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Symbol-Domain Chase Combining on Fourier-Curve Constellations: Exact Penalties of Per-Round Bit Reduction

Bin Han, Muxia Sun, Hans D. Schotten

arXiv:2609.03262v2eess.SP

TL;DR

The paper studies how receivers should combine Chase-repeated observations for repeated M-ary symbols when each candidate has its own covariance. It derives exact and max-log gaps between joint and per-round bit reduction, then shows that accumulating matched candidate metrics and reducing to bits once avoids substantial penalties.

  • Problem

    The missing account is how per-round bit reduction changes Chase combining and when exact or max-log reduction commutes with accumulation for repeated M-ary symbols.

  • Method

    The paper derives channel-independent consistency identities and equality conditions for exact and max-log reduction, then evaluates matched Fourier receivers and separating controls.

  • Results

    Per-round reduction incurs substantial SNR losses, including 1.6 dB at L = 4 for a 5G NR LDPC code on Gray 64-QAM, whereas joint max-log matches joint exact.

  • Takeaways & Limitations

    Receivers should retain the normalized M-candidate likelihood state across Chase rounds and form bit LLRs only after enforcing one common repeated symbol.

  • Takeaways & Limitations

    Imperfect channel-state information, rate-compatible incremental redundancy, and stop-on-success throughput remain for future work.

Abstract

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A Fourier-curve constellation places $M$ symbols on a closed curve in $\R^{2k}$ and injects artificial noise along the tangent at the transmitted symbol, so every symbol candidate carries its own rank-one noise covariance; a Chase retransmission repeats one such $M$-ary symbol. How should the covariance-aware receiver combine the repeated observations? It can accumulate the $M$ candidate metrics and form bit log-likelihood ratios (LLRs) once, or it can form bit LLRs in every round and add them. The rounds are independent given the symbol but not given a single label bit, so even exact per-round bit LLRs do not add up to the joint-round LLR. We derive exact identities for the gap under log-sum-exp and max-log reduction, with their equality conditions; they hold for any repeated $M$-ary symbol, grow with the number of rounds, and vanish for binary signaling. On the Fourier channel with a rate-$1/2$ LDPC code after $L=4$ rounds, per-round max-log reduction needs $1.95$ dB more per-slot SNR at block error rate $10^{-1}$ than even covariance-ignorant Euclidean accumulation. Optimized bit-metric generalized mutual information puts the SNR penalty of per-round exact reduction at the code-rate threshold at $2.7$ dB on the Fourier channel and $1.5$ dB on Gray 64-QAM, joint max-log costs less than $0.1$ dB, and a 5G NR LDPC code on Gray 64-QAM loses $1.6$ dB at $L=4$. Controls with Gray labeling, isotropic noise, and $β=0$ show that the loss does not depend on the symbol-dependent covariance, whose own effect is the separate matched-versus-Euclidean correction. The Fourier receiver should therefore accumulate matched candidate metrics across rounds and reduce to bits once.

I. INTRODUCTION

The paper asks whether Chase observations should be combined at the candidate-symbol level or after per-round bit reduction. It develops a channel-independent account of the difference and applies it to covariance-aware Fourier-curve reception.

  • I. INTRODUCTION: The paper supplies exact and max-log consistency identities, equality conditions, and a minimal likelihood-ratio state for repeated M-ary symbols.The identities allow the known candidate density to vary across rounds.
  • I. INTRODUCTION: Fourier-curve symbols lie on a closed curve in R2k and receive tangent artificial noise, giving each candidate its own rank-one Gaussian covariance.The construction uses k harmonics and M equally spaced symbols.
  • I. INTRODUCTION: Per-round bit reduction is not equivalent to joint symbol-level combining because rounds share the repeated symbol but not a single label bit.Different nuisance candidates can explain different rounds when per-round marginals are multiplied.
  • I. INTRODUCTION: The contribution also separates candidate-consistency loss from max-log approximation and covariance mismatch through exact-reduction, labeling, common-covariance, β = 0, and Gray 64-QAM controls.A 5G NR LDPC control tests whether the penalty persists under a standard code.
  • I. INTRODUCTION: The sufficient state is the accumulated vector of M candidate metrics, equivalently M−1 candidate log-likelihood ratios up to normalization.This state is minimal in likelihood-ratio equivalence classes, though structured channels may admit smaller representations.

