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487
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 409 (17 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
Spectral Efficiency in the Wideband Regime
, 2002
"... The tradeoff of spectral efficiency (b/s/Hz) versus energy perinformation bit is the key measure of channel capacity in the wideband powerlimited regime. This paper finds the fundamental bandwidthpower tradeoff of a general class of channels in the wideband regime characterized by low, but nonz ..."
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Cited by 393 (29 self)
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The tradeoff of spectral efficiency (b/s/Hz) versus energy perinformation bit is the key measure of channel capacity in the wideband powerlimited regime. This paper finds the fundamental bandwidthpower tradeoff of a general class of channels in the wideband regime characterized by low, but nonzero, spectral efficiency and energy per bit close to the minimum value required for reliable communication. A new criterion for optimality of signaling in the wideband regime is proposed, which, in contrast to the traditional criterion, is meaningful for finitebandwidth communication.
Grassmannian beamforming for multipleinput multipleoutput wireless systems
 IEEE TRANS. INFORM. THEORY
, 2003
"... Transmit beamforming and receive combining are simple methods for exploiting the significant diversity that is available in multipleinput and multipleoutput (MIMO) wireless systems. Unfortunately, optimal performance requires either complete channel knowledge or knowledge of the optimal beamformi ..."
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Cited by 329 (39 self)
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Transmit beamforming and receive combining are simple methods for exploiting the significant diversity that is available in multipleinput and multipleoutput (MIMO) wireless systems. Unfortunately, optimal performance requires either complete channel knowledge or knowledge of the optimal beamforming vector which are not always realizable in practice. In this correspondence, a quantized maximum signaltonoise ratio (SNR) beamforming technique is proposed where the receiver only sends the label of the best beamforming vector in a predetermined codebook to the transmitter. By using the distribution of the optimal beamforming vector in independent identically distributed Rayleigh fading matrix channels, the codebook design problem is solved and related to the problem of Grassmannian line packing. The proposed design criterion is flexible enough to allow for side constraints on the codebook vectors. Bounds on the codebook size are derived to guarantee full diversity order. Results on the density of Grassmannian line packings are derived and used to develop bounds on the codebook size given a capacity or SNR loss. Monte Carlo simulations are presented that compare the probability of error for different quantization strategies.
Unitary SpaceTime Modulation for MultipleAntenna Communications in Rayleigh Flat Fading
 IEEE Trans. Inform. Theory
, 1998
"... Motivated by informationtheoretic considerations, we propose a signalling scheme, unitary spacetime modulation, for multipleantenna communication links. This modulation is ideally suited for Rayleigh fastfading environments, since it does not require the receiver to know or learn the propagation ..."
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Cited by 307 (19 self)
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Motivated by informationtheoretic considerations, we propose a signalling scheme, unitary spacetime modulation, for multipleantenna communication links. This modulation is ideally suited for Rayleigh fastfading environments, since it does not require the receiver to know or learn the propagation coefficients. Unitary spacetime modulation uses constellations of T \cross M spacetime signals {\Phi_l, l= 1,...L},where T represents the coherence interval during which the fading is approximately constant, and M > M . We design some multipleantenna signal constellations and simulate their effectiveness as measured by bit error probability with maximum likelihood decoding. We demonstrate that two antennas have a 6dB diversity gain over one antenna at 15db SNR.
Communication on the Grassmann Manifold: A Geometric Approach to the Noncoherent MultipleAntenna Channel
 IEEE Trans. Inform. Theory
, 2002
"... In this paper, we study the capacity of multipleantenna fading channels. We focus on the scenario where the fading coefficients vary quickly; thus an accurate estimation of the coefficients is generally not available to either the transmitter or the receiver. We use a noncoherent block fading model ..."
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Cited by 271 (8 self)
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In this paper, we study the capacity of multipleantenna fading channels. We focus on the scenario where the fading coefficients vary quickly; thus an accurate estimation of the coefficients is generally not available to either the transmitter or the receiver. We use a noncoherent block fading model proposed by Marzetta and Hochwald. The model does not assume any channel side information at the receiver or at the transmitter, but assumes that the coefficients remain constant for a coherence interval of length symbol periods. We compute the asymptotic capacity of this channel at high signaltonoise ratio (SNR) in terms of the coherence time , the number of transmit antennas , and the number of receive antennas . While the capacity gain of the coherent multiple antenna channel is min bits per second per hertz for every 3dB increase in SNR, the corresponding gain for the noncoherent channel turns out to be (1 ) bits per second per herz, where = min 2 . The capacity expression has a geometric interpretation as sphere packing in the Grassmann manifold.
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 271 (14 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.
Differential spacetime modulation
 IEEE Trans. Inform. Theory
, 2000
"... Abstract—Space–time coding and modulation exploit the presence of multiple transmit antennas to improve performance on multipath radio channels. Thus far, most work on space–time coding has assumed that perfect channel estimates are available at the receiver. In certain situations, however, it may b ..."
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Cited by 262 (1 self)
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Abstract—Space–time coding and modulation exploit the presence of multiple transmit antennas to improve performance on multipath radio channels. Thus far, most work on space–time coding has assumed that perfect channel estimates are available at the receiver. In certain situations, however, it may be difficult or costly to estimate the channel accurately, in which case it is natural to consider the design of modulation techniques that do not require channel estimates at the transmitter or receiver. We propose a general approach to differential modulation for multiple transmit antennas based on group codes. This approach can be applied to any number of transmit and receive antennas, and any signal constellation. We also derive lowcomplexity differential receivers, error bounds, and modulator design criteria, which we use to construct optimal differential modulation schemes for two transmit antennas. These schemes can be demodulated with or without channel estimates. This permits the receiver to exploit channel estimates when they are available. Performance degrades by approximately 3 dB when estimates are not available. Index Terms—Differential modulation, group codes, multipath channels, noncoherent communication, space–time coding, transmit diversity. I.
Distributed spacetime coding in wireless relay networks
, 2006
"... We apply the idea of spacetime coding devised for multipleantenna systems to the problem of communications over a wireless relay network with Rayleigh fading channels. We use a twostage protocol, where in one stage the transmitter sends information and in the other, the relays encode their rece ..."
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Cited by 223 (15 self)
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We apply the idea of spacetime coding devised for multipleantenna systems to the problem of communications over a wireless relay network with Rayleigh fading channels. We use a twostage protocol, where in one stage the transmitter sends information and in the other, the relays encode their received signals into a “distributed ” linear dispersion (LD) code, and then transmit the coded signals to the receive node. We show that for high SNR, the pairwise error probability (PEP) behaves as (log P/P) min{T,R} , with T the coherence interval, that is, the number of symbol periods during which the channels keep constant, R the number of relay nodes, and P the total transmit power. Thus, apart from the log P factor, the system has the same diversity as a multipleantenna system with R transmit antennas, which is the same as assuming that the R relays can fully cooperate and have full knowledge of the transmitted signal. We further show that for a network with a large number of relays and a fixed total transmit power across the entire network, the optimal power allocation is for the transmitter to expend half the power and for the relays to collectively expend the other half. We also show that at low and high SNR, the coding gain is the same as that of a multipleantenna system with R antennas. However, at intermediate SNR, it can be quite different, which has implications for the design of distributed spacetime codes.