Cyclic-by-row approximation of iterative polynomial EVD algorithms

A recent class of sequential matrix diagonalisation (SMD) algorithms have been demonstrated to provide a fast converging solution to iteratively approximating the polynomial eigenvalue decomposition of a parahermitian matrix. However, the calculation of an EVD, and the application of a full unitary matrix to every time lag of the parahermitian matrix in the SMD algorithm results in a high numerical cost. In this paper, we replace the EVD with a limited number of Givens rotations forming a cyclic-by-row Jacobi sweep. Simulations indicate that a considerable reduction in computational complexity compared to SMD can be achieved with a negligible sacrifice in diagonalisation performance, such that the benefits in applying the SMD are maintained.

[1]  John G. McWhirter,et al.  Multiple shift maximum element sequential matrix diagonalisation for parahermitian matrices , 2014, 2014 IEEE Workshop on Statistical Signal Processing (SSP).

[2]  Jürgen Götze,et al.  Monitoring the stage of diagonalization in Jacobi-type methods , 1994, ICASSP.

[3]  John G. McWhirter,et al.  An approximate polynomial matrix eigenvalue decomposition algorithm for para-Hermitian matrices , 2011, 2011 IEEE International Symposium on Signal Processing and Information Technology (ISSPIT).

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

[5]  Stephan D. Weiss,et al.  An extension of the MUSIC algorithm to broadband scenarios using a polynomial eigenvalue decomposition , 2011, 2011 19th European Signal Processing Conference.

[6]  John G. McWhirter,et al.  An EVD Algorithm for Para-Hermitian Polynomial Matrices , 2007, IEEE Transactions on Signal Processing.

[7]  P. Vaidyanathan.  Multirate Systems And Filter Banks , 1992 .

[8]  John G. McWhirter,et al.  Sequential Matrix Diagonalisation Algorithms for Polynomial EVD of Parahermitian Matrices , 2014 .

[9]  Andre Tkacenko,et al.  Approximate eigenvalue decomposition of para-Hermitian systems through successive FIR paraunitary transformations , 2010, 2010 IEEE International Conference on Acoustics, Speech and Signal Processing.

[10]  John G. McWhirter,et al.  Design of FIR Paraunitary Filter Banks for Subband Coding Using a Polynomial Eigenvalue Decomposition , 2011, IEEE Transactions on Signal Processing.

[11]  P. P. Vaidyanathan,et al.  Theory of optimal orthonormal subband coders , 1998, IEEE Trans. Signal Process..

[12]  John G. McWhirter,et al.  Causality-constrained multiple shift sequential matrix diagonalisation for parahermitian matrices , 2014, 2014 22nd European Signal Processing Conference (EUSIPCO).

[13]  Pierre Comon,et al.  Some Properties of Laurent Polynomial Matrices , 2012 .