Optimal and Quasi-optimal Relaxation Parameter for Massive MIMO Detector Based on SOR Method

Abstract

The linear Minimum Mean Square Error (MMSE) detector uses the inversion matrix to detect in massive Multiple-Input Multiple-Output (MIMO) systems, which involves high cubic complexity in reference to the number of transmitting antennas. Thus, the low-complexity detector using the successive over-relaxation (SOR) method was employed in order to avoid the MIMO matrix inversion. In the SOR method, its relaxation parameter was achieved by intensive computational simulations and only for a given massive MIMO system configuration. Thus, in this paper, we develop and propose an approach to obtain the optimal and the quasi-optimal relaxation parameter of the SOR-based detector by using the massive MIMO channel properties such as near orthogonality and large number theory. It is shown that the quasi-optimal relaxation parameter has been obtained as a relation of the number of receiving and the number of transmitting antennas of the MIMO system where the computation of this quasi-optimal relaxation parameter is feasible.

A Property “A”

A given matrix $\mathbf {A}$ has Property “A” if there exist two disjoint sub-sets S and T of V, the set of the first N integers, such that $S \cup T = V$ and if $a_{ij}\ne 0$ then either $i=j$ or $i\in S$ and $j\in T$, or $i\in T$ and $j\in S$. An equivalent definition is as follows: the matrix $\mathbf {A}$ has Property “A” if there exists a vector $\mathbf {q}=\left( q_1, q_2, \dots , q_N\right)$ with integral components such that if $a_{ij}\ne 0$ and $i\ne j$ then $\left| q_i-q_j \right| =1$.