Abstract
Achievable error exponents for the one-way with noisy feedback and two-way AWGN channels are derived for the transmission of a finite number of messages M under almost sure (AS) and expected block (EXP) transmit power constraints. In the one-way setting under noisy AWGN feedback, under an AS power constraint, known linear and non-linear passive schemes are modified to incorporate AS constraints in the feedback link as well. In addition, a new active feedback scheme is presented in which the receiver feeds back the most likely pair of codewords, and the transmitter re-transmits which of these two was originally sent. This active feedback scheme outperforms one of the passive feedback schemes for all channel parameters; the linear scheme outperforms the others for low feedback noise variance. Under the EXP constraint, a known achievable error exponent for the transmission of two messages is generalized to any arbitrary but finite number of messages M through the use of simplex codes and erasure decoding. In the two-way AWGN setting, each user has its own message to send in addition to (possibly) aiding in the transmission of feedback for the opposite direction. Two-way error exponent regions are defined and achievable error exponent regions are derived for the first time under both AS and EXP power constraints. For the presented achievability schemes, feedback or interaction leads to error exponent gains in one direction, possibly at the expense of a decrease in the error exponents attained in the other direction. The relationship between M and n supported by our achievable strategies is explored.
Highlights
T HE reliability function [1]–[3], or error exponent, of a one-way channel characterizes the rate of decay of the probability of error when communicating one of 2nR messages as E(R) = lim n→∞ − 1 n logP(en), Manuscript received December 3, 2018; revised February 16, 2020; accepted February 16, 2021
For the two-way additive white Gaussian noise (AWGN) channel: 5) Theorems 5, 6 and 7 demonstrate – for the first time – achievable error exponent regions for the two-way AWGN channel under the almost sure (AS) power constraint, provided that one channel’s SNR is better than the other. This non-symmetric SNR scenario is of interest since for the one-way AWGN channel under the AS constraint, only a feedback link significantly stronger than the forward has been shown to lead to error exponent gains over feedback-free transmissions
Definition 2: The error exponent region (EER) for the two-way AWGN channel and the transmission of M messages corresponds to the union over all achievable error exponent pairs (E12, E21)Ψ, where we will often drop the arguments of Eij for simplicity and sometimes we may refer to (E12, E21)AS as (E1A2S, E2A1S)
Summary
T HE reliability function [1]–[3], or error exponent, of a one-way channel characterizes the rate of decay of the probability of error when communicating one of 2nR messages as. While even perfect feedback cannot increase the capacity of non-anticipatory DMC channels [7], it may greatly improve the error exponents achieved This was first demonstrated for the AWGN channel under the EXP power constraint in [8], in which the probability of error was made to decay double-exponentially in the block length n; [9] later demonstrated a decay rate equal to any number of exponential levels. In this article we continue this line of work and study error exponents of one-way additive white Gaussian noise (AWGN) channels with noisy AWGN feedback for the transmission of a finite number of messages (zero-rate). We show that the same may not be said about error exponent regions: that is, adaptation does improve the error exponents of this two-way AWGN channel for the transmission of a finite number of messages. The authors of [13] comment that this “trick” can not be used for general DMCs, but is useful for continuous-valued channels characterized by additive noise
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