Achievable Error Exponents in the Gaussian Channel With Rate-Limited Feedback

Abstract

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].

Reliability of the Gaussian broadcast channel with common message and feedback

“…Proof: In Section IV-A we present a bursty-feedback scheme achieving the desired performance; it is based on the scheme in [10], see also [11]. Its analysis is omitted.…”

“…In fact, we present a sequence of coding schemes that uses the feedback in a bursty way (that means the feedback signals are used only in very few transmissions as in [10]) and that can achieve double-exponential decay of the probability of error for all rates up to capacity. Moreover, in our scheme it suffices to have rate-limited feedback with feedback rate R fb no smaller than the forward rate R.…”

Energy-Efficient Communication via Feedback

“…see [9], [10] and [11]). In particular, [10] considers communication over AWGN forward and feedback links each of which is allocated a certain transmit power.…”

“…For a point-to-point communication system, the presence of a feedback link can lead to large improvements in reliability and efficient use of the transmitter's resources. For example, given an AWGN channel with a transmit energy constraint, the use of feedback can boost the error exponent and simplify the capacity achieving encoding/decoding scheme (see [3]- [11]). However, the benefits of feedback come at the cost of extra energy consumption on the feedback link.…”

Signaling over the Gaussian channel with intermittent feedback

“…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.…”