Jacobi method for signal subspace computation

The Jacobi method for singular value decomposition is well-suited for parallel architectures. Its application to signal subspace computations is well known. Basically the subspace spanned by singular vectors of large singular values are separated from subspace spanned by those of small singular values. The Jacobi algorithm computes the singular values and the corresponding vectors in random order. This requires sorting the result after convergence of the algorithm to select the signal subspace. A modification of the Jacobi method based on a linear objective function merges the sorting into the SVD-algorithm at little extra cost. In fact, the complexity of the diagonal processor cells in a triangular array get slightly larger. In this paper we present these extensions, in particular the modified algorithm for computing the rotation angles and give an example of its usefulness for subspace separation.

[1]  G. Stewart.  A Jacobi-Like Algorithm for Computing the Schur Decomposition of a Nonhermitian Matrix , 1985 .

[2]  U. Helmke,et al.  Optimization and Dynamical Systems , 1994, Proceedings of the IEEE.

[3]  Aaas News,et al.  Book Reviews , 1893, Buffalo Medical and Surgical Journal.

[4]  E. Deprettere.  SVD and signal processing: algorithms, applications and architectures , 1989 .

[5]  S.,et al.  SUBSPACE SEPARATION BY DISCRETIZATIONS OF DOUBLE BRACKET FLOWS , .

[6]  John G. Proakis,et al.  Digital Signal Processing: Principles, Algorithms, and Applications , 1992 .

[7]  Knut Hüper.  Structure and convergence of Jacobi type methods for matrix computations , 1996 .

[8]  Tomás Lang,et al.  Redundant and On-Line CORDIC: Application to Matrix Triangularization and SVD , 1990, IEEE Trans. Computers.

[9]  Paul Van Dooren,et al.  On efficient implementations of Kogbetliantz's algorithm for computing the singular value decomposition , 1987 .

[10]  Gene H. Golub,et al.  Matrix computations , 1983 .

[11]  Joseph R. Cavallaro,et al.  CORDIC arithmetic for an SVD processor , 1987, 1987 IEEE 8th Symposium on Computer Arithmetic (ARITH).

[12]  R. O. Schmidt,et al.  Multiple emitter location and signal Parameter estimation , 1986 .

[13]  James Demmel,et al.  Jacobi's Method is More Accurate than QR , 1989, SIAM J. Matrix Anal. Appl..

[14]  R. Brent,et al.  The Solution of Singular-Value and Symmetric Eigenvalue Problems on Multiprocessor Arrays , 1985 .

[15]  Patrick Dewilde,et al.  Subspace model identification Part 1. The output-error state-space model identification class of algorithms , 1992 .

[16]  Eldon R. Hansen,et al.  On Cyclic Jacobi Methods , 1963 .

[17]  Thomas Kailath,et al.  ESPRIT-estimation of signal parameters via rotational invariance techniques , 1989, IEEE Trans. Acoust. Speech Signal Process..

[18]  R. Brockett.  Dynamical systems that sort lists, diagonalize matrices, and solve linear programming problems , 1991 .

[19]  H. T. Kung,et al.  Matrix Triangularization By Systolic Arrays , 1982, Optics & Photonics.