Efficient Square-Root and Division Free Algorithms for Inverse LDLT Factorization and the Wide-Sense Givens Rotation with Application to V-BLAST

Abstract

We propose efficient square-root and division free algorithms for inverse LDL <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">T</sup> factorization and the wide-sense Givens rotation, mainly for conventional multiply-add digital computers. The algorithms are then employed to propose a fast algorithm for vertical Bell Laboratories Layered Space-Time architecture (V-BLAST). The proposed inverse LDL <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">T</sup> factorization avoids the conventional back substitution of the Cholesky factor, and then has the speedups of 1.19 over the square root and division free alternative Cholesky factorization plus the back substitution. The proposed wide-sense Givens rotation requires the same computational complexity as the efficient Givens rotation. The speedups of the proposed V-BLAST algorithm over the recursive V-BLAST algorithm are 1.05~1.4. The difference between the complexity of the proposed V-BLAST algorithm and that of the square-root V-BLAST algorithm is trivial and even negligible. However, the square-root V-BLAST algorithm requires many square-root and division operations, while the proposed V-BLAST algorithm requires no square-root operations, and requires only one division for each transmit signal, which can be postponed until the signal estimation is required.