Polynomial Matrix SVD via Generalized Sequential Matrix Diagonalisation

The singular value decomposition (SVD) of polynomial matrices serves as a cornerstone in the analysis and optimization of broadband multi-input multi-output (MIMO) systems. This paper introduces novel algorithms for performing the SVD of polynomial matrices, leveraging a sequential matrix diagonalization (SMD) framework. The proposed methodology begins by identifying the column or row with the highest off-diagonal energy using a maximum search procedure. Subsequently, this energy is transferred to the zero-lag coefficient matrix through a delay operation, which is then diagonalized using a conventional SVD. This iterative process continues until the maximum off-diagonal element falls below a predefined threshold. The proposed framework encompasses multiple algorithmic variants, each designed to offer distinct convergence speeds, thereby addressing diverse computational and accuracy requirements. Rigorous proofs of convergence are provided, alongside a thorough comparative analysis of the computational efficiency and diagonalization accuracy of the algorithms. Extensive simulations, conducted on ensembles of randomly generated polynomial matrices, demonstrate that the proposed algorithms consistently outperform state-of-the-art polynomial SVD (PSVD) methods across all evaluated performance metrics. Furthermore, the application of the proposed algorithm to decouple broadband or convolutive MIMO channels validates its accuracy and effectiveness in practical scenarios.