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.