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

Abstract

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 $\beta=0$ 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.