On the probability of error using a repeat-request strategy on the additive white Gaussian noise channel

An upper bound on the error probability is obtained for digital communication (with average power Po and no bandwidth constraint) in the presence of additive white gaussian noise (with one-sided spectral density No) with the use of a noiseless feedback link. A repeat-request strategy is used: the receiver decodes a signal only when it is relatively sure that one particular message was actually transmitted, otherwise it requests (via the feedback channel) a retransmission. We show that as the coding delay T becomes large, we can transmit at an effective rate $\bar{R} \lt C = P_\deg / N_\deg,$ the channel capacity, with error probability Pe approximately exp ${- T[(\sqrt{C} - \sqrt{\bar{R}})^2 + C - \bar{R}]}$$, which is a considerable improvement over the reliability attainable with a one-way channel. These results parallel those obtained earlier by Forney for the discrete memoryless channel.