On the Optimality of MMSE-GDFE Pre-Processed Sphere Decoding

In this work, we consider maximum likelihood (ML) sequence detection in MIMO linear channels corrupted by additive white Gaussian noise. While sphere decoding (SpD) algorithms have been developed to reduce the average complexity of ML detection, the average complexity of classical SpD can itself be impractical in low-SNR settings or when the channel is ill-conditioned. In response, sequential decoding algorithms that employ a preprocessing stage based on minimum mean-squared error generalized decision-feedback equalization (MMSE-GDFE) have been proposed. They are capable of near-ML detection at a complexity that remains low over a wide SNR range and/or with ill-conditioned channels. While it has always been assumed that MMSE-GDFE pre-processing compromises the ML-optimality of the downstream minimum-distance detector, we establish, in this work, that MMSE-GDFE pre-processing preserves ML-optimality under uncoded BPSK/QPSK signaling, regardless of channel dimension and rank. The implication is that, when BPSK/QPSK signaling is used, MMSE-GDFE pre-processing can be used in conjunction with efficient SpD algorithms for true ML detection. This is particularly attractive in moderate-to-low SNR ranges or with ill-conditioned or under-determined linear channels.