation of uncertain discrete time systems the robust estimation problem for uncertain discrete time systems is treated in this paper considering parametric uncertainties in the polynomial framework the estimator is the minimiser of a cost function written in a epsilon contaminated form uncertainties are not formally modelled it is shown that in the present context the proposed approach can be connected to stochastic modelling of uncertainties the optimal robust estimator is obtained by computing a spectral factorisation and by solving a single diophantine equation an example shows the efficiency of the proposed method

solution of the h sub infinity optimal linear filtering problem for discrete time systems the solution of l sub 2 minimum variance and h sub infinity estimation problems is considered using a polynomial systems approach the results for the l sub 2 filtering problem which corresponds with wiener or kalman filtering prediction are first presented in polynomial matrix form attention then turns to the solution of the h sub infinity estimation problem for scalar systems numerous examples are presented to illustrate the computational procedures the two types of estimator are appropriate to very different estimation problems and the new h sub infinity devices should be valuable in certain application areas

optimal and self tuning deconvolution in time domain this paper is concerned with both the optimal minimum mean square error variance and self tuning deconvolution problems for discrete time systems when the signal model measurement model and noise statistics are known a novel approach for the design of the optimal deconvolution filter predictor and smoother is proposed based on projection theory and innovation analysis in the time domain the estimators are given in terms of an autoregressive moving average arma innovation model and one unilateral linear polynomial equation where the arma innovation model is obtained by performing one spectral factorization a self tuning scheme can be incorporated when the noise statistics the input model and or colored noise model are unknown the self tuning estimator is designed by identifying two arma innovation models

a potential reduction algorithm allowing column generation using the dantzig wolfe decomposition technique a potential reduction algorithm allowing column generation for the linear feasibility lf problem is developed the point of departure is a simple containing polytope and its analytic center in each iteration an inequality violated at the current center is selected used to cut the polytope and then used to find the new center for the shrunken polytope the potential value associated with the containing polytope is reduced by a constant at each step and the algorithm is terminated in a polynomial time that depends only on the number and data length of the inequalities generated in the iterative process

computation of minimum volume covering ellipsoids we present a practical algorithm for computing the minimum volume n dimensional ellipsoid that must contain m given points a1 am n this convex constrained problem arises in a variety of applied computational settings particularly in data mining and robust statistics its structure makes it particularly amenable to solution by interior point methods and it has been the subject of much theoretical complexity analysis here we focus on computation we present a combined interior point and active set method for solving this problem our computational results demonstrate that our method solves very large problem instances m 30 000 and n 30 to a high degree of accuracy in under 30 seconds on a personal computer

polynomial systems approach to continuous time weighted optimal linear filtering and prediction the solution of the optimal weighted minimum variance estimation problem is considered using a polynomial matrix description for the continuous time linear system description which allows for the possible presence of transport delays on the measurements the filter or predictor is given by the solution of two diophantine equations and is equivalent in the delay free case to the state equation form of the steady state kalman filter or the transfer function matrix form of the wiener filter the pole zero properties of the optimal filter are more obvious in the polynomial representation and useful insights into the disturbance rejection properties of the filter are obtained allowance is made for both control and disturbance input subsystems and white and colored measurement noise or an output disturbance subsystem the model structure was determined by the needs of filtering and prediction problems in the metal processing industries where for example there are delays between the x ray gauge and the roll gap of the mill

a general chebyshev complex function approximation procedure and an application to beamforming a new computational technique is described for the chebyshev or minimax approximation of a given complex valued function by means of linear combinations of given complex valued basis functions the domain of definition of all functions can be any finite set whatever neither the basis functions nor the function approximated need satisfy any special hypotheses beyond the requirement that they be defined on a common domain theoretical upper and lower bounds on the accuracy of the computed chebyshev error are derived these bounds permit both a priori and a posteriori error assessments efforts to extend the method to functions whose domain of definition is a continuum are discussed an application is presented involving re shading a 50 element antenna array to minimize the effects of a 10 element failure rate while maintaining full steering capability and mainlobe beamwidth

a novel derivation methodology for polynomial lq controller design a simple derivation method for optimization of linear quadratic lq controllers is presented in the discrete time polynomial systems framework a control law variation regarded as a potential feedforward from the innovations is used orthogonality is evaluated in the frequency domain by collectively cancelling unstable poles by zeros the suggested method summarized as a three step scheme is exemplified on a disturbance measurement feedforward and an output feedback problem it is a simple more direct alternative to the completing the squares approach

optimal design of fir filters with the complex chebyshev error criteria we present an improved algorithm for an established filter design problem the design of an fir filter that best approximates in the complex chebyshev sense a desired complex valued frequency response the algorithm is a variant of the simplex algorithm of linear programming which has an interpretation as an implicit multiple exchange it is iterative robust and exhibits good convergence speed global optimum convergence is guaranteed both complex and real valued impulse responses can be designed with it the design of complex coefficient filters is new an example is given for each case the design of noncausal filters is new in addition to these new applications we conjecture that this new algorithm may have important advantages over existing techniques with respect to the maximum filter length possible speed and stability of convergence accuracy and memory requirements the ability to design long filters is among the more significant improvements over previous work filters of length 1000 have been designed with the new method

an interior point cutting plane method for the convex feasibility problem with second order cone inequalities the convex feasibility problem in general is a problem of finding a point in a convex set that contains a full dimensional ball and is contained in a compact convex set we assume that the outer set is described by second order cone inequalities and propose an analytic center cutting plane technique to solve this problem we discuss primal and dual settings simultaneously two complexity results are reported the complexity of restoration procedure and complexity of the overall algorithm we prove that an approximate analytic center is updated after adding a second order cone cut socc ino 1 newton step and that the analytic center cutting plane method accpm with socc is a fully polynomial algorithm

