Relay positioning for multicast relay networks

The problem of relay positioning to maximize the multicast rate is addressed for a discrete-time Gaussian multicast relay channel with a line-of-sight path-loss model. Cut-set upper bounds, decode-and-forward lower bounds and quantize-and-forward lower bounds on the capacity of the Gaussian multicast relay channel are derived. For the low signal-to-noise ratio regime two rate strategies are presented, namely a routing based decode-and-forward strategy and a network coding based decode-and-forward strategy. A hypergraph inter-pretation is given for the routing based strategy and it is shown that the relay positioning problem can be solved by solving a series of convex optimization problems. For the network coding strategy the relay positioning problem is shown to be a quasi-concave optimization problem. Lastly, it is shown that for all signal-to-noise ratio regimes the cut-set upper bounds, decode-and-forward lower bounds and quantize-and-forward lower bounds are quasi-concave.