TenSIM: Tensor-Based Channel Estimation for MIMO Systems with Stacked Intelligent Metasurfaces

Stacked intelligent metasurfaces (SIMs) are emerging as a promising architecture for the sixth generation (6G) and beyond of wireless systems, enabling richer electromagnetic-wave manipulation than conventional single-layer metasurfaces. However, realizing these gains requires accurate and scalable channel estimation under the strong inter-layer coupling and multilinear parameter interactions introduced by the stacked programmable metasurface layers. This paper proposes TenSIM, a tensor-based channel-estimation framework for SIM-assisted multiple-input multiple-output (MIMO) systems. By exploiting a structured SIM training protocol, TenSIM derives two parity-dependent observation models: a PARAllel FACtor (PARAFAC) model for odd-layer SIMs and a Tucker model for even-layer SIMs. These formulations decouple the transmitter-SIM and SIM-receiver channels while explicitly accounting for inter-layer wave coupling. Based on the resulting tensor models, we develop alternating least squares estimators, establish identifiability conditions using the associated design matrices, and characterize practical sufficient conditions for full-column-rank training designs, including those involving scaling ambiguities. The proposed framework is validated through extensive numerical experiments and reveals the main operating trade-offs. We show that both TenSIM-PARAFAC and TenSIM-Tucker outperform unstructured least squares baselines by exploiting the tensor structure of the SIM cascade. Moreover, TenSIM-PARAFAC offers better scalability, lower computational complexity, and stronger robustness to inter-layer spacing, while TenSIM-Tucker can provide more accurate channel reconstruction when sufficient training and strong layer coupling are available. Finally, it is shown that the proposed TenSIM framework remains effective under imperfect or blind SIM training when additional pilot diversity is available.