Geometric Mean Decomposition and Generalized Triangular Decomposition Yi

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Geometric Mean Decomposition and Generalized Triangular Decomposition Yi Jiang yjiang@dsp. ufl. edu William W.

Geometric Mean Decomposition and Generalized Triangular Decomposition Yi Jiang yjiang@dsp. ufl. edu William W. Hager hager@math. ufl. edu Jian Li li@dsp. ufl. edu Department of Electrical & Computer Engineering Department of Mathematics University of Florida July 14, 2004

H = QRP* n Matrix Decomposition H = QRP* – Q, P: matrices with

H = QRP* n Matrix Decomposition H = QRP* – Q, P: matrices with orthonormal columns – R: upper triangular n Some Special Cases – Singular value decomposition R: diagonal matrix containing singular values of H – Schur decomposition R: upper triangular with eigenvalues of H on the diagonal – QR decomposition P: identity matrix July 14, 2004 SIAM 2004 2

Motivation– Joint Transceiver Design for MIMO Communications n Multi-input Multi-output Communications Received data Channel

Motivation– Joint Transceiver Design for MIMO Communications n Multi-input Multi-output Communications Received data Channel matrix July 14, 2004 SIAM 2004 3

MIMO Transceiver n MIMO Transceiver – Decomposition H=QRP* – Linear precoder P , x

MIMO Transceiver n MIMO Transceiver – Decomposition H=QRP* – Linear precoder P , x = P s – Linear equalizer Q, v = Q*y n Equivalent Channel – v = R s + Q* z n Overall System Performance Limited by July 14, 2004 SIAM 2004 4

Problem Formulation n Generalized Maximin problem n The Solution is Geometric Mean Decomposition –

Problem Formulation n Generalized Maximin problem n The Solution is Geometric Mean Decomposition – P, Q: matrices with orthonormal columns – R: upper triangular with equal diagonal elements July 14, 2004 SIAM 2004 5

GMD Algorithm n n Starts with SVD Applies Givens Rotations and Permutations to –

GMD Algorithm n n Starts with SVD Applies Givens Rotations and Permutations to – If – – – Illustration of k-th step K-1 iterations Non-unique July 14, 2004 SIAM 2004 6

A Numerical Research Problem n A numerical research problem by Higham (1996) – Develop

A Numerical Research Problem n A numerical research problem by Higham (1996) – Develop an efficient algorithm for computing a unit upper triangular K x K matrix with prescribed singular values n A solution was given by [Kosowski and Smoktunowicz, Computing, 2000] n GMD is a new solution to Higham’s problem July 14, 2004 SIAM 2004 7

Advantages n Advantages of GMD ü GMD transceiver has superior performance compared with any

Advantages n Advantages of GMD ü GMD transceiver has superior performance compared with any other published transceiver schemes ü Computationally efficient – needs only an additional O((M+N)K) flops compared with SVD ü Numerically stable – involves 2 K-2 Givens rotations ü The technique can be easily extended to the generalized triangular decomposition (GTD) July 14, 2004 SIAM 2004 8

GTD : H = QRP* n Two Questions Q 1: What is the achievable

GTD : H = QRP* n Two Questions Q 1: What is the achievable set for the diagonal of R ? Q 2: Is there a systematic approach to get any achievable decomposition? n Two Observations O 1: H and R share the same singular values O 2: The diagonal are the eigenvalues of R July 14, 2004 SIAM 2004 9

Weyl-Horn Theorem n Weyl-Horn Theorem July 14, 2004 SIAM 2004 10

Weyl-Horn Theorem n Weyl-Horn Theorem July 14, 2004 SIAM 2004 10

GTD Theorem n Generalized Triangular Decomposition [Jiang et. al. 2004] July 14, 2004 SIAM

GTD Theorem n Generalized Triangular Decomposition [Jiang et. al. 2004] July 14, 2004 SIAM 2004 11

GTD Algorithm n n Starts with SVD Applies Givens Rotations and Permutations to –

GTD Algorithm n n Starts with SVD Applies Givens Rotations and Permutations to – If – – – Illustration of k-th step K-1 iterations Key difference from GMD is the permutation matrices July 14, 2004 SIAM 2004 12

Applications of GTD n A New Solution to Inverse Eigenvalue Problem – Constructing matrices

Applications of GTD n A New Solution to Inverse Eigenvalue Problem – Constructing matrices with prescribed eigenvalues and singular values [Chu, SIAM J. Numer. Anal. 2000] n Design MIMO Transceiver with Quality of Service (Qo. S) Constraints [Jiang, et. al. , Asilomar, 2004] July 14, 2004 SIAM 2004 13

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