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Performance of Hybrid-ARQ in Block-Fading Channels: A Fixed Outage Probability Analysis

Peng Wu, Nihar Jindal

arXiv:0811.4191v3cs.IT

TL;DR

The paper addresses H-ARQ performance when fast fading prevents instantaneous channel-quality feedback and rate adaptation relies only on channel statistics. It analyzes long-term average transmitted rate under a maximum-round and fixed-outage constraint, finding that H-ARQ significantly outperforms non-H-ARQ systems and can approach ergodic capacity. Its conclusions are scoped to the assumed error-free mechanisms and block-fading setting.

  • Problem

    Prior H-ARQ research focused mainly on code design, leaving relatively little performance analysis for fast block-fading systems with statistics-based rate adaptation.

  • Method

    The paper uses a mutual-information-based analysis of long-term average transmitted rate with a maximum number of H-ARQ rounds and fixed target outage probability.

  • Results

    H-ARQ provides a significant rate advantage over equivalent non-H-ARQ systems and achieves rates reasonably close to ergodic capacity in many practical settings.

  • Takeaways & Limitations

    A few H-ARQ rounds provide implicit adaptation to instantaneous channel quality while retaining statistics-based rate selection.

  • Takeaways & Limitations

    The analysis assumes error-free mechanisms and independent block fading across H-ARQ rounds.

Abstract

from arXiv · show

This paper studies the performance of hybrid-ARQ (automatic repeat request) in Rayleigh block fading channels. The long-term average transmitted rate is analyzed in a fast-fading scenario where the transmitter only has knowledge of channel statistics, and, consistent with contemporary wireless systems, rate adaptation is performed such that a target outage probability (after a maximum number of H-ARQ rounds) is maintained. H-ARQ allows for early termination once decoding is possible, and thus is a coarse, and implicit, mechanism for rate adaptation to the instantaneous channel quality. Although the rate with H-ARQ is not as large as the ergodic capacity, which is achievable with rate adaptation to the instantaneous channel conditions, even a few rounds of H-ARQ make the gap to ergodic capacity reasonably small for operating points of interest. Furthermore, the rate with H-ARQ provides a significant advantage compared to systems that do not use H-ARQ and only adapt rate based on the channel statistics.

I. INTRODUCTION

The paper analyzes H-ARQ in fast block-fading channels when the transmitter knows only channel statistics and maintains a target outage probability. It finds that H-ARQ implicitly adapts to channel quality, outperforming non-H-ARQ systems and approaching ergodic capacity in many practical settings.

  • Problem and scope: When instantaneous channel quality is unavailable, H-ARQ provides implicit rate adaptation by terminating transmission once decoding becomes possible.This is useful when the amount or quality of received information is uncertain.
  • Problem and scope: H-ARQ analysis targets long-term average transmitted rate under a maximum number of rounds and a fixed post-H-ARQ outage probability.The rate is adapted to average SNR so the target outage remains constant across SNR values.
  • Main findings: H-ARQ generally provides a significant advantage over systems without H-ARQ operating with equivalent channel selectivity.The comparison uses the same value of M for H-ARQ and non-H-ARQ systems.
  • Main findings: H-ARQ rates are reasonably close to ergodic capacity in many practical settings, despite lacking instantaneous channel information at the transmitter.Ergodic capacity is achievable when instantaneous channel information is available to the transmitter.
  • Main findings: H-ARQ rates are much less sensitive to the desired outage probability than equivalent systems without H-ARQ.The fixed-outage setting reflects contemporary wireless systems, where an outage level near 1% is typical.
  • Metric choice: For fixed post-H-ARQ outage, studying attempted transmission rate is effectively equivalent to studying successful rate.The successful rate is the attempted rate multiplied by one minus the post-H-ARQ outage probability.

II. SYSTEM MODEL

The system models fast Rayleigh block fading with receiver CSI and statistical-only transmitter knowledge, using independent H-ARQ rounds and a bounded round budget. Fixed outage targets, power, coding, and selectivity assumptions define the comparison setting.

  • Channel and H-ARQ assumptions: The model uses Rayleigh fading with unit-power Gaussian inputs, i.i.d. complex-Gaussian block coefficients, and independent additive noise.The transmitted symbols satisfy a unit average-power constraint.
  • Scope: The paper focuses on Rayleigh fading and single-antenna systems, while noting that its basic insights can extend to other fading distributions and MIMO.Frequency-domain variation within an H-ARQ round is treated separately.
  • Channel and H-ARQ assumptions: The receiver has perfect CSI, while the transmitter knows only the channel distribution and not instantaneous channel quality.This open-loop setting models fading too fast for instantaneous channel feedback.
  • Channel and H-ARQ assumptions: The channel is constant within each H-ARQ round but independent across rounds, with at most M rounds before outage is declared.The no-H-ARQ baseline instead spans L fading blocks; comparisons set L = M to equalize maximum selectivity.
  • Channel and H-ARQ assumptions: The number of H-ARQ rounds is limited to control decoder memory complexity and delay.The decoder retains information from prior rounds, so larger round budgets increase these costs.

