High-Order Cumulants Based Sparse Array Design Via Fractal Geometries—Part I: Structures and DOFs
Zixiang Yang, Qing Shen, Wei Liu, Yonina C. Eldar, Wei Cui
Abstract
Array structures based on the high-order difference co-array concept provide a large number of degrees of freedom, but are typically difficult to design under multiple optimality criteria. In this paper, we present a joint across-order (across different cumulant order <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$q$</tex-math></inline-formula> ) and inner-order (within the same cumulant order <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$q$</tex-math></inline-formula> ) fractal framework to form a fractal array based on the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$2q$</tex-math></inline-formula> th-order difference co-array ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$2q$</tex-math></inline-formula> th-O-Fractal) by recursively using a simple generator. We show that multiple properties of interest, including large consecutive difference co-array, closed-form sensor positions, hole-free difference co-array, robustness to sensor failures, and resilience to mutual coupling, are inherited from the generator under appropriate conditions. Part I of the work focuses on array structures with a large uniform or hole-free (higher-order) difference co-array. First, we show that for an array of size <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$N$</tex-math></inline-formula> , <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathcal {O}(N^{2q})$</tex-math></inline-formula> consecutive co-array lags can be provided by optimizing the generator. In addition, the generated structure outperforms existing structures in terms of the number of consecutive lags offered. Then, proof is provided that under given requirements on the generator, the hole-free property is inherited for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$q = 2$</tex-math></inline-formula> , and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathcal {O}(N^{4})$</tex-math></inline-formula> hole-free fourth-order difference co-array lags can be achieved by the proposed framework, which is larger than those of existing structures. Simulation results verify the superiority of the proposed framework in terms of estimation accuracy and resolution capability. Part II of this work focuses on the properties of array robustness and mutual coupling.