On the AWGN channel with noisy feedback and peak energy constraint

Optimal coding over the additive white Gaussian noise channel under the peak energy constraint is studied when there is noisy feedback over an orthogonal additive white Gaussian noise channel. Previously, Shepp, Wolf, Wyner, and Ziv, and Pinsker showed that under the peak energy constraint the best error exponent for transmission of two messages is achieved by antipodal signaling, regardless of the presence of feedback. This negative result might lead to an impression that under the peak energy constraint, even noise-free feedback does not improve the reliability of communication. Pinsker proved the contrary by showing that the best error exponent for sending M messages does not depend on M, and hence can be strictly larger than the best error exponent without feedback. This paper further extends this and shows that if the noise level in the feedback link is sufficiently small, then the best error exponent for transmission of three messages can be strictly larger than the one without feedback. This result is motivated by a series of recent papers of Burnashev and Yamamoto who considered a similar problem over binary symmetric channels.