An Analysis of Valid Nodes Distribution for Sphere Decoding in the MIMO Wireless Communication System

Sphere Decoding (SD) algorithms can achieve a quasi-maximum likelihood (ML) decoder performance over Gaussian multiple input-multiple output (MIMO) channels with much lower complexity compared to the exhaustive search method. The SD algorithm is based on a closest lattice point search over a limited search space (hypersphere). On top of that, QR-decomposition simplifies the SD linear system's matrix to be an upper triangle matrix. The solution solver then is done by searching in the exponentially expanding search tree, started from the top with only a single node then increases by M times every level (in $N_T\times N_R$ MIMO system). Fortunately, the SD algorithm shrinks its hypersphere at every level (once the level node is determined) and phases out a vast number of the candidates, remaining only specific valid nodes in the current considered level. In this work, we proposed the statistical approach for evaluating the adequate number of valid search nodes at every level of the search tree, aiming to optimize the overall computational workload. We use a massive number of inputs patterns and extensive simulation to project the number of remaining valid nodes during the searching process. The simulations have been conducted for $4\times 4$ and $8\times 8$ MIMO systems. Our results indicate that for a particular targeted BER, choosing an appropriate sphere radius is essentially important and the number of necessary calculations increases only at the middle layer and can be generically quantified regardless of the system characteristics. This finding is beneficial for the hardware implementation of the SD, where the number of computational units has to be fixed in advance.