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

Abstract

A Fourier-curve constellation adds artificial noise along the tangent at the transmitted symbol, giving each symbol candidate its own rank-one noise covariance. A Chase retransmission repeats such a symbol, and the receiver can either accumulate the candidate metrics before forming bit log-likelihood ratios (LLRs) or add bit LLRs formed in every round. The rounds are independent given the symbol but not necessarily given one label bit, so summed exact per-round LLRs do not in general equal the joint LLR. For a fixed labeling, equiprobable symbols, and conditionally independent rounds, we derive exact identities for this difference under exact and max-log reduction, with equality conditions: each difference equals the gap between two nonnegative class corrections, which are nondecreasing in the number of rounds and vanish for binary signaling. With a rate-1/2 LDPC code after four rounds, the per-round matched Fourier receiver needs 1.97 dB more per-slot SNR at block error rate 0.1 than the covariance-ignorant joint Euclidean receiver. After four rounds, per-round exact reduction costs 2.7 dB on the Fourier link and 1.5 dB on Gray 64-QAM in bit-metric generalized mutual information, and 1.6 dB in block error rate with a 5G NR LDPC code, whereas joint max-log reduction costs less than 0.1 dB. Controls with Gray labeling, isotropic noise, and no tangent noise show that this loss does not require candidate-dependent covariance. The Fourier receiver should therefore accumulate the matched candidate metrics and reduce them to bits at each decoding attempt.