Beating the aliasing limit with aperiodic monotile arrays

TL;DR

The study introduces MAS arrays built from aperiodic Hat monotiles, focusing on Tile($p$) tilings with Specter ($p=\tfrac{1}{2}$) as a standout. Through ARF and beamforming analyses, it demonstrates that certain $p$ ranges, particularly near $p=0.5$, can beat the WNS aliasing limit by distributing sampling in a way that reduces prominent aliasing peaks and yields high SNR for both single and distributed sources. Large-N implementations reveal near-uniform sampling of interstation azimuths and distances, making MAS arrays competitive or superior to regular and other aperiodic tilings, with robust performance under sensor perturbations. The findings suggest a general design principle for wave-field sampling across domains, with potential applications in seismology, photonics, and telecommunications.

Abstract

Finding optimal wave sampling methods has far-reaching implications in wave physics, such as seismology, acoustics, and telecommunications. A key challenge is surpassing the Whittaker-Nyquist-Shannon (WNS) aliasing limit, establishing a frequency below which the signal cannot be faithfully reconstructed. However, the WNS limit applies only to periodic sampling, opening the door to bypass aliasing by aperiodic sampling. In this work, we investigate the efficiency of a recently discovered family of aperiodic monotile tilings, the Hat family, in overcoming the aliasing limit when spatially sampling a wavefield. By analyzing their spectral properties, we show that monotile aperiodic seismic (MAS) arrays, based on a subset of the Hat tiling family, are efficient in surpassing the WNS sampling limit. Our investigation leads us to propose MAS arrays as a novel design principle for seismic arrays. We show that MAS arrays can outperform regular and other aperiodic arrays in realistic beamforming scenarios using single and distributed sources, including station-position noise. While current seismic arrays optimize beamforming or imaging applications using spiral or regular arrays, MAS arrays can accommodate both, as they share properties with both periodic and aperiodic arrays. More generally, our work suggests that aperiodic monotiles can be an efficient design principle in various fields requiring wave sampling.

Figures (11)

• Figure 1: The Tile$(p)$ family of tilings and other quasicrystals a) We highlight five examples of Tile$(p)$ tilings, from left to right: $p = 0, \frac{1}{1+\sqrt{3}}, 1/2, \frac{\sqrt{3}}{1+\sqrt{3}}, 1$. These particular tilings are formed by tiles whose names are written on the corresponding tiles. Next to each tile, we show a small portion of the tiling where the mirrored tile needed to generate the tiling is outlined in a darker color. The fourteen vertices of each tile are highlighted by black dots for the Hat and the Specter, and the two length scales, $a=p$ and $b=1-p$, are shown around the Hat tile. In the central semicircular diagram, the green shaded area highlights the range of $p$ for which seismic arrays bypass aliasing. b) The Specter tile can also tile the plane aperiodically without using the reflected tile. Instead, it requires the Mystic composite tile, composed of two Specter tiles highlighted with darker colors. c) An example of a Penrose tiling using thick and thin rhombi. d) An example of the aperiodic Square-Triangle tiling.
• Figure 2: Single Tile(p) ARF analysis. a) ARF of the Hat, b) the Specter, and c) the Turtle as a function of the normalized wavenumber $|\mathbf{\overline{k}}|$. d) and f) show the ARF of a Triangular lattice (hexagonal lattice plus its centers) array and the Square-Triangle metatile, respectively. For each $p$, e) shows the peak sidelobe level as a colormap, calculated using Eq. \ref{['eq:ARFmax']} for each $|\mathbf{\overline{k}}|$. The white horizontal dashed lines, indicated by red arrows show the positions of the maxima of the Triangular array ARF (d): $\left[\frac{2n}{\sqrt{3}}, 2n \right], n \in \mathbb{N}$, and their combinations. The white horizontal dotted lines show the maxima of the Square-Triangle metatile ARF (f), indicated by the purple arrows. The black dotted horizontal lines show the positions of the two main sidelobes in common between the Specter and the Square-Triangle ARFs. g) Illustration of the shared vertices between a Triangular lattice and some Tile($p$) vertices. The open-colored circles show the shared vertices. The solid circles show the Tile($p$) vertices falling outside of the Triangular lattice. h) Re-arrangement of the tiles of the Square-Triangle metatile (g) to form a pseudo-Specter. i) Correspondence between a Square-Triangle aperiodic tiling and the Specter (delineated in black).
• Figure 3: Synthetic beamforming for single Tile($p$) arrays. The first row shows the maps of the arrays. The grey dots for the Hat, Specter, and Turtle tiles are the randomized positions of the stations. The second row shows the beamforming results for a single source coming from the northwest at 2 km/s, highlighted by the black circle. The third row shows the beamforming results for a homogeneous illumination with sources of 2 km/s plane waves distributed around the arrays. The black dashed circles delineate the "signal" ring, and the white circle, the "noise" ring used to compute the signal-to-noise ratio (SNR) in f). The different columns show a) a triangular array, b) the Hat, c) the Specter, d) the Turtle, and e) a Square-Triangle metatile. f) SNR of the beamforming results for the distributed sources as a function of the parameter $p$. The gray curve and shaded area show the Tile($p$) SNR's mean and standard deviation, respectively, when perturbing the station positions (gray dots in the first row). Vertical dashed lines highlight the $p$ values of the Hat, Specter, and Turtle arrays. The thick red line shows the SNR for the triangular array. The mean SNR of the triangular array with perturbed positions is shown by the thin red line with the light red shaded area representing the standard deviation. The Square-Triangle array SNR is displayed by the purple horizontal line. The disordered curves are averaged over 100 disorder realizations.
• Figure 4: Beamforming SNR as a function of the wavelength. Beamforming SNR for distributed sources as a function of the normalized wavelength for regular square and triangular arrays and for the Specter and Square-Triangle aperiodic arrays. The vertical dashed line indicates the aliasing limit: when the wavelength $\lambda$ equals twice the minimum interstation distance $r_{\mathrm{min}}$.
• Figure 5: Beamforming performances of Large-N aperiodic arrays. a) from left to right: Station map of the triangular array, ARF (intensity in log scale), beamforming for distributed sources, beamforming for a single source. The rest of the columns show the same as in a) for the b) Specter array, c) the Square-Triangle array, and d) the Penrose array. The black circles in the distributed sources beamforming panels (third column) delineate the areas used to compute the SNR. The black circle in the single source beamforming panels delineates the areas used to compute the SNR. e) SNR of the distributed sources beamforming for the Tile($p$) arrays (black curve) as a function of $p$. The colored horizontal lines show the SNR level for the triangular, the Square-Triangle, and the Penrose arrays. The vertical dashed lines show the $p$ values of the Hat, the Specter, and the Turtle arrays. f) Same as e) for the single source beamforming results. In e) and f), SNR $\leq$ 1 when the signal is aliased and larger if not.