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72
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 200 (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.
MIMO Broadcast Channels With FiniteRate Feedback
, 2006
"... Multiple transmit antennas in a downlink channel can provide tremendous capacity (i.e., multiplexing) gains, even when receivers have only single antennas. However, receiver and transmitter channel state information is generally required. In this correspondence, a system where each receiver has per ..."
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Cited by 184 (1 self)
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Multiple transmit antennas in a downlink channel can provide tremendous capacity (i.e., multiplexing) gains, even when receivers have only single antennas. However, receiver and transmitter channel state information is generally required. In this correspondence, a system where each receiver has perfect channel knowledge, but the transmitter only receives quantized information regarding the channel instantiation is analyzed. The wellknown zeroforcing transmission technique is considered, and simple expressions for the throughput degradation due to finiterate feedback are derived. A key finding is that the feedback rate per mobile must be increased linearly with the signaltonoise ratio (SNR) (in decibels) in order to achieve the full multiplexing gain. This is in sharp contrast to pointtopoint multipleinput multipleoutput (MIMO) systems, in which it is not necessary to increase the feedback rate as a function of the SNR.
MIMO broadcast channels with finite rate feedback
 IEEE Trans. on Inform. Theory
, 2006
"... Multiple transmit antennas in a downlink channel can provide tremendous capacity (i.e. multiplexing) gains, even when receivers have only single antennas. However, receiver and transmitter channel state information is generally required. In this paper, a system where each receiver has perfect channe ..."
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Cited by 148 (10 self)
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Multiple transmit antennas in a downlink channel can provide tremendous capacity (i.e. multiplexing) gains, even when receivers have only single antennas. However, receiver and transmitter channel state information is generally required. In this paper, a system where each receiver has perfect channel knowledge, but the transmitter only receives quantized information regarding the channel instantiation is analyzed. The well known zero forcing transmission technique is considered, and simple expressions for the throughput degradation due to finite rate feedback are derived. A key finding is that the feedback rate per mobile must be increased linearly with the SNR (in dB) in order to achieve the full multiplexing gain, which is in sharp contrast to pointtopoint MIMO systems in which it is not necessary to increase the feedback rate as a function of the SNR. I.
Transmit beamforming in multipleantenna systems with finite rate feedback: A VQbased approach
 IEEE Trans. Inform. Theory
, 2006
"... Abstract—This paper investigates quantization methods for feeding back the channel information through a lowrate feedback channel in the context of multipleinput singleoutput (MISO) systems. We propose a new quantizer design criterion for capacity maximization and develop the corresponding iterat ..."
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Cited by 62 (4 self)
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Abstract—This paper investigates quantization methods for feeding back the channel information through a lowrate feedback channel in the context of multipleinput singleoutput (MISO) systems. We propose a new quantizer design criterion for capacity maximization and develop the corresponding iterative vector quantization (VQ) design algorithm. The criterion is based on maximizing the meansquared weighted inner product (MSwIP) between the optimum and the quantized beamforming vector. The performance of systems with quantized beamforming is analyzed for the independent fading case. This requires finding the density of the squared inner product between the optimum and the quantized beamforming vector, which is obtained by considering a simple approximation of the quantization cell. The approximate density function is used to lowerbound the capacity loss due to quantization, the outage probability, and the bit error probability. The resulting expressions provide insight into the dependence of the performance of transmit beamforming MISO systems on the number of transmit antennas and feedback rate. Computer simulations support the analytical results and indicate that the lower bounds are quite tight. Index Terms—Bit error probability, channel capacity, channel state information, multiple antennas, transmit beamforming, outage probability, vector quantization (VQ). I.
BER criterion and codebook construction for finiterate precoded spatial multiplexing with linear receivers
 IEEE TRANS. SIGNAL PROCESS
, 2006
"... Precoded spatial multiplexing systems with ratelimited feedback have been studied recently based on various precoder selection criteria. Instead of those based on indirect performance indicators, we in this paper propose a new criterion directly based on the exact bit error rate (BER) that is appl ..."
