Upper bounds on error probabilities for continuous-time white Gaussian channels with feedback

In information transmission over additive white Gaussian channels with feedback, the use of feedback link to improve the performance of communication systems has been studied by a number of authors. It is well known that the error probability in information transmission can be substantially reduced by using feedback, namely, under the average power constraint, the error probability decreases more rapidly than the exponential of any order. Recently, for discrete-time additive white Gaussian channels, Gallager and Nakiboğlu proposed a feedback coding scheme such that the resulting error probability P_e(N) decreases with an exponential order αN which is linearly increasing with block length N, where α is a positive constant. In this paper, we consider continuous-time additive white Gaussian channels with feedback. The aim is to prove a stronger result on the multiple-exponential decay of the error probability. More precisely, for any positive constant α, there exists a feedback coding scheme such that the resulting error probability P_e(T) at time T decreases more rapidly than the exponential of order αT as T → ∞.