  This paper considers belief propagation algorithm over pair-wise graphical
models to develop low complexity, iterative multiple-input multiple-output
(MIMO) detectors. The pair-wise graphical model is a bipartite graph where a
pair of variable nodes are related by an observation node represented by the
bivariate Gaussian function obtained by marginalizing the posterior joint
probability density under the Gaussian input assumption. Specifically, we
consider two types of pair-wise models, the fully connected and ring-type. The
pair-wise graphs are sparse, compared to the conventional graphical model in
[18], insofar as the number of edges connected to an observation node (edge
degree) is only two. Consequently the computations are much easier than those
of maximum likelihood (ML) detection, which are similar to the belief
propagation (BP) that is run over the fully connected bipartite graph. The link
level performance for non-Gaussian input is evaluated via simulations, and the
results show the validity of the proposed algorithms. We also customize the
algorithm with Gaussian input assumption to obtain the Gaussian BP run over the
two pair-wise graphical models and, for the ring-type, we prove its convergence
in mean to the linear minimum mean square error (MMSE) estimates. Since the
maximum a posterior (MAP) estimator for Gaussian input is equivalent to the
linear MMSE estimator, it shows the optimality, in mean, of the scheme for
Gaussian input.