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Cited by 36 (1 self)
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Precoded spatial multiplexing systems with ratelimited feedback have been studied recently based on various precoder selection criteria. Instead of those based on indirect performance indicators, we in this paper propose a new criterion directly based on the exact bit error rate (BER) that is applicable to systems with linear receivers and rectangular/square quadratureamplitudemodulation constellations. The BER criterion outperforms any other alternative in terms of optimizing the BER performance for an uncoded system with linear receivers. We then develop a precoder codebook construction method based on the generalized Lloyd algorithm from the vector quantization literature. This construction is not directly based on the BER criterion. Hence, it is suboptimal in the BER sense. However, relative to those currently available, our newfound codebooks improve considerably various minimum distances between any pair of codewords of the codebook. Finally, we analyze the BERoptimal precoder in the asymptotic case with infiniterate feedback that amounts to perfect channel knowledge at the transmitter. The infiniterate optimal precoder based on the BER criterion is drastically different from the counterparts with other criteria, and it leads to a benchmark performance for finiterate precoded spatial multiplexing systems. We observe from numerical results that the BER performance of finiterate feedback with suboptimal codebooks approaches quickly the benchmark performance of infiniterate feedback. This suggests that i) the number of feedback bits in practical systems need not be large and ii) the room for performance improvement via further codebook optimization shrinks quickly as the codebook size increases.
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 34 (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 30 (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
Efficient feedback methods for mimo channels based on parameterization
 IEEE Trans. on Wireless Commun
, 2007
"... Abstract — In this paper, we propose two efficient lowcomplexity quantization methods for multipleinput multipleoutput (MIMO) systems with finiterate feedback based on proper parameterization of the information to be fed back followed by quantization in the new parameter domain. For a MIMO channel ..."
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Cited by 28 (0 self)
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Abstract — In this paper, we propose two efficient lowcomplexity quantization methods for multipleinput multipleoutput (MIMO) systems with finiterate feedback based on proper parameterization of the information to be fed back followed by quantization in the new parameter domain. For a MIMO channel which has multiple orthonormal vectors as channel spatial information, we exploit the geometrical structure of orthonormality while quantizing the spatial information matrix. The parameterization is of two types: one is in terms of a set of unitnorm vectors with different lengths, and the other is in terms of a minimal number of scalar parameters. These parameters are shown to be independent for the i.i.d. flatfading Rayleigh channel, facilitating efficient quantization. In the first scheme, each of the unitnorm vectors is independently quantized with a finite number of bits using an optimal vector quantization (VQ) technique. Bit allocation is needed between the vectors, and the optimum bit allocation depends on the operating SNR of the system. In the second scheme, the scalar parameters are quantized. In slowly timevarying channels, the scalar parameters are also found to be smoothly changing over time, leading to the development of a simple quantization and feedback method using adaptive delta modulation. The results show that the proposed feedback scheme has a channel tracking feature and achieves a capacity very close to perfect feedback with a reasonable feedback rate. Index Terms — Channel information feedback, channel state information, MIMO systems, multiple antennas, parameterization, quantization, transmit beamforming. I.
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 22 (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.
P.,: MIMO transmit beamforming under uniform elemental power constraint
 IEEE Trans. Signal Process
, 2007
"... Abstract—We consider multiinput multioutput (MIMO) transmit beamforming under the uniform elemental power constraint. This is a nonconvex optimization problem, and it is usually difficult to find the optimal transmit beamformer. First, we show that for the multiinput singleoutput (MISO) case, th ..."
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Cited by 19 (1 self)
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Abstract—We consider multiinput multioutput (MIMO) transmit beamforming under the uniform elemental power constraint. This is a nonconvex optimization problem, and it is usually difficult to find the optimal transmit beamformer. First, we show that for the multiinput singleoutput (MISO) case, the optimal solution has a closedform expression. Then we propose a cyclic algorithm for the MIMO case which uses the closedform MISO optimal solution iteratively. The cyclic algorithm has a low computational complexity and is locally convergent under mild conditions. Moreover, we consider finiterate feedback methods needed for transmit beamforming. We propose a simple scalar quantization method, as well as a novel vector quantization method. For the latter method, the codebook is constructed under the uniform elemental power constraint and the method is referred as VQUEP. We analyze VQUEP performance for the MISO case. Specifically, we obtain an approximate expression for the average degradation of the receive signaltonoise ratio (SNR) caused by VQUEP. Numerical examples are provided to demonstrate the effectiveness of our proposed transmit beamformer designs and the finiterate feedback techniques. Index Terms—Finiterate feedback, multiinput multioutput (MIMO), multiinput singleoutput (MISO), quantization, transmit beamforming, uniform elemental power constraint. I.