A fixed-point implementation of matrix inversion using Cholesky decomposition

Fast Matrix Inversion Updates for Massive MIMO Detection and Precoding

“…Note that most of the computational burden in (2) is in computing the inverses of the Hermitian matrix Z. In particular, the computation of Z −1 using exact inversion methods, such as Cholesky decomposition [10], [11], requires O(M 3 ) operations, which might be cumbersome to implement for the case where a large number of users are being served. Using the fact that in massive MIMO systems Z is an almost diagonal matrix, a hardware-efficient method based on the Neumann series was first proposed in [5] to approximate the required inverse in (2).…”

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“…We can use (17) and (11) to compute F from F −1 iteratively till we get F . The iterations start from F 1 satisfying (14), which can be computed by…”

“…Now from (13), (7) and (4), it can be seen that (9) (proposed in [4]) and (11) actually reveal the relation between the ℎ and the ( − 1) ℎ order inverse Cholesky factor of the matrix R. This relation is also utilized to implement adaptive filters in [15], [16], where the ℎ order inverse Cholesky factor is obtained from the ℎ order Cholesky factor [15, equation (12)], [16, equation (16)]. Thus the algorithms in [15], [16] are still similar to the conventional matrix inversion algorithm [17] using Cholesky factorization, where the inverse Cholesky factor is computed from the Cholesky factor by the backsubstitution (for triangular matrix inversion), an inherent serial process unsuitable for the parallel implementation [18]. Contrarily, the proposed algorithm computes the inverse Cholesky factor of R from R directly by (17) and (11).…”

“…Thus the algorithms in [15], [16] are still similar to the conventional matrix inversion algorithm [17] using Cholesky factorization, where the inverse Cholesky factor is computed from the Cholesky factor by the backsubstitution (for triangular matrix inversion), an inherent serial process unsuitable for the parallel implementation [18]. Contrarily, the proposed algorithm computes the inverse Cholesky factor of R from R directly by (17) and (11). Then it avoids the conventional back substitution of the Cholesky factor.…”

“…(9) and (11)) has been mentioned [4], [15], [16], our contributions in this letter include substituting this relation into (14) to find (18) and (17). Specifically, to compute the ℎ order inverse Cholesky factor, the conventional matrix inversion algorithm using Cholesky factorization [17] usually requires 2 divisions (i.e. divisions for Cholesky factorization and the other divisions for the back-substitution), while the proposed algorithm only requires divisions to compute (19) and (17a).…”