The Capacity of the Gaussian Channel with Feedback

When Shannon first showed that feedback could not increase the capacity of a memoryless channel, he mentioned that the capacity could be increased when the channel had memory.1 One example of such a channel is the additive colored gaussian noise channel with an average power limitation on the transmitted signal. We prove here that the capacity of this channel is never more than twice the capacity without feedback and as the noise becomes white the capacity approaches the forward capacity. The limiting case has been attributed to Shannon for years and has only recently been rigorously proven.2 We derive an exact expression for the mutual information between the input and output of the channel. The application of different bounds to this expression produces twice the forward capacity with the weakest bound, or the forward capacity plus the normalized correlation of the signal and noise with a slightly stronger bound. It is shown that a gaussian signal maximizes the information, and consequently the optimum feedback technique is linear. Our results are based 011 the model shown in Fig. 1. The added noise 1705