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Finite Block-length Analysis of the Incremental Redundancy HARQ
Behrooz Makki, Tommy Svensson, Michele Zorzi
TL;DR
Finite-length codewords create a need to reassess INR HARQ throughput because conventional analyses assume asymptotically long codes, while several applications require short codes. The paper combines finite-blocklength achievable-rate results with throughput and outage analysis, including feedback delay. It finds throughput improvements over fixed-length INR and open-loop transmission, especially at high SNR and across a large range of feedback delays.
Problem
Existing HARQ performance analyses assume asymptotically long codewords, although applications such as vehicle communications and augmented-reality video processing require short codewords.
Method
The paper uses finite-blocklength achievable-rate results to derive INR HARQ outage probabilities and throughput expressions, then analyzes variable-length coding and feedback delay.
Results
Finite-length variable-length INR HARQ increases throughput compared with fixed-length INR HARQ and open-loop transmission, particularly at high SNRs.
Takeaways & Limitations
Finite-length INR HARQ improves throughput over a large range of feedback delays when sub-codeword lengths are properly adapted.
Abstract
from arXiv · showhide
This letter studies the power-limited throughput of a communication system utilizing incremental redundancy (INR) hybrid automatic repeat request (HARQ). We use some recent results on the achievable rates of finite-length codes to analyze the system performance. With codewords of finite length, we derive closed-form expressions for the outage probabilities of INR HARQ and study the throughput in the cases with variable-length coding. Moreover, we evaluate the effect of feedback delay on the throughput and derive sufficient conditions for the usefulness of the HARQ protocols, in terms of power-limited throughput. The results show that, for a large range of HARQ feedback delays, the throughput is increased by finite-length coding INR HARQ, if the sub-codeword lengths are properly adapted.
I. INTRODUCTION
The paper addresses HARQ performance when practical applications require short codewords rather than the asymptotically long codewords assumed in prior analyses. It studies finite-length INR HARQ throughput and feedback-delay effects.
- HARQ is commonly used in wireless networks to combat packet loss caused by channel fading.
- Earlier HARQ analyses generally assume asymptotically long codewords.
- Vehicle communications and augmented-reality video processing require codewords of roughly 100 channel uses.
- The paper maximizes power-limited throughput for finite-length INR HARQ and derives outage expressions across retransmission rounds.
- For a large range of feedback delays, finite-length INR HARQ improves throughput when sub-codeword lengths are properly adapted.
II. SYSTEM MODEL
The system is a point-to-point quasi-static fading channel using INR HARQ with a maximum of M transmissions. The receiver knows the channel in each slot, while the transmitter receives only HARQ feedback bits as channel information.
- The model considers point-to-point transmission with transmit power P, unit-variance input X, fading coefficient h, and iid complex Gaussian noise Z.
- Each INR HARQ packet is transmitted over at most M rounds.
- Channel coefficients remain constant during a packet transmission and change between packets according to the fading distribution.
- The receiver knows the channel coefficient in each slot, but the transmitter has no instantaneous channel state information beyond HARQ feedback bits.
III. ANALYTICAL RESULTS
The analysis derives finite-blocklength INR HARQ throughput by combining renewal-reward expressions with achievable-rate approximations, outage approximations, and bounds. It also incorporates feedback delay and identifies coding requirements for realizing HARQ gains.
- The throughput expression includes feedback delay and depends on probabilities determined by the sub-codeword lengths.
- A parent codeword is punctured into M successive sub-codewords, and the receiver combines them until decoding succeeds or M rounds are reached.
- The equivalent rate after round m is determined by the information payload divided by the accumulated sub-codeword length.
- Feedback delay D contributes to the total channel uses, with mD added when transmission stops before the maximum round and (M−1)D at the maximum round.
- Finite-blocklength achievable-rate results are used to calculate outage probabilities, while approximation and bounding techniques handle probabilities lacking closed forms.
- The approximation is assumed exact for analysis because it is very tight for sufficiently large blocklengths.
- Useful INR HARQ requires a puncturable parent code with rate-optimized sub-codewords and a decoder approaching the finite-length performance expression across retransmissions.
A. On the Effect of Feedback Cost
HARQ can save transmission resources when decoding succeeds early, but feedback delay can offset those savings. Variable-length coding tolerates a larger range of feedback delays than fixed-length coding.
- Feedback-cost trade-off: When channel quality is low, HARQ uses all transmissions and approaches non-HARQ performance apart from additional feedback delays.At high channel quality, early decoding can save the channel uses allocated to later rounds.
- Feedback-delay conditions: The acceptable feedback-delay range is found by optimizing throughput for fixed power and sweeping the relative feedback delay.The resulting maximum delay is the threshold below which HARQ exceeds non-HARQ throughput.
- Feedback-cost trade-off: HARQ and non-HARQ have the same outage probability under fixed packet length, so HARQ improves throughput when its expected delay is lower.This yields a sufficient usefulness condition based on comparing expected delay with the open-loop packet length.
- Feedback-delay conditions: Sufficient conditions for useful HARQ are obtained by bounding the probability terms governing expected delay.The analysis uses Lemma 2 and the decreasing dependence of the throughput expression on Ω_m.
- Feedback-delay conditions: Variable-length HARQ tolerates a larger range of feedback delays than fixed-length HARQ.The fixed-length condition provides a sufficient bound, while variable-length coding performs better according to the paper.
IV. NUMERICAL RESULTS AND CONCLUSIONS
The numerical study compares variable- and fixed-length INR HARQ with open-loop communication under optimized finite-length settings. Variable-length coding increases throughput, especially at high SNR, while feedback-delay effects are strongest at medium SNR and worsen as K increases.
- Numerical results: The finite-length approximations are sufficiently tight for sub-codewords with l_m ≥100 channel uses, the minimum length considered numerically.The approximation tightness increases with sub-codeword length.
- Numerical results: Variable-length INR increases throughput over fixed-length INR and open-loop communication, especially at high SNRs.Figure 1a compares throughput, while Figure 1b reports the variable-length HARQ throughput gain.
- Numerical results: The acceptable feedback-delay range is very low at medium SNR, whereas its effect is relaxed at low and high SNRs.The bounds and sufficient condition are reported as very tight over a large SNR range.
- Numerical results: Figure 1 evaluates throughput, throughput gain, and acceptable relative feedback delay versus SNR for M = 2 transmissions.Throughput optimization jointly selects the number of information nats and sub-codeword lengths in subplot (a).
- Numerical results: The acceptable feedback-delay range decreases as K increases.The figure uses K = 600 nats for throughput gain and K = 300, 600 nats for acceptable relative feedback delay.
- Conclusions: Finite-length INR HARQ increases throughput for a large range of feedback delays, particularly when SNR increases.This is the paper’s concluding numerical finding when sub-codeword lengths are appropriately selected.