Upper Bounds of Error Probabilities for Stationary Gaussian Channels With Feedback

In this paper, we discuss the coding schemes and error probabilities in information transmission over additive Gaussian noise channels with feedback, where the Gaussian noise processes are stationary but not necessarily white. In the case of the white Gaussian channel it is known that the minimum error probability, under the average power constraint, decreases faster than the exponential of any order. Recently Gallager and Nakiboğlu (2010) proposed a coding scheme for the white Gaussian channel and successfully showed the multiple-exponential decay of the error probability for all rates below capacity. This paper aims to prove that, without any special assumptions on the noise of the stationary Gaussian channel, the minimum error probability decreases multiple-exponentially fast. In general, no explicit formulas are known for the capacity of the stationary Gaussian channel. In this paper, we introduce a lower bound C^* on the capacity C. Then we prove that the minimum error probability decreases multiple-exponentially fast for all rates below C*, but not for all rates below C. In the process of proving, the scheme proposed by Gallager and Nakiboğlu proves itself to be quite useful, even though the Gaussian channel is not white.