Channel Estimation for Beyond Diagonal RIS via Tensor Decomposition

TL;DR

This paper introduces tensor decomposition techniques for efficient channel estimation in beyond diagonal reconfigurable intelligent surfaces, providing methods with improved performance and reduced training overhead over traditional approaches.

Contribution

It develops two tensor-based solutions, a closed-form block Tucker Kronecker factorization and an iterative block Tucker ALS algorithm, for decoupled channel estimation in BD-RIS.

Findings

BTKF offers fast, parallel channel estimation with noise rejection.

BTALS provides flexible training design with lower overhead.

Both methods outperform conventional LS estimation in accuracy and efficiency.

Abstract

This paper addresses the channel estimation problem for beyond diagonal reconfigurable intelligent surface (BD-RIS) from a tensor decomposition perspective. We first show that the received pilot signals can be arranged as a three-way tensor, allowing us to recast the cascaded channel estimation problem as a block Tucker decomposition problem that yields decoupled estimates for the involved channel matrices while offering a substantial performance gain over the conventional (matrix-based) least squares (LS) estimation method. More specifically, we develop two solutions to solve the problem. The first one is a closed-form solution that extracts the channel estimates via a block Tucker Kronecker factorization (BTKF), which boils down to solving a set of parallel rank-one matrix approximation problems. Exploiting such a low-rank property yields a noise rejection gain compared to the standard LS estimation scheme while allowing the two involved channels to be estimated separately. The second solution is based on a block Tucker alternating least squares (BTALS) algorithm that directly estimates the involved channel matrices using an iterative estimation procedure. We discuss the uniqueness and identifiability issues and their implications for training design. We also propose a tensor-based design of the BD-RIS training tensor for each algorithm that ensures unique decoupled channel estimates under trivial scaling ambiguities. Our numerical results shed light on the tradeoffs offered by BTKF and BTALS methods. Specifically, while the first enjoys fast and parallel extraction of the channel estimates in closed form, the second has a more flexible training design, allowing for a significantly reduced training overhead compared to the state-of-the-art LS method.