III. PERFORMANCE WITHOUT H-ARQ: FIXED-LENGTH CODING

Without H-ARQ, each codeword spans L fading blocks and operates at the largest rate satisfying the outage constraint. The paper characterizes this fixed-length benchmark exactly when possible and with high- and moderate-SNR approximations otherwise.

  • Fixed-length benchmark: The ε-outage capacity is the largest rate whose outage probability does not exceed ε.For L = 1 it can be obtained in closed form, whereas larger L generally requires numerical computation.
  • Approximations: For L > 1, the paper uses a high-SNR affine approximation and a Gaussian approximation based on the mean and variance of accumulated mutual information.The affine approximation is asymptotically tight at high SNR; the Gaussian approximation targets moderate and low SNRs.
  • Approximation accuracy: The Gaussian approximation becomes more accurate as L increases and is reasonably accurate over 0.01 ≤ ε ≤ 0.2 and 0 ≤ SNR ≤ 20 dB.Its accuracy is reduced for very small ε and at low SNR, where the mutual-information distribution is less Gaussian.
  • Approximation accuracy: The affine approximation provides the correct high-SNR rate offset, while the Gaussian approximation is reasonably accurate at moderate SNR and improves with larger L.These behaviors are illustrated for L = 3 and L = 10 at ε = 0.01.

A. Ergodic Capacity Gap

The fixed-length outage rate approaches ergodic capacity as diversity order L grows. The paper quantifies the convergence through a capacity-gap bound and a Gaussian approximation.

  • Convergence with diversity: The ε-outage capacity converges to the ergodic capacity µ(SNR) as L →∞, typically from below for ε < 0.5.This follows from concentration of accumulated mutual information across independent fading blocks.
  • Capacity gap: The capacity gap is defined as the difference between ergodic and ε-outage capacities.The paper uses this quantity to measure convergence speed.
  • Capacity gap: The rate gap decreases at least as fast as O(1/√L).A Gaussian approximation is then used to characterize this quantity more explicitly.

IV. PERFORMANCE WITH HYBRID-ARQ

Hybrid-ARQ exploits incremental redundancy and early termination to raise the long-term average rate under a fixed outage constraint. Its advantage over non-H-ARQ is substantial across a broad SNR range but disappears at high SNR.

  • Rate definition: The long-term H-ARQ rate is the attempted rate divided by the expected number of rounds per message.This captures the savings from early termination when decoding succeeds before M rounds.
  • H-ARQ operation: Incremental-redundancy H-ARQ accumulates mutual information across rounds and decodes at the smallest round count whose accumulated information exceeds Rinit.The number of rounds is capped at M, with outage when decoding remains impossible after the final round.
  • Outage relation: The H-ARQ outage expression matches M-order diversity without H-ARQ except that mutual information is summed rather than averaged across rounds.Defining Rinit per round and dividing by E[X] makes the rate expressions consistent.
  • Rate comparison: H-ARQ achieves at least the non-H-ARQ rate because E[X] ≤ M, with the benchmark advantage expressed through the corresponding multiplicative factor.The same information bits require MT symbols without H-ARQ but only E[X]T symbols on average with H-ARQ.
  • Rate comparison: H-ARQ with 6 rounds outperforms H-ARQ with 2 rounds.The comparison includes M = 1, 2, and 6, with ergodic capacity shown as a reference.
  • Rate comparison: H-ARQ provides a significant advantage over non-H-ARQ across a wide SNR range for the same M, but the advantage vanishes at high SNR.Increasing M raises the rate through additional time diversity and more early-termination opportunities.

A. Gaussian Approximation

The H-ARQ rate is approximated by replacing accumulated mutual information with Gaussian variables, producing a simpler and reasonably accurate computation.

  • The resulting expression approximates the H-ARQ rate through the inverse function A_M(·) and the outage constraint.
  • The accumulated mutual information in k rounds is approximated by a Gaussian with mean µ_k and variance σ^2_k.
  • The approximation is easier to compute than the actual H-ARQ rate and is reasonably accurate.It is also useful for obtaining analytical insights.

B. Scaling with H-ARQ Rounds M

Increasing the maximum number of H-ARQ rounds drives the rate toward ergodic capacity, with an approximately O(1/M) capacity gap caused primarily by round-level termination.

  • The H-ARQ rate converges to the ergodic capacity as M →∞ for any SNR.
  • The dominant finite-M penalty is the 0.5(1−ǫ) term caused by using only an integer number of H-ARQ rounds.
  • The rate gap decreases roughly as O(1/M), and exact numerical results exhibit the same order-1/M decay.
  • The Gaussian and simplified capacity-gap approximations are reasonably accurate, especially for large M.
  • If termination could occur precisely when enough mutual information was received, ergodic capacity would be achieved; H-ARQ instead incurs within-round transmission waste.
  • The analysis extends to multiple-antenna and within-round frequency- or time-diverse channels when their mutual-information mean and variance are accounted for.

