A Majorization-Minimization Framework for Activity Detection in Mixed Near-and-Far-Field Random Access

Abstract

Grant-free random access in massive machine-type communications requires detecting a small active set from length-<inline-formula> <tex-math notation="LaTeX">$L$ </tex-math></inline-formula> uplink pilots received at an <inline-formula> <tex-math notation="LaTeX">$M$ </tex-math></inline-formula>-antenna base station. Classical covariance-based detectors are largely built on the far-field (FF) model, where the <inline-formula> <tex-math notation="LaTeX">$M$ </tex-math></inline-formula> antenna-domain observations are treated as independent snapshots, and inference reduces to an <inline-formula> <tex-math notation="LaTeX">$L\times L$ </tex-math></inline-formula> covariance description. This model becomes inadequate in mixed near-field (NF) and FF access, where NF devices induce device-specific structured spatial covariances and the aggregate observation no longer fits the FF snapshot structure. In this paper, we develop a unified covariance-aware Rician framework for activity detection that treats FF and NF devices by a single likelihood model. Within this framework, we propose a majorization-minimization projected gradient descent (MM-PGD) detector for the resulting relaxed likelihood. For scalable exact likelihood evaluation, we further derive a Kronecker–Woodbury implementation that exploits the mixed NF/FF covariance structure, and avoids factorizing the full <inline-formula> <tex-math notation="LaTeX">$LM\times LM$ </tex-math></inline-formula> covariance matrix. Numerical results show that MM-PGD is on par with the strongest coordinate-wise baseline in the all-FF limit, while its advantage becomes more pronounced as the fraction of NF users increases under the tested regimes.