Optimal Detection over Multi-Hop Relays with Unknown Source Distribution: Over-the-Air Diffusion

Multi-hop amplify-and-forward (AF) relaying inherently amplifies and propagates noise along with the desired signal, leading to an accumulation of noise power over successive hops. At the receiver, an optimal detector is desirable but challenging to realize for non-Gaussian or unknown source distributions (e.g., video or voice signals), for which the Bayesian minimum mean-square error (MMSE) estimator typically does not admit a closed-form expression. Learning-based detectors, in turn, raise key questions regarding the appropriate learning framework, generalization across propagation environments and relay depths, and whether channel state information (CSI) and noise statistics at intermediate relays are needed for optimal detection. This paper leverages information theory to establish a formal mapping between AF relay chains and variance-preserving diffusion processes, i.e., over-the-air diffusion. The analysis further shows that intermediate CSI does not improve optimal detection beyond the end-to-end sufficient statistics. The equivalence suggests that detection under unknown or non-Gaussian source distributions can be reformulated as learning the inverse of the effective noise-injection mechanism, rather than requiring an explicit specification of the source prior, i.e., reversal of a Markovian diffusion process. Moreover, it is shown that the reverse denoising detector can be trained under one multi-hop configuration and instantiated under another, provided that the effective signal-to-noise ratio (SNR) that determines the starting point of the reverse process is preserved. Numerical simulations show that the proposed decoder substantially reduces the mean-square error, the symbol error rate, and the bit error rate compared to conventional symbol-by-symbol maximum likelihood detection.