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28
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 216 (10 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
From theory to practice: an overview of MIMO spacetime coded wireless systems
 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS
, 2003
"... This paper presents an overview of recent progress in the area of multipleinput–multipleoutput (MIMO) space–time coded wireless systems. After some background on the research leading to the discovery of the enormous potential of MIMO wireless links, we highlight the different classes of technique ..."
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Cited by 199 (5 self)
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This paper presents an overview of recent progress in the area of multipleinput–multipleoutput (MIMO) space–time coded wireless systems. After some background on the research leading to the discovery of the enormous potential of MIMO wireless links, we highlight the different classes of techniques and algorithms proposed which attempt to realize the various benefits of MIMO including spatial multiplexing and space–time coding schemes. These algorithms are often derived and analyzed under ideal independent fading conditions. We present the state of the art in channel modeling and measurements, leading to a better understanding of actual MIMO gains. Finally, the paper addresses current questions regarding the integration of MIMO links in practical wireless systems and standards.
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 59 (13 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 51 (2 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.
Quantifying the Power Loss when Transmit Beamforming Relies on Finite Rate Feedback
, 2003
"... Transmit beamforming achieves optimal performance in systems with multiple transmitantennas and a single receiveantenna, from both the capacity and the received signalto noise ratio (SNR) perspectives, but ideally requires perfect channel knowledge at the transmitter. In practical systems where t ..."
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Cited by 35 (7 self)
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Transmit beamforming achieves optimal performance in systems with multiple transmitantennas and a single receiveantenna, from both the capacity and the received signalto noise ratio (SNR) perspectives, but ideally requires perfect channel knowledge at the transmitter. In practical systems where the feedback link can only convey a finite number of bits, transmitbeamformer designs have been extensively investigated using either the outage probability, or the average SNR, as the figure of merit. In this paper, we study the symbol error rate (SER) for transmit beamforming with finiterate feedback, in a multiinput singleoutput (MISO) setting. We derive a SER lower bound, which is tight for good beamformer designs. Comparing this bound with the SER corresponding to the ideal case, we quantify the power loss due to the finite rate constraint, across the entire SNR range.
Capacityachieving input covariance for singleuser multiantenna channels
 IEEE Trans. Wireless Commun
, 2006
"... Abstract — We characterize the capacityachieving input covariance for multiantenna channels known instantaneously at the receiver and in distribution at the transmitter. Our characterization, valid for arbitrary numbers of antennas, encompasses both the eigenvectors and the eigenvalues. The eigenv ..."
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Cited by 24 (9 self)
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Abstract — We characterize the capacityachieving input covariance for multiantenna channels known instantaneously at the receiver and in distribution at the transmitter. Our characterization, valid for arbitrary numbers of antennas, encompasses both the eigenvectors and the eigenvalues. The eigenvectors are found for zeromean channels with arbitrary fading profiles and a wide range of correlation and keyhole structures. For the eigenvalues, in turn, we present necessary and sufficient conditions as well as an iterative algorithm that exhibits remarkable properties: universal applicability, robustness and rapid convergence. In addition, we identify channel structures for which an isotropic input achieves capacity. Index Terms — Capacity, MIMO, input optimization, fading, antenna correlation, Ricean fading, keyhole channel.
Spacetime communication for OFDM with implicit channel feedback
 IEEE Trans. Inf. Theory
, 2004
"... Abstract—We consider wideband communication (e.g., using orthogonal frequencydivision multiplexed (OFDM) systems) over a typical cellular “downlink, ” in which both the base station and the mobile may have multiple antennas, but the number of antennas at the mobile is assumed to be small. Implicit ..."
