Error Exponent for Gaussian Channels With Partial Sequential Feedback

Abstract

This paper studies the error exponent of block coding over an additive white Gaussian noise channel where a fraction ( ) of the channel output symbols are revealed to the transmitter through noiseless feedback. If the code rate exceeds , where is the channel capacity, then the probability of decoding error cannot decay faster than exponentially with block length. However, if the code rate is below , the error probability can decrease faster than exponentially with the block length, as with full feedback ( ). This is achieved by combining a feedback code and a forward error control code, and jointly decoding them at the receiver. This scheme can attain higher reliability than rate splitting in which feedback and forward codes independently encode separate source messages.Index Terms-Error exponent, feedback, random coding exponent, Schalkwijk-Kailath (SK) coding.

Signaling over the Gaussian channel with intermittent feedback

“…For example, if x = (1, 2, 3, 4, 5) and s = (0, 1, 1, 0, 1), then x s = (2,3,5). The all-ones tuple is denoted 1, and if s is a binary tuple then s c stands for the componentwise difference 1 − s. The Euclidean norm and inner product are denoted · and ·, · .…”

“…Different feedback models have been studied, such as ratelimited [1], noisy [2]- [4] or partial feedback [5]. In [6] we considered the case introduced in [7] where the feedback is "intermittent," i.e., where each channel output is fed back with probability ρ.…”

Coding for the Gaussian channel with intermittent feedback

“…This can only decrease the probability of error. It is shown in [4] that when a fixed fraction f of the output symbols is fed back to the transmitter (the positions of the symbols that are fed back are fixed and known to the transmitter and receiver), and R is larger than f times the capacity, then the probability of error cannot decay faster than exponentially. To apply this result here, let K denote the event that the number of symbols fed back in the first n channel uses is no more than (ρ + ε)n, i.e.,…”

“…Our model is more pessimistic than the one in [4], and indeed our converse for the positive-rate case is based on the converse in [4]. In this paper, most of the effort goes into the achievability proofs.…”

“…Much of the work on feedback has focused on ideal feedback (as in [1] and references therein), but there has recently been growing interest in imperfect feedback, e.g., [4]- [8]. Most relevant to the present work is the model introduced in [2] where the feedback is "intermittent," i.e., where each channel output is fed back with some given probability ρ and the receiver is cognizant of which outputs were fed back.…”