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We consider the broadcast relay channel (BRC), where a single sourcetransmits to multiple destinations with the help of a relay, in the limit of alarge bandwidth. We address the problem of optimal relay positioning and powerallocations at source and relay, to maximize the multicast rate from source toall destinations. To solve such a network planning problem, we develop athree-faceted approach based on an underlying information theoretic model,computational geometric aspects, and network optimization tools. Firstly,assuming superposition coding and frequency division between the source and therelay, the information theoretic framework yields a hypergraph model of thewideband BRC, which captures the dependency of achievable rate-tuples on thenetwork topology. As the relay position varies, so does the set of hyperarcsconstituting the hypergraph, rendering the combinatorial nature of optimizationproblem. We show that the convex hull C of all nodes in the 2-D plane can bedivided into disjoint regions corresponding to distinct hyperarcs sets. Thesesets are obtained by superimposing all k-th order Voronoi tessellation of C. Wepropose an easy and efficient algorithm to compute all hyperarc sets, and provethey are polynomially bounded. Using the switched hypergraph approach, we modelthe original problem as a continuous yet non-convex network optimizationprogram. Ultimately, availing on the techniques of geometric programming and$p$-norm surrogate approximation, we derive a good convex approximation. Weprovide a detailed characterization of the problem for collinearly locateddestinations, and then give a generalization for arbitrarily locateddestinations. Finally, we show strong gains for the optimal relay positioningcompared to seemingly interesting positions.

Optimal relay location and power allocation for low SNR broadcast relay channels