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Coding Versus ARQ in Fading Channels: How reliable should the PHY be?

Peng Wu, Nihar Jindal

arXiv:0904.0226v2cs.IT

TL;DR

The paper asks how channel coding and ARQ should share reliability in Rayleigh block-fading channels. It formulates the choice as a goodput-maximizing optimization over packet error probability and transmitted rate, then shows that the optimum error probability decreases with average SNR and channel diversity. Across a wide range of channel parameters, an error probability around 10% can be near-optimal, so very reliable PHY operation may substantially reduce throughput.

  • Problem

    The paper addresses how to choose the PHY reliability level and transmitted rate when ARQ retransmissions trade off against coding reliability in fading channels.

  • Method

    The paper optimizes long-term successful throughput over the packet error probability and its associated transmitted rate, while analyzing idealized and practical ARQ settings.

  • Results

    An error probability of 10% is optimal or near-optimal for a wide range of channel parameters, and the optimum decreases with average SNR and diversity order.

  • Takeaways & Limitations

    Cross-layer design can favor a relatively unreliable PHY because extra coding requires a conservative rate in fading channels, whereas ARQ handles retransmissions.

Abstract

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This paper studies the tradeoff between channel coding and ARQ (automatic repeat request) in Rayleigh block-fading channels. A heavily coded system corresponds to a low transmission rate with few ARQ re-transmissions, whereas lighter coding corresponds to a higher transmitted rate but more re-transmissions. The optimum error probability, where optimum refers to the maximization of the average successful throughput, is derived and is shown to be a decreasing function of the average signal-to-noise ratio and of the channel diversity order. A general conclusion of the work is that the optimum error probability is quite large (e.g., 10% or larger) for reasonable channel parameters, and that operating at a very small error probability can lead to a significantly reduced throughput. This conclusion holds even when a number of practical ARQ considerations, such as delay constraints and acknowledgement feedback errors, are taken into account.

I. INTRODUCTION

The paper studies how transmitted rate and ARQ retransmissions should be jointly chosen when the transmitter lacks instantaneous channel information. It finds that maximizing goodput often favors a relatively unreliable PHY, with the optimum error probability decreasing as average SNR and fading selectivity increase.

  • I. INTRODUCTION: The central design problem is choosing the transmitted rate that maximizes the rate at which bits are successfully delivered.Higher rates carry more information per packet but require more retransmissions, whereas lower rates reduce retransmissions but carry fewer bits.
  • I. INTRODUCTION: The rate–ARQ tradeoff is nontrivial when the transmitter chooses rate from fading statistics because packet error probability depends on transmitted rate.The setting excludes instantaneous channel-state information at the transmitter, as in high-velocity cellular systems.
  • I. INTRODUCTION: Making the PHY too reliable can significantly reduce achieved goodput, the long-term average successful throughput.The paper frames PHY reliability as an optimization variable rather than an independently minimized error probability.
  • I. INTRODUCTION: The goodput-maximizing packet error probability decreases with average SNR and fading selectivity.Thus, the preferred PHY reliability depends systematically on channel conditions rather than having one universal target.
  • I. INTRODUCTION: For many channel parameters, an error probability of 10% yields near-optimal performance, while practical ARQ constraints are also considered.The practical considerations include limits on ARQ retransmissions and unreliable acknowledgement feedback.

A. Prior Work

The paper distinguishes its unicast, diversity-aware analysis from prior work on packet-level erasure correction and PHY coding. It additionally explains how optimal PHY reliability depends on diversity and average SNR while incorporating ARQ-specific issues.

  • A. Prior Work: The relevant performance metric is the product of transmitted rate and packet success probability, matching the goodput metric in the idealized ARQ setting.
  • A. Prior Work: Unlike prior work focused on multicast or unicast without per-transmission diversity, this paper analyzes unicast with diversity per transmission.
  • A. Prior Work: The analysis provides a general explanation of how PHY reliability depends on both diversity and average SNR.
  • A. Prior Work: The paper extends the analysis to ARQ-specific practical issues, including acknowledgement errors and hybrid-ARQ.

