Multiuser Channel Estimation in FIM Systems: A Structured Block Term Decomposition Approach
Flexible intelligent metasurfaces (FIMs) create geometry-dependent array responses that provide spatial diversity but complicate multiuser channel estimation. This paper studies uplink estimation using pilot observations collected under multiple prescribed and calibrated morphing configurations. Each user contributes a matrix--vector block that couples its diagonal reference-array response with its morphing-dependent phases, while its third-mode vector is the known pilot scaled by the complex channel gain. We develop a pilot-constrained structured block term decomposition (BTD) estimator whose alternating least-squares updates exploit both the known pilot directions and the diagonal spatial factors. The undeformed configuration sets the columnwise scaling convention, and a unit first-entry normalization assigns the remaining block scale to the channel gain. Matching the recovered factors to the calibrated spatial--morphing manifold then yields the physical angles and configuration-dependent channels. We characterize the rank conditions for unique conditional least-squares updates, distinguish these conditions from global factor and physical-angle identifiability, and summarize the training, coherence-time, and computational requirements. Simulations show accurate two-dimensional angle and channel recovery in the tested settings, equal total pilot-symbol overhead in rigid-panel comparisons, improved steering-vector estimates as morphing diversity increases, and degradation as user loading or pilot correlation increases.