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54
Capacity Limits of MIMO Channels
 IEEE J. SELECT. AREAS COMMUN
, 2003
"... We provide an overview of the extensive recent results on the Shannon capacity of singleuser and multiuser multipleinput multipleoutput (MIMO) channels. Although enormous capacity gains have been predicted for such channels, these predictions are based on somewhat unrealistic assumptions about t ..."
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Cited by 341 (13 self)
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We provide an overview of the extensive recent results on the Shannon capacity of singleuser and multiuser multipleinput multipleoutput (MIMO) channels. Although enormous capacity gains have been predicted for such channels, these predictions are based on somewhat unrealistic assumptions about the underlying timevarying channel model and how well it can be tracked at the receiver, as well as at the transmitter. More realistic assumptions can dramatically impact the potential capacity gains of MIMO techniques. For timevarying MIMO channels there are multiple Shannon theoretic capacity definitions and, for each definition, different correlation models and channel information assumptions that we consider. We first provide a comprehensive summary of ergodic and capacity versus outage results for singleuser MIMO channels. These results indicate that the capacity gain obtained from multiple antennas heavily depends
Transmitter Optimization and Optimality of Beamforming for Multiple Antenna Systems with Imperfect Feedback
"... We solve the transmitter optimization problem and determine a necessary and sucient condition under which beamforming achieves Shannon capacity in a narrowband point to point communication system employing multiple transmit and receive antennas. We assume perfect channel state information at the rec ..."
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Cited by 111 (6 self)
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We solve the transmitter optimization problem and determine a necessary and sucient condition under which beamforming achieves Shannon capacity in a narrowband point to point communication system employing multiple transmit and receive antennas. We assume perfect channel state information at the receiver (CSIR) and imperfect channel state feedback from the receiver to the transmitter. We consider the cases of mean and covariance feedback. The channel is modeled at the transmitter as a matrix of complex jointly Gaussian random variables with either a zero mean and a known covariance matrix (covariance feedback), or a nonzero mean and a white covariance matrix (mean feedback). For both cases we develop a necessary and sucient condition for when the Shannon capacity is achieved through beamforming, i.e. the channel can be treated like a scalar channel and onedimensional codes can be used to achieve capacity. We also provide a waterpouring interpretation of our results and nd that less channel uncertainty not only increases the system capacity but may also allow this higher capacity to be achieved with scalar codes which involves signi cantly less complexity in practice than vector coding.
HighSNR power offset in multiantenna communication
 IEEE Transactions on Information Theory
, 2005
"... Abstract—The analysis of the multipleantenna capacity in the high regime has hitherto focused on the high slope (or maximum multiplexing gain), which quantifies the multiplicative increase as a function of the number of antennas. This traditional characterization is unable to assess the impact of ..."
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Cited by 79 (17 self)
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Abstract—The analysis of the multipleantenna capacity in the high regime has hitherto focused on the high slope (or maximum multiplexing gain), which quantifies the multiplicative increase as a function of the number of antennas. This traditional characterization is unable to assess the impact of prominent channel features since, for a majority of channels, the slope equals the minimum of the number of transmit and receive antennas. Furthermore, a characterization based solely on the slope captures only the scaling but it has no notion of the power required for a certain capacity. This paper advocates a more refined characterization whereby, as a function of �f, the high capacity is expanded as an affine function where the impact of channel features such as antenna correlation, unfaded components, etc., resides in the zeroorder term or power offset. The power offset, for which we find insightful closedform expressions, is shown to play a chief role for levels of practical interest. Index Terms—Antenna correlation, channel capacity, coherent communication, fading channels, high analysis, multiantenna arrays, Ricean channels.
Impact of antenna correlation on the capacity of multiantenna channels
 IEEE TRANS. INFORM. THEORY
, 2005
"... This paper applies random matrix theory to obtain analytical characterizations of the capacity of correlated multiantenna channels. The analysis is not restricted to the popular separable correlation model, but rather it embraces a more general representation that subsumes most of the channel model ..."
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Cited by 76 (4 self)
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This paper applies random matrix theory to obtain analytical characterizations of the capacity of correlated multiantenna channels. The analysis is not restricted to the popular separable correlation model, but rather it embraces a more general representation that subsumes most of the channel models that have been treated in the literature. For arbitrary signaltonoise ratios @ A, the characterization is conducted in the regime of large numbers of antennas. For the low and high regions, in turn, we uncover compact capacity expansions that are valid for arbitrary numbers of antennas and that shed insight on how antenna correlation impacts the tradeoffs among power, bandwidth, and rate.
