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Nuclear Atomic Norm for parametric estimation of sparse channels

Álvaro Callejas-Ramos, Matilde Sánchez-Fernández, Antonia Tulino, Jaime Llorca

TL;DR

This work addresses gridless parametric estimation of sparse mmWave channels in MIMO systems by formulating channel estimation as a continuous spectral problem and solving it with an atomic-norm framework. It develops a unified model for d-D heterogeneous arrays, derives a generalized measurement model y = Q h_u(ℓ1:K), and establishes recovery guarantees through multi-level Toeplitz (MLT) structures and Vandermonde decompositions to extract continuous AoD/AoA parameters. Both noiseless and noisy scenarios are treated via rank-based and nuclear-norm (convex) relaxations, with rigorous conditions relating sparsity K, array reconstruction degrees κ_L, and pilot design to guarantee unique recovery and parameter estimation. Benchmarks against on-grid methods show that the proposed gridless atomic-norm approach achieves superior accuracy at comparable complexity, enabling high-dimensional, gridless propagation parameter estimation for practical mmWave MIMO deployments.

Abstract

Parametric channel estimation in mmWave not only enables the anticipated large spectral efficiency gains of \acs{MIMO} systems but also reveals important propagation parameters, allowing for a low complexity representation of the channel matrix. In this work, we propose to use atomic norm as a gridless multidimensional spectral estimation approach to address parametric channel estimation where both AoD and AoA are identified. The conditions for recovery of the propagation parameters are given depending on properties of the measurement matrix, and on structural features such as the antenna geometry or the number of scatters to resolve. The proposed methodology is compared against several state-of-the-art parametric approaches.

Nuclear Atomic Norm for parametric estimation of sparse channels

TL;DR

This work addresses gridless parametric estimation of sparse mmWave channels in MIMO systems by formulating channel estimation as a continuous spectral problem and solving it with an atomic-norm framework. It develops a unified model for d-D heterogeneous arrays, derives a generalized measurement model y = Q h_u(ℓ1:K), and establishes recovery guarantees through multi-level Toeplitz (MLT) structures and Vandermonde decompositions to extract continuous AoD/AoA parameters. Both noiseless and noisy scenarios are treated via rank-based and nuclear-norm (convex) relaxations, with rigorous conditions relating sparsity K, array reconstruction degrees κ_L, and pilot design to guarantee unique recovery and parameter estimation. Benchmarks against on-grid methods show that the proposed gridless atomic-norm approach achieves superior accuracy at comparable complexity, enabling high-dimensional, gridless propagation parameter estimation for practical mmWave MIMO deployments.

Abstract

Parametric channel estimation in mmWave not only enables the anticipated large spectral efficiency gains of \acs{MIMO} systems but also reveals important propagation parameters, allowing for a low complexity representation of the channel matrix. In this work, we propose to use atomic norm as a gridless multidimensional spectral estimation approach to address parametric channel estimation where both AoD and AoA are identified. The conditions for recovery of the propagation parameters are given depending on properties of the measurement matrix, and on structural features such as the antenna geometry or the number of scatters to resolve. The proposed methodology is compared against several state-of-the-art parametric approaches.
Paper Structure (25 sections, 15 theorems, 42 equations, 3 figures, 1 table, 1 algorithm)

This paper contains 25 sections, 15 theorems, 42 equations, 3 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Channel Model for sparse mmWave propagation
  3. Propagation sensing with $\mathsf d$-D arrays
  4. Heterogeneous array deployments
  5. Parametrical MIMO Channel with heterogenous arrays
  6. Parametrical channel estimation using Atomic Norm
  7. Structural features of the measurement vector
  8. Sparse model recovery in the spectral domain
  9. Previous results for Atomic Norm-based optimization
  10. Multi-level Toeplitz matrices and the Vandermonde decomposition
  11. Enforcing a model ${\mathbf l}_u$ to match a sparse model in ${\mathcal{U}}_{\boldsymbol{\mathsf L}}^{\mathcal{F}}$
  12. Unique decomposition of a sparse model in ${\mathcal{U}}^{\mathcal{F}}_{\boldsymbol{\mathsf L}}$
  13. Atomic Norm-based model recovery for generalized linear measurements
  14. Unique recovery of a linear observation of a sparse model
  15. Noiseless sparse model recovery under generalized linear measurements
  16. ...and 10 more sections

Key Result

Proposition 1

The ${N}{M} \times {L_u}$-matrix ${\mathbf A}^\mathsf{L}=({\mathbf A}^\mathsf{tx}\otimes{\mathbf A}^\mathsf{rx}){\bf \Pi}_u$, in Q_def, has reconstruction degree, $\kappa_{\mathsf L} = \kappa_\mathsf{tx} + \kappa_\mathsf{rx}$ where $\kappa_\mathsf{tx}$ and $\kappa_\mathsf{rx}$ are the reconstructi

Figures (3)

  • Figure 1: MSE performance in a noiseless scenario solving \ref{['ec_Thm2']} for ${\boldsymbol{\mathsf L}}=[4,4,6]$ and different pilot alphabets with $P=\{1,3,4,6\}$
  • Figure 2: MSE performance in a noisy scenario solving \ref{['eq:Tracenoisy']} for ${\boldsymbol{\mathsf L}}=[4,4,6]$, and different pilot alphabets with (a)-(d) $P=3$ and ${L}=72$ (${L}$ < ${L_u}$), (b)-(e) $P=4$ and ${L}=96$ (${L}$ = ${L_u}$), (c)-(f) $P=6$ and ${L}=144$ (${L}$ > ${L_u}$). The noiseless baseline performance, when visible, is shown in gray with $\color{gray}{\star}$ marker.
  • Figure 3: MSE performance in a noisy scenario solving \ref{['eq:Tracenoisy']} for ${\boldsymbol{\mathsf L}}=[4,4,6]$, and ${\mathcal{P}}_\text{QPSK}$ pilot alphabet with (a)-(d) $P=3$ and ${L}=72$ (${L}$ < ${L_u}$), (b)-(e) $P=4$ and ${L}=96$ (${L}$ = ${L_u}$), (c)-(f) $P=6$ and ${L}=144$ (${L}$ > ${L_u}$).

Theorems & Definitions (35)

  • Remark 1
  • Definition 1
  • Definition 2
  • Remark 2
  • Proposition 1
  • proof
  • Definition 3
  • Definition 4
  • Definition 5
  • Lemma 1
  • ...and 25 more