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Joint TxRx beamforming design for multicarrier MIMO channels: a unified framework for convex optimization
 IEEE TRANS. SIGNAL PROCESSING
, 2003
"... This paper addresses the joint design of transmit and receive beamforming or linear processing (commonly termed linear precoding at the transmitter and equalization at the receiver) for multicarrier multipleinput multipleoutput (MIMO) channels under a variety of design criteria. Instead of consid ..."
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Cited by 155 (17 self)
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This paper addresses the joint design of transmit and receive beamforming or linear processing (commonly termed linear precoding at the transmitter and equalization at the receiver) for multicarrier multipleinput multipleoutput (MIMO) channels under a variety of design criteria. Instead of considering each design criterion in a separate way, we generalize the existing results by developing a unified framework based on considering two families of objective functions that embrace most reasonable criteria to design a communication system: Schurconcave and Schurconvex functions. Once the optimal structure of the transmitreceive processing is known, the design problem simplifies and can be formulated within the powerful framework of convex optimization theory, in which a great number of interesting design criteria can be easily accommodated and efficiently solved, even though closedform expressions may not exist. From this perspective, we analyze a variety of design criteria, and in particular, we derive optimal beamvectors in the sense of having minimum average bit error rate (BER). Additional constraints on the peaktoaverage ratio (PAR) or on the signal dynamic range are easily included in the design. We propose two multilevel waterfilling practical solutions that perform very close to the optimal in terms of average BER with a low implementation complexity. If cooperation among the processing operating at different carriers is allowed, the performance improves significantly. Interestingly, with carrier cooperation, it turns out that the exact optimal solution in terms of average BER can be obtained in closed form.
Robust Solutions To Uncertain Semidefinite Programs
 SIAM J. OPTIMIZATION
, 1998
"... In this paper we consider semidefinite programs (SDPs) whose data depend on some unknown but bounded perturbation parameters. We seek "robust" solutions to such programs, that is, solutions which minimize the (worstcase) objective while satisfying the constraints for every possible value ..."
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Cited by 86 (8 self)
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In this paper we consider semidefinite programs (SDPs) whose data depend on some unknown but bounded perturbation parameters. We seek "robust" solutions to such programs, that is, solutions which minimize the (worstcase) objective while satisfying the constraints for every possible value of parameters within the given bounds. Assuming the data matrices are rational functions of the perturbation parameters, we show how to formulate sufficient conditions for a robust solution to exist as SDPs. When the perturbation is "full," our conditions are necessary and sufficient. In this case, we provide sufficient conditions which guarantee that the robust solution is unique and continuous (Hölderstable) with respect to the unperturbed problem's data. The approach can thus be used to regularize illconditioned SDPs. We illustrate our results with examples taken from linear programming, maximum norm minimization, polynomial interpolation, and integer programming.
A robust maximin approach for MIMO communications with imperfect channel state information based on convex optimization
 IEEE Trans. Signal Processing
, 2006
"... Abstract—This paper considers a wireless communication system with multiple transmit and receive antennas, i.e., a multipleinputmultipleoutput (MIMO) channel. The objective is to design the transmitter according to an imperfect channel estimate, where the errors are explicitly taken into account ..."
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Cited by 25 (4 self)
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Abstract—This paper considers a wireless communication system with multiple transmit and receive antennas, i.e., a multipleinputmultipleoutput (MIMO) channel. The objective is to design the transmitter according to an imperfect channel estimate, where the errors are explicitly taken into account to obtain a robust design under the maximin or worst case philosophy. The robust transmission scheme is composed of an orthogonal space–time block code (OSTBC), whose outputs are transmitted through the eigenmodes of the channel estimate with an appropriate power allocation among them. At the receiver, the signal is detected assuming a perfect channel knowledge. The optimization problem corresponding to the design of the power allocation among the estimated eigenmodes, whose goal is the maximization of the signaltonoise ratio (SNR), is transformed to a simple convex problem that can be easily solved. Different sources of errors are considered in the channel estimate, such as the Gaussian noise from the estimation process and the errors from the quantization of the channel estimate, among others. For the case of Gaussian noise, the robust power allocation admits a closedform expression. Finally, the benefits of the proposed design are evaluated and compared with the pure OSTBC and nonrobust approaches. Index Terms—Antenna arrays, beamforming, convex optimization theory, maximum optimization problems, multipleinput multipleoutput (MIMO) systems, saddle point, space–time coding, worstcase robust designs. I.
An Adaptive Dynamic Programming Algorithm for Dynamic Fleet Management, I: Single Period Travel Times
, 2002
"... We consider a stochastic version of a dynamic resource allocation problem. In this setting, reusable resources must be assigned to tasks that arise randomly over time. We solve the problem using an adaptive dynamic programming algorithm that uses nonlinear functional approximations that give the val ..."
