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69
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 232 (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.
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 203 (28 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.
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 185 (13 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.
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 154 (9 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.
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 153 (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.
Just relax: Convex programming methods for subset selection and sparse approximation
, 2004
"... Abstract. 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 electric ..."
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Cited by 90 (4 self)
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Abstract. 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. 1.
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 51 (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.
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 39 (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.
A GroupTheoretic Framework for the Construction of Packings in Grassmannian Spaces
, 2002
"... By using totally isotropic subspaces in an orthogonal space Ω + (2i,2), several infinite families of packings of 2 kdimensional subspaces of real 2 idimensional space are constructed, some of which are shown to be optimal packings. A certain Clifford group underlies the construction and links this ..."
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Cited by 33 (12 self)
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By using totally isotropic subspaces in an orthogonal space Ω + (2i,2), several infinite families of packings of 2 kdimensional subspaces of real 2 idimensional space are constructed, some of which are shown to be optimal packings. A certain Clifford group underlies the construction and links this problem with BarnesWall lattices, Kerdock sets and quantumerrorcorrecting codes.
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 33 (6 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,