optimal differentiation based on stochastic signal models the problem of estimating the time derivative of a signal from sampled measurements is addressed the measurements may be corrupted by colored noise a key idea is to use stochastic models of the signal to be differentiated and of the measurement noise two approaches are suggested the first is based on a continuous time stochastic process as a model of the signal the second uses a discrete time arma model of the signal and a discrete time approximation of the derivative operator digital differentiators are presented in a shift operator polynomial form they minimize the mean square estimation error and are calculated from a linear polynomial equation and a polynomial spectral factorization the three obstacles to perfect differentiation namely a finite smoothing lag measurement noise and aliasing effects due to sampling are discussed

fast minimum variance deconvolution we propose a statistical estimation approach to the deconvolution problem which is optimal in the minimum variance sense when the a priori knowledge on the signal to be restored is strictly limited to its first two moments by viewing the estimation problem as a degenerate case of a kalman filter applied to a static system with time varying measurements and by developing the corresponding chandrasekhar type equations the estimator may be obtained recursively and implemented with a fast practical algorithm

a cutting plane algorithm for convex programming that uses analytic centers anoracle for a convex sets n accepts as input any pointz in n and ifz s then it returns yes while ifz s then it returns no along with a separating hyperplane we give a new algorithm that finds a feasible point ins in cases where an oracle is available our algorithm uses the analytic center of a polytope as test point and successively modifies the polytope with the separating hyperplanes returned by the oracle the key to establishing convergence is that hyperplanes judged to be unimportant are pruned from the polytope if a ball of radius 2 l is contained ins ands is contained in a cube of side 2l 1 then we can show our algorithm converges after o nl2 iterations and performs a total of o n4l3 tnl2 arithmetic operations wheret is the number of arithmetic operations required for a call to the oracle the bound is independent of the number of hyperplanes generated in the algorithm an important application in which an oracle is available is minimizing a convex function overs

interior point methods an old and new approach to nonlinear programming in this paper we discuss the main concepts of structural optimization a field of nonlinear programming which was formed by the intensive development of modern interior point schemes

brief papers new results in state estimation and regulation an alternative technique to design linear state estimators and regulators is presented this technique is based on transfer matrix considerations the optimal regulator or estimator gain is obtained via spectral factorization and the solution of a simple equation in polynomial matrices this approach provides further insight displays the duality of estimation and control nicely and bridges the state space and frequency domain techniques the resulting design procedure is computationally attractive and particularly simple for system matrices in the observer or controller canonical form

brief recursive algorithms for inner ellipsoidal approximation of convex polytopes in this paper fast recursive algorithms for the approximation of an n dimensional convex polytope by means of an inscribed ellipsoid are presented these algorithms consider at each step a single inequality describing the polytope and under mild assumptions they are guaranteed to converge in a finite number of steps for their recursive nature the proposed algorithms are better suited to treat a quite large number of constraints than standard off line solutions and have their natural application to problems where the set of constraints is iteratively updated as on line estimation problems nonlinear convex optimization procedures and set membership identification

a fast algorithm for linear complex chebyshev approximations we propose a new algorithm for finding best minimax polynomial approximations in the complex plane the algorithm is the first satisfactory generalization of the well known remez algorithm for real approximations among all available algorithms ours is the only quadratically convergent one numerical examples are presented to illustrate rapid convergence

on the minimum volume covering ellipsoid of ellipsoids let cal s denote the convex hull of m full dimensional ellipsoids in mathbb r n given epsilon 0 and delta 0 we study the problems of computing a 1 epsilon approximation to the minimum volume covering ellipsoid of cal s and a 1 delta n rounding of cal s we extend the first order algorithm of kumar and j optim theory appl 126 2005 pp 1 21 that computes an approximation to the minimum volume covering ellipsoid of a finite set of points in mathbb r n which in turn is a modification of khachiyan s algorithm l g khachiyan math oper res 21 1996 pp 307 320 our algorithm can also compute a 1 delta n rounding of cal s for fixed epsilon 0 and delta 0 we establish polynomial time complexity results for the respective problems each of which is linear in the number of ellipsoids m in particular our algorithm can approximate the minimum volume covering ellipsoid of cal s in asymptotically the same number of iterations as that required by the algorithm of kumar and to approximate the minimum volume covering ellipsoid of a set of m points the main ingredient in our analysis is the extension of polynomial time complexity of certain subroutines in the algorithm from a set of points to a set of ellipsoids as a byproduct our algorithm returns a finite core set cal x subseteq cal s with the property that the minimum volume covering ellipsoid of k provides a good approximation to the minimum volume covering ellipsoid of cal s furthermore the size of the core set depends only on the dimension n and the approximation parameter e but not on the number of ellipsoids m we also discuss the extent to which our algorithm can be used to compute an approximate minimum volume covering ellipsoid and an approximate n rounding of the convex hull of other sets in mathbb r n we adopt the real number model of computation in our analysis

solving lp problems via weighted centers the feasibility problem for a system of linear inequalities can be converted into an unconstrained optimization problem by using ideas from the ellipsoid method which can be viewed as a very simple minimization technique for the resulting nonlinear function this function is related to the volume of an ellipsoid containing all feasible solutions which is parametrized by certain weights which we choose to minimize the function the center of the resulting ellipsoid turns out to be a feasible solution to the inequalities using more sophisticated nonlinear minimization algorithms we develop and investigate more efficient methods which lead to two kinds of weighted centers for the feasible set using these centers we develop new algorithms for solving linear programming problems

