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Structured Tensor Decomposition Based Channel Estimation and Double Refinements for Active RIS Empowered Broadband Systems

Yirun Wang, Yongqing Wang, Yuyao Shen, Gongpu Wang, Chintha Tellambura

TL;DR

This work tackles tensor-based channel parameter estimation for active RIS-aided broadband systems in fully multipath environments. It introduces a fifth-order CP tensor model with a Vandermonde-factor and four rank-deficient factors, and resolves decomposition via spatial smoothing and Vandermonde-structured CPD (VSCPD). A triple-stage CE algorithm (coarse estimation, then two refinements) converts a multidimensional problem into effectively 1D searches, with a CRLB that accounts for active-RIS noise. Numerical results show that active RIS substantially improves estimation performance over passive RIS, while the VSCPD framework delivers robust, efficient parameter recovery across LOS and multipath scenarios, enabling improved system design and sensing capabilities.

Abstract

Channel parameter recovery is critical for the next-generation reconfigurable intelligent surface (RIS)-empowered communications and sensing. Tensor-based mechanisms are particularly effective, inherently capturing the multi-dimensional nature of wireless channels. However, existing studies assume either a line-of-sight (LOS) scenario or a blocked TX-RX channel. This paper solves a novel problem: tensor-based channel parameter estimation for active RIS-aided multiple-antenna broadband connections in fully multipath environments with the TX-RX link. System settings are customized to construct a fifth-order canonical polyadic (CP) signal tensor that matches the five-dimensional channel. Four tensor factors contain redundant columns, rendering the classical Kruskal's condition for decomposition uniqueness unsatisfied. The fifth-order Vandermonde structured CP decomposition (VSCPD) is developed to address this challenge, making the tensor factorization problem solvable using only linear algebra and offering a relaxed general uniqueness condition. With VSCPD as a perfect decoupling scheme, a sequential triple-stage channel estimation algorithm is proposed based on one-dimensional parameter estimation. The first stage enables multipath identification and algebraic coarse estimation. The following two stages offer optional successive refinements at the cost of increased complexity. The closed-form Cramer-Rao lower bound (CRLB) is derived to assess the estimation performance. Herein, the noise covariance matrix depends on multipath parameters in our active-RIS scenario. Numerical results are provided to verify the effectiveness of proposed algorithms under various evaluation metrics. Our results also show that active RIS can significantly improve channel estimation performance compared to passive RIS.

Structured Tensor Decomposition Based Channel Estimation and Double Refinements for Active RIS Empowered Broadband Systems

TL;DR

This work tackles tensor-based channel parameter estimation for active RIS-aided broadband systems in fully multipath environments. It introduces a fifth-order CP tensor model with a Vandermonde-factor and four rank-deficient factors, and resolves decomposition via spatial smoothing and Vandermonde-structured CPD (VSCPD). A triple-stage CE algorithm (coarse estimation, then two refinements) converts a multidimensional problem into effectively 1D searches, with a CRLB that accounts for active-RIS noise. Numerical results show that active RIS substantially improves estimation performance over passive RIS, while the VSCPD framework delivers robust, efficient parameter recovery across LOS and multipath scenarios, enabling improved system design and sensing capabilities.

Abstract

Channel parameter recovery is critical for the next-generation reconfigurable intelligent surface (RIS)-empowered communications and sensing. Tensor-based mechanisms are particularly effective, inherently capturing the multi-dimensional nature of wireless channels. However, existing studies assume either a line-of-sight (LOS) scenario or a blocked TX-RX channel. This paper solves a novel problem: tensor-based channel parameter estimation for active RIS-aided multiple-antenna broadband connections in fully multipath environments with the TX-RX link. System settings are customized to construct a fifth-order canonical polyadic (CP) signal tensor that matches the five-dimensional channel. Four tensor factors contain redundant columns, rendering the classical Kruskal's condition for decomposition uniqueness unsatisfied. The fifth-order Vandermonde structured CP decomposition (VSCPD) is developed to address this challenge, making the tensor factorization problem solvable using only linear algebra and offering a relaxed general uniqueness condition. With VSCPD as a perfect decoupling scheme, a sequential triple-stage channel estimation algorithm is proposed based on one-dimensional parameter estimation. The first stage enables multipath identification and algebraic coarse estimation. The following two stages offer optional successive refinements at the cost of increased complexity. The closed-form Cramer-Rao lower bound (CRLB) is derived to assess the estimation performance. Herein, the noise covariance matrix depends on multipath parameters in our active-RIS scenario. Numerical results are provided to verify the effectiveness of proposed algorithms under various evaluation metrics. Our results also show that active RIS can significantly improve channel estimation performance compared to passive RIS.

Paper Structure

This paper contains 28 sections, 2 theorems, 63 equations, 9 figures, 4 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Channel Model
  4. Signal Model
  5. Proposed Tensor-Based CE Framework
  6. Tensor Construction
  7. ESPRIT Preliminaries
  8. VSCPD with Spatial Smoothing
  9. CE Stage I for Path Identification and Coarse Estimation
  10. CE Stage II for Estimate Refinement
  11. CE Stage III for Further Refinement
  12. Performance Analysis
  13. CRLB Analysis
  14. Complexity Analysis
  15. Numerical Results
  16. ...and 13 more sections

Key Result

Theorem 1

Let $\mathbf{A}_1\in\mathbb{C}^{K\times R}$, $\mathbf{B}_2\in\mathbb{C}^{G_1\times R}$, $\mathbf{B}_3\in\mathbb{C}^{G_2\times R}$, $\mathbf{B}_4\in\mathbb{C}^{N_1\times R}$, and $\mathbf{B}_5\in\mathbb{C}^{N_2\times R}$ be the factor matrices of tensor $\bm{\mathcal{Y}} \in\mathbb{C}^{K\times G_1\ti

Figures (9)

  • Figure 1: A UAV-mounted RIS-assisted uplink wireless system with multipath channels.
  • Figure 2: A flowchart of the proposed tensor-based CE framework.
  • Figure 3: Illustration of reshaping Kronecker product of vectors into a tensor. We consider an example with $\mathbf{a}\otimes\mathbf{b}\otimes\mathbf{c}$ resulting in a third-order rank-$1$ tensor, where both $\mathbf{a}$ and $\mathbf{b}$ only contain two elements for ease of illustration.
  • Figure 4: RMSEs of $\tau_{\textrm{L}}$, $\tau_{\textrm{R}}$, $\psi_{2}$, $\psi_{3}$, $\bm{\theta}_{\textrm{L}}$, and $\bm{\theta}_{\textrm{R}}$ versus SNR in LOS scenario.
  • Figure 5: Average computation time with varying SNR in LOS scenario.
  • ...and 4 more figures

Theorems & Definitions (2)

  • Theorem 1
  • Theorem 2