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MIMO Beam Map Reconstruction via Toeplitz-Structured Matrix-Vector Tensor Decomposition

Hao Sun, Junting Chen, Xianghao Yu

TL;DR

The paper tackles reconstructing high-dimensional MIMO beam maps from sparse measurements by exploiting a polar-coordinate transformation that reveals a Toeplitz-structured matrix-vector outer product in the beam-space gains. It introduces a Toeplitz-structured tensor decomposition with regularization to jointly model LOS, reflection, and obstruction through multiple propagation components, solved via an alternating minimization scheme. The approach yields more than 20% NMSE improvement over baselines under sparse sampling and various propagation conditions, validating the effectiveness of leveraging angular shift-invariance and distance attenuation priors. The proposed framework offers a data-efficient, structure-aware method for beam management and link optimization in 6G scenarios, with potential for real-time deployment given the reduced degrees of freedom in LOS-dominated cases.

Abstract

As wireless networks progress toward sixthgeneration (6G), understanding the spatial distribution of directional beam coverage becomes increasingly important for beam management and link optimization. Multiple-input multipleoutput (MIMO) beam map provides such spatial awareness, yet accurate construction under sparse measurements remains difficult due to incomplete spatial coverage and strong angular variations. This paper presents a tensor decomposition approach for reconstructing MIMO beam map from limited measurements. By transforming measurements from a Cartesian coordinate system into a polar coordinate system, we uncover a matrix-vector outer-product structure associated with different propagation conditions. Specifically, we mathematically demonstrate that the matrix factor, representing beam-space gain, exhibits an intrinsic Toeplitz structure due to the shift-invariant nature of array responses, and the vector factor captures distance-dependent attenuation. Leveraging these structural priors, we formulate a regularized tensor decomposition problem to jointly reconstruct line-of-sight (LOS), reflection, and obstruction propagation conditions. Simulation results confirm that the proposed method significantly enhances data efficiency, achieving a normalized mean square error (NMSE) reduction of over 20% compared to state-of-the-art baselines, even under sparse sampling regimes.

MIMO Beam Map Reconstruction via Toeplitz-Structured Matrix-Vector Tensor Decomposition

TL;DR

The paper tackles reconstructing high-dimensional MIMO beam maps from sparse measurements by exploiting a polar-coordinate transformation that reveals a Toeplitz-structured matrix-vector outer product in the beam-space gains. It introduces a Toeplitz-structured tensor decomposition with regularization to jointly model LOS, reflection, and obstruction through multiple propagation components, solved via an alternating minimization scheme. The approach yields more than 20% NMSE improvement over baselines under sparse sampling and various propagation conditions, validating the effectiveness of leveraging angular shift-invariance and distance attenuation priors. The proposed framework offers a data-efficient, structure-aware method for beam management and link optimization in 6G scenarios, with potential for real-time deployment given the reduced degrees of freedom in LOS-dominated cases.

Abstract

As wireless networks progress toward sixthgeneration (6G), understanding the spatial distribution of directional beam coverage becomes increasingly important for beam management and link optimization. Multiple-input multipleoutput (MIMO) beam map provides such spatial awareness, yet accurate construction under sparse measurements remains difficult due to incomplete spatial coverage and strong angular variations. This paper presents a tensor decomposition approach for reconstructing MIMO beam map from limited measurements. By transforming measurements from a Cartesian coordinate system into a polar coordinate system, we uncover a matrix-vector outer-product structure associated with different propagation conditions. Specifically, we mathematically demonstrate that the matrix factor, representing beam-space gain, exhibits an intrinsic Toeplitz structure due to the shift-invariant nature of array responses, and the vector factor captures distance-dependent attenuation. Leveraging these structural priors, we formulate a regularized tensor decomposition problem to jointly reconstruct line-of-sight (LOS), reflection, and obstruction propagation conditions. Simulation results confirm that the proposed method significantly enhances data efficiency, achieving a normalized mean square error (NMSE) reduction of over 20% compared to state-of-the-art baselines, even under sparse sampling regimes.
Paper Structure (33 sections, 1 theorem, 44 equations, 16 figures)

This paper contains 33 sections, 1 theorem, 44 equations, 16 figures.

Table of Contents

  1. Introduction
  2. System Model
  3. Propagation and Beam Model Complex scenario showing a direct path from the BS, a reflected path via the mirror BS, and an obstructed path affected by physical blockings.
  4. Measurement Model
  5. Toeplitz-Structured Tensor Model via Coordinate Transformation
  6. Coordinate Transformation
  7. Transformation from Cartesian to Polar Coordinates
  8. Toeplitz Structure
  9. Tensor Model under the Presence of Reflection and Obstruction
  10. Pure LOS Region
  11. Coexistence of LOS and Reflection
  12. Coexistence of LOS and Obstruction
  13. Coexistence of LOS, Reflection, and Obstruction
  14. Toeplitz-Structured Matrix-Vector Tensor Decomposition
  15. Matrix-Vector Tensor Decomposition Formulation
  16. ...and 18 more sections

Key Result

Lemma 1

If the angles $\theta_{i}$ and $\phi_{j}$ are chosen such that their sines form uniform grids, i.e., $\sin(\theta_{i})=i/I$ and $\sin(\phi_{j})=j/J$, and $I=J$, then the matrix $\bm{G}$ exhibits a Toeplitz and symmetric structure.

Figures (16)

  • Figure 2: (a) The MIMO beam map under the Cartesian coordinate system. (b) The transformed MIMO beam map under the polar coordinate system, revealing Toeplitz and symmetric structures that can be exploited for structured reconstruction.
  • Figure 3: Three common beam propagation scenarios: (a) Pure LOS Region. (b) Coexistence of LOS and Reflection. (c) Coexistence of LOS and Obstruction.
  • Figure 4: Visible plots of $\bm{G}_{1}$ and $\bm{G}_{2}$. $\bm{G}_{1}$ exhibits a Toeplitz structure corresponding to the LOS region, while $\bm{G}_{2}$ displays a quasi-Toeplitz pattern with a dark region indicating signal obstruction.
  • Figure 5: Propagation from the mirror BS $\bm{s}_{1}$ yields three distance regimes: (i) an LOS region for $d\le d_{1}$; (ii) a single-obstruction region for $d_{1}<d\le d_{2}$; and (iii) a double-obstruction region for $d>d_{2}$.
  • Figure 6: NMSE comparison between the Toeplitz constraint and Toeplitz regularization methods under different sampling ratios. The constraint method performs better than using Toeplitz as regularization.
  • ...and 11 more figures

Theorems & Definitions (2)

  • Lemma 1: Toeplitz and Symmetric Structure
  • proof