On optimal policies for control and estimation over a Gaussian relay channel

The problem of causal transmission of a memoryless Gaussian source over a two-hop memoryless Gaussian relay channel is considered. The source and the relay encoders have average transmit power constraints, and the performance criterion is mean squared distortion. The main contribution of this paper is to show that unlike in the case of a point-to-point channel, linear encoding schemes are not optimal over a two-hop relay channel in general, extending the sub-optimality results which are known for more than two hops. In some cases, simple three level quantization policies employed at the source and at the relay can outperform the best linear policies. Further a lower bound on the distortion is derived and it is shown that the distortion bounds derived using cut-set arguments are not tight in general for sensor networks.