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Cited by 17 (4 self)
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Abstract—We consider wideband communication (e.g., using orthogonal frequencydivision multiplexed (OFDM) systems) over a typical cellular “downlink, ” in which both the base station and the mobile may have multiple antennas, but the number of antennas at the mobile is assumed to be small. Implicit channel feedback can play a powerful role in such systems, especially for outdoor channels, which typically exhibit narrow spatial spreads. A summary of our findings is as follows. a) Implicit channel feedback regarding the covariance matrix for the downlink space–time channel can be obtained, without any power or bandwidth overhead, by suitably averaging uplink channel measurements across frequency. Since this approach relies on statistical reciprocity, it applies to both timedivision duplex (TDD) and frequencydivision duplex (FDD) systems. Using such covariance feedback yields significantly better performance at lower complexity than conventional space–time or space–frequency codes, which do not employ feedback. b) We provide guidelines for optimizing antenna spacing in systems with covariance feedback. Theoretical investigation of a hypothetical system with completely controllable channel eigenvalues shows that the optimal number of channel eigenmodes is roughly matched to the (small) number of receive antenna elements. Thus, while antenna elements in conventional systems without feedback should be spaced far apart in order to ensure uncorrelated responses, the optimal antenna spacing with covariance feedback is much smaller, thereby concentrating the channel energy into a small number of eigenmodes. Index Terms—Diversity methods, fading channels, feedback communication, information rates, multipleinput multipleoutput (MIMO) systems. I.
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 14 (10 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.
Capacity of a multipleantenna fading channel with a quantized precoding matrix
 IEEE Trans. Inf. Theory
, 2009
"... channel, feedback from the receiver can be used to specify a transmit precoding matrix, which selectively activates the strongest channel modes. Here we analyze the performance of Random Vector Quantization (RVQ), in which the precoding matrix is selected from a random codebook containing independen ..."
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Cited by 11 (6 self)
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channel, feedback from the receiver can be used to specify a transmit precoding matrix, which selectively activates the strongest channel modes. Here we analyze the performance of Random Vector Quantization (RVQ), in which the precoding matrix is selected from a random codebook containing independent, isotropically distributed entries. We assume that channel elements are i.i.d. and known to the receiver, which relays the optimal (ratemaximizing) precoder codebook index to the transmitter using B bits. We first derive the large system capacity of beamforming (rankone precoding matrix) as a function of B, where large system refers to the limit as B and the number of transmit and receive antennas all go to infinity with fixed ratios. RVQ for beamforming is asymptotically optimal, i.e., no other quantization scheme can achieve a larger asymptotic rate. We subsequently consider a precoding matrix with arbitrary rank, and approximate the asymptotic RVQ performance with optimal and linear receivers (matched filter and Minimum Mean Squared Error (MMSE)). Numerical examples show that these approximations accurately predict the performance of finitesize systems of interest. Given a target spectral efficiency, numerical examples show that the amount of feedback required by the linear MMSE receiver is only slightly more than that required by the optimal receiver, whereas the matched filter can require significantly more feedback. Index Terms—Beamforming, large system analysis, limited feedback, MultiInput MultiOutput (MIMO), precoding, vector quantization. I.
Capacity of Beamforming with Limited Training and Feedback
"... We examine the capacity of beamforming over a MultiInput/SingleOutput block Rayleigh fading channel with finite training for channel estimation and limited feedback. A fixedlength packet is assumed, which is spanned by ¢ training symbols, £ feedback bits, and the data symbols. The training symb ..."
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Cited by 10 (5 self)
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We examine the capacity of beamforming over a MultiInput/SingleOutput block Rayleigh fading channel with finite training for channel estimation and limited feedback. A fixedlength packet is assumed, which is spanned by ¢ training symbols, £ feedback bits, and the data symbols. The training symbols are used to obtain a Minimum Mean Squared Error (MMSE) estimate of the channel vector. Given this estimate, the receiver selects a transmit beamforming vector from a codebook containing ¤¦ ¥ i.i.d. random vectors, and relays the corresponding £ bits back to the transmitter. We derive bounds on the capacity and show that for a large number of transmit antennas §© ¨ , the optimal ¢ and £ , which maximize the bounds, are approximately equal and both increase as §�¨������¦��§© ¨. We conclude that with limited training and feedback, the optimal number of antennas to activate also increases as § ¨ �����¦�� § ¨.