a probabilistic approach to multivariable robust filtering and open loop control a new approach to robust filtering prediction and smoothing of discrete time signal vectors is presented linear time invariant filters are designed to be insensitive to spectral uncertainty in signal models the goal is to obtain a simple design method leading to filters which are not overly conservative modeling errors are described by sets of models parameterized by random variables with known covariances these covariances could either be estimated from data or be used as robustness tuning knobs a robust design is obtained by minimizing the spl hscrsub 2 norm or equivalently the mean square estimation error averaged with respect to the assumed model errors a polynomial solution based on an averaged spectral factorization and a unilateral diophantine equation is derived the robust estimator is referred to as a cautious wiener filter it turns out to be only slightly more complicated to design than an ordinary wiener filter the methodology can be applied to any open loop filtering or control problem in particular we illustrate this for the design of robust multivariable feedforward regulators decoupling and model matching filters

uniqueness of local minima for linear quadratic control design the problem of minimizing a weighted sum of the input and output variances from a linear scalar system is considered this is viewed as an optimization problem with the parameters of the regulator as unknowns and it is proved that if the regulator is flexible enough then every local minimum to this problem is a global optimum this result is useful if a gradient method is used to find the optimal regulator

complex chebyshev approximation for fir digital filters an adaptive approach to fir finite impulse response filter design is applied to a fully general complex specification showing it to be capable of giving an optimum chebyshev solution it is thought to be one of the first practical approaches to achieve this in the case of a complex function design examples include a bandpass filter with nearly linear phase and all pass filters with general shaping phase comparisons are drawn with equivalent results from other algorithms

duality of ellipsoidal approximations via semi infinite programming in this paper we develop duality of the minimum volume circumscribed ellipsoid and the maximum volume inscribed ellipsoid problems we present a unified treatment of both problems using convex semi infinite programming we establish the known duality relationship between the minimum volume circumscribed ellipsoid problem and the optimal experimental design problem in statistics the duality results are obtained using convex duality for semi infinite programming developed in a functional analysis setting

tracking of time varying mobile radio channels 1 the wiener lms algorithm adaptation algorithms with constant gains are designed for tracking smoothly time varying parameters of linear regression models in particular channel models occurring in mobile radio communications in a companion paper an application to channel tracking in the is 136 tdma system is discussed the proposed algorithms are based on two key concepts first the design is transformed into a wiener filtering problem second the parameters are modeled as correlated arima processes with known dynamics this leads to a new framework for systematic and optimal design of simple adaptation laws based on prior information the algorithms can be realized as wiener filters called learning filters or as lms newton updates complemented by filters that provide predictions or smoothing estimates the simplest algorithm named the wiener lms is presented all parameters are here assumed governed by the same dynamics and the covariance matrix of the regressors is assumed known the computational complexity is of the same order of magnitude as that of lms for regressors which are either white or have autoregressive statistics the tracking performance is however substantially improved

mmse deconvolution via polynomial methods and its dual lqg regulation abstract the problem of linear minimum mean square error mmse multichannel deconvolution of sampled signals from noisy observations is approached via matrix polynomial equations the general solution is given in terms of a left spectral factorization and a pair of bilateral diophantine equations the first diophantine equation is obtained by imposing optimality of the deconvolution filter whereas the second ensures stability of the filter should the signal model be unstable the proposed solution encompasses classical wiener as well as stationary kalman filtering prediction and fixed lag smoothing the duality with the polynomial equations for lqg regulation is also discussed

the structure and design of realizable decision feedback equalizers for iir channels with colored noise a simple algorithm for optimizing decision feedback equalizers dfes by minimizing the mean square error mse is presented a complex baseband channel and correct past decisions are assumed the dispersive channel may have infinite impulse response and the noise may be colored consideration is given to optimal realizable stable and finite lag smoothing forward and feedback filters in discrete time they are parameterized as recursive filters in the special case of transmission channels with finite impulse response and autoregressive noise the minimum mse can be attained with transversal feedback and forward filters in general the forward part should include a noise whitening filter the inverse noise model the finite realizations of the filters are calculated using a polynomial equation approach to the linear quadratic optimization problem the equalizer is optimized essentially by solving a system of linear equations ax b where a contains transfer function coefficients from the channel and noise model no calculation of correlations is required with this method a simple expression for the minimal mse is presented the dfe is compared to mse optimal linear recursive equalizers expressions for the equalizer in the limiting case of infinite smoothing lags are also discussed

complexity estimates of some cutting plane methods based on the analytic barrier in this paper we establish the efficiency estimates for two cutting plane methods based on the analytic barrier we prove that the rate of convergence of the second method is optimal uniformly in the number of variables we present a modification of the second method in this modified version each test point satisfies an approximate centering condition we also use the standard strategy for updating approximate hessians of the logarithmic barrier function we prove that the rate of convergence of the modified scheme remains optimal and demonstrate that the number of newton steps in the auxiliary minimization processes is bounded by an absolute constant we also show that the approximate hessian strategy significantly improves the total arithmetical complexity of the method

optimal and self tuning white noise estimators with applications to deconvolution and filtering problems using the innovation analysis method in the time domain based on the autoregressive moving average arma innovation model this paper presents a unified white noise estimation theory that includes both input and measurement white noise estimators and presents a new steady state optimal state estimation theory non recursive optimal state estimators are given whose recursive version gives a steady state kalman filter where a new algorithm of the kalman filter gain is proposed two new algorithms of the kalman predictor gain are also derived local asymptotic stability of the kalman filter is proved the classical kalman filtering theory is extended and modified the method used covers unstable systems with correlation noises and singular transition matrix and also covers the self tuning white noise signal and state estimators when the noise statistics is unknown to illustrate three simulation examples are given

