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30
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 199 (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 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.
Multiantenna capacity of sparse multipath channels
 IEEE TRANS. INFORM. THEORY
, 2006
"... Existing results on multiinput multioutput (MIMO) channel capacity implicitly assume a rich scattering environment in which the channel power scales quadratically with the number of antennas, resulting in linear capacity scaling with the number of antennas. While this assumption may be justified ..."
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Cited by 23 (6 self)
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Existing results on multiinput multioutput (MIMO) channel capacity implicitly assume a rich scattering environment in which the channel power scales quadratically with the number of antennas, resulting in linear capacity scaling with the number of antennas. While this assumption may be justified in systems with few antennas, it leads to violation of fundamental power conservation principles in the limit of large number of antennas. Furthermore, recent measurement results have shown that physical MIMO channels exhibit a sparse multipath structure, even for relatively few antenna dimensions. Motivated by these observations, we propose a framework for modeling sparse channels and study the coherent capacity of sparse MIMO channels from two perspectives: 1) capacity scaling with the number of antennas, and 2) capacity as a function of transmit SNR for a fixed number of antennas. The statistically independent degrees of freedom (DoF) in sparse channels are less than the number of signalspace dimensions and, as a result, sparse channels afford a fundamental new degree of freedom over which channel capacity can be optimized: the distribution of the DoF’s in the available signalspace dimensions. Our investigation is based on a family of sparse channel configurations whose capacity admits a simple and intuitive closedform approximation and reveals a new tradeoff between the multiplexing gain and the received SNR. We identify an ideal channel
Why Does the Kronecker Model Result in Misleading Capacity Estimates?
, 808
"... Many recent works that study the performance of multiinput multioutput (MIMO) systems in practice assume a Kronecker model where the variances of the channel entries, upon decomposition on to the transmit and the receive eigenbases, admit a separable form. Measurement campaigns, however, show tha ..."
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Cited by 5 (4 self)
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Many recent works that study the performance of multiinput multioutput (MIMO) systems in practice assume a Kronecker model where the variances of the channel entries, upon decomposition on to the transmit and the receive eigenbases, admit a separable form. Measurement campaigns, however, show that the Kronecker model results in poor estimates for capacity. Motivated by these observations, a channel model that does not impose a separable structure has been recently proposed and shown to fit the capacity of measured channels better. In this work, we show that this recently proposed modeling framework can be viewed as a natural consequence of channel decomposition on to its canonical coordinates, the transmit and/or the receive eigenbases. Using tools from random matrix theory, we then establish the theoretical basis behind the Kronecker mismatch at the low and the highSNR extremes: 1) Sparsity of the dominant statistical degrees of freedom (DoF) in the true channel at the lowSNR extreme, and 2) Nonregularity of the sparsity structure (disparities in the distribution of the DoF across the rows and the columns) at the highSNR extreme. Index Terms Correlation, fading channels, information rates, MIMO systems, multiplexing, random matrix theory, sparse systems. I.
Impact of Spatial Correlation on Statistical Precoding
 in MIMO Channels with Linear Receivers,” Proc. Annual Allerton Conf. Commun. Control and Computing, Sept. 2006, Available: [Online]. http://www.ifp.uiuc.edu/∼vasanth
"... Abstract — Multiple antennas at the transmitter and the receiver can be used to achieve diversity or multiplexing gain or a combination of the two gains. Most works to date study schemes that achieve either of the two extremes in this diversitymultiplexing tradeoff. In this work, we assume communi ..."
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Cited by 4 (4 self)
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Abstract — Multiple antennas at the transmitter and the receiver can be used to achieve diversity or multiplexing gain or a combination of the two gains. Most works to date study schemes that achieve either of the two extremes in this diversitymultiplexing tradeoff. In this work, we assume communication with a fixed number (M) of independent datastreams and a linear receiver architecture. Motivated by a limited feedback paradigm, we focus on precoding with a semiunitary matrix. We first provide evidence that the optimal semiunitary precoder consists of the M dominant right singular vectors of the channel. Then, we study the average relative enhancement in error probability and relative loss in mutual information with statistical precoding and show that these quantities vanish in the receive antenna asymptotics. For a given transmit and receive antenna dimension, Nt and Nr, we also show that these quantities are minimized if the eigenvalues of the transmit covariance matrix can be partitioned into two components: a dominant component of M eigenvalues that is wellconditioned and a subdominant component of Nt −M eigenvalues that is illconditioned away from the dominant component. I.
LowComplexity Structured Precoding for Spatially Correlated MIMO Channels
, 805
"... The focus of this paper is on spatial precoding in correlated multiantenna channels, where the number of independent datastreams is adapted to tradeoff the datarate with the transmitter complexity. Towards the goal of a lowcomplexity implementation, a structured precoder is proposed, where the ..."
