[Paper Review] Variable-Length Coding with Stop-Feedback for the Common-Message Broadcast Channel in the Nonasymptotic Regime

Kasper Fløe Trillingsgaard, Wei Yang+2 more|arXiv (Cornell University)|Jan 1, 2016

Wireless Communication Security Techniques18 references2 citations

TL;DR

This paper studies variable-length coding with stop-feedback for common-message transmission over K-user discrete memoryless broadcast channels in the nonasymptotic regime. It derives tighter achievability and converse bounds than prior work, showing that zero dispersion—implying fast convergence to capacity—is unattainable for certain channels like the binary symmetric broadcast channel, and that convergence is slower than in point-to-point settings.

ABSTRACT

We investigate the maximum coding rate for a given average blocklength and error probability over a K-user discrete memoryless broadcast channel for the scenario where a common message is transmitted using variable-length stop-feedback codes. For the point-to-point case, Polyanskiy et al. (2011) demonstrated that variable-length coding combined with stop-feedback significantly increase the speed of convergence of the maximum coding rate to capacity. This speed-up manifests itself in the absence of a square-root penalty in the asymptotic expansion of the maximum coding rate for large blocklengths, i.e., zero dispersion. In this paper, we present nonasymptotic achievability and converse bounds on the maximum coding rate of the common-message K-user discrete memoryless broadcast channel, which strengthen and generalize the ones reported in Trillingsgaard et al. (2015) for the two-user case. An asymptotic analysis of these bounds reveals that zero dispersion cannot be achieved for certain common-message broadcast channels (e.g., the binary symmetric broadcast channel). Furthermore, we identify conditions under which our converse and achievability bounds are tight up to the second order. Through numerical evaluations, we illustrate that our second-order expansions approximate accurately the maximum coding rate and that the speed of convergence to capacity is indeed slower than for the point-to-point case.

Motivation & Objective

• To analyze the maximum coding rate for common-message transmission over K-user discrete memoryless broadcast channels under nonasymptotic constraints.
• To extend prior two-user results to the K-user case using nonasymptotic bounds on rate, blocklength, and error probability.
• To investigate whether zero dispersion—implying fast convergence to capacity—can be achieved with stop-feedback in broadcast channels.
• To identify conditions under which the derived bounds are tight up to the second order in blocklength.
• To numerically evaluate the accuracy of second-order approximations to the maximum coding rate.

Proposed method

• Derives nonasymptotic achievability and converse bounds on the maximum coding rate for variable-length stop-feedback codes in the K-user common-message broadcast channel.
• Generalizes the approach of Polyanskiy et al. (2011) from point-to-point to K-user broadcast settings, incorporating stop-feedback to reduce average blocklength.
• Performs asymptotic analysis of the bounds to determine the dispersion term, revealing non-zero dispersion in certain channels like the binary symmetric broadcast channel.
• Uses second-order asymptotic expansions to approximate the maximum coding rate, comparing them to numerical evaluations.
• Applies techniques from finite-blocklength information theory, including the use of information spectrum methods and error exponent analysis.
• Validates the accuracy of the bounds through numerical evaluations across various channel parameters.

Experimental results

Research questions

• RQ1 Can variable-length coding with stop-feedback achieve zero dispersion in the K-user common-message broadcast channel?
• RQ2 How does the convergence speed of the maximum coding rate to capacity compare between point-to-point and K-user broadcast channels under stop-feedback?
• RQ3 Under what channel conditions are the derived achievability and converse bounds tight up to the second order?
• RQ4 How accurately do second-order asymptotic expansions approximate the maximum coding rate in the nonasymptotic regime?
• RQ5 What is the impact of feedback on the trade-off between blocklength, error probability, and rate in broadcast channels?

Key findings

• Zero dispersion cannot be achieved in certain common-message broadcast channels, such as the binary symmetric broadcast channel, indicating slower convergence to capacity than in the point-to-point case.
• The convergence speed of the maximum coding rate to capacity is slower under stop-feedback in the K-user broadcast setting compared to the point-to-point case.
• The derived nonasymptotic bounds are tighter and generalize prior results from the two-user case to the K-user scenario.
• Second-order asymptotic expansions of the bounds accurately approximate the maximum coding rate in the nonasymptotic regime, as confirmed by numerical evaluations.
• The bounds are tight up to the second order under specific channel conditions, providing a refined characterization of finite-blocklength performance.
• The presence of a non-zero dispersion term in the asymptotic expansion confirms that feedback does not eliminate the fundamental delay penalty in broadcast channels as it does in point-to-point links.