ellipsoidal approximations of convex sets based on the volumetric barrier let c rn be a convex set we assume that x 1 for all x c and that c contains a ball of radius 1 r for x rn r r and b and n n symmetric positive definite matrix let ex b r y y xtby x r2 a rounding of c is an ellipsoid ex b r such that ex b r c ex b r in the case that c is characterized by a separation oracle it is well known that an on3 2 rounding of c can be obtained using the shallow cut ellipsoid method in on3 lnnr oracle calls we show that a modification of the volumetric cutting plane method obtains an on3 2 rounding of c in on2 lnnr oracle calls we also consider the problem of obtaining an on rounding of c when c has an explicit polyhedral description our analysis uses a new characterization of circumscribing ellipsoids centered at or near the volumetric center of a polyhedral set

a newton raphson method for moving average spectral factorization using the euclid algorithm an implementation of the newton raphson approach to compute the minimum phase moving average spectral factor of a finite positive definite correlation sequence is presented each step in the successive approximation method involves a system of linear equations that is solved using either the levinson algorithm backwards the jury stability test or a symmetrized version of the euclid algorithm various properties of the newton raphson map are studied the algorithm is generalized to other symmetries other than with respect to the unit circle the special case of the symmetry with respect to the imaginary axis is presented and related to the routh hurwitz stability test for continuous time transfer function

paper efficient algorithm for matrix spectral factorization an algorithm is presented for the spectral factorization of polynomial or rational matrices arising in optimal control and filtering theory as well as in network theory there are two versions of the algorithm one applicable to continuous time problems the other to discrete time ones both versions are based on newton s method feature quadratic convergence and provide a significant improvement in efficiency over the existing methods

realizable mimo decision feedback equalizers structure and design we present and discuss the structure and design of optimum multivariable decision feedback equalizers dfes the equalizers are derived under the constraint of realizability that is causal and stable filters and finite decision delay the design is based on a known dispersive discrete time multivariable channel model with infinite impulse response the additive noise is described by a multivariate arma model equations for obtaining minimum mean square error mmse and zero forcing dfes are derived under the assumption of correct past decisions the design of a realizable mmse dfe requires the solution of a linear system of equations in the model parameters no spectral factorization is required we derive novel necessary and sufficient conditions for the existence of zero forcing dfes and point out the significance of these conditions for the design of multiuser detectors an optimal mmse dfe will contain iir filters in general simulations indicate that the performance improvement obtained with an iir dfe is reduced more than for a suboptimal fir dfe when error propagation is taken into account since the use of iir feedback filters tends to worsen the error propagation

linear antenna array pattern synthesis with prescribed broad nulls a technique of synthesizing an antenna array pattern with prescribed broad nulls is presented the array pattern synthesis problem is formulated as a least square null constrained optimization problem numerical techniques based on the matrix factorization method are developed for reducing the computational complexity of determining the optimal weight vector subsequently a set of eigenvector constraints are used to approximate the effect of the quadratic constraint numerical results are presented to illustrate the performance achievable

wiener design of adaptation algorithms with time invariant gains a design method is presented that extends least mean squared lms adaptation of time varying parameters by including general linear time invariant filters that operate on the instantaneous gradient vector the aim is to track time varying parameters of linear regression models in situations where the regressors are stationary or have slowly time varying properties the adaptation law is optimized with respect to the steady state parameter error covariance matrix for time variations modeled as vector arima processes the design method systematically uses prior information about time varying parameters to provide filtering prediction or fixed lag smoothing estimates for arbitrary lags the method is based on a transformation of the adaptation problem into a wiener filter design problem the filter works in open loop for slow parameter variations whereas a time varying closed loop has to be considered for fast variations in the latter case the filter design is performed iteratively the general form of the solution at each iteration is obtained by a bilateral diophantine polynomial matrix equation and a spectral factorization for white gradient noise the diophantine equation has a closed form solution further structural constraints result in very simple design equations under certain model assumptions the wiener designed adaptation laws reduce to lms adaptation compared with kalman estimators the channel tracking performance becomes nearly the same in mobile radio applications whereas the complexity is in general much lower

robust filtering and feedforward control based on probabilistic descriptions of model errors abstract a new approach to robust estimation of signals prediction of time series and robust feedforward control is considered modelling errors are parametrized by random variables with known covariances a robust design is obtained by minimizing the squared estimation error averaged both with respect to model errors and the noise a polynomial equations approach based on averaged spectral factorizations and averaged diophantine equations is derived mild solvability conditions guarantee the existence of stable optimal filters and feedforward regulators the robust design turns out to be no more complicated than the design of an ordinary wiener filter or lqg regulator the proposed approach avoids two drawbacks of minimax design first probabilistic descriptions of model uncertainties may have soft bounds these are more readily obtainable in a noisy environment than the hard bounds required for minimax design furthermore not only the range of uncertainties but also their likelihood is taken into account common model deviations will have a greater impact on an estimator design than do very rare worst cases the conservativeness is thus reduced

a new algorithm for minimizing convex functions over convex sets let s subseteq mathbb r n be a convex set for which there is an oracle with the following property given any pointz n the oracle returns a yes ifz s whereas ifz s then the oracle returns a no together with a hyperplane that separatesz froms the feasibility problem is the problem of finding a point ins the convex optimization problem is the problem of minimizing a convex function overs we present a new algorithm for the feasibility problem the notion of a volumetric center of a polytope and a related ellipsoid of maximum volume inscribable in the polytope are central to the algorithm our algorithm has a significantly better global convergence rate and time complexity than the ellipsoid algorithm the algorithm for the feasibility problem easily adapts to the convex optimization problem

optimal deconvolution based on polynomial methods the problem of estimating the input to a known linear system is treated in a shift operator polynomial formulation the mean square estimation error is minimized the input and a colored measurement noise are described by independent arma autoregressive moving average processes the filter is calculated by performing a spectral factorization and solving a polynomial equation the approach can be applied to input prediction filtering and smoothing problems as well as to the use of prefilters in the quadratic criterion it applies to nonminimum phase as well as unstable systems as illustrated by two examples

