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115
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 282 (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.
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.
What Is the Value of Limited Feedback for MIMO Channels?
, 2004
"... Feedbackinacommunicationssystemcan enablethetransmittertoexploitchannelcondi  tionsandavoidinterference.Inthecaseofa multipleinputmultipleoutputchannel,feedback canbeusedtospecifyaprecodingmatrixatthe transmitter,whichactivatesthestrongestchan  nelmodes.Insituationswherethefeedbackis severelylim ..."
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Cited by 105 (22 self)
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Feedbackinacommunicationssystemcan enablethetransmittertoexploitchannelcondi  tionsandavoidinterference.Inthecaseofa multipleinputmultipleoutputchannel,feedback canbeusedtospecifyaprecodingmatrixatthe transmitter,whichactivatesthestrongestchan  nelmodes.Insituationswherethefeedbackis severelylimited,importantissuesarehowto quantizetheinformationneededatthetransmitterandhowmuchimprovementinassociated performancecanbeobtainedasafunctionof theamountoffeedbackavailable.Wegivean overviewofsomerecentworkinthisarea.Meth  odsarepresentedforconstructingasetofpossibleprecodingmatrices, fromwhichaparticular choicecanberelayedtothetransmitter.Perfor  manceresultsshowthatevenafewbitsoffeedbackcanprovideperformanceclosetothatwith fullchannelknowledgeatthetransmitter.
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 80 (5 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.
Channel capacity and beamforming for multiple transmit and receive antennas with covariance feedback
, 2001
"... Abstract—We consider the capacity of a narrowband point to point communication system employing multipleelement antenna arrays at both the transmitter and the receiver with covariance feedback. Under covariance feedback the receiver is assumed to have perfect Channel State Information (CSI) while a ..."
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Cited by 62 (5 self)
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Abstract—We consider the capacity of a narrowband point to point communication system employing multipleelement antenna arrays at both the transmitter and the receiver with covariance feedback. Under covariance feedback the receiver is assumed to have perfect Channel State Information (CSI) while at the transmitter the channel matrix is modeled as consisting of zero mean complex jointly Gaussian random variables with known covariances. Specifically we assume a channel matrix with i.i.d. rows and correlated columns, a common model for downlink transmission. We determine the optimal transmit precoding strategy to maximize the Shannon capacity of such a system. We also derive closed form necessary and sufficient conditions on the spatial covariance for when the maximum capacity is achieved by beamforming. The conditions for optimality of beamforming agree with the notion of waterfilling over multiple degrees of freedom. I.
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 60 (14 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.
Limited feedback unitary precoding for spatial multiplexing systems
 IEEE Trans. Info. Theory
, 2005
"... Abstract—Multipleinput multipleoutput (MIMO) wireless systems use antenna arrays at both the transmitter and receiver to provide communication links with substantial diversity and capacity. Spatial multiplexing is a common space–time modulation technique for MIMO communication systems where indepe ..."
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Cited by 54 (13 self)
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Abstract—Multipleinput multipleoutput (MIMO) wireless systems use antenna arrays at both the transmitter and receiver to provide communication links with substantial diversity and capacity. Spatial multiplexing is a common space–time modulation technique for MIMO communication systems where independent information streams are sent over different transmit antennas. Unfortunately, spatial multiplexing is sensitive to illconditioning of the channel matrix. Precoding can improve the resilience of spatial multiplexing at the expense of full channel knowledge at the transmitter—which is often not realistic. This correspondence proposes a quantized precoding system where the optimal precoder is chosen from a finite codebook known to both receiver and transmitter. The index of the optimal precoder is conveyed from the receiver to the transmitter over a lowdelay feedback link. Criteria are presented for selecting the optimal precoding matrix based on the error rate and mutual information for different receiver designs. Codebook design criteria are proposed for each selection criterion by minimizing a bound on the average distortion assuming a Rayleighfading matrix channel. The design criteria are shown to be equivalent to packing subspaces in the Grassmann manifold using the projection twonorm and Fubini–Study distances. Simulation results showthat the proposed system outperforms antenna subset selection and performs close to optimal unitary precoding with a minimal amount of feedback. Index Terms—Diversity methods, Grassmannian subspace packing, multipleinput multipleoutput (MIMO) systems, quantized precoding, Rayleigh channels, spatial multiplexing, vertical Bell Labs layered space– time (VBLAST) architecture. I.
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 52 (3 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.
An overview of limited feedback in wireless communication systems
 IEEE J. SEL. AREAS COMMUN
, 2008
"... It is now well known that employing channel adaptive signaling in wireless communication systems can yield large improvements in almost any performance metric. Unfortunately, many kinds of channel adaptive techniques have been deemed impractical in the past because of the problem of obtaining channe ..."
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Cited by 46 (8 self)
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It is now well known that employing channel adaptive signaling in wireless communication systems can yield large improvements in almost any performance metric. Unfortunately, many kinds of channel adaptive techniques have been deemed impractical in the past because of the problem of obtaining channel knowledge at the transmitter. The transmitter in many systems (such as those using frequency division duplexing) can not leverage techniques such as training to obtain channel state information. Over the last few years, research has repeatedly shown that allowing the receiver to send a small number of information bits about the channel conditions to the transmitter can allow near optimal channel adaptation. These practical systems, which are commonly referred to as limited or finiterate feedback systems, supply benefits nearly identical to unrealizable perfect transmitter channel knowledge systems when they are judiciously designed. In this tutorial, we provide a broad look at the field of limited feedback wireless communications. We review work in systems using various combinations of single antenna, multiple antenna, narrowband, broadband, singleuser, and multiuser technology. We also provide a synopsis of the role of limited feedback in the standardization of next generation wireless systems.