MIMO decoding based on stochastic reconstruction from multiple projections

Least squares (LS) fitting is one of the most fundamental techniques in science and engineering. It is used to estimate parameters from multiple noisy observations. In many problems the parameters are known a-priori to be bounded integer valued, or they come from a finite set of values on an arbitrary finite lattice. In this case finding the closest vector becomes NP-Hard problem. In this paper we propose a novel algorithm, the Tomographic Least Squares Decoder (TLSD), that not only solves the ILS problem, better than other sub-optimal techniques, but also is capable of providing the a-posteriori probability distribution for each element in the solution vector. The algorithm is based on reconstruction of the vector from multiple two-dimensional projections. The projections are carefully chosen to provide low computational complexity. Unlike other iterative techniques, such as the belief propagation, the proposed algorithm has ensured convergence. We also provide simulated experiments comparing the algorithm to other sub-optimal algorithms.

💡 Research Summary

The paper tackles the integer least‑squares (ILS) problem that lies at the heart of many modern communication and signal‑processing tasks, especially MIMO detection where transmitted symbols belong to a finite lattice (e.g., QAM constellations). Classical optimal solvers such as sphere decoding guarantee the maximum‑likelihood (ML) solution but suffer from exponential complexity as the number of transmit antennas (or the dimension N) grows, making them unsuitable for real‑time large‑scale systems. Sub‑optimal alternatives—lattice‑reduction‑aided linear detectors, belief‑propagation (BP) on factor graphs, approximate message passing—offer lower complexity but either lack convergence guarantees, provide only point estimates, or cannot easily deliver per‑symbol posterior probabilities needed for higher‑layer processing (e.g., channel coding).

To overcome these limitations, the authors propose the Tomographic Least Squares Decoder (TLSD). The core idea is to view the unknown integer vector x as a high‑dimensional object that can be reconstructed from a set of carefully chosen two‑dimensional projections. Each projection corresponds to a pair of columns (i, j) of the channel matrix H, and is represented by a projection matrix P₍ᵢⱼ₎ that extracts the sub‑space spanned by those two columns. By projecting the received vector y onto this sub‑space, the original ILS problem collapses into a small 2‑D integer LS problem: \