A Short Report on Tensors for Wireless Communications and Array Processing

This extended single-column survey reviews tensor decompositions used in signal processing (SP) with a primary focus on two application pillars: (i) wireless communications and (ii) array processing (radar/sonar/acoustic arrays and related localization/beamforming problems). We cover modeling pathways from physical measurement equations to low-rank tensor structures (CP/PARAFAC, structured CP, Tucker, and coupled models), identifiability and ill-conditioning phenomena, and algorithmic building blocks (ALS/first-order methods, MTTKRP, regularization/constraints) that recur across both pillars. We also highlight the practical bridge from factor estimates to physical parameters (angles, delays, Dopplers) under Vandermonde/shift-invariance structure, and we discuss common failure modes such as collinearity, degeneracy, and protocol-driven rank deficiencies. Compared with earlier drafts, we add a detailed treatment of CP identifiability (Kruskal-type conditions, generic rank and typical ranks, and degeneracy/illconditioning) as well as an appendix deriving ALS updates and common regularized variants with complexity analysis.