[Paper Review] Modelling Decoding Errors in HARQ

Redouane Sassioui, Etienne Pierre-Doray+2 more|arXiv (Cornell University)|Dec 8, 2015

Advanced Wireless Network Optimization17 references7 citations

TL;DR

This paper proposes a deterministic errors model for hybrid ARQ (HARQ) that provides a more accurate and analytically tractable upper bound on the sequence of decoding errors compared to the widely used independent error assumption. By leveraging Gaussian vector projections and recursive probability bounds, the model achieves closed-form expressions for outage probability and average transmissions in block-fading channels, significantly improving performance evaluation and system-level simulation accuracy.

ABSTRACT

In this work we address the issues of probabilistic modelling of the decoding errors in hybrid ARQ (HARQ) rounds. In particular we i) claim that the assumption of independence of decoding errors, used implicitly in various works on this subject, is an approximation, and ii) propose equally simple but much more accurate method to calculate the probability of the sequence of decoding errors. The model we propose is useful from the point of view of performance evaluation, system-level simulation, and/or link adaptation. Its simplicity leads also to closed form expression for the outage probability and for the average number of transmissions in block-fading channel.

Motivation & Objective

• Address the inaccuracy of assuming independent decoding errors in HARQ protocols, which is commonly used but flawed in practical systems.
• Develop a simple yet precise analytical model for the sequence of decoding errors in repetition redundancy HARQ (RR-HARQ) with maximum ratio combining.
• Provide a formal upper bound on the error sequence probability that accounts for cumulative SNR and channel variations across rounds.
• Enable accurate performance evaluation, system-level simulation, and link adaptation by replacing complex multidimensional lookup tables with analytical expressions.
• Demonstrate that existing models relying on error independence can severely underestimate the true error sequence probability, especially in finite-length code scenarios.

Proposed method

• Propose a deterministic errors model based on the joint distribution of Gaussian random variables representing received signals across HARQ rounds.
• Use vector projections and conditional probability bounds to derive a recursive inequality that upper-bounds the error sequence probability.
• Leverage the independence of intermediate Gaussian variables (e.g., $y_l$) to decouple dependencies and simplify the probability calculation.
• Establish a recursive lower bound on the pairwise error probability (PEP) using the inequality $P_{l:k}(d) \geq \frac{1}{2} P_{l+1:k}(d)$, which enables closed-form analysis.
• Derive closed-form expressions for key metrics such as outage probability and average number of transmissions in i.i.d. block-fading channels.
• Validate the model’s accuracy by comparing it to the independent error assumption and showing its superiority in capturing real decoder behavior.

Experimental results

Research questions

• RQ1 How does the assumption of independent decoding errors affect the accuracy of HARQ performance modeling in practical systems?
• RQ2 What is a more accurate and analytically tractable model for the sequence of decoding errors in RR-HARQ with MRC combining?
• RQ3 Can a deterministic error model be derived that serves as a formal upper bound on the error sequence probability without relying on multidimensional lookup tables?
• RQ4 How does the proposed model compare to the widely used independent error assumption in terms of predicting outage probability and average number of transmissions?
• RQ5 What are the implications of the proposed model for system-level simulation and link adaptation in block-fading channels?

Key findings

• The proposed deterministic errors model provides a formal upper bound on the sequence of decoding errors, $f_k = \Pr\{\mathsf{ERR}_1, \ldots, \mathsf{ERR}_k\}$, which is significantly more accurate than the independent error assumption.
• The model reveals that the independent error assumption used in prior works can severely underestimate the true error sequence probability, especially in systems with finite-length codewords.
• The recursive inequality $P_{l:k}(d) \geq \frac{1}{2} P_{l+1:k}(d)$ enables a closed-form expression for the outage probability in i.i.d. block-fading channels.
• The average number of transmissions in block-fading channels can be computed in closed form using the proposed model, enabling efficient system design and optimization.
• The model’s simplicity allows for direct integration into system-level simulations and link adaptation algorithms without requiring complex lookup tables.
• The theoretical bounds are validated through probabilistic analysis of Gaussian projections and conditional independence, ensuring mathematical rigor and practical relevance.