Error exponents for block Markov superposition encoding with varying decoding latency

Abstract

Abstract-Block Markov superposition encoding has been used on a number of channels to enable transmitter cooperation, including the decode-and-forward (DF) relaying scheme on the full-duplex relay channel. We analyze the error performance of DF with regular encoding and sliding window decoding as the window size of the decoder is allowed to grow. Specifically, we use Gallager's random coding exponent to analyze the behavior of DF in the finite block length regime where the error probability cannot be made arbitrarily small for a fixed rate and block length. Although using a larger decoding window may not result in a better achievable rate in the infinite block length regime, doing so for finite block lengths enables a higher rate of transmission for a given error probability. In particular, these rate enhancements can lead to a larger range of operating scenarios in which relaying can outperform direct transmission.

Achievable Rates for Gaussian Degraded Relay Channels With Non-Vanishing Error Probabilities

“…rate of decay of the error probability; (ii) The non-vanishing error regime where one is also possibly concerned with the second-order asymptotics in addition to the ε-capacity. For the former, Bradford and Laneman [12] and Tan [13] derived bounds on the error exponent (reliability function) of the discrete memoryless RC. Also see [14]- [16] for other related works on error exponents for RCs.…”

On the Reliability Function of the Discrete Memoryless Relay Channel

“…As in [9], we use block-Markov coding to send a message M representing N R eff = nbR eff bits of information over the DM-RC. We use the channel N times and this total blocklength is partitioned into b correlated blocks each of length n. We term n, a large integer, as the per-block blocklength.…”

Error exponents for the relay channel

“…Note that F (R ) is the exponent at the relay and G(R ) andG(R ) are the exponents at the decoder. Setting U = X 1 recovers DF for which the exponent is provided in [7]. Note that (3) indicates a tradeoff between rate and error probability: as b increases, R eff increases but the error exponent decreases.…”

“…The work that is most closely related to this paper is [7] in which the authors derived the error exponent for slidingwindow DF based on Gallager's Chernoff-bounding techniques [8]. We generalize their result to PDF and we use MMI decoding [4].…”

“…The average error probability is P(ϕ(Y n 3 ) = M ). As in [7], for both PDF and CF, we will use block-Markov coding to send a message M representing NR eff bits of information. We use the channel N = nb times and this total blocklength is partitioned into b blocks each of blocklength n. The number of blocks b is fixed and regarded as a constant (does not grow with n).…”