Chernoff Bounds for Analysis of Rate-Compatible Sphere-Packing with Numerous Transmissions

TL;DR

This paper develops Chernoff bound techniques to analyze rate-compatible sphere-packing with many retransmissions, providing bounds that are computationally efficient and useful for optimizing throughput and latency.

Contribution

It introduces Chernoff bounds for RCSP analysis with numerous retransmissions, extending previous exact methods and enabling more practical computations.

Findings

Derived upper and lower bounds for two-transmission RCSP

Extended bounds to scenarios with multiple retransmissions (m ≥ 1)

Bounds effectively estimate throughput and latency for optimization

Abstract

Recent results by Chen et al. and Polyanskiy et al. explore using feedback to approach capacity with short blocklengths. This paper explores Chernoff bounding techniques to extend the rate-compatible sphere-packing (RCSP) analysis proposed by Chen et al. to scenarios involving numerous retransmissions and different step sizes in each incremental retransmission. Williamson et al. employ exact RCSP computations for up to six transmissions. However, exact RCSP computation with more than six retransmissions becomes unwieldy because of joint error probabilities involving numerous chi-squared distributions. This paper explores Chernoff approaches for upper and lower bounds to provide support for computations involving more than six transmissions. We present two versions of upper and lower bounds for the two-transmission case. One of the versions is extended to the general case of $m$ transmissions where $m \geq 1$. Computing the general bounds requires minimization of exponential functions with the auxiliary parameters, but is less complex and more stable than multiple rounds of numerical integration. These bounds also provide a good estimate of the expected throughput and expected latency, which are useful for optimization purposes.