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58
Linear precoding via conic optimization for fixed mimo receivers
 IEEE Trans. on Signal Processing
, 2006
"... We consider the problem of designing linear precoders for fixed multiple input multiple output (MIMO) receivers. Two different design criteria are considered. In the first, we minimize the transmitted power subject to signal to interference plus noise ratio (SINR) constraints. In the second, we maxi ..."
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Cited by 154 (3 self)
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We consider the problem of designing linear precoders for fixed multiple input multiple output (MIMO) receivers. Two different design criteria are considered. In the first, we minimize the transmitted power subject to signal to interference plus noise ratio (SINR) constraints. In the second, we maximize the worst case SINR subject to a power constraint. We show that both problems can be solved using standard conic optimization packages. In addition, we develop conditions for the optimal precoder for both of these problems, and propose two simple fixed point iterations to find the solutions which satisfy these conditions. The relation to the well known downlink uplink duality in the context of joint downlink beamforming and power control is also explored. Our precoder design is general, and as a special case it solves the beamforming problem. In contrast to most of the existing precoders, it is not limited to full rank systems. Simulation results in a multiuser system show that the resulting precoders can significantly outperform existing linear precoders. 1
Transmitter Optimization for the MultiAntenna Downlink with PerAntenna Power Constraints
 IEEE TRANSACTIONS ON SIGNAL PROCESSING
, 2007
"... This paper considers the transmitter optimization problem for a multiuser downlink channel with multiple transmit antennas at the basestation. In contrast to the conventional sumpower constraint on the transmit antennas, this paper adopts a more realistic perantenna power constraint, because in ..."
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Cited by 135 (7 self)
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This paper considers the transmitter optimization problem for a multiuser downlink channel with multiple transmit antennas at the basestation. In contrast to the conventional sumpower constraint on the transmit antennas, this paper adopts a more realistic perantenna power constraint, because in practical implementations each antenna is equipped with its own power amplifier and is limited individually by the linearity of the amplifier. Assuming perfect channel knowledge at the transmitter, this paper investigates two different transmission schemes under the perantenna power constraint: a minimumpower beamforming design for downlink channels with a single antenna at each remote user and a capacityachieving transmitter design for downlink channels with multiple antennas at each remote user. It is shown that in both cases, the perantenna downlink transmitter optimization problem may be transformed into a dual uplink problem with an uncertain noise. This generalizes previous uplink–downlink duality results and transforms the perantenna transmitter optimization problem into an equivalent minimax optimization problem. Further, it is shown that various notions of uplink–downlink duality may be unified under a Lagrangian duality framework. This new interpretation of duality gives rise to efficient numerical optimization techniques for solving the downlink perantenna transmitter optimization problem.
Exploiting multiantennas for opportunistic spectrum sharing in cognitive radio networks
 IEEE J. Select. Topics in Signal Processing
, 2008
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Optimum linear joint transmitreceive processing for MIMO channels with QoS constraints
 IEEE Transactions on Signal Processing
, 2004
"... Abstract—This paper considers vector communications through multipleinput multipleoutput (MIMO) channels with a set of quality of service (QoS) requirements for the simultaneously established substreams. Linear transmitreceive processing (also termed linear precoder at the transmitter and linear ..."
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Cited by 56 (7 self)
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Abstract—This paper considers vector communications through multipleinput multipleoutput (MIMO) channels with a set of quality of service (QoS) requirements for the simultaneously established substreams. Linear transmitreceive processing (also termed linear precoder at the transmitter and linear equalizer at the receiver) is designed to satisfy the QoS constraints with minimum transmitted power (the exact conditions under which the problem becomes unfeasible are given). Although the original problem is a complicated nonconvex problem with matrixvalued variables, with the aid of majorization theory, we reformulate it as a simple convex optimization problem with scalar variables. We then propose a practical and efficient multilevel waterfilling algorithm to optimally solve the problem for the general case of different QoS requirements. The optimal transmitreceive processing is shown to diagonalize the channel matrix only after a very specific prerotation of the data symbols. For situations in which the resulting transmit power is too large, we give the precise way to relax the QoS constraints in order to reduce the required power based on a perturbation analysis. We also propose a robust design under channel estimation errors that has an important interest for practical systems. Numerical results from simulations are given to support the mathematical development of the problem. Index Terms—Array signal processing, beamforming, joint transmitreceive equalization, linear precoding, MIMO channels, spacetime filtering, waterfilling. I.
Zero Forcing Precoding and Generalized Inverses
"... We consider the problem of linear zero forcing precoding design, and discuss its relation to the theory of generalized inverses in linear algebra. Special attention is given to a specific generalized inverse known as the pseudoinverse. We begin with the standard design under the assumption of a tot ..."
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Cited by 52 (0 self)
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We consider the problem of linear zero forcing precoding design, and discuss its relation to the theory of generalized inverses in linear algebra. Special attention is given to a specific generalized inverse known as the pseudoinverse. We begin with the standard design under the assumption of a total power constraint and prove that precoders based on the pseudoinverse are optimal in this setting. Then, we proceed to examine individual perantenna power constraints. In this case, the pseudoinverse is not necessarily the optimal generalized inverse. In fact, finding the optimal inverse is nontrivial and depends on the specific performance measure. We address two common criteria, fairness and throughput, and show that the optimal matrices may be found using standard convex optimization methods. We demonstrate the improved performance offered by our approach using computer simulations.
