# Sparse Array Design via Fractal Geometries

arXiv: https://arxiv.org/abs/2001.01217 · HTML: https://mlsanity.com/paper/2001.01217/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 5 Jan 2020.

## Built on by (most cited first, 3 of 3)

- [Deep Unfolded Recovery of Sub-Nyquist Sampled Ultrasound Image](https://mlsanity.com/paper/2103.01263/) (2021)
- [Ultrasound Signal Processing: From Models to Deep Learning](https://mlsanity.com/paper/2204.04466/) (2022)
- [Deep Learning for Ultrasound Beamforming](https://mlsanity.com/paper/2109.11431/) (2021)

## Abstract (arXiv)

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. Thus, the design of sparse arrays with large difference coarrays is of great interest. In addition, other array properties such as symmetry, robustness and array economy are important in different applications. Numerous studies have proposed diverse sparse geometries, focusing on certain properties while lacking others. Incorporating multiple properties into the design task leads 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 propose a scalable systematic way to design large sparse arrays considering multiple properties. To that end, we introduce a fractal array design in which a generator array is recursively expanded according to its difference coarray. Our main result states 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 paves the way to efficient design of large arrays of hundreds or thousands of elements with specific properties.