Numerically Optimized Uniformly Most Powerful Alphabets for Hierarchical-Decode-and-Forward Two-Way Relaying

Abstract

We address the issue of the parametric performance of the Hierarchical-Decode-and-Forward (HDF) strategy in a wireless 2-way relay channel. Promising HDF, representing the concept of wireless network coding, performs well with a pre-coding strategy that requires Channel State Information (CSI) on the transceiver side. Assuming a practical case when CSI is available only on the receiver side and the channel conditions do not allow adaptive strategies, the parametrization causes significant HDF performance degradation for some modulation alphabets. Alphabets that are robust to the parametrization (denoted Uniformly Most Powerful (UMP)) have already been proposed restricting on the class of non-linear multi-dimensional frequency modulations. In this work, we focus on the general design of unrestricted UMP alphabets. We formulate an optimization problem which is solved by standard non-linear convex constrained optimization algorithms, particularly by Nelder-Mead global optimization search, which is further refined by the local interior-pointsmethod.