Second Order Sequential Best Rotation Algorithm with Householder Reduction for Polynomial Matrix Eigenvalue Decomposition

Abstract

The Second-order Sequential Best Rotation (SBR2) algorithm, used for Eigenvalue Decomposition (EVD) on para-Hermitian polynomial matrices typically encountered in wideband signal processing applications like multichannel Wiener filtering and channel coding, involves a series of delay and rotation operations to achieve diagonalisation. In this paper, we proposed the use of Householder transformations to reduce polynomial matrices to tridiagonal form before zeroing the dominant element with rotation. Similar to performing Householder reduction on conventional matrices, our method enables SBR2 to converge in fewer iterations with smaller order of polynomial matrix factors because more off-diagonal Frobeniusnorm (F-norm) could be transferred to the main diagonal at every iteration. A reduction in the number of iterations by 12.35% and 0.1% improvement in reconstruction error is achievable.

Speech Enhancement Using Polynomial Eigenvalue Decomposition

“…where the rows of U(z) are the eigenvectors with corresponding eigenvalues on the diagonal polynomial matrix, Λ(z). The decomposition is computed using an iterative algorithm [19,16,17,18] based on similarity transforms involving L para-unitary polynomial matrices, U(z) = UL(z) . .…”

“…The zeroing unitary matrix computed at iteration can take the form of a Givens rotation in SBR2 [19], that targets the dominant element, or Householder-like optimization procedure as in [18]. A combination of Householder reflection and Givens rotation matrices is used in [17] and the sequential matrix diagonalization (SMD) algorithm [16], that targets the dominant column, uses the eigenvector matrix.…”

“…Rxx(z) ← Z{Rxx(τ )} // see (5) U(z), Λ(z) ← PEVD {Rxx(z), δ, µ, L} // use any PEVD algorithm [16,17,18,19] x(z) ← x(n) // see (5) y(z) ← U(z)x(z) // speech enhancement return y(z).…”

“…The modelling of spacetime correlations for broadband signal processing can be achieved using polynomial matrices. This has motivated the development of a family of polynomial eigenvalue decomposition (PEVD) algorithms [16,17,18], based on the pioneering second-order sequential best rotation (SBR2) algorithm [19]. Since then, PEVD has been widely used in polynomial multiple signal classification (MUSIC) [20], blind source separation [21], source identification [22] and adaptive beamforming [23].…”

Mathematical Tools for Processing Broadband Multi-Sensor Signals

“…The past decade has seen the development of a number of algorithms that generate the McWhirter decomposition, which includes the second order sequential best rotation algorithm (SBR2 [23]) and the sequential matrix diagonalisation (SMD) families of algorithms [26]. Various versions of SBR2 [7,27] and SMD [28,29,30] have since emerged that enhance convergence in one aspect or another. Additionally, the computational complexity of these algorithms has been addressed by various means, including linear algebraic approximations of the EVD [31,32], the truncation of large polynomials [33,34,35,36], reduction of the optimisation parameter space [37,38,39,40] as well as the exploitation of the symmetry of R(z) [41], and the parallelisation of algorithms in [41,42,43,44].…”