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251
Duality, achievable rates, and sumrate capacity of Gaussian MIMO broadcast channels
 IEEE TRANS. INFORM. THEORY
, 2003
"... We consider a multiuser multipleinput multipleoutput (MIMO) Gaussian broadcast channel (BC), where the transmitter and receivers have multiple antennas. Since the MIMO BC is in general a nondegraded BC, its capacity region remains an unsolved problem. In this paper, we establish a duality between ..."
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Cited by 210 (19 self)
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We consider a multiuser multipleinput multipleoutput (MIMO) Gaussian broadcast channel (BC), where the transmitter and receivers have multiple antennas. Since the MIMO BC is in general a nondegraded BC, its capacity region remains an unsolved problem. In this paper, we establish a duality between what is termed the “dirty paper” achievable region (the Caire–Shamai achievable region) for the MIMO BC and the capacity region of the MIMO multipleaccess channel (MAC), which is easy to compute. Using this duality, we greatly reduce the computational complexity required for obtaining the dirty paper achievable region for the MIMO BC. We also show that the dirty paper achievable region achieves the sumrate capacity of the MIMO BC by establishing that the maximum sum rate of this region equals an upper bound on the sum rate of the MIMO BC.
Sum Capacity of a Gaussian Vector Broadcast Channel
 IEEE Trans. Inform. Theory
, 2002
"... This paper characterizes the sum capacity of a class of nondegraded Gaussian vectB broadcast channels where a singletransmitter with multiple transmit terminals sends independent information to multiple receivers. Coordinat+[ is allowed among the transmit teminals, but not among the different recei ..."
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Cited by 193 (21 self)
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This paper characterizes the sum capacity of a class of nondegraded Gaussian vectB broadcast channels where a singletransmitter with multiple transmit terminals sends independent information to multiple receivers. Coordinat+[ is allowed among the transmit teminals, but not among the different receivers. The sum capacity is shown t be a saddlepoint of a Gaussian mu al informat]R game, where a signal player chooses a tansmit covariance matrix to maximize the mutual information, and a noise player chooses a fictitious noise correlation to minimize the mutual information. This result holds fort he class of Gaussian channels whose saddlepoint satisfies a full rank condition. Furt her,t he sum capacity is achieved using a precoding method for Gaussian channels with additive side information noncausally known at the transmitter. The optimal precoding structure is shown t correspond to a decisionfeedback equalizer that decomposes t e broadcast channel into a series of singleuser channels with intk ference presubtract] at the transmiter.
On the capacity of MIMO broadcast channel with partial side information
 IEEE Trans. Inform. Theory
, 2005
"... Abstract—In multipleantenna broadcast channels, unlike pointtopoint multipleantenna channels, the multiuser capacity depends heavily on whether the transmitter knows the channel coefficients to each user. For instance, in a Gaussian broadcast channel with transmit antennas and singleantenna use ..."
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Cited by 173 (6 self)
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Abstract—In multipleantenna broadcast channels, unlike pointtopoint multipleantenna channels, the multiuser capacity depends heavily on whether the transmitter knows the channel coefficients to each user. For instance, in a Gaussian broadcast channel with transmit antennas and singleantenna users, the sum rate capacity scales like log log for large if perfect channel state information (CSI) is available at the transmitter, yet only logarithmically with if it is not. In systems with large, obtaining full CSI from all users may not be feasible. Since lack of CSI does not lead to multiuser gains, it is therefore of interest to investigate transmission schemes that employ only partial CSI. In this paper, we propose a scheme that constructs random beams and that transmits information to the users with the highest signaltonoiseplusinterference ratios (SINRs), which can be made available to the transmitter with very little feedback. For fixed and increasing, the throughput of our scheme scales as log log, where is the number of receive antennas of each user. This is precisely the same scaling obtained with perfect CSI using dirty paper coding. We furthermore show that a linear increase in throughput with can be obtained provided that does not not grow faster than log. We also study the fairness of our scheduling in a heterogeneous network and show that, when is large enough, the system becomes interference dominated and the probability of transmitting to any user converges to 1, irrespective of its path loss. In fact, using = log transmit antennas emerges as a desirable operating point, both in terms of providing linear scaling of the throughput with as well as in guaranteeing fairness. Index Terms—Broadcast channel, channel state information (CSI), multiuser diversity, wireless communications. I.
