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75
Capacity of a Mobile MultipleAntenna Communication Link in Rayleigh Flat Fading
 IEEE Trans. Inform. Theory
, 1999
"... We analyze a mobile wireless link comprising M transmitter and N receiver antennas operating in a Rayleigh flatfading environment. The propagation coefficients between pairs of transmitter and receiver antennas are statistically independent and unknown; they remain constant for a coherence interv ..."
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Cited by 375 (20 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 pairs 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 1 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 2M transmitted signal matrix is equal to the product of two statistically independent matrices: a T 2 T isotropically distributed unitary matrix times a certain T 2M 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. Index TermsMultielement antenna arrays, spacetime modulation, wireless communications. I.
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 ...
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 considered clusteri ..."
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Cited by 84 (20 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...
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 44 (1 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.
Incremental condition estimation
 SIAM J. Matrix Anal. Appl
, 1990
"... Abstract. This paper presents an improved version of incremental condition estimation, a technique for tracking the extremal singular values of a triangular matrix as it is being constructed one column at a time. We present a new motivation for this estimation technique using orthogonal projections. ..."
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Cited by 41 (2 self)
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Abstract. This paper presents an improved version of incremental condition estimation, a technique for tracking the extremal singular values of a triangular matrix as it is being constructed one column at a time. We present a new motivation for this estimation technique using orthogonal projections. The paper focuses on an implementation of this estimation scheme in an accurate and consistent fashion. In particular, we address the subtle numerical issues arising in the computation of the eigensystem of a symmetric rankone perturbed diagonal 2 2 matrix. Experimental results show that the resulting scheme does a good job in estimating the extremal singular values of triangular matrices, independent of matrix size and matrix condition number, and that it performs qualitatively in the same fashion as some of the commonly used nonincremental condition estimation schemes. AMS(MOS) subject classi cations. 65F35, 65F05 Key words. Condition number, singular values, incremental condition estimation. 1
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 39 (9 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...
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 37 (3 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.
Randomized algorithms for probabilistic robustness with real and complex structured uncertainty
 IEEE Trans. Autom. Control
, 2000
"... Abstract—In recent years, there has been a growing interest in developing randomized algorithms for probabilistic robustness of uncertain control systems. Unlike classical worst case methods, these algorithms provide probabilistic estimates assessing, for instance, if a certain design specification ..."
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Cited by 29 (3 self)
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Abstract—In recent years, there has been a growing interest in developing randomized algorithms for probabilistic robustness of uncertain control systems. Unlike classical worst case methods, these algorithms provide probabilistic estimates assessing, for instance, if a certain design specification is met with a given probability. One of the advantages of this approach is that the robustness margins can be often increased by a considerable amount, at the expense of a small risk. In this sense, randomized algorithms may be used by the control engineer together with standard worst case methods to obtain additional useful information. The applicability of these probabilistic methods to robust control is presently limited by the fact that the sample generation is feasible only in very special cases which include systems affected by real parametric uncertainty bounded in rectangles or spheres. Sampling in more general uncertainty sets is generally performed through overbounding, at the expense of an exponential rejection rate. In this paper, randomized algorithms for stability and performance of linear time invariant uncertain systems described by a general1 configuration are studied. In particular, efficient polynomialtime algorithms for uncertainty structures 1 consisting of an arbitrary number of full complex blocks and uncertain parameters are developed. Index Terms—Random matrices, randomized algorithms, robust control, uncertainty. I.
Fast linear algebra is stable
 In preparation
, 2006
"... In [23] we showed that a large class of fast recursive matrix multiplication algorithms is stable in a normwise sense, and that in fact if multiplication of nbyn matrices can be done by any algorithm in O(n ω+η) operations for any η> 0, then it can be done stably in O(n ω+η) operations for any η> ..."
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Cited by 25 (15 self)
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In [23] we showed that a large class of fast recursive matrix multiplication algorithms is stable in a normwise sense, and that in fact if multiplication of nbyn matrices can be done by any algorithm in O(n ω+η) operations for any η> 0, then it can be done stably in O(n ω+η) operations for any η> 0. Here we extend this result to show that essentially all standard linear algebra operations, including LU decomposition, QR decomposition, linear equation solving, matrix inversion, solving least squares problems, (generalized) eigenvalue problems and the singular value decomposition can also be done stably (in a normwise sense) in O(n ω+η) operations. 1