On the Structure of the Optimal Detector for Sub-THz Multi-Hop Relays with Unknown Prior: Over-the-Air Diffusion

Amplify and forward (AF) relaying is a viable strategy to extend the coverage of sub-terahertz (sub-THz) links, but inevitably propagates noise, leading to cumulative degradation across multiple hops. At the receiver, optimal decoding is desirable, yet challenging under non-Gaussian input distributions (video, voice, etc), for which neither the Minimum Mean Square Error (MMSE) estimator nor the mutual information admits a closed form. A further open question is whether knowledge of Channel State Information (CSI) and noise statistics at the intermediate relays is necessary for optimal detection. Aiming for an optimal decoder, this paper introduces a new framework that interprets the AF relay chain as a variance-preserving diffusion process and employs denoising diffusion implicit models (DDIMs) for signal recovery. We show that each AF hop is mathematically equivalent to a diffusion step with hop-dependent attenuation and noise injection. Consequently, the entire multi-hop chain collapses to an equivalent Gaussian channel fully described by only three real scalars per block: the cumulative complex gain and the effective noise variance. At the receiver, these end-to-end sufficient statistics define a matched reverse schedule that guides the DDIM-based denoiser, enabling near-optimal Bayesian decoding without per-hop CSI. We establish the information-theoretic foundation of this equivalence, proving that decoding performance depends solely on the final effective Signal-to-Noise-Ratio (SNR), regardless of intermediate noise/channel allocation or prior distribution. Simulations under AWGN and Rician fading confirm that the proposed AF-DDIM decoder reduces mean-squared error, symbol error rate, and bit error rate, particularly at moderate SNRs and for higher-order constellations.