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435
Decoding by Linear Programming
, 2004
"... This paper considers the classical error correcting problem which is frequently discussed in coding theory. We wish to recover an input vector f ∈ Rn from corrupted measurements y = Af + e. Here, A is an m by n (coding) matrix and e is an arbitrary and unknown vector of errors. Is it possible to rec ..."
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Cited by 662 (15 self)
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This paper considers the classical error correcting problem which is frequently discussed in coding theory. We wish to recover an input vector f ∈ Rn from corrupted measurements y = Af + e. Here, A is an m by n (coding) matrix and e is an arbitrary and unknown vector of errors. Is it possible to recover f exactly from the data y? We prove that under suitable conditions on the coding matrix A, the input f is the unique solution to the ℓ1minimization problem (‖x‖ℓ1:= i xi) min g∈R n ‖y − Ag‖ℓ1 provided that the support of the vector of errors is not too large, ‖e‖ℓ0: = {i: ei ̸= 0}  ≤ ρ · m for some ρ> 0. In short, f can be recovered exactly by solving a simple convex optimization problem (which one can recast as a linear program). In addition, numerical experiments suggest that this recovery procedure works unreasonably well; f is recovered exactly even in situations where a significant fraction of the output is corrupted. This work is related to the problem of finding sparse solutions to vastly underdetermined systems of linear equations. There are also significant connections with the problem of recovering signals from highly incomplete measurements. In fact, the results introduced in this paper improve on our earlier work [5]. Finally, underlying the success of ℓ1 is a crucial property we call the uniform uncertainty principle that we shall describe in detail.
Diversity and Multiplexing: A Fundamental Tradeoff in Multiple Antenna Channels
 IEEE Trans. Inform. Theory
, 2002
"... Multiple antennas can be used for increasing the amount of diversity or the number of degrees of freedom in wireless communication systems. In this paper, we propose the point of view that both types of gains can be simultaneously obtained for a given multiple antenna channel, but there is a fund ..."
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Cited by 642 (16 self)
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Multiple antennas can be used for increasing the amount of diversity or the number of degrees of freedom in wireless communication systems. In this paper, we propose the point of view that both types of gains can be simultaneously obtained for a given multiple antenna channel, but there is a fundamental tradeo# between how much of each any coding scheme can get. For the richly scattered Rayleigh fading channel, we give a simple characterization of the optimal tradeo# curve and use it to evaluate the performance of existing multiple antenna schemes.
Fading correlation and its effect on the capacity of multielement antenna systems
 IEEE Trans. Commun
, 2000
"... Abstract—We investigate the effects of fading correlations in multielement antenna (MEA) communication systems. Pioneering studies showed that if the fades connecting pairs of transmit and receive antenna elements are independently, identically distributed, MEA’s offer a large increase in capacity c ..."
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Cited by 309 (4 self)
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Abstract—We investigate the effects of fading correlations in multielement antenna (MEA) communication systems. Pioneering studies showed that if the fades connecting pairs of transmit and receive antenna elements are independently, identically distributed, MEA’s offer a large increase in capacity compared to singleantenna systems. An MEA system can be described in terms of spatial eigenmodes, which are singleinput singleoutput subchannels. The channel capacity of an MEA is the sum of capacities of these subchannels. We will show that the fading correlation affects the MEA capacity by modifying the distributions of the gains of these subchannels. The fading correlation depends on the physical parameters of MEA and the scatterer characteristics. In this paper, to characterize the fading correlation, we employ an abstract model, which is appropriate for modeling narrowband Rayleigh fading in fixed wireless systems. I.
On the distribution of the largest eigenvalue in principal components analysis
 Ann. Statist
, 2001
"... Let x �1 � denote the square of the largest singular value of an n × p matrix X, all of whose entries are independent standard Gaussian variates. Equivalently, x �1 � is the largest principal component variance of the covariance matrix X ′ X, or the largest eigenvalue of a pvariate Wishart distribu ..."
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Cited by 197 (2 self)
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Let x �1 � denote the square of the largest singular value of an n × p matrix X, all of whose entries are independent standard Gaussian variates. Equivalently, x �1 � is the largest principal component variance of the covariance matrix X ′ X, or the largest eigenvalue of a pvariate Wishart distribution on n degrees of freedom with identity covariance. Consider the limit of large p and n with n/p = γ ≥ 1. When centered by µ p = � √ n − 1 + √ p � 2 and scaled by σ p = � √ n − 1 + √ p��1 / √ n − 1 + 1 / √ p � 1/3 � the distribution of x �1 � approaches the Tracy–Widom lawof order 1, which is defined in terms of the Painlevé II differential equation and can be numerically evaluated and tabulated in software. Simulations showthe approximation to be informative for n and p as small as 5. The limit is derived via a corresponding result for complex Wishart matrices using methods from random matrix theory. The result suggests that some aspects of large p multivariate distribution theory may be easier to apply in practice than their fixed p counterparts. 1. Introduction. The
Communication on the Grassmann Manifold: A Geometric Approach to the Noncoherent MultipleAntenna Channel
 IEEE Trans. Inform. Theory
, 2002
"... In this paper, we study the capacity of multipleantenna fading channels. We focus on the scenario where the fading coefficients vary quickly; thus an accurate estimation of the coefficients is generally not available to either the transmitter or the receiver. We use a noncoherent block fading model ..."
