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62
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 205 (41 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.
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 35 (8 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.
Systematic Codebook Designs for Quantized Beamforming in Correlated MIMO
 Channels,” IEEE Journ. Sel. Areas in Commun
, 2007
"... Abstract — The full diversity gain provided by a multiantenna channel can be achieved by transmit beamforming and receive combining. This requires the knowledge of channel state information (CSI) at the transmitter which is difficult to obtain in practice. Quantized beamforming where fixed codebook ..."
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Cited by 32 (13 self)
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Abstract — The full diversity gain provided by a multiantenna channel can be achieved by transmit beamforming and receive combining. This requires the knowledge of channel state information (CSI) at the transmitter which is difficult to obtain in practice. Quantized beamforming where fixed codebooks known at both the transmitter and the receiver are used to quantize the CSI has been proposed to solve this problem. Most recent works focus attention on limited feedback codebook design for the uncorrelated Rayleigh fading channel. Such designs are suboptimal when used in correlated channels. In this paper, we propose systematic codebook design for correlated channels when channel statistical information is known at the transmitter. This design is motivated by studying the performance of pure statistical beamforming in correlated channels and is implemented by maps that can rotate and scale spherical caps on the Grassmannian manifold. Based on this study, we show that even statistical beamforming is nearoptimal if the transmitter covariance matrix is illconditioned and receiver covariance matrix is wellconditioned. This leads to a partitioning of the transmit and receive covariance spaces based on their conditioning with variable feedback requirements to achieve an operational performance level in the different partitions. When channel statistics are difficult to obtain at the transmitter, we propose a universal codebook design (also implemented by the rotationscaling maps) that is robust to channel statistics. Numerical studies show that even few bits of feedback, when applied with our designs, lead to near perfect CSI performance in a variety of correlated channel conditions. Index Terms — Diversity methods, fading channels, Grassmannian line packing, limited feedback, MIMO systems, quantization
Bit interleaved coded multiple beamforming
 IEEE Trans. Commun
"... Abstract — This paper addresses the performance of bitinterleaved coded multiple beamforming (BICMB) with imperfect knowledge of beamforming vectors. Various wireless standards become equivalent to BICMB when they are operated in beamforming mode. In BICMB, the invariance of the precoding matrix und ..."
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Cited by 23 (12 self)
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Abstract — This paper addresses the performance of bitinterleaved coded multiple beamforming (BICMB) with imperfect knowledge of beamforming vectors. Various wireless standards become equivalent to BICMB when they are operated in beamforming mode. In BICMB, the invariance of the precoding matrix under an arbitrary unitary transform widely studied in the literature is not applicable. On the other hand, the optimum precoder and detector are not unique because of invariance under a diagonal unitary transform. We propose an optimal Euclidean distortion measure and a new linear detector. In addition, a new codebook design is proposed via the generalized Lloyd algorithm based on the new distortion measure. We provide simulation results demonstrating the performance improvement achieved with the proposed distortion measure and the linear detector. I.
Compression of Feedback for Adaptive Transmission and Scheduling
, 2007
"... In multiuser wireless radio systems, it may be possible to increase throughput by reducing the feedbackrate of channel quality data while meeting quality of service requirements. By Thomas Eriksson and Tony Ottosson ABSTRACT  For wireless systems with adaptive modulation and/or scheduling, feedbac ..."
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Cited by 23 (6 self)
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In multiuser wireless radio systems, it may be possible to increase throughput by reducing the feedbackrate of channel quality data while meeting quality of service requirements. By Thomas Eriksson and Tony Ottosson ABSTRACT  For wireless systems with adaptive modulation and/or scheduling, feedback of channel quality information is often necessary. It has been questioned whether the increased system performance is worth the additional feedback rate and the increased algorithm complexity. In this paper, we study how the feedback rate can be minimized, without losing the gains due to adaptive modulation and multiuser diversity. We present an indepth study of the literature in the area, and evaluate the performance of several stateoftheart channel quality feedback schemes. By illustrating the compromise between system throughput and feedback channel rate for various schemes, we are able to give valuable insight in choice of method for feedback rate reduction. A major conclusion is that for multicarrier systems, a lossy compression scheme is the best choice, while for singlecarrier systems, schemes limiting feedback to only highSNR users show good performance. Another conclusion is that there are still many issues to study before the schemes can be used in practice.