II. SYSTEM MODEL

The system model uses a Rayleigh block-fading channel without instantaneous transmitter channel information and evaluates simple ARQ through long-term average goodput. Packet error probability is linked to rate, average SNR, and diversity experienced across each codeword.

  • II. SYSTEM MODEL: Each codeword spans L independent Rayleigh fading blocks, so L represents the time/frequency selectivity experienced per transmission.The model uses a single-antenna channel with complex Gaussian channel gains and additive Gaussian noise.
  • II. SYSTEM MODEL: For a strong code, packet error probability is approximated by mutual-information outage probability and depends on average SNR, selectivity order L, and transmitted rate R.For fixed SNR and L, the rate can be expressed as a function of the target error probability ε.
  • II. SYSTEM MODEL: Simple ARQ retransmits incorrectly decoded packets using only the most recent transmission, with ACKs advancing to the next packet and NACKs triggering retransmission.
  • II. SYSTEM MODEL: The system’s performance metric is long-term average goodput, defined as the rate at which successfully received packets deliver bits.Fading is assumed independent across retransmissions under the no-CSIT, fast-fading model.

III. OPTIMAL PHY RELIABILITY IN THE IDEAL SETTING

Under idealized ARQ assumptions, the paper optimizes PHY reliability by maximizing goodput over the rate–error-probability tradeoff. It finds that both excessively reliable and excessively unreliable operation perform poorly, with the optimum error probability decreasing in SNR and diversity order.

  • III. OPTIMAL PHY RELIABILITY IN THE IDEAL SETTING: The idealized analysis assumes mutual-information-limit codes, perfect error detection, unlimited retransmissions, and perfect ACK/NACK feedback.
  • III. OPTIMAL PHY RELIABILITY IN THE IDEAL SETTING: The goodput-maximizing operating point is found by optimizing over ε, which determines the corresponding transmitted rate for fixed SNR and L.This defines the optimal packet error probability and PHY reliability level as functions of the channel parameters.
  • III. OPTIMAL PHY RELIABILITY IN THE IDEAL SETTING: For L > 1, the optimum is obtained numerically because the outage probability lacks a closed-form expression, while a Gaussian approximation provides an analytical approximation.
  • III. OPTIMAL PHY RELIABILITY IN THE IDEAL SETTING: Making the PHY too reliable or too unreliable yields poor goodput, while the optimal outage probability decreases with SNR and L.These are identified as the key behaviors of the coding–ARQ tradeoff.
  • III. OPTIMAL PHY RELIABILITY IN THE IDEAL SETTING: With transmitter channel-state information, the optimization becomes trivial because the transmitter can choose a rate just below instantaneous capacity.

B. Optimization of Goodput Approximation

The paper characterizes the goodput-maximizing PHY reliability through a fixed-point optimization and examines how it varies with channel conditions. The optimum error probability decreases with diversity and SNR, while overly reliable operation can impose substantial performance penalties.

  • The PHY reliability maximizing Gaussian-approximated goodput is the unique solution of a fixed-point equation.The optimized quantity depends on channel parameters through κ, and the solution function g is increasing in κ.
  • The optimum error probability decreases as channel diversity order L and SNR increase.The paper reports that g decreases in both L and SNR, and explains the SNR trend through a steeper success-probability curve versus normalized rate.
  • At SNR = 10 dB, goodput for L = 5 peaks at transmitted rate R = 2.3 bits/symbol because higher rates thereafter cause too many retransmissions.Before the peak, increased transmission rate compensates for reduced success probability; beyond it, the retransmission increase dominates.
  • Increasing diversity steepens the success-probability curve and produces a larger optimum rate and optimum success probability.The paper links this to the goodput curve moving closer to the transmitted-rate curve.
  • The optimum error probabilities remain quite large, because reducing errors beyond the optimum sacrifices transmission rate faster than it saves ARQ retransmissions.This tradeoff makes operating the PHY too reliably inefficient even at high SNR and large selectivity.
  • Choosing ε = 0.001 instead of the optimum incurs approximately 10 dB and 2 dB power penalties for L = 2 and L = 10, respectively.By contrast, ε = 0.1 provides near-optimal performance for both selectivity values.
  • The main conclusion remains valid when finite blocklength, imperfect error detection, limited ARQ rounds, and imperfect feedback are considered.