a method for the optimal pattern synthesis of linear arrays with prescribed nulls a new l sub spl infin optimal method for the synthesis of equispaced linear array functions with asymmetrical far field pattern functions is proposed this iterative method provides for the exact specification of the beam width while at the same time allowing for the specification of the relative levels of individual sidelobes by index or as a function of bearing e g angularly extended nulls as well as the realization of specified narrowband nulls the resulting array factors are optimal in the weighted l sub spl infin sense and in general have complex coefficients this new remez type method employs multiple objective functions to provide the degrees of freedom that are required for exact null placement examples which demonstrate the design flexibility offered by the method are included for various sum and difference patterns including superdirective and shaped beam arrays

voronoi cells probabilistic bounds and hypothesis testing in mixed integer linear models although real valued linear models whether or not of full rank have been thoroughly investigated and are well documented very little is known about statistical and probabilistic aspects of a mixed integer linear model which arose from space geodesy and serves as the standard starting model for precise positioning using the global positioning system gps voronoi cells play a fundamental role in the least squares estimation of the integer unknowns of the model in this paper we first develop a method to construct voronoi cells and study how to fit figures of simple shape to a voronoi cell both from inside and outside we then derive a number of new lower and upper bounds on the probability that the integers of the model are correctly estimated finally we discuss the tests of two hypotheses on the integer mean

design of fir filters in the complex domain the design of fir digital filters with a complex valued desired frequency response using the chebyshev error is investigated the complex approximation problem is converted into a real approximation problem which is nearly equivalent to the complex problem a standard linear programming algorithm for the chebyshev solution of overdetermined equations is used to solve the real approximation problem additional constraints are introduced which allow weighting of the phase and or group delay of the approximation digital filters are designed which have nearly constant group delay in the passbands the desired constant group delay which gives the minimum chebyshev error is found to be smaller than that of a linear phase filter of the same length these filters in addition to having a smaller approximately constant group delay have better magnitude characteristics than exactly linear phase filters with the same length the filters have nearly equiripple magnitude and group delay

fast iterative method of power synthesis for antenna arrays a fast and simple method of power synthesis for antenna arrays of arbitrary geometry is presented the method adopts an auxiliary function having the desired amplitude pattern and a generic phase pattern and determines in closed form the array pattern approximating the auxiliary function in both amplitude and phase a cost function involving the auxiliary function and the approximating pattern is minimized by modifying the phase pattern of the auxiliary function using a simple and fast iterative technique once the optimal phase pattern has been found the array pattern approximating the corresponding auxiliary function is the synthesized pattern the proposed method allows to satisfy stringent requirements on the amplitude pattern including multibeam synthesis and radiation suppression within large angular regions numerical results show the effectiveness of the algorithm and are compared to those obtained using other existing methods in order to achieve a more significant comparison such methods have been properly modified to improve their performances such comparison shows that the presented algorithm allows a significant reduction of cpu time with respect to the other considered methods as well as a weaker dependence on the starting point finally the presented method is extended in such a way as to reduce the dynamic range ratio drr of the array excitations

realizable spatio temporal tomlinson harashima precoders theory and fast computation we derive the optimal realizable tomlinson harashima precoder for frequency selective multiple input multiple output mimo channels with respect to a simplified mean square error mse criterion realizability means that in contrast to other works we do not restrict the internal filters of the precoder to have finite impulse responses firs but nevertheless ensure that the precoder can be operated in real time subject to a finite latency time which can be chosen by the system designer in particular this allows us to consider channels with infinite impulse responses iirs as they occur e g in digital subscriber lines the feedforward filter is located at the transmitter in our system model and an additional scalar gain is employed at the receiver the relocation of the feedforward filter allows us to consider decentralized receivers however it also makes it necessary to impose a transmit power constraint the power constraint often has the effect that the input signals at the receiver have to be rescaled we argue that the inclusion of the additional scalar gain allows us to incorporate the effect of the rescaling operation into the optimization special emphasis is put on the fast computation of the realizable tomlinson harashima precoder via displacement structure theory we propose a fast algorithm which includes a successive construction of a close to optimal ordering of the data streams the complexity of this algorithm is only cubic in the number of channel inputs and quadratic in the latency time the algorithm can also be applied in situations where certain fir constraints have to be fulfilled because we find that the optimal realizable tomlinson harashima precoder involves only fir filters as soon as the channel is fir our results are reproducible

design and characterization of optimal fir filters with arbitrary phase an algorithm for designing a chebyshev optimal fir filter that approximates an arbitrary complex valued frequency response is presented this algorithm computes the optimal filter by solving the dual to the filter design problem it is guaranteed to converge theoretically and requires o n sup 2 computations per iteration for a filter of length n for the first time properties of the optimal filter are derived and the case where the desired filter has arbitrary constant group delay is studied in detail

a new method for d dimensional exact deconvolution deconvolution is an important problem of signal processing and conventional approaches including fourier methods have stability problems due to the zeros of the convolution kernel we present a new method of multidimensional exact deconvolution this method is always stable even when the convolution kernel h n has zeros on the unit circle and there exist closed form solutions for the one dimensional 1 d case d 1 for the multidimensional case d 1 the proposed method yields stable solutions when det h d this solution set covers a portion of all possible convolution kernels including the ones that have zeros on the multidimensional unit circle this novel time domain method is based on the fact that the convolution inverse of a first order kernel can be found exactly in multidimensional space convolution inverses for higher order kernels are obtained using this fact and the zeros of the convolution kernel the presented method is exact stable and computationally efficient several examples are given in order to show the performance of this method in 1 d and multidimensional cases