The capacity region of the Gaussian multipleinput multipleoutput broadcast channel
 IEEE Trans. Inf. Theory
, 2006
"... (MIMO) broadcast channel (BC) is considered. The dirtypaper coding (DPC) rate region is shown to coincide with the capacity region. To that end, a new notion of an enhanced broadcast channel is introduced and is used jointly with the entropy power inequality, to show that a superposition of Gaussia ..."
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Cited by 156 (3 self)
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(MIMO) broadcast channel (BC) is considered. The dirtypaper coding (DPC) rate region is shown to coincide with the capacity region. To that end, a new notion of an enhanced broadcast channel is introduced and is used jointly with the entropy power inequality, to show that a superposition of Gaussian codes is optimal for the degraded vector broadcast channel and that DPC is optimal for the nondegraded case. Furthermore, the capacity region is characterized under a wide range of input constraints, accounting, as special cases, for the total power and the perantenna power constraints. Index Terms—Broadcast channel, capacity region, dirtypaper coding (DPC), enhanced channel, entropy power inequality, Minkowski’s inequality, multipleantenna. I.
Zeroforcing methods for downlink spatial multiplexing in multiuser MIMO channels
 IEEE Trans. Signal Processing
, 2004
"... Abstract—The use of spacedivision multiple access (SDMA) in the downlink of a multiuser multipleinput, multipleoutput (MIMO) wireless communications network can provide a substantial gain in system throughput. The challenge in such multiuser systems is designing transmit vectors while considering ..."
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Cited by 117 (4 self)
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Abstract—The use of spacedivision multiple access (SDMA) in the downlink of a multiuser multipleinput, multipleoutput (MIMO) wireless communications network can provide a substantial gain in system throughput. The challenge in such multiuser systems is designing transmit vectors while considering the cochannel interference of other users. Typical optimization problems of interest include the capacity problem—maximizing the sum information rate subject to a power constraint—or the power control problem—minimizing transmitted power such that a certain qualityofservice metric for each user is met. Neither of these problems possess closedform solutions for the general multiuser MIMO channel, but the imposition of certain constraints can lead to closedform solutions. This paper presents two such constrained solutions. The first, referred to as “blockdiagonalization,” is a generalization of channel inversion when there are multiple antennas at each receiver. It is easily adapted to optimize for either maximum transmission rate or minimum power and approaches the optimal solution at high SNR. The second, known as “successive optimization, ” is an alternative method for solving the power minimization problem one user at a time, and it yields superior results in some (e.g., low SNR) situations. Both of these algorithms are limited to cases where the transmitter has more antennas than all receive antennas combined. In order to accommodate more general scenarios, we also propose a framework for coordinated transmitterreceiver processing that generalizes the two algorithms to cases involving more receive than transmit antennas. While the proposed algorithms are suboptimal, they lead to simpler transmitter and receiver structures and allow for a reasonable tradeoff between performance and complexity. Index Terms—Antenna arrays, array signal processing, MIMO systems, signal design, space division multiaccess (SDMA), wireless LAN. I.
On the optimality of multiantenna broadcast scheduling using zeroforcing beamforming
 IEEE J. SELECT. AREAS COMMUN
, 2006
"... Although the capacity of multipleinput/multipleoutput (MIMO) broadcast channels (BCs) can be achieved by dirty paper coding (DPC), it is difficult to implement in practical systems. This paper investigates if, for a large number of users, simpler schemes can achieve the same performance. Specifica ..."
