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Resolution-Aliasing Trade-off in Near-Field Localisation

Baptiste Sambon, Gilles Monnoyer, Luc Vandendorpe, Claude Oestges

TL;DR

This work addresses the problem of localisation in Near-Field XL-MIMO systems under sub-Nyquist sampling, where increased aperture improves resolution but risks aliasing. It develops a unified chirp-based framework that links array geometry and sampling to the spatial bandwidth and to aliasing artefacts, introducing the Non-Contributive Zone and Critical Antenna Elements as geometric tools. The study provides explicit aliasing-free region and resolution conditions for rectangular and circular arrays, and reveals that resolution and aliasing are not strictly coupled, enabling design guidelines that can achieve higher resolution without necessarily increasing aliasing. The results offer practical insights for NF array design, including spacing, aperture, dimensionality, and radius, to balance resolution and aliasing in XL-MIMO deployments.

Abstract

Extremely Large-scale MIMO (XL-MIMO) systems operating in Near-Field (NF) introduce new degrees of freedom for accurate source localisation, but make dense arrays impractical. Sparse or distributed arrays can reduce hardware complexity while maintaining high resolution, yet sub-Nyquist spatial sampling introduces aliasing artefacts in the localisation ambiguity function. This paper presents a unified framework to jointly characterise resolution and aliasing in NF localisation and study the trade-off between the two. Leveraging the concept of local chirp spatial frequency, we derive analytical expressions linking array geometry and sampling density to the spatial bandwidth of the received field. We introduce two geometric tools--Critical Antenna Elements (CAEs) and the Non-Contributive Zone (NCZ)--to intuitively identify how individual antennas contribute to resolution and/or aliasing. Our analysis reveals that resolution and aliasing are not always strictly coupled, e.g., increasing the array aperture can improve resolution without necessarily aggravating aliasing. These results provide practical guidelines for designing NF arrays that optimally balance resolution and aliasing, supporting efficient XL-MIMO deployment.

Resolution-Aliasing Trade-off in Near-Field Localisation

TL;DR

This work addresses the problem of localisation in Near-Field XL-MIMO systems under sub-Nyquist sampling, where increased aperture improves resolution but risks aliasing. It develops a unified chirp-based framework that links array geometry and sampling to the spatial bandwidth and to aliasing artefacts, introducing the Non-Contributive Zone and Critical Antenna Elements as geometric tools. The study provides explicit aliasing-free region and resolution conditions for rectangular and circular arrays, and reveals that resolution and aliasing are not strictly coupled, enabling design guidelines that can achieve higher resolution without necessarily increasing aliasing. The results offer practical insights for NF array design, including spacing, aperture, dimensionality, and radius, to balance resolution and aliasing in XL-MIMO deployments.

Abstract

Extremely Large-scale MIMO (XL-MIMO) systems operating in Near-Field (NF) introduce new degrees of freedom for accurate source localisation, but make dense arrays impractical. Sparse or distributed arrays can reduce hardware complexity while maintaining high resolution, yet sub-Nyquist spatial sampling introduces aliasing artefacts in the localisation ambiguity function. This paper presents a unified framework to jointly characterise resolution and aliasing in NF localisation and study the trade-off between the two. Leveraging the concept of local chirp spatial frequency, we derive analytical expressions linking array geometry and sampling density to the spatial bandwidth of the received field. We introduce two geometric tools--Critical Antenna Elements (CAEs) and the Non-Contributive Zone (NCZ)--to intuitively identify how individual antennas contribute to resolution and/or aliasing. Our analysis reveals that resolution and aliasing are not always strictly coupled, e.g., increasing the array aperture can improve resolution without necessarily aggravating aliasing. These results provide practical guidelines for designing NF arrays that optimally balance resolution and aliasing, supporting efficient XL-MIMO deployment.
Paper Structure (37 sections, 5 theorems, 60 equations, 10 figures, 2 tables)

This paper contains 37 sections, 5 theorems, 60 equations, 10 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Contributions
  3. Paper Structure
  4. Notations
  5. System Model
  6. Ambiguity Function Analysis
  7. Discrete Antenna Arrays and Non-Aliasing Conditions
  8. Localisation Resolution
  9. Chirp-based framework
  10. Chirp-based Modelling of the Received Field
  11. Chirp-based Non-Aliasing Conditions
  12. Geometrical Tools
  13. Non-Contributive Zone
  14. Critical Antenna Elements
  15. Linear Array
  16. ...and 22 more sections

Key Result

Proposition 1

Given two antenna arrays $\mathcal{Z}^{(1)} \subseteq \mathcal{Z}^{(2)}$ with the same inter-element spacings $\Delta_{i}$ along each dimension $i$, the corresponding aliasing-free regions, $\mathcal{S}^{(1)}$ and $\mathcal{S}^{(2)}$ satisfy for all sources locations $\boldsymbol{x}_s \in \mathbb{R}^{d}$.

Figures (10)

  • Figure 1: Configuration with a source location $\boldsymbol{x}_s$, receive antenna set $\mathcal{Z}$, in a near-field configuration.
  • Figure 2: Ambiguity functions of a point source at $\boldsymbol{x}_s = (0, 2000) \lambda$ for linear arrays of $N=1200$ antennas over a $600\lambda$ aperture (first row), $N=300$ antennas over a $1200\lambda$ aperture (second row), $N=400$ antennas over a $1600\lambda$ aperture (third row) .
  • Figure 3: Illustration of the spectral folding phenomenon and its impact on aliasing in the $AF$.
  • Figure 4: Local spatial frequencies $\boldsymbol{\mathrm{k}}^{h}(\boldsymbol{z}; \boldsymbol{x}_s)$ of the received field $h(\boldsymbol{z}; \boldsymbol{x}_s)$ at various antenna positions $\boldsymbol{z}$ for a source at $\boldsymbol{x}_s$.
  • Figure 5: Amplitude of the spectrum $H(\bar{\boldsymbol{k}}^{h} ; \tilde{\boldsymbol{x}}_s, \boldsymbol{x}_s)$ for two-dimensional rectangular and circular arrays. The maximum and minimum chirp frequencies along each axis, derived from \ref{['eq:ki_max_min_chirp']}, are indicated by the vertical and horizontal lines.
  • ...and 5 more figures

Theorems & Definitions (12)

  • Definition 1
  • Definition 2: Non-contributive Zone
  • Proposition 1: Inclusion Principle
  • Definition 3: Critical Antenna Elements
  • Proposition 2: Removal of Antennas
  • proof
  • Proposition 3: Addition of Antennas
  • proof
  • Proposition 4
  • proof
  • ...and 2 more