Resolution-Aliasing Trade-off in Near-Field Localisation

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.

💡 Research Summary

This paper investigates the fundamental trade‑off between spatial resolution and aliasing in near‑field (NF) localisation using extremely large‑scale MIMO (XL‑MIMO) arrays. In the NF region, the spherical‑wave propagation model makes each antenna’s received signal a non‑linear function of both range and angle, thereby providing a richer spatial frequency content—referred to as the local chirp spatial frequency—than in the far‑field. The authors first formulate a continuous ambiguity function (AF) that quantifies how well a hypothesised source location matches the true one. When the array is discretised, spatial sampling creates periodic replicas of the underlying spatial spectrum. By analysing the Fourier transform of the product of steering vectors, they derive a relaxed non‑aliasing condition: the maximum local spatial frequency (\bar K_i) along each axis must satisfy (\bar K_i \le 2\pi/\Delta_i), where (\Delta_i) is the inter‑element spacing. This condition is less stringent than the classic Nyquist rule ((\Delta_i \le \lambda/2)) because aliasing only harms localisation when the zero‑frequency component of the spectrum (which directly yields the AF peak) is corrupted.

Resolution is defined as the full‑width‑half‑maximum (FWHM) of the AF peak and is shown to be inversely proportional to the spatial bandwidth (\bar B_i) captured by the array: (\delta x_i = 2\pi/\bar B_i). The spatial bandwidth is the extent of the spatial‑frequency support of the received field across the aperture. Larger apertures increase (\bar B_i) and thus improve resolution, but they also tend to increase (\bar K_i) because the phase variation across the array becomes steeper, potentially violating the non‑aliasing condition.

To make these relationships intuitive, the authors introduce two geometric tools:

1. Critical Antenna Elements (CAEs) – antennas that, for a given test location, push the maximum spatial frequency to its limit. Adding CAEs enlarges the spatial bandwidth (improving resolution) but may also raise (\bar K_i) and cause aliasing.

2. Non‑Contributive Zone (NCZ) – regions of the aperture where additional antennas do not increase the spatial bandwidth. Placing antennas in the NCZ reduces the spacing (\Delta_i) without affecting (\bar B_i), thereby helping to satisfy the non‑aliasing condition.

Closed‑form expressions for (\bar B_i) and (\bar K_i) are derived for two canonical array geometries: rectangular and circular. For a rectangular array, the bandwidth scales with the product of the number of elements along each axis and inversely with the element spacing. For a circular array of radius (R) with (N) uniformly spaced elements, the bandwidth behaves roughly as (N/(2R)) while the spacing is (2\pi R/N). Consequently, increasing the radius (i.e., the aperture) improves resolution but also enlarges the spacing, potentially violating the alias‑free condition unless more elements are added.

Based on these analyses, the paper proposes a practical design methodology:

1. Specify target resolution → compute required spatial bandwidth.
2. Determine minimal number of antennas and aperture that can deliver this bandwidth.
3. Check the relaxed non‑aliasing condition for the chosen geometry.
4. If violated, identify the NCZ and add antennas there to shrink (\Delta_i) without altering (\bar B_i), or redesign the aperture to balance the two effects.

Simulation results validate the theory. Various array configurations (dense vs. sparse, rectangular vs. circular) are evaluated for different source distances and angles. The results show that sparsely populated arrays, when designed according to the proposed framework, achieve resolution comparable to dense arrays while drastically reducing hardware complexity, and that the relaxed non‑aliasing condition accurately predicts the presence or absence of spurious peaks in the AF.

In conclusion, the study demonstrates that resolution and aliasing in NF localisation are not inherently coupled; with careful geometric design and an understanding of local chirp spatial frequencies, one can simultaneously attain high resolution and suppress aliasing. The introduced CAE/NCZ concepts provide intuitive visual tools for engineers, and the derived analytical expressions serve as a solid foundation for future work on multi‑source scenarios, wideband signals, and adaptive array reconfiguration in XL‑MIMO systems.