Optimal linear precoding and postcoding for MIMO multi-sensor noisy observation problem

This paper proposes an efficient method for optimal joint precoding-postcoding design in a multi-input multi-output (MIMO) multi-sensor noisy observation context - a problem that is of great interest to the multi-relay MIMO transmission system. A set of wireless sensors, each provisioned with a different number of antennas and a different power constraint, precode and send their noisy observations of the same data to a common fusion center, which postcodes the data to make a best estimate of the original data. Taking the mean square error as a performance metric, we show that the optimal joint precoding-postcoding design problem is non-convex. Leveraging the alternative minimization framework, we are able to decompose it to two convex subproblems, one of which promises closed-form solutions. However, unlike previous studies that assume a total power constraint, the condition of individual power constraint and individual noise uncertainty at each sensor has tremendously complicated the second convex subproblem. Rather than numerically solve it via conventional convex optimization tools, we attack it analytically by transforming, approximating, and decomposing it to a set of new problems. We show that the new problems can be efficiently tackled via the Karush-Kuhn-Tucker conditions in an iterative manner. Simultions show that it leads to a convergence much faster and more robust than the conventional convex optimization tools.