B. Exact reduction

Exact per-round bit LLR accumulation admits cross-round candidate tuples that joint reduction excludes. The resulting candidate-consistency penalty has explicit equality conditions and increases with repeated rounds for nonbinary classes.

  • B. Exact reduction: When L ≥ 2 and a bit class contains at least two symbols, its exact candidate-consistency penalty is strictly positive for positive likelihoods.The numerator contains positive cross tuples while the joint denominator retains only diagonal tuples.
  • B. Exact reduction: For each bit class, the exact penalty is the log-likelihood mass assigned to cross-round candidate tuples that no single repeated symbol could produce.Joint reduction retains only diagonal tuples using one candidate in every round.
  • B. Exact reduction: The joint and per-round exact bit LLRs coincide exactly when the two bit classes have equal candidate-consistency penalties, even if neither penalty is zero.The equality concerns the difference between the class penalties, not elimination of cross-round mass.
  • B. Exact reduction: With identical within-class posteriors across rounds, the penalty equals (L − 1) times the order-L Rényi entropy and grows linearly with L for a uniform class.For a uniform class of size K, it is (L − 1) log K.

C. Max-log reduction

Max-log reduction has a parallel candidate-consistency penalty based on roundwise minima. Its equality conditions depend on shared minimizers, and it vanishes for binary signaling or a single round.

  • C. Max-log reduction: The max-log penalty is the relaxation gap produced when per-round minima are combined instead of minimizing jointly over one repeated candidate.It is the max-log counterpart of the exact cross-round consistency penalty.
  • C. Max-log reduction: The max-log penalty is zero exactly when the roundwise minimizers within each bit class share a candidate, including ties.This condition applies separately to the relevant class minima.
  • C. Max-log reduction: Both exact and max-log reduction gaps vanish for one round and for binary signaling, where every bit class is a singleton.The identities concern repeated symbols and do not apply to incremental redundancy with new parity symbols.

A. Channel and branch metrics

The Fourier channel uses a keyed closed curve with tangent artificial noise, so each candidate has its own covariance and matched branch metric. Repeated observations are combined through joint maximum-likelihood candidate metrics before bit reduction.

  • A. Channel and branch metrics: Each Fourier symbol is generated from a keyed closed curve in R^{2k}, with symbols x_i = x(2πi/M) and unit tangents b_ti.The phase vector is shared with the legitimate receiver and held fixed across the HARQ session.
  • A. Channel and branch metrics: Tangent artificial noise makes Y_l given candidate i Gaussian with mean x̄_i and candidate-dependent covariance Σ_i.The covariance includes the tangent-noise direction and post-equalization AWGN variance; key refresh and fading are outside scope.
  • A. Channel and branch metrics: The matched branch metric uses each candidate’s covariance, whereas the Euclidean metric ignores that covariance and acts as an auxiliary AWGN model.The Euclidean receiver uses this metric unchanged for its soft outputs.
  • A. Channel and branch metrics: Because rounds are independent conditioned on the repeated symbol, their candidate metrics accumulate to form the joint maximum-likelihood metric.This is the channel-level operation to which the paper’s consistency identities apply.

B. L-round pairwise laws

The paper derives pairwise error laws for joint matched and Euclidean receivers over repeated Fourier observations, then connects their scaling to uncoded and coded performance. The results separate covariance-aware metric mismatch from the distinct penalty caused by reducing candidate metrics independently across rounds.

  • B. L-round pairwise laws: The pairwise metric difference is Gaussian for the joint Euclidean receiver, and independent round contributions add across L observations.For the matched receiver, the corresponding increment is represented through a Gaussian quadratic-form transform and a Chernoff-type rate.
  • B. L-round pairwise laws: A positive exponent gap separates the two log error probabilities linearly in L, while fixed-target SNR penalties require separate analysis.The uncoded Fourier figure brackets simulated SER with best-pair and union bounds for both receivers.
  • B. L-round pairwise laws: The Euclidean-to-matched SER ratio grows with L and β, but equals one at β = 0 when the two metrics coincide.This identifies covariance awareness as a mechanism distinct from candidate-consistency effects.
  • B. L-round pairwise laws: At L = 4, the controls compare joint and per-round exact or max-log reduction using GMI, paired GMI advantage, and block-error outcomes across Fourier and 64-QAM channels.The code-rate crossing table reports SNR penalties relative to joint exact with bootstrap intervals.