A. Finite Codeword Block-length

The finite-blocklength analysis incorporates atypical noise realizations into the outage approximation and compares the resulting reliability tradeoff with the infinite-blocklength case. Finite blocklength reduces curve steepness and generally increases the optimal error probability, while the two regimes converge at high SNR.

  • The finite-blocklength outage model adds a noise-dependent term to the infinite-blocklength mutual-information expression.The term vanishes as n grows but cannot be ignored for finite n.
  • Finite blocklength reduces the steepness of the success-probability versus transmitted-rate curve because of atypical noise realizations.
  • Finite-blocklength coding produces a larger optimal error probability than the infinite-blocklength case.The paper attributes this to the reduced steepness of the success-rate curves.
  • At high SNR, the finite- and infinite-blocklength optimal-reliability curves almost overlap.The unusual noise term becomes negligible at large SNR, so finite blocklength does not significantly change the idealized reliability level.

B. Non-ideal Error Detection

The paper analyzes imperfect error detection by constraining undetected errors while maximizing goodput. It finds that CRC overhead is a more efficient way to satisfy the reliability constraint than forcing the PHY packet error probability below its goodput-optimal value.

  • The overall undetected-error probability is approximated by ε · 2^-k, where ε is packet error probability and k is the number of CRC parity bits.
  • The design problem maximizes goodput subject to a constraint on undetected error probability.The constraint can be met by increasing CRC overhead k or reducing the PHY packet error probability ε.
  • The constraint should therefore be satisfied by choosing k sufficiently large while leaving the PHY transmitted rate nearly unchanged.The corresponding choice is k⋆ = ⌈−log2(p/ε⋆)⌉.
  • Reducing ε below the perfect-error-detection optimum decreases goodput because the rate loss exceeds the small CRC-overhead reduction.

C. End-to-End Delay Constraint

The delay-constrained analysis limits ARQ retransmissions and packet discards, then extends the setting to unreliable acknowledgement feedback. Goodput favors the largest error probability allowed by the constraints, while weaker forward reliability must be compensated by more reliable feedback.

  • A delay constraint does not affect goodput because it changes which packets are delivered in different slots, not the long-term delivery rate.
  • Under strict delay and reliability constraints, goodput is maximized by selecting the largest allowed packet error probability.Thus ARQ should be used to the maximum extent permitted, even though strict constraints limit aggressive retransmission.
  • The noisy-feedback problem jointly optimizes forward-data and reverse acknowledgement reliability.The feedback channel is modeled with Rayleigh fading, diversity order Lfb, repeated BPSK symbols, and acknowledgement overhead.

2) Performance Analysis:

The analysis models packet delivery under bounded-round ARQ with decoding and acknowledgement errors, then optimizes goodput under reliability and feedback-overhead tradeoffs. It shows that feedback reliability and channel diversity strongly affect the error probability and PHY operating point that maximize throughput.

  • NACK→ACK errors can lose packets under the delay constraint, whereas ACK→NACK errors primarily waste one ARQ round.A NACK→ACK error advances the transmitter to the next packet before recovery, while an ACK→NACK error causes retransmission of an already decoded packet.
  • ξ_d increases with both forward-channel error probability ε and feedback error probability ε_fb, so a target packet-loss probability can be met through different reliability combinations.A less reliable forward channel requires a more reliable feedback channel.
  • The joint reliability constraint is governed by the combined forward and feedback channels because ε·ε_fb is the probability of an initial decoding failure followed by a NACK→ACK error.This yields a general design principle requiring sufficient reliability across both channels.
  • With feedback errors, improving the forward channel can require an inefficient reliability increase, while increasing feedback reliability can impose substantial feedback overhead.The preferred balance depends on feedback coding and channel diversity.
  • For L_fb=2, the optimum balances PHY reliability against feedback overhead, while larger feedback diversity shifts the optimum toward less conservative forward-channel reliability.Making the PHY more reliable can lose more goodput than the saved feedback overhead recovers; with L_fb=5, the optimum moves left.
  • With diversity, goodput rises nearly to q^(1/d), whereas without diversity it peaks far below q^(1/d) and requires ε somewhat below that target.Without diversity, feedback error probability decreases only on the order of 1/f, requiring large feedback coding effort.

