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77
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 222 (11 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
On Beamforming with Finite Rate Feedback in Multiple Antenna Systems
, 2003
"... In this paper, we study a multiple antenna system where the transmitter is equipped with quantized information about instantaneous channel realizations. Assuming that the transmitter uses the quantized information for beamforming, we derive a universal lower bound on the outage probability for any f ..."
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Cited by 185 (13 self)
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In this paper, we study a multiple antenna system where the transmitter is equipped with quantized information about instantaneous channel realizations. Assuming that the transmitter uses the quantized information for beamforming, we derive a universal lower bound on the outage probability for any finite set of beamformers. The universal lower bound provides a concise characterization of the gain with each additional bit of feedback information regarding the channel. Using the bound, it is shown that finite information systems approach the perfect information case as (t 1)2 , where B is the number of feedback bits and t is the number of transmit antennas. The geometrical bounding technique, used in the proof of the lower bound, also leads to a design criterion for good beamformers, whose outage performance approaches the lower bound. The design criterion minimizes the maximum inner product between any two beamforming vectors in the beamformer codebook, and is equivalent to the problem of designing unitary space time codes under certain conditions. Finally, we show that good beamformers are good packings of 2dimensional subspaces in a 2tdimensional real Grassmannian manifold with chordal distance as the metric.
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 174 (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.
MIMO Channel Modelling and the Principle of Maximum Entropy
, 2004
"... In this paper , we devise theoretical grounds for constructing channel models for Multiinput Multioutput (MIMO) systems based on information theoretic tools. The paper provides a general method to derive a channel model which is consistent with one's state of knowledge. The framework we give her ..."
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Cited by 43 (23 self)
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In this paper , we devise theoretical grounds for constructing channel models for Multiinput Multioutput (MIMO) systems based on information theoretic tools. The paper provides a general method to derive a channel model which is consistent with one's state of knowledge. The framework we give here has already been fruitfully explored with success in the context of Bayesian spectrum analysis and parameter estimation. For each channel model, we conduct an asymptotic analysis (in the number of antennas) of the achievable transmission rate using tools from random matrix theory. A central limit theorem is provided on the asymptotic behavior of the mutual information and validated in the finite case by simulations. The results are both useful in terms of designing a system based on criteria such as quality of service and in optimizing transmissions in multiuser networks .
Dirtypaper coding versus TDMA for MIMO broadcast channels
 IEEE Trans. Inf. Theory
, 2005
"... Abstract—We compare the capacity of dirtypaper coding (DPC)to that of timedivision multiple access (TDMA)for a multipleantenna (multipleinput multipleoutput (MIMO)) Gaussian broadcast channel (BC). We find that the sumrate capacity (achievable using DPC)of the multipleantenna BC is at most ��� ..."
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Cited by 42 (3 self)
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Abstract—We compare the capacity of dirtypaper coding (DPC)to that of timedivision multiple access (TDMA)for a multipleantenna (multipleinput multipleoutput (MIMO)) Gaussian broadcast channel (BC). We find that the sumrate capacity (achievable using DPC)of the multipleantenna BC is at most ��� @ A times the largest singleuser capacity (i.e., the TDMA sumrate)in the system, where is the number of transmit antennas and is the number of receivers. This result is independent of the number of receive antennas and the channel gain matrix, and is valid at all signaltonoise ratios (SNRs). We investigate the tightness of this bound in a timevarying channel (assuming perfect channel knowledge at receivers and transmitters)where the channel experiences uncorrelated Rayleigh fading and in some situations we find that the dirty paper gain is upperbounded by the ratio of transmittoreceive antennas. We also show that ��� @ A upperbounds the sumrate gain of successive decoding over TDMA for the uplink channel, where is the number of receive antennas at the base station and is the number of transmitters. Index Terms—Broadcast channel (BC), channel capacity, dirtypaper coding (DPC), multipleinput multipleoutput (MIMO) systems, timedivision multiple access (TDMA). I.
A comparison of timesharing, DPC, and beamforming for MIMO broadcast channels with many users
 IEEE Trans. Commun
, 2007
"... In this paper, we derive the scaling laws of the sum rate for fading MIMO Gaussian broadcast channels using timesharing to the strongest user, dirty paper coding (DPC), and beamforming when the number of users (receivers) n is large. Throughout the paper, we assume a fix average transmit power and ..."
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Cited by 32 (0 self)
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In this paper, we derive the scaling laws of the sum rate for fading MIMO Gaussian broadcast channels using timesharing to the strongest user, dirty paper coding (DPC), and beamforming when the number of users (receivers) n is large. Throughout the paper, we assume a fix average transmit power and consider a block fading Rayleigh channel. First, we show that for a system with M transmit antennas and users equipped with N antennas, the sum rate scales like M log log nN for DPC and beamforming when M is fixed and for any N (either growing to infinity or not). On the other hand, when both M and N are fixed, the sum rate of timesharing to the strongest user scales like min(M, N) log log n. Therefore, the asymptotic gain of DPC over timesharing for the sum rate is M min(M,N) when M and N are fixed. It is also shown that if M grows as log n, the sum rate of DPC and beamforming will grow linearly in M, but with different constant multiplicative factors. In this region, the sum rate capacity of timesharing scales like N log log n.