A. Setup and coded comparison

The Fourier coded comparison evaluates matched and Euclidean receivers under Chase combining, using BLER 10^-1 SNR penalties relative to joint matched accumulation. At L = 4, per-round matched reduction is substantially worse than even joint Euclidean accumulation.

  • A. Setup and coded comparison: The Fourier experiment uses (k, M) = (20, 64), natural-binary labeling, β = 0.3, and a rate R = 506/1008 Gallager LDPC code.Each symbol occupies 20 complex slots and carries six coded bits.
  • A. Setup and coded comparison: 1.95 dB more per-slot SNR is required by per-round matched reduction than joint Euclidean accumulation at BLER 10^-1 and L = 4.The 95% confidence interval is [1.92, 1.99] dB.
  • A. Setup and coded comparison: 2.69 dB is the L = 4 penalty of per-round matched reduction versus joint matched accumulation, compared with 0.74 dB for joint Euclidean reduction.At L = 2, the corresponding penalties are 1.39 dB and 1.52 dB.
  • A. Setup and coded comparison: The per-round penalty grows with L, while the measured Euclidean penalty shrinks as combining moves the operating point into heavier noise.The exponent comparison at fixed noise variance does not predict these operating-point penalties.

B. Exact-reduction, labeling, and channel controls

Controls show that the dominant loss arises from reducing each round before accumulation, rather than from max-log approximation, labeling, or candidate-dependent covariance. Joint reduction remains close to exact, whereas per-round reduction retains a substantial gap across Fourier and 64-QAM controls.

  • B. Exact-reduction, labeling, and channel controls: Joint max-log nearly matches joint exact, while per-round max-log falls to 2.88 bits/symbol and produces 397/400 coded errors in the main Fourier row.Per-round exact reduction already produces 391/400 errors, so exact per-round LLRs do not remove the loss.
  • B. Exact-reduction, labeling, and channel controls: The loss persists with Gray labeling, candidate-independent isotropic covariance, and β = 0, separating reduction inconsistency from covariance mismatch.Gray labeling leaves a 1.51-bit GMI gap, while ordinary Gray 64-QAM retains a 0.45-bit loss.
  • B. Exact-reduction, labeling, and channel controls: Candidate-dependent covariance supplies a separate matched-versus-Euclidean gain, whereas accumulation and reduction fail to commute without relying on that covariance.The controls use channel-specific operating points and rank receivers within each channel.
  • B. Exact-reduction, labeling, and channel controls: 2.2–2.8 dB is the per-round exact-reduction penalty on Fourier rows at the code-rate crossing, versus 1.5 dB on Gray 64-QAM.Joint max-log costs less than 0.1 dB in every row.

C. Standard-code control

A 5G NR LDPC control on Gray 64-QAM reproduces the ordering observed with the Fourier code: joint exact and joint max-log remain close, while per-round reduction degrades with retransmissions.

  • C. Standard-code control: 1.61 dB is the L = 4 BLER 10^-1 penalty of per-round exact reduction versus joint exact on the 5G NR LDPC Gray 64-QAM link.The 95% interval is [1.55, 1.67] dB, close to the 1.47 dB GMI penalty in Table II.
  • C. Standard-code control: The per-round exact penalty is 0.58 dB at L = 2 and 1.61 dB at L = 4, while joint max-log stays within 0.02 dB of joint exact.At L = 1, the receivers coincide and max-log is within 0.01 dB.
  • C. Standard-code control: The ordering and its growth with L survive changing the code, constellation, and interleaver.The control repeats the same 160 symbols each round and uses 1000 paired frames per point.

V. CONCLUSION

For repeated M-ary symbols, per-round bit reduction admits inconsistent nuisance candidates across rounds, creating exact and max-log penalties that vanish for binary signaling. The recommended Fourier receiver therefore accumulates matched candidate metrics across rounds and forms bit LLRs only once.

  • V. CONCLUSION: Per-round exact and max-log reduction both admit cross-round nuisance-candidate combinations, so their penalties grow with retransmissions and vanish for binary signaling.The identities do not require candidate-dependent covariance or Fourier geometry.
  • V. CONCLUSION: The receiver should retain the normalized M-candidate likelihood state across Chase rounds and reduce to bits only after enforcing one common repeated symbol.For the Fourier link, this means accumulating matched candidate metrics and forming bit LLRs after the last round.
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