SISO APP Searches in Lattices With Tanner Graphs

An efficient, low-complexity, soft-output detector for general lattices is presented, based on their Tanner graph (TG) representations. Closest-point searches in lattices can be performed as nonbinary belief propagation on associated TGs; soft-information output is naturally generated in the process; the algorithm requires no backtrack (cf. classic sphere decoding), and extracts extrinsic information. A lattice's coding gain enables equivalence relations between lattice points, which can be thereby partitioned in cosets. Total and extrinsic a posteriori probabilities at the detector's output further enable the use of soft detection information in iterative schemes. The algorithm is illustrated via two scenarios that transmit a 32-point, uncoded super-orthogonal (SO) constellation for multiple-input multiple-output (MIMO) channels, carved from an 8-dimensional nonorthogonal lattice
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: it achieves maximum likelihood performance in quasistatic fading; and, performs close to interference-free transmission, and identically to list sphere decoding, in independent fading with coordinate interleaving and iterative equalization and detection. Latter scenario outperforms former despite absence of forward error correction coding-because the inherent lattice coding gain allows for the refining of extrinsic information. The lattice constellation is the same as the one employed in the SO space-time trellis codes first introduced for 2 × 2 MIMO by Ionescu et al., then independently by Jafarkhani and Seshadri. Algorithmic complexity is log-linear in lattice dimensionality versus cubic in classic sphere decoders.