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44
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 224 (20 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.
An InteriorPoint Algorithm For Nonconvex Nonlinear Programming
 COMPUTATIONAL OPTIMIZATION AND APPLICATIONS
, 1997
"... The paper describes an interiorpoint algorithm for nonconvex nonlinear programming which is a direct extension of interiorpoint methods for linear and quadratic programming. Major modifications include a merit function and an altered search direction to ensure that a descent direction for the mer ..."
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Cited by 186 (14 self)
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The paper describes an interiorpoint algorithm for nonconvex nonlinear programming which is a direct extension of interiorpoint methods for linear and quadratic programming. Major modifications include a merit function and an altered search direction to ensure that a descent direction for the merit function is obtained. Preliminary numerical testing indicates that the method is robust. Further, numerical comparisons with MINOS and LANCELOT show that the method is efficient, and has the promise of greatly reducing solution times on at least some classes of models.
Robust minimum variance beamforming
 IEEE Transactions on Signal Processing
, 2005
"... Abstract—This paper introduces an extension of minimum variance beamforming that explicitly takes into account variation or uncertainty in the array response. Sources of this uncertainty include imprecise knowledge of the angle of arrival and uncertainty in the array manifold. In our method, uncerta ..."
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Cited by 88 (10 self)
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Abstract—This paper introduces an extension of minimum variance beamforming that explicitly takes into account variation or uncertainty in the array response. Sources of this uncertainty include imprecise knowledge of the angle of arrival and uncertainty in the array manifold. In our method, uncertainty in the array manifold is explicitly modeled via an ellipsoid that gives the possible values of the array for a particular look direction. We choose weights that minimize the total weighted power output of the array, subject to the constraint that the gain should exceed unity for all array responses in this ellipsoid. The robust weight selection process can be cast as a secondorder cone program that can be solved efficiently using Lagrange multiplier techniques. If the ellipsoid reduces to a single point, the method coincides with Capon’s method. We describe in detail several methods that can be used to derive an appropriate uncertainty ellipsoid for the array response. We form separate uncertainty ellipsoids for each component in the signal path (e.g., antenna, electronics) and then determine an aggregate uncertainty ellipsoid from these. We give new results for modeling the elementwise products of ellipsoids. We demonstrate the robust beamforming and the ellipsoidal modeling methods with several numerical examples. Index Terms—Ellipsoidal calculus, Hadamard product, robust beamforming, secondorder cone programming.
FIR Filter Design via Spectral Factorization and Convex Optimization
, 1997
"... We consider the design of finite impulse response (FIR) filters subject to upper and lower bounds on the frequency response magnitude. The associated optimization problems, with the filter coefficients as the variables and the frequency response bounds as constraints, are in general nonconvex. Usin ..."
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Cited by 46 (6 self)
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We consider the design of finite impulse response (FIR) filters subject to upper and lower bounds on the frequency response magnitude. The associated optimization problems, with the filter coefficients as the variables and the frequency response bounds as constraints, are in general nonconvex. Using a change of variables and spectral factorization, we can pose such problems as linear or nonlinear convex optimization problems. As a result we can solve them efficiently (and globally) by recently developed interiorpoint methods. We describe applications to filter and equalizer design, and the related problem of antenna array weight design.
Optimal Downlink Beamforming Using Semidefinite Optimization
, 1999
"... When using antenna arrays at the base station of a cellular system, one critical aspect is the transmission strategy. An optimal choice of beamformers for simultaneous transmission to several cochannel users must be solved jointly for all users and base stations in an area. We formulate an optimal ..."
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Cited by 40 (4 self)
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When using antenna arrays at the base station of a cellular system, one critical aspect is the transmission strategy. An optimal choice of beamformers for simultaneous transmission to several cochannel users must be solved jointly for all users and base stations in an area. We formulate an optimal transmit strategy and show how the solution can be calculated efficiently using interior point methods for semidefinite optimization. The algorithm minimizes the total transmitted power under certain constraints to guarantee a specific quality of service. The method provides large flexibility in the choice of constraints and can be extended to be robust to channel perturbations.
Disciplined convex programming
 Global Optimization: From Theory to Implementation, Nonconvex Optimization and Its Application Series
, 2006
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Using LOQO To Solve SecondOrder Cone Programming Problems
 PRINCETON UNIVERSITY
, 1998
"... Many nonlinear optimization problems can be cast as secondorder cone programming problems. In this paper, we discuss a broad spectrum of such applications. For each application, we consider various formulations, some convex some not, and study which ones are amenable to solution using a generalpur ..."
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Cited by 13 (1 self)
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Many nonlinear optimization problems can be cast as secondorder cone programming problems. In this paper, we discuss a broad spectrum of such applications. For each application, we consider various formulations, some convex some not, and study which ones are amenable to solution using a generalpurpose interiorpoint solver LOQO. We also compare with other commonly available nonlinear programming solvers and specialpurpose codes for secondorder cone programming.
Design Of FarField And NearField Broadband Beamformers Using Eigenfilters
, 2003
"... This paper discusses two novel noniterative design procedures based on eigenfilters for designing broadband beamformers with an arbitrary spatial directivity pattern for an arbitrary microphone configuration. In the conventional eigenfilter technique a reference frequencyangle point is required, w ..."
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Cited by 10 (2 self)
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This paper discusses two novel noniterative design procedures based on eigenfilters for designing broadband beamformers with an arbitrary spatial directivity pattern for an arbitrary microphone configuration. In the conventional eigenfilter technique a reference frequencyangle point is required, whereas in the eigenfilter technique based on a TLS (Total Least Squares) error criterion, no reference point is required. It is shown how to design broadband beamformers in the farfield, nearfield and mixed nearfield farfield of the microphone array. Both eigenfilter techniques are compared with other broadband beamformer design procedures (leastsquares, maximum energy array, nonlinear criterion). It will be shown by simulations that among the considered noniterative design procedures the TLS eigenfilter technique has the best performance, i.e. best resembling the performance of the nonlinear design procedure but having a significantly lower computational complexity.
Beamforming with uncertain weights
 IEEE Signal Processing Letters
, 2007
"... Abstract—In this letter, we show that worstcase robust beamforming, with uncertain weights subject to multiplicative variations, can be cast as a convex optimization problem. We interpret this problem as a weighted complex Iregularization of the nominal beamforming problem, and show that it can be ..."
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Cited by 6 (3 self)
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Abstract—In this letter, we show that worstcase robust beamforming, with uncertain weights subject to multiplicative variations, can be cast as a convex optimization problem. We interpret this problem as a weighted complex Iregularization of the nominal beamforming problem, and show that it can be solved with the same computational complexity as nominal beamforming, ignoring the variations. We derive a simple lower bound on how much worse the robust beamformer will be compared to the nominal beamformer solution with no weight uncertainty. We demonstrate the robust approach with a simple narrowband beamformer. Index Terms—Regularization, robust beamforming, robust optimization, robust sensor array signal processing. I. BEAMFORMING WE consider an array of sensor elements. Let be the array response to a wave of unit amplitude
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 6 (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