On Digital Communication Over a Discrete-Time Gaussian Channel with Noisy Feedback

We consider the problem of transmission of digital data over a discrete-time Gaussian channel with the use of a Gaussian feedback channel. We are particularly interested in the case where the signal-to-noise ratio in the feedback channel is finite. In Sections II and I I I we consider Elias' scheme and a simple extension for transmitting analog data over this channel with feedback. 1,2 In Section IV we apply this extended Elias scheme to the digital transmission problem. The main result is that for any rate R R*, a number less than the channel capacity, it is possible to transmit digital data at a rate R with error probability P. = exp [ -- E*n,, + o(n0)], as n0 --» oo, where n 0 is the encoding-decoding delay, and E* > Ei, the "one-way" exponent estimated by Shannon. 3 In particular, when R = 0, E x = p/4 and E* = (p/4)[l + p(l + p) -1 ], where p and p are the forward and 3173 3174 T H E BELL SYSTEM TECHNICAL JOURNAL, DECEMBER 1969 feedback signal-to-noise ratios respectively. Finally, we suggest a modfication of this scheme which will probably permit extending R* to capacity. Stimulated by the work of Schalkwijk and Kailath, a great deal of research has been done on this problem (see for example Refs. 4-11). To the present author's knowledge, however, the result in this paper is the first to show that a noisy feedback channel can improve the errorexponent for digital communication on a band-limited channel. (References 4 and 8 treat the infinite band case.) Like the optimal coding schemes for the one-way channel, our scheme is not constructive.