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Cited by 4 (3 self)
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The focus of this paper is on spatial precoding in correlated multiantenna channels, where the number of independent datastreams is adapted to tradeoff the datarate with the transmitter complexity. Towards the goal of a lowcomplexity implementation, a structured precoder is proposed, where the precoder matrix evolves fairly slowly at a rate comparable with the statistical evolution of the channel. Here, the eigenvectors of the precoder matrix correspond to the dominant eigenvectors of the transmit covariance matrix, whereas the power allocation across the modes is fixed, known at both the ends, and is of lowcomplexity. A particular case of the proposed scheme (semiunitary precoding), where the spatial modes are excited with equal power, is shown to be nearoptimal in matched channels. A matched channel is one where the dominant eigenvalues of the transmit covariance matrix are wellconditioned and their number equals the number of independent datastreams, and the receive covariance matrix is also wellconditioned. In mismatched channels, where the above conditions are not met, it is shown that the loss in performance with semiunitary precoding when compared with a perfect channel information benchmark is substantial. This loss needs to be mitigated via limited feedback techniques that provide partial channel information to the transmitter. More importantly, we develop matching metrics that
Limited feedback multiuser MISO systems with differential codebooks in correlated channels
 in Proc. IEEE Int. Conf. on Commun
, 2013
"... Abstract—In this paper, we present a differential codebook design by modifying the Grassmannian codebook for a singlecell multiuser (MU) multipleinput singleoutput (MISO) system operating under spatially and temporally correlated channels. The differential codebook design involves scaling and rot ..."
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Cited by 4 (3 self)
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Abstract—In this paper, we present a differential codebook design by modifying the Grassmannian codebook for a singlecell multiuser (MU) multipleinput singleoutput (MISO) system operating under spatially and temporally correlated channels. The differential codebook design involves scaling and rotation methods that help a codebook to track the slow varying channel. We propose an adaptive scaling technique that improves the performance of the system in correlated channels without any additional feedback information. Monte Carlo simulations show that the proposed differential codebook reduces quantization errors and improves sumrate performance as compared to other differential codebooks designed for spatially and temporally correlated MISO channels. I.
Limited feedback precoder design for spatially correlated MIMO channels
 in Proc. Conf. Info. Sciences and Systems
, 2007
"... Abstract — It is wellknown that perfect channel state information (CSI) at the transmitter and the receiver (CSIT/CSIR) can be used to decompose a multiantenna channel into a bank of parallel channels. While perfect CSIR maybe a reasonable assumption for practical systems, perfect CSIT is general ..."
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Cited by 4 (0 self)
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Abstract — It is wellknown that perfect channel state information (CSI) at the transmitter and the receiver (CSIT/CSIR) can be used to decompose a multiantenna channel into a bank of parallel channels. While perfect CSIR maybe a reasonable assumption for practical systems, perfect CSIT is generally difficult to achieve. Recent attention in communication systems design has thus shifted towards limited feedback schemes where partial CSI is fed back from the receiver to the transmitter. In this work, we consider a precoding scheme which excites a subset of the transmit dimensions with independent data. The main focus is on systematic quantized precoder designs that bridge the gap between statistical precoding and perfect CSIT precoding in spatially correlated channels. In this work, we propose an asymptotic perturbation theoryinspired codebook design obtained from a quantization of the local neighborhood around the statistically dominant precoding direction(s). This design is implemented in practice by maps that can rotate and shrink sets on the Grassmannian manifold. Numerical results show substantial gains can be achieved with the proposed design over statistical precoding. Index Terms — Diversity, limited feedback, MIMO systems, multiplexing, precoding Type of Paper: Regular I.
To Code or Not to Code Across Time: SpaceTime Coding with Feedback
, 2007
"... Spacetime codes leverage the availability of multiple antennas to enhance the reliability of communication over wireless channels. While spacetime codes have initially been designed with a focus on openloop systems, recent technological advances have enabled the possibility of lowrate feedback f ..."
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Cited by 3 (2 self)
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Spacetime codes leverage the availability of multiple antennas to enhance the reliability of communication over wireless channels. While spacetime codes have initially been designed with a focus on openloop systems, recent technological advances have enabled the possibility of lowrate feedback from the receiver to the transmitter. The focus of this paper is on the implications of this feedback in a singleuser multiantenna system with a general model for spatial correlation. We assume a limited feedback model, that is, a coherent receiver and statistics along with B bits of quantized channel information at the transmitter. We study spacetime coding with a family of linear dispersion (LD) codes that meet an additional orthogonality constraint so as to ensure lowcomplexity decoding. Our results show that, when the number of bits of feedback (B) is small, a spacetime coding scheme that is equivalent to beamforming and does not code across time is optimal in a weak sense in that it maximizes the average received SNR. As B increases, this weak optimality transitions to optimality in a strong sense which is characterized by the maximization of the average mutual information. Thus, from a system designer’s perspective, our work suggests that beamforming may not only be attractive from a lowcomplexity viewpoint, but also from an informationtheoretic viewpoint.