digital differentiation of noisy data measured through a dynamic system the design of discrete time differentiating filters in the presence of colored noise and nonneglectable transducer dynamics is investigated the signal and noise are described by autoregressive moving average arma models possibly with poles on mod z mod 1 the mean square error mse optimal filter based on a discrete time approximation of the derivative operator is given by a spectral factorization and a linear polynomial equation

complexity of circumscribed and inscribed ellipsoid methods for solving equilibrium economical models an exchange production model and the arrow debreu model with linear utility functions are considered these models are reduced to special convex programming problems these problems can be solved by the inscribed and circumscribed ellipsoid methods it is proved that the inscribed ellipsoid method s complexity in this case is less than the complexity of the circumscribed ellipsoid method

brief paper a predictive controller with artificial lyapunov function for linear systems with input state constraints this paper copes with the problem of satisfying input and or state hard constraints in set point tracking problems stability is guaranteed by synthesizing a lyapunov quadratic function for the system and by imposing that the terminal state lies within a level set of the function procedures to maximize the volume of such an ellipsoidal set are provided and interiorpoint methods to solve on line optimization are considered

single versus double diophantine equation debate comments on a polynomial approach to wiener filtering comments made on a paper by grimble concerning the number of diophantine equations which are needed to solve the optimal filtering problem are shown not to be valid for the system description employed a comment regarding the proof of optimality is also discussed

semidefinite programming based preconditioning for more robust near separable nonnegative matrix factorization nonnegative matrix factorization nmf under the separability assumption can provably be solved efficiently even in the presence of noise and has been shown to be a powerful technique in document classification and hyperspectral unmixing this problem is referred to as near separable nmf and requires that there exists a cone spanned by a small subset of the columns of the input nonnegative matrix approximately containing all columns in this paper we propose a preconditioning based on the minimum volume ellipsoid and semidefinite programming making the input matrix well conditioned this in turn can improve significantly the performance of near separable nmf algorithms which is illustrated on the popular successive projection algorithm spa the new preconditioned spa is provably more robust to noise and outperforms spa on several synthetic data sets we also show how an active set method allows us to apply the preconditioning on large scale real world hyperspectral images

optimal parallel model solutions using the polynomial approach solutions to the optimal filtering and open loop control problems using a generalized parallel model scheme are presented the solutions are based on the algebra of polynomial matrices different c

consistency of constraints on nulls and on dynamic range ratio in pattern synthesis for antenna arrays given an antenna array of n elements we deal with the problem of finding an array pattern that approximates a desired pattern while a null constraints are imposed in m les n 2 assigned directions b an upper bound d is imposed on the dynamic range ratio drr of the excitations at first we show that the system of constraint equations has solutions if and only if d ges d0 where d0 is a parameter depending on the null directions and describe a method to evaluate d0 then we propose two simple and fast algorithms to solve the synthesis problem and show their effectiveness by numerical examples statistical results show that usually d0 1 when m is small compared to n thus exact nulls can be formed for any d ges 1 this result provides an insight into the numerical results

on the complexity of approximating the maximal inscribed ellipsoid for a polytope we give a new polynomial bound on the complexity of approximating the maximal inscribed ellipsoid for a polytope

improved complexity for maximum volume inscribed ellipsoids let cal p x vert ax le b where a is an m n matrix we assume that cal p contains a ball of radius one centered at the origin and is itself contained in a ball of radius r centered at the origin we consider the problem of approximating the maximum volume ellipsoid inscribed in cal p such ellipsoids have a number of interesting applications including the inscribed ellipsoid method for convex optimization we describe an optimization algorithm that obtains an ellipsoid whose volume is at least a factor e epsilon of the maximum possible in o m 3 5 ln mr epsilon operations our result provides an alternative to a saddlepoint based approach with the same complexity developed by nemirovskii we also show that a further reduction in complexity can be obtained by first computing an approximation of the analytic center of cal p

l infinity norm design of linear phase robust broadband beamformers using constrained optimization a new method for the design of linear phase robust far field broadband beamformers using constrained optimization is proposed in the method the maximum passband ripple and minimum stopband attenuation are ensured to be within prescribed levels while at the same time maintaining a good linear phase characteristic at a prescribed group delay in the passband since the beamformer is intended primarily for small sized microphone arrays where the microphone spacing is small relative to the wavelength at low frequencies the beamformer can become highly sensitive to spatial white noise and array imperfections if a direct minimization of the error is performed therefore to limit the sensitivity of the beamformer the optimization is carried out by constraining a sensitivity parameter namely the white noise gain wng to be above prescribed levels across the frequency band two novel design variants have been developed the first variant is formulated as a convex optimization problem where the maximum error in the passband is minimized while the second variant is formulated as an iterative optimization problem and has the advantage of significantly improving the linear phase characteristics of the beamformer under any prescribed group delay or linear array configuration in the second variant the passband group delay deviation is minimized while ensuring that the maximum passband ripple and stopband attenuation are within prescribed levels to reduce the computational effort in carrying out the optimization a nonuniform variable sampling approach over the frequency and angular dimensions is used to compute the required parameters experiment results show that beamformers designed using the proposed methods have much smaller passband group delay deviation for similar passband ripple and stopband attenuation than a modified version of an existing method

rounding of polytopes in the real number model of computation let a be a set of m points in rn we show that the problem of 1 en rounding of a i e the problem of computing an ellipsoid e rn such that 1 en 1e conv hulla e can be solved in omn2e 1 ln n ln ln m arithmetic operations and comparisons this result implies that the problem of approximating the minimum volume ellipsoid circumscribed about a can be solved in om3 5 lnme 1 operations to a relative accuracy of e in the volume the latter bound also applies to the 1 en rounding problem our bounds hold for the real number model of computation

