We investigate the achievable error probability in communication over an AWGN discrete time memoryless channel with noiseless delay-less rate-limited feedback. For the case where the feedback rate R F B is lower than the data rate R transmitted over the forward channel, we show that the decay of the probability of error is at most exponential in blocklength, and obtain an upper bound for increase in the error exponent due to feedback. Furthermore, we show that the use of feedback in this case results in an error exponent that is at least R F B higher than the error exponent in the absence of feedback. For the case where the feedback rate exceeds the forward rate (R F B ≥ R), we propose a simple iterative scheme that achieves a probability of error that decays doubly exponentially with the codeword blocklength n. More generally, for some positive integer L, we show that a L th order exponential error decay is achievable if R F B ≥ (L − 1)R. We prove that the above results hold whether the feedback constraint is expressed in terms of the average feedback rate or per channel use feedback rate. Our results show that the error exponent as a function of R F B has a strong discontinuity at R, where it jumps from a finite value to infinity. Ω(n) times(Ω(n)))). 2 However, once the feedback channel is corrupted with some noise, the benefits of feedback in terms of the error probability decay rate can drop. In fact, when this corruption corresponds to an additive white Gaussian noise on the feedback channel, the Schalkwijk-Kailath communication scheme (or any other linear scheme) fails to achieve any nonzero rate with vanishing error probability [4]. Furthermore, in this case, the achievable error decay for any coding scheme can be no better than exponential in blocklength [5], similar to the case without feedback [1].

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We propose a feedback optimization framework to minimize the total energy consumption in point-to-point wireless communication links. The energy cost of both the forward link and the feedback link are taken into account. Given the energy consumption profile of both links, we minimize error probability subject to the total energy budget and a delay constraint. The proposed framework is based on a multi-phase feedback scheme in which a transmission, if decoded incorrectly, is followed by a retransmission with boosted energy. We use this framework to show that the gain of utilizing feedback is highly dependent on the energy consumption profile of the links and the total available energy. In particular, we identify scenarios in which the use of feedback significantly increases the energy efficiency, as well as scenarios where, surprisingly, the use of feedback is strictly suboptimal as compared to communication without feedback.Index Terms-Energy efficiency, feedback communication, interactive communication.