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128
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.
Quantum Error Correction Via Codes Over GF(4)
, 1997
"... The problem of finding quantumerrorcorrecting codes is transformed into the problem of finding additive codes over the field GF(4) which are selforthogonal with respect to a certain trace inner product. Many new codes and new bounds are presented, as well as a table of upper and lower bounds on s ..."
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Cited by 304 (18 self)
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The problem of finding quantumerrorcorrecting codes is transformed into the problem of finding additive codes over the field GF(4) which are selforthogonal with respect to a certain trace inner product. Many new codes and new bounds are presented, as well as a table of upper and lower bounds on such codes of length up to 30 qubits.
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 272 (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.
Grassmannian frames with applications to coding and communication
 Appl. Comp. Harmonic Anal
, 2003
"... For a given class F of unit norm frames of fixed redundancy we define a Grassmannian frame as one that minimizes the maximal correlation 〈fk, fl〉  among all frames {fk}k∈I ∈ F. We first analyze finitedimensional Grassmannian frames. Using links to packings in Grassmannian spaces and antipodal sph ..."
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Cited by 227 (13 self)
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For a given class F of unit norm frames of fixed redundancy we define a Grassmannian frame as one that minimizes the maximal correlation 〈fk, fl〉  among all frames {fk}k∈I ∈ F. We first analyze finitedimensional Grassmannian frames. Using links to packings in Grassmannian spaces and antipodal spherical codes we derive bounds on the minimal achievable correlation for Grassmannian frames. These bounds yield a simple condition under which Grassmannian frames coincide with unit norm tight frames. We exploit connections to graph theory, equiangular line sets, and coding theory in order to derive explicit constructions of Grassmannian frames. Our findings extend recent results on unit norm tight frames. We then introduce infinitedimensional Grassmannian frames and analyze their connection to unit norm tight frames for frames which are generated by grouplike unitary systems. We derive an example of a Grassmannian Gabor frame by using connections to sphere packing theory. Finally we discuss the application of Grassmannian frames to wireless communication and to multiple description coding.
Systematic design of unitary spacetime constellations
 IEEE TRANS. INFORM. THEORY
, 2000
"... We propose a systematic method for creating constellations of unitary space–time signals for multipleantenna communication links. Unitary space–time signals, which are orthonormal in time across the antennas, have been shown to be welltailored to a Rayleigh fading channel where neither the transm ..."
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Cited by 201 (10 self)
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We propose a systematic method for creating constellations of unitary space–time signals for multipleantenna communication links. Unitary space–time signals, which are orthonormal in time across the antennas, have been shown to be welltailored to a Rayleigh fading channel where neither the transmitter nor the receiver knows the fading coefficients. The signals can achieve low probability of error by exploiting multipleantenna diversity. Because the fading coefficients are not known, the criterion for creating and evaluating the constellation is nonstandard and differs markedly from the familiar maximumEuclideandistance norm. Our construction begins with the first signal in the constellation—an oblong complexvalued matrix whose columns are orthonormal—and systematically produces the remaining signals by successively rotating this signal in a highdimensional complex space. This construction easily produces large constellations of highdimensional signals. We demonstrate its efficacy through examples involving one, two, and three transmitter antennas.
Just relax: Convex programming methods for subset selection and sparse approximation
, 2004
"... Subset selection and sparse approximation problems request a good approximation of an input signal using a linear combination of elementary signals, yet they stipulate that the approximation may only involve a few of the elementary signals. This class of problems arises throughout electrical enginee ..."
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Cited by 103 (5 self)
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Subset selection and sparse approximation problems request a good approximation of an input signal using a linear combination of elementary signals, yet they stipulate that the approximation may only involve a few of the elementary signals. This class of problems arises throughout electrical engineering, applied mathematics and statistics, but small theoretical progress has been made over the last fifty years. Subset selection and sparse approximation both admit natural convex relaxations, but the literature contains few results on the behavior of these relaxations for general input signals. This report demonstrates that the solution of the convex program frequently coincides with the solution of the original approximation problem. The proofs depend essentially on geometric properties of the ensemble of elementary signals. The results are powerful because sparse approximation problems are combinatorial, while convex programs can be solved in polynomial time with standard software. Comparable new results for a greedy algorithm, Orthogonal Matching Pursuit, are also stated. This report should have a major practical impact because the theory applies immediately to many realworld signal processing problems.
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 75 (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.
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
Universally optimal distribution of points on spheres
 Journal of the American Mathematical Society
"... Abstract. We study configurations of points on the unit sphere that minimize potential energy for a broad class of potential functions (viewed as functions of the squared Euclidean distance between points). Call a configuration sharp if there are m distances between distinct points in it and it is a ..."
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Cited by 61 (13 self)
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Abstract. We study configurations of points on the unit sphere that minimize potential energy for a broad class of potential functions (viewed as functions of the squared Euclidean distance between points). Call a configuration sharp if there are m distances between distinct points in it and it is a spherical (2m −1)design. We prove that every sharp configuration minimizes potential energy for all completely monotonic potential functions. Examples include the minimal vectors of the E8 and Leech lattices. We also prove the same result for the vertices of the 600cell, which do not form a sharp configuration. For most known cases, we prove that they are the unique global minima for energy, as long as the potential function is strictly completely monotonic. For certain potential functions, some of these configurations were previously analyzed by Yudin, Kolushov, and Andreev; we build on their techniques. We also generalize our results to other compact twopoint homogeneous spaces, and we
MultipleAntenna Signal Constellations for Fading Channels
 IEEE Trans. Inform. Theory
, 1999
"... In this paper we show that the problem of designing eficient multipleantenna signal constellations for fading channels can be related to the problem of finding packings with large minimum distance in the complex Grassmannian space. We describe a numerical optimization procedure for finding good pac ..."
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Cited by 61 (0 self)
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In this paper we show that the problem of designing eficient multipleantenna signal constellations for fading channels can be related to the problem of finding packings with large minimum distance in the complex Grassmannian space. We describe a numerical optimization procedure for finding good packings in the complex Grassmannian space and report the best signal constellations found by this procedure. These constellations improve significantly upon previously known results.