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Cited by 15 (5 self)
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We consider a stochastic version of a dynamic resource allocation problem. In this setting, reusable resources must be assigned to tasks that arise randomly over time. We solve the problem using an adaptive dynamic programming algorithm that uses nonlinear functional approximations that give the value of resources in the future. Our functional approximations are piecewise linear and naturally provide integer solutions. We show that the approximations provide nearoptimal solutions to deterministic problems and solutions that significantly outperform deterministic rollinghorizon methods on stochastic problems.
Convex Optimization Theory Applied to Joint TransmitterReceiver Design in MIMO Channels
 in SpaceTime Processing for MIMO Communications, Chapter 8
, 2005
"... Multiantenna MIMO channels have recently become a popular means to increase the spectral efficiency and quality of wireless communications by the use of spatial diversity at both sides of the link [1–4]. In fact, the MIMO concept is much more general and embraces many other scenarios such as wirelin ..."
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Cited by 3 (1 self)
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Multiantenna MIMO channels have recently become a popular means to increase the spectral efficiency and quality of wireless communications by the use of spatial diversity at both sides of the link [1–4]. In fact, the MIMO concept is much more general and embraces many other scenarios such as wireline digital subscriber line (DSL) systems [5] and singleantenna
On Reoptimizing MultiClass Classifiers ∗
, 2006
"... Significant changes in the instance distribution or associated cost function of a learning problem require one to reoptimize a previously learned classifier to work under new conditions. We study the problem of reoptimizing a multiclass classifier based on its ROC hypersurface and a matrix describi ..."
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Cited by 2 (0 self)
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Significant changes in the instance distribution or associated cost function of a learning problem require one to reoptimize a previously learned classifier to work under new conditions. We study the problem of reoptimizing a multiclass classifier based on its ROC hypersurface and a matrix describing the costs of each type of prediction error. For a binary classifier, it is straightforward to find an optimal operating point based on its ROC curve and the relative cost of true positive to false positive error. However, the corresponding multiclass problem (finding an optimal operating point based on a ROC hypersurface and cost matrix) is more challenging and until now, it was unknown whether an efficient algorithm existed that found an optimal solution. We answer this question by first proving that the decision version of this problem is NPcomplete. As a complementary positive result, we give an algorithm that finds an optimal solution in polynomial time if the number of classes n is a constant. We also present several heuristics for this problem, including linear, nonlinear, and quadratic programming formulations, genetic algorithms, and a customized algorithm. Empirical results suggest that under uniform costs several methods exhibit significant improvements while genetic algorithms and margin maximization quadratic programs fare the best under nonuniform cost models.
Optimal Linear Precoding Strategies for Wideband Noncooperative Systems Based on Game
"... Abstract—In this twopart paper, we propose a decentralized strategy, based on a gametheoretic formulation, to find out the optimal precoding/multiplexing matrices for a multipointtomultipoint communication system composed of a set of wideband links sharing the same physical resources, i.e., tim ..."
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Abstract—In this twopart paper, we propose a decentralized strategy, based on a gametheoretic formulation, to find out the optimal precoding/multiplexing matrices for a multipointtomultipoint communication system composed of a set of wideband links sharing the same physical resources, i.e., time and bandwidth. We assume, as optimality criterion, the achievement of a Nash equilibrium and consider two alternative optimization problems: 1) the competitive maximization of mutual information on each link, given constraints on the transmit power and on the spectral mask imposed by the radio spectrum regulatory bodies; and 2) the competitive maximization of the transmission rate, using finite order constellations, under the same constraints as above, plus a constraint on the average error probability. In this first part of the paper, we start by showing that the solution set of both noncooperative games is always nonempty and contains only pure strategies. Then, we prove that the optimal precoding/multiplexing scheme for both games leads to a channel diagonalizing structure, so that both matrixvalued problems can be recast in a simpler unified vector power control game, with no performance penalty. Thus, we study this simpler game and derive sufficient conditions ensuring the uniqueness of the Nash equilibrium. Interestingly, although derived under stronger constraints, incorporating for example spectral mask constraints, our uniqueness conditions have broader validity than previously known conditions. Finally, we assess the goodness of the proposed decentralized strategy by comparing its performance with the performance of a Paretooptimal centralized scheme. To reach the Nash equilibria of the game, in Part II, we propose alternative distributed algorithms, along with their convergence conditions. Index Terms—Ad hoc networks, cognitive radio, game theory, linear precoding, mutual information, Nash equilibrium. I.
A Derivation of the SourceChannel Error Exponent Using Nonidentical Product Distributions
"... Abstract — This paper studies the randomcoding exponent of joint sourcechannel coding for a scheme where source messages are assigned to disjoint subsets (referred to as classes), and codewords are independently generated according to a distribution that depends on the class index of the source m ..."
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Abstract — This paper studies the randomcoding exponent of joint sourcechannel coding for a scheme where source messages are assigned to disjoint subsets (referred to as classes), and codewords are independently generated according to a distribution that depends on the class index of the source message. For discrete memoryless systems, two optimally chosen classes and product distributions are found to be sufficient to attain the spherepacking exponent in those cases where it is tight. Index Terms — Joint sourcechannel coding, reliability function, random coding, product distributions, spherepacking bound. I.