傅里叶曲线星座的符号域合并：逐轮比特缩减的精确代价
Symbol-Domain Chase Combining on Fourier-Curve Constellations: Exact Penalties of Per-Round Bit Reduction
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- RPTU University Kaiserslautern-Landau（凯泽斯劳滕-兰多应用技术大学）
- Tsinghua University（清华大学）
- German Research Center for Artificial Intelligence (DFKI)（德国人工智能研究中心）
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中文总结 AI 辅助
该研究针对带切向人工噪声的傅里叶曲线星座，推导了逐轮比特缩减的精确代价，建议接收机跨轮次累加匹配候选度量并一次性缩减为比特以降低SNR损失。
中文摘要 AI 辅助
在带切向人工噪声的相位键控傅里叶曲线星座中，每个符号候选携带自身的秩1噪声协方差，Chase重传会重复一个M元符号。协方差感知型接收机应如何合并重复观测值？它可一次性累加M个候选度量并形成比特对数似然比（LLR），也可每轮形成比特LLR后相加。给定符号时各轮独立，但给定单个标签比特时并非如此，因此即使是精确的逐轮比特LLR也无法加和为联合轮次LLR。我们推导了对数求和指数（log-sum-exp）和最大对数（max-log）缩减下的间隙精确恒等式及其等式条件，该恒等式对任意重复M元符号成立，随轮次增加而增大，在二进制信号传输时消失。在L=4、速率1/2的LDPC码的傅里叶信道上，当块错误率为10^-1时，逐轮max-log缩减每时隙信噪比（SNR）比不考虑协方差的欧氏累加多需1.95dB。优化后的比特度量广义互信息在傅里叶信道的码率阈值处将逐轮精确缩减的SNR代价定为2.7dB，在格雷64-QAM上为1.5dB；联合max-log代价小于0.1dB，且L=4时格雷64-QAM上的5G NR LDPC码损失1.6dB。格雷标签、各向同性噪声及β=0的对照实验表明，该损失不依赖于符号相关协方差，协方差自身的效应是独立的匹配-欧氏校正。因此傅里叶接收机应跨轮次累加匹配的候选度量并一次性缩减为比特。
英文摘要
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.
