Uniformly most powerful alphabet for HDF two-way relaying designed by non-linear optimization tools

Hierarchical-Decode-and-Forward (HDF) strategy, representing wireless/physical-layer network coding, is a promising 2-way relay strategy due to its potential to operate outside the classical multiple access capacity region. We consider a practical scenario assuming channel state information at the receiver side and no channel-adaptation, then HDF performance is unavoidably parametrized by channel fading coefficients. While some modulations have hierarchical minimal distance reaching its upper-bound for all parameter values (e.g. BPSK), some other modulations have the distance much lower (e.g. QPSK), which appears as it undergoes considerably stronger fading. Design of modulations with performance similar to BPSK (denoted Uniformly Most Powerful (UMP)) have already been investigated restricting on non-linear frequency modulations. In this paper, we formulate UMP alphabet design as a bi-quadratically-constrained linear-program on which non-linear optimization methods are applied. Optimized linear UMP alphabets perform several dB gain over canonical linear alphabets and optimized multi-dimensional UMP alphabets do not significantly overcome BPSK.