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61
Grassmannian beamforming for multipleinput multipleoutput wireless systems
 IEEE TRANS. INFORM. THEORY
, 2003
"... Transmit beamforming and receive combining are simple methods for exploiting the significant diversity that is available in multipleinput and multipleoutput (MIMO) wireless systems. Unfortunately, optimal performance requires either complete channel knowledge or knowledge of the optimal beamformi ..."
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Cited by 329 (38 self)
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Transmit beamforming and receive combining are simple methods for exploiting the significant diversity that is available in multipleinput and multipleoutput (MIMO) wireless systems. Unfortunately, optimal performance requires either complete channel knowledge or knowledge of the optimal beamforming vector which are not always realizable in practice. In this correspondence, a quantized maximum signaltonoise ratio (SNR) beamforming technique is proposed where the receiver only sends the label of the best beamforming vector in a predetermined codebook to the transmitter. By using the distribution of the optimal beamforming vector in independent identically distributed Rayleigh fading matrix channels, the codebook design problem is solved and related to the problem of Grassmannian line packing. The proposed design criterion is flexible enough to allow for side constraints on the codebook vectors. Bounds on the codebook size are derived to guarantee full diversity order. Results on the density of Grassmannian line packings are derived and used to develop bounds on the codebook size given a capacity or SNR loss. Monte Carlo simulations are presented that compare the probability of error for different quantization strategies.
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.
Universal SpaceTime Coding
 IEEE Trans. Inform. Theory
, 2003
"... A universal framework is developed for constructing fullrate and fulldiversity coherent spacetime codes for systems with arbitrary numbers of transmit and receive antennas. The proposed framework combines spacetime layering concepts with algebraic component codes optimized for singleinputsi ..."
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Cited by 143 (7 self)
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A universal framework is developed for constructing fullrate and fulldiversity coherent spacetime codes for systems with arbitrary numbers of transmit and receive antennas. The proposed framework combines spacetime layering concepts with algebraic component codes optimized for singleinputsingleoutput (SISO) channels. Each component code is assigned to a "thread" in the spacetime matrix, allowing it thus full access to the channel spatial diversity in the absence of the other threads. Diophantine approximation theory is then used in order to make the different threads "transparent" to each other. Within this framework, a special class of signals which uses algebraic numbertheoretic constellations as component codes is thoroughly investigated. The lattice structure of the proposed numbertheoretic codes along with their minimal delay allow for polynomial complexity maximumlikelihood (ML) decoding using algorithms from lattice theory. Combining the design framework with the Cayley transform allows to construct full diversity differential and noncoherent spacetime codes. The proposed framework subsumes many of the existing codes in the literature, extends naturally to timeselective and frequency selective channels, and allows for more flexibility in the tradeoff between power efficiency, bandwidth efficiency, and receiver complexity. Simulation results that demonstrate the significant gains offered by the proposed codes are presented in certain representative scenarios.
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 125 (18 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.
Designing Structured Tight Frames via an Alternating Projection Method
, 2003
"... Tight frames, also known as general WelchBoundEquality sequences, generalize orthonormal systems. Numerous applicationsincluding communications, coding and sparse approximationrequire finitedimensional tight frames that possess additional structural properties. This paper proposes an alterna ..."
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Cited by 87 (10 self)
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Tight frames, also known as general WelchBoundEquality sequences, generalize orthonormal systems. Numerous applicationsincluding communications, coding and sparse approximationrequire finitedimensional tight frames that possess additional structural properties. This paper proposes an alternating projection method that is versatile enough to solve a huge class of inverse eigenvalue problems, which includes the frame design problem. To apply this method, one only needs to solve a matrix nearness problem that arises naturally from the design specifications. Therefore, it is fast and easy to develop versions of the algorithm that target new design problems. Alternating projection will often succeed even if algebraic constructions are unavailable. To demonstrate
Quantization bounds on Grassmann manifolds and applications in MIMO systems
 IEEE Trans. Inf. Theory
, 2008
"... Abstract This paper considers the quantization problem on the Grassmann manifold with dimension n and p. The unique contribution is the derivation of a closedform formula for the volume of a metric ball in the Grassmann manifold when the radius is sufficiently small. This volume formula holds for ..."
