基于分形几何的稀疏阵列设计
Sparse Array Design via Fractal Geometries
AI总结：
针对现有稀疏阵列设计难以兼顾多特性且大规模阵列组合优化NP难的问题，提出基于生成元差共阵递归扩展的分形阵列设计方法，可继承生成元特性并实现大差共阵，为大规模阵列高效设计提供方案。
摘要：
稀疏传感器阵列已在雷达、阵列处理、超声成像和通信等诸多领域受到广泛关注。在基于相关的处理场景下，这类阵列能够分辨的不相关源数量多于其物理传感器数量。稀疏阵列的这一特性源于其差共阵的规模，差共阵定义为阵元位置的差值集合。因此，设计具有大差共阵的稀疏阵列具有重要研究价值。此外，阵列的对称性、鲁棒性和阵列经济性等其他特性在不同应用中也十分关键。已有大量研究提出了多种稀疏阵列构型，它们往往侧重某些特性却存在其他方面的不足。将多种特性纳入设计任务会产生通常属于NP难的组合优化问题。对于小型阵列，这类优化问题可通过暴力求解解决，但在大规模场景下会变得难以处理。本文提出了一种可扩展的系统化方法，用于设计兼顾多种特性的大型稀疏阵列。为此，我们提出了一种分形阵列设计方案，其中生成元阵列会根据其差共阵进行递归扩展。我们的核心结果表明，只要合适选择生成元，这类分形阵列就能具备大差共阵。此外，我们证明分形阵列会继承其生成元的特性。因此，可以根据需求优化小型生成元，再通过扩展构建出满足相同指标的分形阵列。该方法为高效设计包含数百或数千个阵元、具备特定特性的大型阵列提供了可行路径。
英文摘要：
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.
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