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40
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 351 (14 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 340 (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.
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 236 (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.
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 113 (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.
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 76 (6 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.
Design and analysis of transmitbeamforming based on limitedrate feedback
, 2006
"... This paper deals with design and performance analysis of transmit beamformers for multipleinput multipleoutput (MIMO) systems based on bandwidthlimited information that is fed back from the receiver to the transmitter. By casting the design of transmit beamforming based on limitedrate feedback ..."
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Cited by 69 (1 self)
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This paper deals with design and performance analysis of transmit beamformers for multipleinput multipleoutput (MIMO) systems based on bandwidthlimited information that is fed back from the receiver to the transmitter. By casting the design of transmit beamforming based on limitedrate feedback as an equivalent sphere vector quantization (SVQ) problem, multiantenna beamformed transmissions through independent and identically distributed (i.i.d.) Rayleigh fading channels are first considered. The ratedistortion function of the vector source is upperbounded, and the operational ratedistortion performance achieved by the generalized Lloyd’s algorithm is lowerbounded. Although different in nature, the two bounds yield asymptotically equivalent performance analysis results. The average signaltonoise ratio (SNR) performance is also quantified. Finally, beamformer codebook designs are studied for correlated Rayleigh fading channels, and a lowcomplexity codebook design that achieves nearoptimal performance is derived.
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 61 (8 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.
How Accurate Channel Prediction needs to be for TransmitBeamforming with Adaptive Modulation over Rayleigh MIMO Channels?
, 2004
"... Adaptive modulation improves the system throughput considerably by matching transmitter parameters to timevarying wireless fading channels. Crucial to adaptive modulation is the quality of channel state information (CSI) at the transmitter. In this paper, we first present a channel predictor based o ..."
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Cited by 51 (3 self)
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Adaptive modulation improves the system throughput considerably by matching transmitter parameters to timevarying wireless fading channels. Crucial to adaptive modulation is the quality of channel state information (CSI) at the transmitter. In this paper, we first present a channel predictor based on pilot symbol assisted modulation (PSAM) for multiinput multioutput (MIMO) Rayleigh fading channels. We then analyze the impact of the channel prediction error on the bit error rate (BER) performance of a transmitbeamformer with adaptive modulation that treats the predicted channels as perfect. Our numerical results reveal the critical value of the normalized prediction error, below which the predicted channels can be treated as perfect by the adaptive modulator; otherwise, explicit consideration of the channel imperfection must be accounted for at the transmitter.
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 18 (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.