C. Scaling with SNR

The H-ARQ advantage persists over a substantial SNR range but eventually vanishes as early termination becomes unlikely; optimizing the initial rate helps only over finite SNR ranges.

  • With fixed M and SNR tending to infinity, the expected number of H-ARQ rounds converges to M and the H-ARQ rate converges to C_M^ǫ(SNR).
  • As SNR increases, early termination becomes unlikely because accumulated mutual-information distributions overlap less and µ(SNR)/σ(SNR) grows.
  • The H-ARQ advantage persists across a large SNR range, and larger M extends the range over which it remains significant.
  • At 30 dB, the H-ARQ rate can behave non-monotonically with the initial rate, unlike the monotonic behavior observed at 10 dB.
  • Optimizing the initial rate can improve performance over a certain SNR range but does not improve the high-SNR offset.
  • For M = 2, initial-rate optimization helps around 25 dB but its advantage vanishes around 55 dB; for M = 6, the advantage begins at higher SNR.

D. Scaling with Outage Constraint ǫ

H-ARQ is generally less sensitive to the outage constraint than non-H-ARQ transmission, particularly at smaller outage probabilities.

  • H-ARQ rates are generally less sensitive to the desired outage probability ǫ than equivalent non-H-ARQ rates.
  • When ǫ is roughly 0.5, H-ARQ provides almost no advantage because the large initial rate makes early termination rare.
  • For more reasonable ǫ values, the H-ARQ rate is roughly constant while the non-H-ARQ rate decreases sharply as ǫ →0.
  • As the outage target becomes smaller, H-ARQ partially compensates for the required rate decrease through a lower expected number of rounds.

E. Chase Combining

Chase combining retransmits packets after NACKs and accumulates SNR across rounds, yielding a closed-form average rate but weaker high-SNR behavior than incremental redundancy.

  • Operation: Chase combining retransmits a packet after a NACK, and the receiver performs maximal-ratio combining across received packets.SNR, rather than mutual information, is accumulated over H-ARQ rounds.
  • Rate analysis: The expected number of Chase-combining rounds is independent of SNR, and its average rate for outage ǫ has a closed-form expression.The denominator of the rate expression is E[X].
  • High-SNR behavior: E[X] log2 SNR + 1 describes the high-SNR affine approximation for the average rate.
  • High-SNR behavior: Because E[X] > 1 for any positive outage value, Chase combining has pre-log factor 1/E[X], which is less than one and causes poor high-SNR performance.
  • Comparison with incremental redundancy: Even after rate optimization, Chase combining is far inferior to incremental redundancy at moderate and high SNR, while performing reasonably well at low SNR.At low SNR, log(1 + x) ≈ x makes SNR accumulation nearly equivalent to mutual-information accumulation.

V. CONCLUSION

The paper analyzes open-loop, fast-fading H-ARQ with rate selected from average SNR to meet a target outage probability. It finds substantial rate gains over non-H-ARQ and rates close to ergodic capacity despite limited channel selectivity.

  • Scope and findings: The study considers open-loop, fast-fading H-ARQ with transmission rate adjusted as a function of average SNR.The adjustment maintains a target outage probability that is not exceeded.
  • Scope and findings: H-ARQ provides a significant rate advantage over systems without H-ARQ at reasonable SNR levels.
  • Scope and findings: H-ARQ achieves a rate quite close to ergodic capacity even when channel selectivity is limited.
  • Extensions and limitations: Balancing simple ARQ layered on H-ARQ remains insufficiently understood in contemporary cellular systems.The paper also identifies erroneous ACK/NACK feedback as important for further consideration.
  • Extensions and limitations: Extending the results to discrete constellations and comparing them with actual-code performance are identified as interesting future directions.

APPENDIX I PROOF OF THEOREM 2

The appendix proves asymptotic results governing the expected number of H-ARQ rounds and the high-SNR pre-log and offset of rate choices. It shows that only specific initial-rate scaling avoids an asymptotic penalty.

  • Asymptotic approximation: For large numbers of rounds, a continuous approximation and the central limit theorem are used to evaluate E[X] asymptotically.The approximation replaces the discrete finite random variable with a continuous variable whose CDF uses the standard normal CDF.
  • High-SNR limits: As SNR tends to infinity, E[X] tends to M when the initial rate has pre-log M.
  • Initial-rate choices: An initial-rate pre-log r < M yields E[X] → ⌈r⌉ for non-integer r, producing a strictly suboptimal average-rate pre-log.
  • Initial-rate choices: For integer r < M with an O(1) rate offset, E[X] → r + δ with δ > 0, so the average-rate pre-log becomes r/(r + δ) < 1.
  • Initial-rate choices: Choosing Rinit = r log SNR − o(log SNR) achieves pre-log one but drives the rate offset to negative infinity.
  • Theorem conclusion: Except for the specified initial-rate choice, alternatives achieve either a strictly suboptimal pre-log or the correct pre-log with a strictly negative offset.
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