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 the case of a point-to-point scalar Gaussian 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 three 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.