Outage Mutual Information of SpaceTime MIMO Channels
 IEEE Trans. Inform. Theory
, 2004
"... We present analytical expressions for the probability density function (PDF) of the random mutual information between transmitted and received vector signals of a random spacetime independent and identically distributed (i.i.d.) multipleinput multipleoutput (MIMO) channel, assuming that the tra ..."
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Cited by 24 (0 self)
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We present analytical expressions for the probability density function (PDF) of the random mutual information between transmitted and received vector signals of a random spacetime independent and identically distributed (i.i.d.) multipleinput multipleoutput (MIMO) channel, assuming that the transmitted signals from the multiple antennas are Gaussian i.i.d.. We show that this PDF can be well approximated by a Gaussian distribution, and such a Gaussian approximation is based on expressions for the given PDF's mean and variance that we derive. We prove that at high SNR, every 3 dB increase in signal to noise ratio (SNR) leads to an increase in outage rate approximately equal to min(M,N ), where M and N denote the number of transmit and receiveantennas, respectively. A simple expression for the moment generating function of the mutual information PDF is also provided, based on which we establish normality of the PDF, when both M and N are large, and the SNR is large.
From Single user to Multiuser Communications: Shifting the MIMO paradigm
 IEEE Sig. Proc. Magazine
, 2007
"... In multiuser MIMO networks, the spatial degrees of freedom offered by multiple antennas can be advantageously exploited to enhance the system capacity, by scheduling multiple users to simultaneously share the spatial channel. This entails a fundamental paradigm shift from single user communications, ..."
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Cited by 24 (8 self)
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In multiuser MIMO networks, the spatial degrees of freedom offered by multiple antennas can be advantageously exploited to enhance the system capacity, by scheduling multiple users to simultaneously share the spatial channel. This entails a fundamental paradigm shift from single user communications, since multiuser systems can experience substantial benefit from channel state information at the transmitter and, at the same time, require more complex scheduling strategies and transceiver methodologies. This paper reviews multiuser MIMO communication from an algorithmic perspective, discussing performance gains, tradeoffs, and practical considerations. Several approaches including nonlinear and linear channelaware precoding are reviewed, along with more practical limited feedback schemes that require only partial channel state information. The interaction between precoding and scheduling is discussed. Several promising strategies for limited multiuser feedback design are looked at, some of which are inspired from the single user MIMO precoding scenario while others are fully specific to the multiuser setting. 1 DRAFT
Exploiting multiuser diversity with only 1–bit feedback
 in Proc. IEEE Wireless Commun. and Networking Conf
, 2005
"... Abstract — In a system with n users, the sumrate capacity of the downlink channel grows as log log n, assuming optimal scheduling. However, optimal scheduling requires that the downlink channel state information (CSI) for all users be fully available at the base station. In this paper we show that ..."
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Cited by 17 (2 self)
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Abstract — In a system with n users, the sumrate capacity of the downlink channel grows as log log n, assuming optimal scheduling. However, optimal scheduling requires that the downlink channel state information (CSI) for all users be fully available at the base station. In this paper we show that the same capacity growth holds even if the feedback rate from the mobiles to the base station is reduced to one bit. We propose a simple scheduling method to achieve this multiuser capacity and furthermore we show that by a judicious choice of the onebit quantizer, not only the growth rate, but also most of the capacity of a fully informed system can be preserved. I.
Opportunistic downlink transmission with limited feedback
 IEEE TRANS. INFORM. THEORY
, 2007
"... Opportunistic scheduling provides attractive sumrate capacities in a multiuser network when the basestation has transmitside channel state information (CSI), which is often estimated at the mobiles and provided to the base station via a feedback channel. This correspondence investigates opportun ..."
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Cited by 15 (1 self)
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Opportunistic scheduling provides attractive sumrate capacities in a multiuser network when the basestation has transmitside channel state information (CSI), which is often estimated at the mobiles and provided to the base station via a feedback channel. This correspondence investigates opportunistic methods in the presence of limited feedback. For flat Rayleighfading channels, strategies with only onebit feedback per user are demonstrated that capture the doublelogarithmic capacity growth (with number of users) of fullCSI systems. Furthermore, for a given system configuration, it is shown that if the onebit feedback is chosen judiciously, there is little to be gained by increasing the feedback rate. Our results provide optimal methods of calculating the onebit feedback, as well as expressions for the sumrate capacity in the onebit feedback regime. It is shown that one may achieve proportional fairness of scheduling in this regime with no loss of throughput. For OFDM multiuser systems, the motivation for limited feedback is even more pronounced. An extension of the onebit technique is presented for subchannel/user selection under both correlated and uncorrelated subchannel conditions, and optimal growth in capacity is demonstrated.