a cutting plane method for solving kyp sdps semidefinite programs originating from the kalman yakubovich popov lemma are convex optimization problems and there exist polynomial time algorithms that solve them however the number of variables is often very large making the computational time extremely long algorithms more efficient than general purpose solvers are thus needed to this end structure exploiting algorithms have been proposed based on the dual formulation in this paper a cutting plane algorithm is proposed in a comparison with a general purpose solver and a structure exploiting solver it is shown that the cutting plane based solver can handle optimization problems of much higher dimension

design of sum and difference patterns with common nulls and low slls simultaneously in the presence of array errors sum sigma and difference delta beampatterns are commonly utilized to estimate the bearings of targets in the radar system however the nulls of delta pattern are not always aligned with those of sigma pattern and this phenomenon would become more severe in practical scenario since the array errors exist meanwhile the sidelobe level sll of the array would also increase due to the unavoidable errors the above described factors would dramatically decrease the performance of radar in the adaptive suppression of interference to increase the output signal to interference plus noise ratio of a radar system this paper presents an approach to design the low sll sigma delta patterns where the nulls of the delta pattern excluding the central null are always aligned with those of the sigma pattern in the presence of errors two schemes are addressed to obtain the covariance matrix for null alignment constraint which is exploited to force nulling in desired angles accordingly in the sequel the matrix tapering method is adopted to broaden the nulls and enhance the robustness for the possible errors with the proposed approach the low sll and nulls alignment can be achieved simultaneously numerical examples are provided and discussed to demonstrate the effectiveness of the proposed solution including the results based on a linearly spaced microstrip array antenna

wiener filter design using polynomial equations a simplified way of deriving realizable and explicit wiener filters is presented discrete time problems are discussed in a polynomial equation framework optimal filters predictors and smoothers are calculated by means of spectral factorizations and linear polynomial equations a tool for obtaining these equations for a given problem structure is described it is based on the evaluation of orthogonality in the frequency domain by means of canceling stable poles with zeros comparisons are made to previously known derivation methodologies such as completing the squares for the polynomial systems approach and the classical wiener solution the simplicity of the proposed derivation method is particularly evident in multistage filtering problems to illustrate two examples are discussed a filtering and a generalized deconvolution problem a new solvability condition for linear polynomial equation appearing in scalar problems is also presented

diophantine equations in control a survey abstract this survey is also a tutorial whose aim is to explain the role of diophantine equations in the synthesis of feedback control systems these are linear equations in a ring and result from a fractional representation of the systems involved the cornerstone of the exposition is a simple parametrization of all stabilizing controllers for a given plant one can then choose in principle the best controllers for various applications these ideas evolved from early attempts to use polynomial equations in the design of discrete time linear systems by now they have been extended to continuous time infinite dimensional time varying and non linear systems

a flexible optimization method for the pattern synthesis of equispaced linear arrays with equiphase excitation a synthesis method is proposed for linear arrays having equiphase excitation currents in addition to allowing for the exact specification of the beamwidth and for the specification of individual sidelobe levels by index or as a function of bearing e g angularly extended nulls multiple deep narrow nulls can be specified at arbitrary bearing angles the method is suitable for the design of broadside or endfire arrays with sum or difference patterns and can also be used for the design of superdirective arrays with good radiation efficiencies and sensitivity properties for computational efficiency the method uses a new constrained multivariable remez type l sub infinity approximation technique

multiobjective optimization of real and coupled antenna array excitations via primal dual interior point filter method from spherical mode expansions a novel method for the multiobjective optimization of arbitrary planar array excitations is presented the optimization problem formulation which inherently takes into account every array element pattern as well as all interelement couplings is based on matrix valued functions which are computed from the generalized scattering matrix characterization of an array and spherical mode expansions of its radiated field it allows the maximization of the directive gain subject to a maximum sidelobe level a maximum crosspolar level and a minimum aperture illumination efficiency the setting of null pointing directions and the dynamic range control to solve the resulting non linear optimization problem we have developed a primal dual interior point method specifically adapted to the presented formulation which makes use of novel filtering techniques numerical examples of arrays of microstrip patches and dielectric resonator antennas covering a wide variety of requirements on these parameters are presented

how to compute the volume in high dimension in some areas of theoretical computer science we feel that randomized algorithms are better and in some others we can prove that they are more efficient than the deterministic ones

windows with rapidly decaying sidelobes and steerable sidelobe dips a window in the form of a sum of cosine terms is considered and the temporal weights are evaluated analytically so that it can have rapidly decaying sidelobes as well as steerable sidelobe dips an analytic expression is derived so that the window designer can exploit the trade off between a high sidelobe falloff rate slfor and many steerable sidelobe dips ssld the treatment is based on using the l degrees of freedom inherent in the l temporal weights of the l 1 term window in partially achieving the goals of a high slfor and many ssld in the extreme case where all degrees of freedom are used in achieving the maximum slfor a simple closed form expression is derived for the temporal weights thus formalizing and generalizing the results of nuttall 1981 the other extreme case where all degrees of freedom are used in creating many ssld has been recently treated by zhong et al see ibid vol 40 no 6 p 1452 1459 1992 combining both desired features in one unified mathematical framework is the main objective of this paper simulation results demonstrate the superior performance of a window having both features to that of a window having only one feature in extracting a weak sinusoidal signal dominated by a leakage from another stronger signal

technical communique descriptor wiener state estimators based on the modern time series analysis method a new time domain wiener filtering approach is presented asymptotically stable wiener state estimators are presented for discrete linear stochastic descriptor systems they can be implemented via the autoregressive moving average arma recursive filters they can handle the optimal state filtering smoothing and prediction problems in a unified framework and can simply be obtained based on the arma innovation model the solution of the diophantine equations and riccati equations is avoided so that the computational burden is reduced a simulation example shows the effectiveness of the new approach

