Optimal Communication Systems with Noiseless Feedback for a Binary Source and Band-Limited White Gaussian Channel

A communication system with noiseless feedback, noisy forward band-limited channel, and binary source is investigated. Each binary symbol m±is represented by a sequence of signals , the amplitude of which is a function of the a posteriori likelihood ratio Yk. In a sequential receiver a decision is made whenever the likelihood ratio reaches one of two levels Y±; in a nonsequential receiver a decision is assumed after a fixed number of signals is sent. The problem is to design signals , so that given a probability of error and average power the transmission rate is maximized. The problem is solved for both sequential and nonsequential receivers, provided the peak power exceeds some value which is a function of the average power.