MIMO wireless linear precoding
 IEEE Signal Processing Magazine
, 2006
"... The benefits of using multiple antennas at both the transmitter and the receiver in a wireless system are well established. Multipleinput multipleoutput (MIMO) systems enable a growth in transmission rate linear in the minimum of the number of antennas at either end [1][2]. MIMO techniques also en ..."
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Cited by 40 (0 self)
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The benefits of using multiple antennas at both the transmitter and the receiver in a wireless system are well established. Multipleinput multipleoutput (MIMO) systems enable a growth in transmission rate linear in the minimum of the number of antennas at either end [1][2]. MIMO techniques also enhance link reliability and
Optimal linear precoders for MIMO wireless correlated channels with nonzero mean in spacetime coded systems
 IEEE TRANS. SIGNAL PROCESSING
, 2006
"... This paper proposes linear precoder designs exploiting statistical channel knowledge at the transmitter in a multipleinput multipleoutput (MIMO) wireless system. The paper focuses on channel statistics, since obtaining realtime channel state information at the transmitter can be difficult due to ..."
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Cited by 27 (4 self)
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This paper proposes linear precoder designs exploiting statistical channel knowledge at the transmitter in a multipleinput multipleoutput (MIMO) wireless system. The paper focuses on channel statistics, since obtaining realtime channel state information at the transmitter can be difficult due to channel dynamics. The considered channel statistics consist of the channel mean and transmit antenna correlation. The receiver is assumed to know the instantaneous channel precisely. The precoder operates along with a space–time block code (STBC) and aims to minimize the Chernoff bound on the pairwise error probability (PEP) between a pair of block codewords, averaged over channel fading statistics. Two PEP design criteria are studied—minimum distance and average distance. The optimal precoder with an orthogonal STBC is established, using a convex optimization framework. Different relaxations then extend the solution to systems with nonorthogonal STBCs. In both cases, the precoder is a function of both the channel mean and the transmit correlation. A linear precoder acts as a combination of a multimode beamformer and an input shaping matrix, matching each side to the channel and to the input signal structure, respectively. Both the optimal beam direction and the power of each mode, obtained via a dynamic waterfilling process, depend on the signaltonoise ratio (SNR). Asymptotic analyses of the results reveal that, for all STBCs, the precoder approaches a singlemode beamformer on the dominant right singular vector of the channel mean as the channel factor increases. On the other hand, as the SNR increases, it approaches an equipower multiplemode beamformer, matched to the eigenvectors of the transmit correlation. Design examples and numerical simulation results for both orthogonal and nonorthogonal STBC precoding solutions are provided, illustrating the precoding array gain.
Optimum power allocation for singleuser MIMO and multiuser MIMOMAC with partial CSI
 IEEE Journal on Selected Areas in Communications
, 2007
"... Abstract — We consider both the singleuser and the multiuser power allocation problems in MIMO systems, where the receiver side has the perfect channel state information (CSI), and the transmitter side has partial CSI, which is in the form of covariance feedback. In a singleuser MIMO system, we co ..."
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Cited by 22 (4 self)
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Abstract — We consider both the singleuser and the multiuser power allocation problems in MIMO systems, where the receiver side has the perfect channel state information (CSI), and the transmitter side has partial CSI, which is in the form of covariance feedback. In a singleuser MIMO system, we consider an iterative algorithm that solves for the eigenvalues of the optimum transmit covariance matrix that maximizes the rate. The algorithm is based on enforcing the KarushKuhnTucker (KKT) optimality conditions of the optimization problem at each iteration. We prove that this algorithm converges to the unique global optimum power allocation when initiated at an arbitrary point. We, then, consider the multiuser generalization of the problem, which is to find the eigenvalues of the optimum transmit covariance matrices of all users that maximize the sum rate of the MIMO multiple access channel (MIMOMAC). For this problem, we propose an algorithm that finds the unique optimum power allocation policies of all users. At a given iteration, the multiuser algorithm updates the power allocation of one user, given the power allocations of the rest of the users, and iterates over all users in a roundrobin fashion. Finally, we make several suggestions that significantly improve the convergence rate of the proposed algorithms. Index Terms — Multiuser MIMO, MIMO multiple access channel, partial CSI, covariance feedback, optimum power allocation.
