A Square-Root Free Implementation of the Square-Root V-BLAST Algorithm by a Wide-sense Complex Givens Rotation

We propose a square-root free implementation of the square-root V-BLAST algorithm, to avoid the square-root operations and reduce the number of divisions. The square-root V-BLAST algorithm utilizes the Cholesky factor as the square-root matrix of the estimation error covariance matrix. Then we propose a wide-sense complex Givens rotation for the decoupled Cholesky factors, i.e., the LDL $^{T}$ factors including the matrices L and D. The proposed wide-sense complex Givens rotation triangularizes the permuted L and updates D accordingly. For the proposed wide-sense Givens rotation, a fast square-root free algorithm is proposed to compute the Givens matrix efficiently, which adds 1 floating-point operation (flop) to save 1 square root operation compared to the fast Givens rotation. Assume K transmit antennas. With respect to the square-root VBLAST algorithm, the proposed square-root free implementation averagely adds $\displaystyle \frac{K^{2}}{6}$ flops to save $\displaystyle \frac{K^{2}}{6}$ square root operations, and in the worst case, it adds $\displaystyle \frac{K^{2}}{3}$ flops to save $\displaystyle \frac{K^{2}}{3}$ square root operations.