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134
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 208 (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 171 (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.
A VectorPerturbation technique for NearCapacity . . .
 IEEE TRANS. COMMUN
, 2005
"... Recent theoretical results describing the sum capacity when using multiple antennas to communicate with multiple users in a known rich scattering environment have not yet been followed with practical transmission schemes that achieve this capacity. We introduce a simple encoding algorithm that achi ..."
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Cited by 130 (3 self)
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Recent theoretical results describing the sum capacity when using multiple antennas to communicate with multiple users in a known rich scattering environment have not yet been followed with practical transmission schemes that achieve this capacity. We introduce a simple encoding algorithm that achieves nearcapacity at sum rates of tens of bits/channel use. The algorithm is a variation on channel inversion that regularizes the inverse and uses a “sphere encoder ” to perturb the data to reduce the power of the transmitted signal. This paper is comprised of two parts. In this first part, we show that while the sum capacity grows linearly with the minimum of the number of antennas and users, the sum rate of channel inversion does not. This poor performance is due to the large spread in the singular values of the channel matrix. We introduce regularization to improve the condition of the inverse and maximize the signaltointerferenceplusnoise ratio at the receivers. Regularization enables linear growth and works especially well at low signaltonoise ratios (SNRs), but as we show in the second part, an additional step is needed to achieve nearcapacity performance at all SNRs.
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 115 (4 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 93 (9 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.
On the duality of Gaussian multipleaccess and broadcast channels
 IEEE Trans. Inf. Theory
, 2004
"... Abstract—We define a duality between Gaussian multipleaccess channels (MACs) and Gaussian broadcast channels (BCs). The dual channels we consider have the same channel gains and the same noise power at all receivers. We show that the capacity region of the BC (both constant and fading) can be writt ..."
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Cited by 70 (13 self)
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Abstract—We define a duality between Gaussian multipleaccess channels (MACs) and Gaussian broadcast channels (BCs). The dual channels we consider have the same channel gains and the same noise power at all receivers. We show that the capacity region of the BC (both constant and fading) can be written in terms of the capacity region of the dual MAC, and vice versa. We can use this result to find the capacity region of the MAC if the capacity region of only the BC is known, and vice versa. For fading channels we show duality under ergodic capacity, but duality also holds for different capacity definitions for fading channels such as outage capacity and minimumrate capacity. Using duality, many results known for only one of the two channels can be extended to the dual channel as well. Index Terms—Broadcast channel (BC), channel capacity, duality, fading channels, multipleinput multipleoutput (MIMO) systems, multipleaccess channel (MAC). I.
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 50 (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
MIMO Broadcast Channels With FiniteRate Feedback
, 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 correspondence, a system where each receiver has per ..."
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Cited by 45 (1 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 correspondence, a system where each receiver has perfect channel knowledge, but the transmitter only receives quantized information regarding the channel instantiation is analyzed. The wellknown zeroforcing transmission technique is considered, and simple expressions for the throughput degradation due to finiterate feedback are derived. A key finding is that the feedback rate per mobile must be increased linearly with the signaltonoise ratio (SNR) (in decibels) in order to achieve the full multiplexing gain. This is in sharp contrast to pointtopoint multipleinput multipleoutput (MIMO) systems, in which it is not necessary to increase the feedback rate as a function of the SNR.