Opportunistic space division multiple access with beam selection
 IEEE TRANS. ON COMMUNICATIONS
, 2006
"... In this paper, a novel transmission technique for the multipleinput multipleoutput (MIMO) broadcast channel is proposed that allows simultaneous transmission to multiple users with limited feedback from each user. During a training phase, the base station modulates a training sequence on multiple ..."
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Cited by 17 (13 self)
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In this paper, a novel transmission technique for the multipleinput multipleoutput (MIMO) broadcast channel is proposed that allows simultaneous transmission to multiple users with limited feedback from each user. During a training phase, the base station modulates a training sequence on multiple sets of randomly chosen orthogonal beamforming vectors. Each user sends the index of the best beamforming vector and the corresponding signaltointerfenceplusnoise ratio for that set of orthogonal vectors back to the base station. The base station opportunistically determines the users and corresponding orthogonal vectors that maximize the sum capacity. Based on the capacity expressions, the optimal amount of training to maximize the sum capacity is derived as a function of the system parameters. The main advantage of the proposed system is that it provides throughput gains for the MIMO broadcast channel with a small feedback overhead, and provides these gains even with a small number of active users. Numerical simulations show that a 20 % gain in sum capacity is achieved (for a small number of users) over conventional opportunistic space division multiple access, and a 100 % gain (for a large number of users) over conventional opportunistic beamforming when the number of transmit antennas is four.
Optimality of beamforming in fading MIMO multiple access channels
 IEEE Transactions on Communications
, 2008
"... Abstract—We consider the sum capacity of a multiinput multioutput (MIMO) multiple access channel (MAC) where the receiver has the perfect channel state information (CSI), while the transmitters have either no or partial CSI. When the transmitters have partial CSI, it is in the form of either the c ..."
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Cited by 15 (4 self)
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Abstract—We consider the sum capacity of a multiinput multioutput (MIMO) multiple access channel (MAC) where the receiver has the perfect channel state information (CSI), while the transmitters have either no or partial CSI. When the transmitters have partial CSI, it is in the form of either the covariance matrix of the channel or the mean matrix of the channel. For the covariance feedback case, we mainly consider physical models that result in singlesided correlation structures. For the mean feedback case, we consider physical models that result in inphase received signals. Under these assumptions, we analyze the MIMOMAC from three different viewpoints. First, we consider a finitesized system. We show that the optimum transmit directions of each user are the eigenvectors of its own channel covariance and mean feedback matrices, in the covariance and mean feedback models, respectively. Also, we find the conditions under which beamforming is optimal for all users. Second, in the covariance feedback case, we prove that the region where beamforming is optimal for all users gets larger with the addition of new users into the system. In the mean feedback case, we show through simulations that this is not necessarily true. Third, we consider the asymptotic case where the number of users is large. We show that in both no and partial CSI cases, beamforming is asymptotically optimal. In particular, in the case of no CSI, we show that a simple form of beamforming, which may be characterized as an arbitrary antenna selection scheme, achieves the sum capacity. In the case of partial CSI, we show that beamforming in the direction of the strongest eigenvector of the channel feedback matrix achieves the sum capacity. Finally, we generalize our covariance feedback results to doublesided correlation structures in the Appendix. Index Terms—Multiuser MIMO, MIMO multiple access channel, partial CSI, covariance feedback, mean feedback, optimality of beamforming, large system analysis. I.
Optimal transmit covariance for MIMO channels with statistical transmitter side information
 IEEE Intern. Symp. Inf. Theory
, 2005
"... Abstract — We give an optimality condition for the input covariance for arbitrary ergodic Gaussian vector channels under the condition that the channel gains are independent of the transmit signal, the transmitter has knowledge of the channel gain probability law and the receiver has knowledge of ea ..."
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Cited by 9 (2 self)
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Abstract — We give an optimality condition for the input covariance for arbitrary ergodic Gaussian vector channels under the condition that the channel gains are independent of the transmit signal, the transmitter has knowledge of the channel gain probability law and the receiver has knowledge of each channel realization. Using this optimality condition, we find an iterative algorithm for numerical computation of optimal input covariance matrices. I.