V. HYBRID-ARQ

The hybrid-ARQ analysis studies how to choose the initial transmission rate when incremental redundancy accumulates information across rounds. It finds that HARQ favors a higher initial rate than simple ARQ, but excessive rates cause post-HARQ outages and restart the process.

  • HARQ accumulates mutual information across rounds, matching the effective transmitted rate to instantaneous channel conditions without transmitter CSI.The receiver retains information from previous rounds, unlike simple ARQ.
  • The model permits at most M HARQ rounds per packet; failure after M rounds declares post-HARQ outage and triggers a higher-layer simple-ARQ retransmission.For M=1, the model reduces to simple ARQ.
  • HARQ goodput is maximized at a considerably higher initial rate than simple ARQ for equal diversity order per round.This comparison is shown for L=2 with up to M=2 HARQ rounds at SNR values of 5 and 10 dB.
  • The initial rate should be high enough to use HARQ effectively but not so high that post-HARQ outages force frequent simple-ARQ restarts.Small first-round outage makes HARQ rarely useful, whereas excessive initial rates increase post-HARQ outage.
  • The optimal HARQ initial rate is upper bounded by 1/M times the optimal non-HARQ transmitted rate with diversity order ML.The bound compares HARQ with a non-HARQ system experiencing diversity order ML.

VI. CONCLUSION

The paper concludes that throughput is often maximized with a relatively unreliable PHY, because making the PHY highly reliable requires a conservative transmitted rate in fading channels. Practical constraints alter the optimum but preserve the central tradeoff between PHY reliability, ARQ, delay, and feedback reliability.

  • A packet error probability of 10% is optimal for a wide range of channel parameters under a cross-layer perspective.The conclusion attributes this to the rate cost of making the PHY highly reliable in fading channels without instantaneous transmitter channel knowledge.
  • Making the physical layer very reliable requires a conservative transmitted rate, reducing the benefit of additional PHY reliability.
  • The conclusion remains broad across scenarios, including cases where PHY-level reliability might intuitively seem necessary; additional error detection is more efficient for reducing packet-error misdetection.
  • Delay constraints impose an upper bound on ARQ retransmissions and PHY error probability, with the optimized system operating at that upper error-probability limit.
  • With acknowledgement errors and high end-to-end reliability requirements, reliability should be achieved through the feedback channel rather than a more reliable data channel.
  • ARQ may make traditional diversity metrics less appropriate and can reduce the attractiveness of transmit-diversity techniques that reduce error probability at the expense of rate.

APPENDIX B EXPECTED ARQ ROUNDS WITH ACKNOWLEDGEMENT ERRORS

The appendix enumerates mutually exclusive ways an ARQ process can terminate when acknowledgement errors occur. It sums the corresponding event probabilities to characterize the process duration, including decoding failures, missed acknowledgements, and incorrect feedback.

  • The ARQ process can terminate after an intermediate round because repeated decoding failures produce correctly received NACKs until the round limit is reached.
  • For an intermediate round, a NACK-to-ACK feedback error can terminate the process after unsuccessful decoding attempts and correctly received earlier NACKs.
  • The appendix separately accounts for packets decoded successfully before delayed or incorrect ACK reception, including subsequent ACK-to-NACK errors.
  • An intermediate termination can also occur when a packet is decoded correctly but its ACK is never received correctly.
  • Because the listed termination events are exclusive, their probabilities can be summed to obtain the expected number of ARQ rounds.
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