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Cited by 174 (5 self)
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In this paper, we study the capacity of multipleantenna fading channels. We focus on the scenario where the fading coefficients vary quickly; thus an accurate estimation of the coefficients is generally not available to either the transmitter or the receiver. We use a noncoherent block fading model proposed by Marzetta and Hochwald. The model does not assume any channel side information at the receiver or at the transmitter, but assumes that the coefficients remain constant for a coherence interval of length symbol periods. We compute the asymptotic capacity of this channel at high signaltonoise ratio (SNR) in terms of the coherence time , the number of transmit antennas , and the number of receive antennas . While the capacity gain of the coherent multiple antenna channel is min bits per second per hertz for every 3dB increase in SNR, the corresponding gain for the noncoherent channel turns out to be (1 ) bits per second per herz, where = min 2 . The capacity expression has a geometric interpretation as sphere packing in the Grassmann manifold.
Efficient Use of Side Information in MultipleAntenna Data Transmission over Fading Channels
, 1998
"... We derive performance limits for two closely related communication scenarios involving a wireless system with multipleelement transmitter antenna arrays: a pointtopoint system with partial side information at the transmitter, and a broadcast system with multiple receivers. In both cases, ideal be ..."
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Cited by 153 (2 self)
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We derive performance limits for two closely related communication scenarios involving a wireless system with multipleelement transmitter antenna arrays: a pointtopoint system with partial side information at the transmitter, and a broadcast system with multiple receivers. In both cases, ideal beamforming is impossible, leading to an inherently lower achievable performance as the quality of the side information degrades or as the number of receivers increases. Expected signaltonoise ratio (SNR) and mutual information are both considered as performance measures. In the pointtopoint case, we determine when the transmission strategy should use some form of beamforming and when it should not. We also show that, when properly chosen, even a small amount of side information can be quite valuable. For the broadcast scenario with an SNR criterion, we find the efficient frontier of operating points and show that even when the number of receivers is larger than the number of antenna array ...
On the capacity of OFDMbased spatial multiplexing systems
 IEEE Trans. Commun
, 2002
"... Abstract—This paper deals with the capacity behavior of wireless Orthogonal Frequency Division Multiplexing (OFDM)based spatial multiplexing systems in broadband fading environments for the case where the channel is unknown at the transmitter and perfectly known at the receiver. Introducing a phys ..."
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Cited by 117 (15 self)
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Abstract—This paper deals with the capacity behavior of wireless Orthogonal Frequency Division Multiplexing (OFDM)based spatial multiplexing systems in broadband fading environments for the case where the channel is unknown at the transmitter and perfectly known at the receiver. Introducing a physically motivated multipleinput multipleoutput (MIMO) broadband fading channel model, we study the influence of physical parameters such as the amount of delay spread, cluster angle spread, and total angle spread, and system parameters such as the number of antennas and antenna spacing on ergodic capacity and outage capacity. We find that in the MIMO case, unlike the singleinput singleoutput (SISO) case, delay spread channels may provide advantage over flat fading channels not only in terms of outage capacity but also in terms of ergodic capacity. Therefore, MIMO delay spread channels will in general provide both higher diversity gain and higher multiplexing gain than MIMO flatfading channels.
Phase transition of the largest eigenvalue for nonnull complex sample covariance matrices
, 2008
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On the spheredecoding algorithm I. Expected complexity
 IEEE Trans. Sig. Proc
, 2005
"... Abstract—The problem of finding the leastsquares solution to a system of linear equations where the unknown vector is comprised of integers, but the matrix coefficient and given vector are comprised of real numbers, arises in many applications: communications, cryptography, GPS, to name a few. The ..."
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Cited by 76 (5 self)
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Abstract—The problem of finding the leastsquares solution to a system of linear equations where the unknown vector is comprised of integers, but the matrix coefficient and given vector are comprised of real numbers, arises in many applications: communications, cryptography, GPS, to name a few. The problem is equivalent to finding the closest lattice point to a given point and is known to be NPhard. In communications applications, however, the given vector is not arbitrary but rather is an unknown lattice point that has been perturbed by an additive noise vector whose statistical properties are known. Therefore, in this paper, rather than dwell on the worstcase complexity of the integer leastsquares problem, we study its expected complexity, averaged over the noise and over the lattice. For the “sphere decoding” algorithm of Fincke and Pohst, we find a closedform expression for the expected complexity, both for the infinite and finite lattice.