Sparse Array Design via Fractal Geometries

Abstract

Sparse sensor arrays have attracted considerable attention in various fields such as radar, array processing, ultrasound imaging and communications. In the context of correlation-based processing, such arrays enable to resolve more uncorrelated sources than physical sensors. This property of sparse arrays stems from the size of their difference coarrays, defined as the differences of element locations. Hence, the design of thinned arrays with large difference coarrays is of great interest. In addition to the difference coarray, other array properties such as symmetry, robustness and array economy are important in different applications. Such design tasks lead to combinatorial problems which are generally NP-hard. For small arrays these optimization problems can be solved by brute force, however, in large scale they become intractable. In this paper, we introduce a fractal array design in which a generator array is recursively expanded according to its difference coarray. Our main result is that for an appropriate choice of the generator such fractal arrays exhibit large difference coarrays. Furthermore, we show that the fractal arrays inherit their properties from their generators. Thus, a small generator can be optimized according to desired requirements and then expanded to create a fractal array which meets the same criteria. This approach provides a simple and efficient way for designing large scale sparse arrays with specific properties.

Deep Unfolded Recovery of Sub-Nyquist Sampled Ultrasound Image

“…Using a sparse set of array elements, following COBA, [17], [19], can lead to substantial data reduction, while producing images at least as good as those obtained with DAS. As was shown in [17], using COBA, one can achieve the effective beampattern as would have been obtained using the sum coarray of the given array of transducers [38].…”

“…Therefore, [17] suggested using a sparse array, given by the original array after removing several elements, for generating a desired beamformer beampattern, in a method called sparse COBA (SCOBA). The use of fractal arrays [39]- [43], was discussed in [19]. The geometry of such arrays is given by…”

“…Two of the datasets consist of temporal sub-sampled data using the Xampling mechanism, [12], with different reduction factors. The third dataset consists of spatial reduction as well, following [17], [19], [21]. The subsampled Fourier coefficients are calculated as previously described in Section II-B, for the first two datasets, and following Section II-D for the third dataset.…”

Sparse Symmetric Linear Arrays with Low Redundancy and a Contiguous Sum Co-Array

“…Proof: Under certain conditions, we may obtain a solution to (P5) by considering a relaxed problem. Specifically, solving (16) for N 3 , substituting the result into (15), and relaxing N 1 , N 2 ∈ R leads to the following concave quadratic program: maximize…”

“…For instance, smaller (but computable) MRAs can be extended to larger apertures by repeating regular substructures in the array [11], or by recursively nesting them in a fractal manner [12]. Recent research into such recursive or fractal arrays [13]- [16] has also revived interest in symmetric array configurations. Symmetry or translational invariance in either the physical or coarray domain can further simplify array design [17], calibration [18], and processing [19]- [22], [2, p. 721].…”