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 50 (5 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.
Convex conic formulations of robust downlink precoder designs with quality of service constraints
 IEEE J. Select. Topics Signal Processing
, 2007
"... We consider the design of linear precoders (beamformers) for broadcast channels with Quality of Service (QoS) constraints for each user, in scenarios with uncertain channel state information (CSI) at the transmitter. We consider a deterministicallybounded model for the channel uncertainty of each u ..."
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Cited by 35 (2 self)
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We consider the design of linear precoders (beamformers) for broadcast channels with Quality of Service (QoS) constraints for each user, in scenarios with uncertain channel state information (CSI) at the transmitter. We consider a deterministicallybounded model for the channel uncertainty of each user, and our goal is to design a robust precoder that minimizes the total transmission power required to satisfy the users ’ QoS constraints for all channels within a specified uncertainty region around the transmitter’s estimate of each user’s channel. Since this problem is not known to be computationally tractable, we will derive three conservative design approaches that yield convex and computationallyefficient restrictions of the original design problem. The three approaches yield semidefinite program (SDP) formulations that offer different tradeoffs between the degree of conservatism and the size of the SDP. We will also show how these conservative approaches can be used to derive efficientlysolvable quasiconvex restrictions of some related design problems, including the robust counterpart to the problem of maximizing the minimum signaltointerferenceplusnoiseratio (SINR) subject to a given power constraint. Our simulation results indicate that in the presence of uncertain CSI the proposed approaches can satisfy the users ’ QoS requirements for a significantly larger set of uncertainties than existing methods, and require less transmission power to do so.
Joint beamforming and scheduling for a multiantenna downlink with imperfect transmitter channel knowledge
 IEEE J. Select. Areas Commun
, 2007
"... We consider the downlink of a wireless system where the basestation has M ≥ 1 antennas and K user terminals have one antenna each. We study the weighted rate sum maximization in the case of nonperfect Channel State Information at the Transmitter (CSIT). Some relevant downlink optimization problems ..."
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Cited by 22 (4 self)
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We consider the downlink of a wireless system where the basestation has M ≥ 1 antennas and K user terminals have one antenna each. We study the weighted rate sum maximization in the case of nonperfect Channel State Information at the Transmitter (CSIT). Some relevant downlink optimization problems, such as the stabilization of the transmission queues under random packet arrivals and the proportional fair scheduling for infinite backlogged systems, can be solved as special cases of the proposed problem. We restrict the transmitter strategy to be based on Gaussian coding and beamforming. Even under this simplifying condition, the problem at hand is nonconvex and it does not appear to lend itself to a simple algorithmic solution. Therefore, we introduce some approximations that yield a definition of signaltointerference plus noise ratio (SINR) commonly used in the classical arrayprocessing/beamforming literature. For the simpler (but still nonconvex) approximated problem, we propose a powerful heuristic solution based on greedy user selection and a gradient iteration that converges to a local maximum of the objective function. This method yields very competitive results with relatively low computational complexity. Extensive simulations show that, in the case of perfect CSIT, the proposed heuristic scheme performs very closely to the optimal (dirtypaper coding) strategy while, in the case of nonperfect CSIT, it significantly outperforms previously proposed suboptimal approaches, such as random beamforming and approximated zeroforcing with greedy user selection.
Robust downlink beamforming in multiuser MISO cognitive radio networks with imperfect channelstate information
 IEEE Trans. Veh. Tech
, 2010
"... This paper studies the problem of robust downlink beamforming design in a multiuser MultiInput SingleOutput (MISO) Cognitive Radio Network (CRNet) in which multiple Primary Users (PUs) coexist with multiple Secondary Users (SUs). Unlike conventional designs in CRNets, in this paper it is assumed ..."
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Cited by 13 (2 self)
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This paper studies the problem of robust downlink beamforming design in a multiuser MultiInput SingleOutput (MISO) Cognitive Radio Network (CRNet) in which multiple Primary Users (PUs) coexist with multiple Secondary Users (SUs). Unlike conventional designs in CRNets, in this paper it is assumed that the Channel State Information (CSI) for all relevant channels is imperfectly known, and the imperfectness of the CSI is modeled using an Euclidean ballshaped uncertainty set. Our design objective is to minimize the transmit power of the SUTransmitter (SUTx) while simultaneously targeting a lower bound on the received SignaltoInterferenceplusNoiseRatio (SINR) for the SU’s, and imposing an upper limit on the InterferencePower (IP) at the PUs. The design parameters at the SUTx are the beamforming weights, i.e. the precoder matrix. The proposed methodology is based on a worst case design scenario through which the performance metrics of the design are immune to variations in the channels. We propose three approaches based on convex programming for which efficient numerical solutions exist. Finally, simulation results are provided to validate the robustness of the proposed methods. Index Terms Robust beamforming, cognitive radio network, multiuser MISO communication, worst case design, imperfect CSI
Optimal distributed beamforming for MISO interference channels
 IEEE Trans. Signal Process
, 2011
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