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119
Capacity of a Mobile MultipleAntenna Communication Link in Rayleigh Flat Fading
"... We analyze a mobile wireless link comprising M transmitter and N receiver antennas operating in a Rayleigh flatfading environment. The propagation coefficients between every pair of transmitter and receiver antennas are statistically independent and unknown; they remain constant for a coherence int ..."
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Cited by 494 (23 self)
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We analyze a mobile wireless link comprising M transmitter and N receiver antennas operating in a Rayleigh flatfading environment. The propagation coefficients between every pair of transmitter and receiver antennas are statistically independent and unknown; they remain constant for a coherence interval of T symbol periods, after which they change to new independent values which they maintain for another T symbol periods, and so on. Computing the link capacity, associated with channel coding over multiple fading intervals, requires an optimization over the joint density of T M complex transmitted signals. We prove that there is no point in making the number of transmitter antennas greater than the length of the coherence interval: the capacity for M> Tis equal to the capacity for M = T. Capacity is achieved when the T M transmitted signal matrix is equal to the product of two statistically independent matrices: a T T isotropically distributed unitary matrix times a certain T M random matrix that is diagonal, real, and nonnegative. This result enables us to determine capacity for many interesting cases. We conclude that, for a fixed number of antennas, as the length of the coherence interval increases, the capacity approaches the capacity obtained as if the receiver knew the propagation coefficients.
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 205 (4 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 ...
Clustering in Large Graphs and Matrices
, 1999
"... We consider the problem of dividing a set of m points in Euclidean n\Gammaspace into k clusters (m; n are variable while k is fixed), so as to minimize the sum of distance squared of each point to its "cluster center". This formulation differs in two ways from the most frequently considere ..."
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Cited by 105 (26 self)
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We consider the problem of dividing a set of m points in Euclidean n\Gammaspace into k clusters (m; n are variable while k is fixed), so as to minimize the sum of distance squared of each point to its "cluster center". This formulation differs in two ways from the most frequently considered clustering problems in the literature, namely, here we have k fixed and m;n variable and we use the sum of squared distances as our measure; we will argue that our problem is natural in many contexts. We consider a relaxation of the discrete problem : find the k\Gammadimensional subspace V so that the sum of distances squared to V (of the m points) is minimized. We show : (i) The relaxation can be solved by Singular Value Decomposition (SVD) of Linear Algebra. (ii) The solution of the relaxation can be used to get a 2approximation algorithm for the original problem. More importantly, (iii) we argue that in fact the relaxation provides a generalized clustering which is useful in its own right. Final...
Methods for the Computation of Multivariate tProbabilities
 Computing Sciences and Statistics
, 2000
"... This paper compares methods for the numerical computation of multivariate tprobabilities for hyperrectangular integration regions. Methods based on acceptancerejection, sphericalradial transformations and separationofvariables transformations are considered. Tests using randomly chosen problems ..."
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Cited by 83 (11 self)
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This paper compares methods for the numerical computation of multivariate tprobabilities for hyperrectangular integration regions. Methods based on acceptancerejection, sphericalradial transformations and separationofvariables transformations are considered. Tests using randomly chosen problems show that the most efficient numerical methods use a transformation developed by Genz (1992) for multivariate normal probabilities. These methods allow moderately accurate multivariate tprobabilities to be quickly computed for problems with as many as twenty variables. Methods for the noncentral multivariate tdistribution are also described. Key Words: multivariate tdistribution, noncentral distribution, numerical integration, statistical computation. 1 Introduction A common problem in many statistics applications is the numerical computation of the multivariate t (MVT) distribution function (see Tong, 1990) defined by T(a; b; \Sigma; ) = \Gamma( +m 2 ) \Gamma( 2 ) p j\Sigma...
Random matrix theory
, 2005
"... Random matrix theory is now a big subject with applications in many disciplines of science, engineering and finance. This article is a survey specifically oriented towards the needs and interests of a numerical analyst. This survey includes some original material not found anywhere else. We includ ..."
