Random coding for additive Gaussian channels with feedback

A random coding strategy for discrete-time additive Gaussian noise channels with feedback is analyzed. It has long been known that feedback may increase the capacity of such channels as long as the additive noise process is not white. Here, it is proved that a strictly positive gain is always achieved and that, as the signal power goes to zero, the ratio of feedback capacity to capacity without feedback may be strictly greater than one if the noise spectrum has a null. This is not the case when the noise spectrum is bounded away from zero. It is also demonstrated that random coding, where the codewords are chosen from an ensemble of stationary Gaussian sequences, does not achieve capacity