Rematch and forward for parallel relay networks

The Gaussian parallel relay network problem consists of transmitting a message from a single source node to a single destination node, through a layer of parallel relay nodes. The source is connected to the relays by a Gaussian broadcast channel, while the relays are connected to the destination by a Gaussian multiple access channel. When the channels are all white with the same bandwidth, and the relays cannot decode the message, the best known strategy is "amplify and forward", which achieves the coherence gain of multiple relays. We propose a strategy which achieves this gain even when the noises are colored or the channels have different bandwidths. To that end we use analog modulo-lattice modulation of the codewords in the BC, and then forward the estimated codeword by each of the relays to the MAC. This modulation allows the relays to re-match the signal to the optimal spectrum of the MAC, thus demonstrating how a channel problem can gain from a joint source/channel approach. We show that this strategy is asymptotically optimal in some limiting cases, and that it outperforms the known alternatives in most other cases, where the optimum is unknown. We also demonstrate how to improve the achievable rate in the original white problem, for some signal to noise ratio values.