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Cited by 80 (4 self)
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Random matrix theory is now a big subject with applications in many disciplines of science, engineering and finance. This article is a survey specifically oriented towards the needs and interests of a numerical analyst. This survey includes some original material not found anywhere else. We include the important mathematics which is a very modern development, as well as the computational software that is transforming the theory into useful practice.
STRUCTURAL VECTOR AUTOREGRESSIONS: THEORY OF IDENTIFICATION AND ALGORITHMS FOR INFERENCE
, 2007
"... ABSTRACT. SVARs are widely used for policy analysis and to provide stylized facts for dynamic general equilibrium models. Yet there have been no workable rank conditions to ascertain whether an SVAR is globally identified and no efficient algorithms for smallsample statistical inference when ident ..."
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Cited by 74 (8 self)
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ABSTRACT. SVARs are widely used for policy analysis and to provide stylized facts for dynamic general equilibrium models. Yet there have been no workable rank conditions to ascertain whether an SVAR is globally identified and no efficient algorithms for smallsample statistical inference when identifying restrictions are directly imposed on impulse responses. To fill these important gaps in the literature, this paper makes four contributions. First, we establish a general rank condition for both exactly and overidentified models. Second, we show that this condition can be easily checked analytically and applies to a wide class of identifying restrictions, including linear and certain nonlinear restrictions. Third, we establish a much simpler rank condition for exactly identified models that amounts to a straightforward counting exercise. Fourth, we develop a number of efficient algorithms for smallsample statistical inference. I.
Analysis of the Cholesky decomposition of a semidefinite matrix
 in Reliable Numerical Computation
, 1990
"... Perturbation theory is developed for the Cholesky decomposition of an n × n symmetric positive semidefinite matrix A of rank r. The matrix W = A −1 11 A12 is found to play a key role in the perturbation bounds, where A11 and A12 are r × r and r × (n − r) submatrices of A respectively. A backward er ..."
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Cited by 65 (4 self)
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Perturbation theory is developed for the Cholesky decomposition of an n × n symmetric positive semidefinite matrix A of rank r. The matrix W = A −1 11 A12 is found to play a key role in the perturbation bounds, where A11 and A12 are r × r and r × (n − r) submatrices of A respectively. A backward error analysis is given; it shows that the computed Cholesky factors are the exact ones of a matrix whose distance from A is bounded by 4r(r + 1) � �W �2+1 � 2 u�A�2+O(u 2), where u is the unit roundoff. For the complete pivoting strategy it is shown that �W � 2 2 ≤ 1 3 (n −r)(4r −1), and empirical evidence that �W �2 is usually small is presented. The overall conclusion is that the Cholesky algorithm with complete pivoting is stable for semidefinite matrices. Similar perturbation results are derived for the QR decomposition with column pivoting and for the LU decomposition with complete pivoting. The results give new insight into the reliability of these decompositions in rank estimation. Key words. Cholesky decomposition, positive semidefinite matrix, perturbation theory, backward error analysis, QR decomposition, rank estimation, LINPACK.
Perturbation Theory for the Singular Value Decomposition
 IN SVD AND SIGNAL PROCESSING, II: ALGORITHMS, ANALYSIS AND APPLICATIONS
, 1990
"... The singular value decomposition has a number of applications in digital signal processing. However, the the decomposition must be computed from a matrix consisting of both signal and noise. It is therefore important to be able to assess the effects of the noise on the singular values and singular v ..."
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Cited by 49 (0 self)
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The singular value decomposition has a number of applications in digital signal processing. However, the the decomposition must be computed from a matrix consisting of both signal and noise. It is therefore important to be able to assess the effects of the noise on the singular values and singular vectors  a problem in classical perturbation theory. In this paper we survey the perturbation theory of the singular value decomposition.