on the design of fir filters by complex chebyshev approximation the long standing problem of approximating a complex valued desired function with a finite impulse response fir filter is considered it is formulated as an equalization to be solved using complex valued filters the proposed algorithm deals directly with the complex error function which depends linearly on the coefficients of the filter to be designed the magnitude of this error function is minimized in the chebshev sense using a generalization of the remez exchange algorithm the method can be used to design complex or real valued selective systems as well the well known design of optimal fir filters with linear phase is included here as a special case

polynomial approach to wiener filtering a rigorous derivation is given of the polynomial method for finding wiener filters in both discrete time and continuous time in each case the method involves one spectral factorization and the solution of one diophantine equation the theory is applicable in multivariable situations but simple scalar examples are included here to compare the polynomial approach with the basic wiener method

on the complexity of computing mixed volumes this paper gives various positive and negative results on the complexity of the problem of computing and approximating mixed volumes of polytopes and more general convex bodies in arbitrary dimension on the negative side we present several p hardness results that focus on the difference of computing mixed volumes versus computing the volume of polytopes we show that computing the volume of zonotopes is p hard while each corresponding mixed volume can be computed easily but also give examples showing that computing mixed volumes is hard even when computing the volume is easy on the positive side we derive a randomized algorithm for computing the mixed volumes v overbrace k 1 ld k 1 m 1 overbrace k 2 dots k 2 m 2 dots overbrace k s dots k s m s of well presented convex bodies k 1 dots k s where m 1 dots m s in n 0 and m 1 geq n psi n with psi n o frac log n log log n the algorithm is an interpolation method based on polynomial time randomized algorithms for computing the volume of convex bodies this paper concludes with applications of our results to various problems in discrete mathematics combinatorics computational convexity algebraic geometry geometry of numbers and operations research

a minimum volume covering approach with a set of ellipsoids a technique for adjusting a minimum volume set of covering ellipsoids technique is elaborated solutions to this problem have potential application in one class classification and clustering problems its main original features are 1 it avoids the direct evaluation of determinants by using diagonalization properties of the involved matrices 2 it identifies and removes outliers from the estimation process 3 it avoids binary variables resulting from the combinatorial character of the assignment problem that are replaced by continuous variables in the range 0 1 4 the problem can be solved by a bilevel algorithm that in its first level determines the ellipsoids and in its second level reassigns the data points to ellipsoids and identifies outliers based on an algorithm that forces the karush kuhn tucker conditions to be satisfied two theorems provide rigorous bases for the proposed methods finally a set of examples of application in different fields is given to illustrate the power of the method and its practical performance

polynomial systems approach to optimal linear filtering and prediction the solution of the optimal linear estimation problem is considered using a polynomial matrix description for the discrete system the filter or predictor is given by the solution of two diophantine equations and is equivalent to the state equation form of the steady state kalman filter or the transfer function matrix form of the wiener filter the pole zero properties of the optimal filter are more obvious in the polynomial representation and new insights into the disturbance rejection properties of the filter are obtained the plant or signal model can be stable or unstable and allowance is made for both control and disturbance input subsystems and white and coloured measurement noise or an output disturbance subsystem the model structure was determined by the needs of several industrial filtering problems the polynomial form of filter may easily be included in a self tuning algorithm and a simple adaptive estimator is described

a polynomial approach to bias aware fixed lag smoothing problem in the present technical note a solution to the problem of fixed lag smoothing of siso systems in presence of a dynamical bias is presented in a polynomial framework the bias aware smoothing problem is solved in three steps namely the design of the general structure of the smoother the estimation of the dynamical bias by means of a deconvolution technique and then the combination of the previous results to obtain the bias aware fixed lag smoother applied to an example this approach shows its efficiency

playing billiards in version space a ray tracing method inspired by ergodic billiards is used to estimate the theoretically best decision rule for a given set of linear separable examples for randomly distributed examples the billiard estimate of the single perceptron with best average generalization probability agrees with known analytic results while for real life classification problems the generalization probability is consistently enhanced when compared to the maximal stability perceptron

h sub 2 optimal linear robust sampled data filtering design using polynomial approach a new frequency domain approach to robust multi input multi output mimo linear filter design for sampled data systems is presented the system and noise models are assumed to be represented by polynomial forms that are not perfectly known except that they belong to a certain set the optimal design guarantees that the error variance is kept below an upper bound that is minimized for all admissible uncertainties the design problem is cast in the context of h sub 2 via the polynomial matrix representation of systems with norm bounded unstructured uncertainties the sampled data mix of continuous and discrete time systems is handled by means of a lifting technique however it does not increase the dimensionality or alter the computational cost of the solution the setup adopted allows dealing with several filtering problems a simple deconvolution example illustrates the procedure

an analytic center cutting plane method for semidefinite feasibility problems semidefinite feasibility problems arise in many areas of operations research the abstract form of these problems can be described as finding a point in a nonempty bounded convex body g in the cone of symmetric positive semidefinite matrices assume that g is defined by an oracle which for any givenm xm symmetric positive semidefinite matrix y either confirms that y e g or returns a cut i e a symmetric matrixa such that g is in the half space y a y a y we study an analytic center cutting plane algorithm for this problem at each iteration the algorithm computes an approximate analytic center of a working set defined by the cutting plane system generated in the previous iterations if this approximate analytic center is a solution then the algorithm terminates otherwise the new cutting plane returned by the oracle is added into the system as the number of iterations increases the working set shrinks and the algorithm eventually finds a solution to the problem all iterates generated by the algorithm are positive definite matrices the algorithm has a worst case complexity of o m3 e 2 on the total number of cuts to be used where e is the maximum radius of a ball contained by g