Order-controlled multiple shift SBR2 algorithm for para-Hermitian polynomial matrices

Abstract

In this work we present a new method of controlling the order growth of polynomial matrices in the multiple shift second order sequential best rotation (MS-SBR2) algorithm which has been recently proposed by the authors for calculating the polynomial matrix eigenvalue decomposition (PEVD) for para-Hermitian matrices. In effect, the proposed method introduces a new elementary delay strategy which keeps all the row (column) shifts in the same direction throughout each iteration, which therefore gives us the flexibility to control the polynomial order growth by selecting shifts that ensure non-zero coefficients are kept closer to the zero-lag plane. Simulation results confirm that further order reductions of polynomial matrices can be achieved by using this directionfixed delay strategy for the MS-SBR2 algorithm.

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

“…can also be expressed as a percentage of A(z) F . Several methods for trimming polynomial terms with small coefficients to keep the order compact have been proposed [9,30,31] but the energy-based technique [32] was chosen in SBR2. For a given ratio of squared F-norm or energy permitted to be lost,Ã(z) is approximated by the smallest τmax satisfying the inequality,…”

A robust LU polynomial matrix decomposition for spatial multiplexing

“…Polynomial order truncation is introduced to limit the degrees of the polynomials. But, this can affect the paraunitary property of the pre-and post-filters (see also [20,21] where the order growth problem is mitigated). In this regard, a MIMO beamforming scheme based on a combination of the classical Smith canonical form and LU (Gauss elimination) was presented in [11] as an alternative solution.…”

Complexity and search space reduction in cyclic-by-row PEVD algorithms

“…Based on previous work [20], [23] to reduce the parameter search space in (6), a further cost reduction for the proposed SMDCbR versions is possible by limiting the search of maximum off-diagonal columns to a particular range of lags surrounding lag-zero. The size of this search segment can be determined by estimating the energy distribution in the parahermitian matrix in current or previous executions of the algorithm; using the latter approach can be useful when the data input to the algorithm remains statistically similar between multiple executions of SMDCbR.…”