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Cited by 116 (5 self)
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Although the capacity of multipleinput/multipleoutput (MIMO) broadcast channels (BCs) can be achieved by dirty paper coding (DPC), it is difficult to implement in practical systems. This paper investigates if, for a large number of users, simpler schemes can achieve the same performance. Specifically, we show that a zeroforcing beamforming (ZFBF) strategy, while generally suboptimal, can achieve the same asymptotic sum capacity as that of DPC, as the number of users goes to infinity. In proving this asymptotic result, we provide an algorithm for determining which users should be active under ZFBF. These users are semiorthogonal to one another and can be grouped for simultaneous transmission to enhance the throughput of scheduling algorithms. Based on the user grouping, we propose and compare two fair scheduling schemes in roundrobin ZFBF and proportionalfair ZFBF. We provide numerical results to confirm the optimality of ZFBF and to compare the performance of ZFBF and proposed fair scheduling schemes with that of various MIMO BC strategies.
MIMO broadcast channels with finite rate feedback
 IEEE Trans. on Inform. Theory
, 2006
"... Multiple transmit antennas in a downlink channel can provide tremendous capacity (i.e. multiplexing) gains, even when receivers have only single antennas. However, receiver and transmitter channel state information is generally required. In this paper, a system where each receiver has perfect channe ..."
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Cited by 94 (10 self)
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Multiple transmit antennas in a downlink channel can provide tremendous capacity (i.e. multiplexing) gains, even when receivers have only single antennas. However, receiver and transmitter channel state information is generally required. In this paper, a system where each receiver has perfect channel knowledge, but the transmitter only receives quantized information regarding the channel instantiation is analyzed. The well known zero forcing transmission technique is considered, and simple expressions for the throughput degradation due to finite rate feedback are derived. A key finding is that the feedback rate per mobile must be increased linearly with the SNR (in dB) in order to achieve the full multiplexing gain, which is in sharp contrast to pointtopoint MIMO systems in which it is not necessary to increase the feedback rate as a function of the SNR. I.
Sum power iterative waterfilling for multiantenna Gaussian broadcast channels
 IEEE Trans. Inform. Theory
, 2005
"... In this paper we consider the problem of maximizing sum rate of a multipleantenna Gaussian broadcast channel. It was recently found that dirty paper coding is capacity achieving for this channel. In order to achieve capacity, the optimal transmission policy (i.e. the optimal transmit covariance str ..."
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Cited by 82 (16 self)
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In this paper we consider the problem of maximizing sum rate of a multipleantenna Gaussian broadcast channel. It was recently found that dirty paper coding is capacity achieving for this channel. In order to achieve capacity, the optimal transmission policy (i.e. the optimal transmit covariance structure) given the channel conditions and power constraint must be found. However, obtaining the optimal transmission policy when employing dirty paper coding is a computationally complex nonconvex problem. We use duality to transform this problem into a wellstructured convex multipleaccess channel problem. We exploit the structure of this problem and derive simple and fast iterative algorithms that provide the optimum transmission policies for the multipleaccess channel, which can easily be mapped to the optimal broadcast channel policies.
Precoding in MultiAntenna and MultiUser Communications
"... In this paper, TomlinsonHarashima precoding for multipleinput/multipleoutput systems including multipleantenna and multiuser systems is studied. It is shown that nonlinear preequalization offers significant advantages over linear preequalization which increases average transmit power. Moreover ..."
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Cited by 51 (1 self)
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In this paper, TomlinsonHarashima precoding for multipleinput/multipleoutput systems including multipleantenna and multiuser systems is studied. It is shown that nonlinear preequalization offers significant advantages over linear preequalization which increases average transmit power. Moreover, it outperforms decisionfeedback equalization at the receiver side which is applicable if joint processing at the receiver side is possible, and which suffers from error propagation. A number of aspects of practical importance are studied. Loading, i.e., the optimum distribution of transmit power and rate is discussed in detail. It is shown that the capacity of the underlying MIMO channel can be utilized asymptotically by means of nonlinear precoding.
Transmitter Optimization for the MultiAntenna Downlink with PerAntenna Power Constraints
 IEEE Transactions on Signal Processing
, 2007
"... Abstract—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, bec ..."
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Cited by 50 (5 self)
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Abstract—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. Index Terms—Beamforming, broadcast channel, capacity region, dirtypaper coding, Lagrangian duality. I.