Optimization of Training and Feedback Overhead for Beamforming over Block Fading Channels
, 2009
"... We examine the capacity of beamforming over a singleuser, multiantenna link taking into account the overhead due to channel estimation and limited feedback of channel state information. Multiinput singleoutput (MISO) and multiinput multioutput (MIMO) channels are considered subject to block Ra ..."
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Cited by 17 (0 self)
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We examine the capacity of beamforming over a singleuser, multiantenna link taking into account the overhead due to channel estimation and limited feedback of channel state information. Multiinput singleoutput (MISO) and multiinput multioutput (MIMO) channels are considered subject to block Rayleigh fading. Each coherence block contains L symbols, and is spanned by T training symbols, B feedback bits, and the data symbols. The training symbols are used to obtain a Minimum Mean Squared Error estimate of the channel matrix. Given this estimate, the receiver selects a transmit beamforming vector from a codebook containing 2B i.i.d. random vectors, and sends the corresponding B bits back to the transmitter. We derive bounds on the beamforming capacity for MISO and MIMO channels and characterize the optimal (ratemaximizing) training and feedback overhead (T and B) as L and the number of transmit antennas Nt both become large. The optimal Nt is limited by the coherence time, and increases as L / logL. For the MISO channel the optimal T/L and B/L (fractional overhead due to training and feedback) are asymptotically the same, and tend to zero at the rate 1 / logNt. For the MIMO channel the optimal feedback overhead B/L tends to zero faster (as 1 / log² Nt).
Bit Allocation Laws for MultiAntenna Channel Feedback Quantization: MultiUser Case
"... This paper addresses the optimal design of limitedfeedback downlink multiuser spatial multiplexing systems. A multipleantenna basestation is assumed to serve multiple singleantenna users, who quantize and feed back their channel state information (CSI) through a shared ratelimited feedback cha ..."
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Cited by 15 (5 self)
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This paper addresses the optimal design of limitedfeedback downlink multiuser spatial multiplexing systems. A multipleantenna basestation is assumed to serve multiple singleantenna users, who quantize and feed back their channel state information (CSI) through a shared ratelimited feedback channel. The optimization problem is cast in the form of minimizing the average transmission power at the basestation subject to users’ target signaltointerferenceplusnoise ratios (SINR) and outage probability constraints. The goal is to derive the feedback bit allocations among the users and the corresponding channel magnitude and direction quantization codebooks in a highresolution quantization regime. Toward this end, this paper develops an optimization framework using approximate analytical closedform solutions, the accuracy of which is then verified by numerical results. The results show that, for channels in the real space, the number of channel direction quantization bits should be (M−1) times the number of channel magnitude quantization bits, where M is the number of basestation antennas. Moreover, users with higher requested qualityofservice (QoS), i.e. lower target outage probabilities, and higher requested downlink rates, i.e. higher target SINR’s, should use larger shares of the feedback rate. It is also shown that, for the target QoS parameters to be feasible, the total feedback bandwidth should scale logarithmically with the geometric mean of the target SINR values and the geometric mean of the inverse target outage probabilities. In particular, the minimum required feedback rate is shown to increase if the users ’ target parameters deviate from the corresponding geometric means. Finally, the paper shows that, as the total number of feedback bits B increases, the performance of the limitedfeedback system approaches the perfectCSI system as 2 −B/M2
Optimization of training and feedback for beamforming over a MIMO channel
 in Proc. IEEE Wireless Commun. and Networking Conf. (WCNC). Hong Kong
, 2007
"... Abstract — We examine the capacity of beamforming over a block Rayleigh fading MultiInput/MultiOutput (MIMO) channel with finite training for channel estimation and limited feedback. A fixedlength packet is assumed, which is spanned by T training symbols, B feedback bits, and the data symbols. Th ..."