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Cited by 65 (11 self)
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Abstract This paper considers the quantization problem on the Grassmann manifold with dimension n and p. The unique contribution is the derivation of a closedform formula for the volume of a metric ball in the Grassmann manifold when the radius is sufficiently small. This volume formula holds for Grassmann manifolds with arbitrary dimension n and p, while previous results are only valid for either p = 1 or a fixed p with asymptotically large n. Based on the volume formula, the GilbertVarshamov and Hamming bounds for sphere packings are obtained. Assuming a uniformly distributed source and a distortion metric based on the squared chordal distance, tight lower and upper bounds are established for the distortion rate tradeoff. Simulation results match the derived results. As an application of the derived quantization bounds, the information rate of a MultipleInput MultipleOutput (MIMO) system with finiterate channelstate feedback is accurately quantified for arbitrary finite number of antennas, while previous results are only valid for either MultipleInput SingleOutput (MISO) systems or those with asymptotically large number of transmit antennas but fixed number of receive antennas. Index Terms the Grassmann manifold, distortion rate tradeoff, MIMO communications is the set of all pdimensional planes (through the origin) of the ndimensional Euclidean space L n , where L is either R or C. It forms a compact Riemann manifold of real dimension βp (n − p), where β = 1/2 when L = R/C respectively. The Grassmann manifold provides a useful analysis tool for multiantenna communications (also known as multipleinput multipleoutput (MIMO) communication systems). For noncoherent MIMO systems, sphere packings on the G n,p (L) can be viewed as a generalization of spherical codes [1] The basic quantization problems addressed in this paper are the sphere packing bounds and distortion rate tradeoff. Roughly speaking, a quantization is a representation of a source in the G n,p (L). In particular, it maps an element in the G n,p (L) into a subset of the G n,p (L), known as the code C. Define the minimum distance of a code δ δ (C) as the minimum distance between any two codewords in a code C. The sphere packing bound relates the size of a code and a given minimum distance δ. The rate distortion tradeoff is another important property of quantizations. A distortion metric is a mapping from the set of element pairs in the G n,p (L) into the set of nonnegative real numbers. Given a source distribution and a distortion metric, the rate distortion tradeoff is described by the minimum expected distortion achievable for a given code size or the minimum code size required to achieve a particular expected distortion. There are several papers addressing the quantization problem in the Grassmann manifold. In
Limited feedback unitary precoding for orthogonal spacetime block codes
 IEEE Trans. Signal Processing
, 2005
"... Abstract—Orthogonal spacetime block codes (OSTBCs) are a class of easily decoded spacetime codes that achieve full diversity order in Rayleigh fading channels. OSTBCs exist only for certain numbers of transmit antennas and do not provide array gain like diversity techniques that exploit transmit c ..."
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Cited by 56 (7 self)
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Abstract—Orthogonal spacetime block codes (OSTBCs) are a class of easily decoded spacetime codes that achieve full diversity order in Rayleigh fading channels. OSTBCs exist only for certain numbers of transmit antennas and do not provide array gain like diversity techniques that exploit transmit channel information. When channel state information is available at the transmitter, though, precoding the spacetime codeword can be used to support different numbers of transmit antennas and to improve array gain. Unfortunately, transmitters in many wireless systems have no knowledge about current channel conditions. This motivates limited feedback precoding methods such as channel quantization or antenna subset selection. This paper investigates a limited feedback approach that uses a codebook of precoding matrices known a priori to both the transmitter and receiver. The receiver chooses a matrix from the codebook based on current channel conditions and conveys the optimal codebook matrix to the transmitter over an errorfree, zerodelay feedback channel. A criterion for choosing the optimal precoding matrix in the codebook is proposed that relates directly to minimizing the probability of symbol error of the precoded system. Low average distortion codebooks are derived based on the optimal codeword selection criterion. The resulting design is found to relate to the famous applied mathematics problem of subspace packing in the Grassmann manifold. Codebooks designed by this method are proven to provide full diversity order in Rayleigh fading channels. Monte Carlo simulations show that limited feedback precoding performs better than antenna subset selection. Index Terms—Diversity methods, Grassmannian subspace packing, MIMO systems, orthogonal spacetime block coding,
Unitary Signal Constellations for Differential Space–Time Modulation With Two Transmit Antennas: Parametric Codes, Optimal Designs, and Bounds
, 2002
"... Differential space–time modulation has been recently proposed in the literature for multipleantenna systems over Rayleighfading channels, where neither the transmitter nor the receiver knows the fading coefficients. For the practical success of differential space–time modulation, it has been show ..."
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Cited by 55 (3 self)
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Differential space–time modulation has been recently proposed in the literature for multipleantenna systems over Rayleighfading channels, where neither the transmitter nor the receiver knows the fading coefficients. For the practical success of differential space–time modulation, it has been shown critical to design unitary space–time signal constellations with large diversity product which is a primary property for the signal constellations to have good performance in high signaltonoise ratio (SNR) scenarios. In this paper, we focus on the design of unitary signal constellations for differential space–time modulation with double transmit antennas. By using the parametric form of a twobytwo unitary matrix, we present a class of unitary space–time codes called parametric codes and show that this class of unitary space–time codes leads to a fivesignal constellation with the largest possible diversity
Rayleigh fading multiantenna channels
 EURASIP Journal on Applied Signal Processing
, 2002
"... Information theoretic properties of flat fading channels with multiple antennas are investigated. Perfect channel knowledge at the receiver is assumed. Expressions for maximum information rates and outage probabilities are derived. The advantages of transmitter channel knowledge are determined and a ..."
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Cited by 45 (3 self)
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Information theoretic properties of flat fading channels with multiple antennas are investigated. Perfect channel knowledge at the receiver is assumed. Expressions for maximum information rates and outage probabilities are derived. The advantages of transmitter channel knowledge are determined and a critical threshold is found beyond which such channel knowledge gains very little. Asymptotic expressions for the error exponent are found. For the case of transmit diversity closed form expressions for the error exponent and cutoff rate are given. The use of orthogonal modulating signals is shown to be asymptotically optimal in terms of information rate.