Aliasing in Near-Field Array Ambiguity Functions: a Spatial Frequency-Domain Framework

Next-generation communication and localization systems increasingly rely on extremely large-scale arrays (XL-arrays), which promise unprecedented spatial resolution and new functionalities. These gains arise from their inherent operation in the near field (NF) regime, where the spherical nature of the wavefront can no longer be ignored; consequently, characterizing the ambiguity function – which amounts to the matched beam pattern – is considerably more challenging. Implementing very wide apertures with half-wavelength element spacing is costly and complex. This motivates thinning the array (removing elements), which introduces intricate aliasing structures, i.e., grating lobes. Whereas prior work has addressed this challenge using approximations tailored to specific array geometries, this paper develops a general framework that reveals the fundamental origins and geometric behavior of grating lobes in near-field ambiguity functions. Using a local spatial-frequency analysis of steering signals, we derive a systematic methodology to model NF grating lobes as aliasing artifacts, quantifying their structure on the AF, and providing design guidelines for XL-arrays that operate within aliasing-safe regions. We further connect our framework to established far-field principles. Finally, we demonstrate the practical value of the approach by deriving closed-form expressions for aliasing-free regions in canonical uniform linear arrays and uniform circular arrays.

💡 Research Summary

The paper tackles a pressing challenge in next‑generation communication and localization systems that employ extremely large‑scale arrays (XL‑arrays). When such arrays operate in the near‑field (NF) region, the spherical wavefront cannot be approximated as planar, and the resulting ambiguity function (AF)—the matched‑filter response that quantifies how well two hypothesized source locations can be distinguished—exhibits a far more intricate structure than in the traditional far‑field (FF) case. In particular, sparse or “thinned” XL‑arrays, which are attractive because they reduce hardware cost and complexity, generate grating lobes that manifest as additional peaks in the AF, potentially causing severe localization ambiguities.

Key Contributions

1. Unified Theoretical Formalism – The authors define NF grating lobes as aliasing artifacts arising from spatial sampling of the steering signals. By introducing a bijective mapping ν(ρ) that parameterizes any antenna geometry (linear, circular, curved, etc.) and treating the steering signal h(ν(ρ);x) as a spatial chirp, they compress the signal’s spectral content into a local spatial frequency k(ρ;x)=∂ϕ/∂ρ.
2. Spatial‑Frequency‑Domain Framework – Applying Fourier analysis to the ρ‑domain reveals that uniform sampling with step Δ folds the continuous spectrum H(κ;x) with period 2π/Δ. Whenever the folded spectra overlap, the AF exhibits grating lobes. Thus, the presence of grating lobes is directly linked to the condition |k(ρ;x)| > π/Δ for some ρ.
3. Connection to Far‑Field Theory – When Δ ≤ λ/2, the local spatial frequency becomes approximately constant, and the aliasing condition collapses to the classic FF rule that spacing larger than half a wavelength produces angular repetitions. Hence the new framework naturally extends FF results to NF scenarios.
4. Design Guidelines and Alias‑Free Regions – By solving |k(ρ;x)| ≤ π/Δ for all ρ, the authors derive explicit “alias‑free” regions in the source‑position space (distance‑angle domain). These regions tell a system designer exactly which ranges of range and angle can be safely covered without grating‑lobe‑induced ambiguities, even when the physical element spacing exceeds λ/2.
5. Closed‑Form Expressions for Canonical Arrays – For a uniform linear array (ULA) and a uniform circular array (UCA), the paper provides analytic formulas that delineate the alias‑free boundaries. For a ULA of length L and spacing Δ, the safe condition becomes Δ ≤ λ /