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Cited by 8 (5 self)
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Abstract — We examine the capacity of beamforming over a block Rayleigh fading MultiInput/MultiOutput (MIMO) channel with finite training for channel estimation and limited feedback. A fixedlength packet is assumed, which is spanned by T training symbols, B feedback bits, and the data symbols. The training symbols are used to obtain a Minimum Mean Squared Error (MMSE) estimate of the channel matrix. Given this estimate, the receiver selects a transmit beamforming vector from a codebook containing 2 B i.i.d. random vectors, and relays the corresponding Bbit index back to the transmitter. We derive bounds on the large system capacity, i.e., as the number of transmit antennas Nt → ∞ and receive antennas Nr → ∞ with fixed ratio Nt/Nr. The bounds are used to show that the optimal T, which maximizes the capacity, increases as Nt / log Nt, whereas the optimal B increases as Nt / log 2 Nt. I.
Treestructured Random Vector Quantization for beamforming in a multiantenna channel
 In Proc. Electrical Engineering/Electronics, Computer, Telecomm. and Info. Technology Int. Conf. (ECTICON
, 2008
"... Abstract—A pointtopoint multiantenna wireless channel is considered. Based on channel information, a receiver selects a transmit beamforming vector, which contains transmit antenna gains, from a vector set or codebook. The codebook index for the selected beamforming vector that maximizes channel c ..."
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Cited by 8 (2 self)
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Abstract—A pointtopoint multiantenna wireless channel is considered. Based on channel information, a receiver selects a transmit beamforming vector, which contains transmit antenna gains, from a vector set or codebook. The codebook index for the selected beamforming vector that maximizes channel capacity is relayed to the transmitter via a ratelimited feedback channel. Previously, we have proposed a Random Vector Quantization (RVQ) codebook, which consists of independent isotropically distributed vectors and showed that it performs close to the optimal codebook. However, RVQ requires exhaustive search to locate the desired beamformer. To lessen the search complexity, we propose a treestructured (TS) RVQ. Numerical results show that number of computations required for TSRVQ search can be orders of magnitude fewer than that required for RVQ search for given performance. I.
Joint power control and beamforming codebook design for MISO channels with limited feedback
 Global Telecommun. Conf. (Globecom
"... Abstract — This paper investigates the joint design and optimization of the power control and beamforming codebooks for the singleuser multipleinput singleoutput (MISO) wireless systems with a ratelimited feedback link. The problem is cast in the form of minimizing the outage probability subject ..."
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Cited by 8 (6 self)
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Abstract — This paper investigates the joint design and optimization of the power control and beamforming codebooks for the singleuser multipleinput singleoutput (MISO) wireless systems with a ratelimited feedback link. The problem is cast in the form of minimizing the outage probability subject to the transmit power constraint and cardinality constraints on the beamforming and power codebooks. We show that by appropriately choosing and fixing the beamforming codebook and optimizing the power codebook for that beamforming codebook, it is possible to achieve a performance very close to the optimal joint optimization. Further, this paper investigates the optimal tradeoffs between beamforming and power codebook sizes for different number of feedback bits and transmit antennas. Given a target outage probability, our results provide the optimal codebook sizes independent of the target rate. As the outage probability decreases, we show that the optimal joint design should use fewer feedback bits for beamforming and more feedback bits for power control. The jointly optimized beamforming and power control modules combine the power gain of beamforming and diversity gain of power control, which enable it to approach the performance of the system with perfect channel state information as the feedback link capacity increases to infinity — something that is not possible with beamforming or power control alone. I.