High-Order Cumulants Based Sparse Array Design Via Fractal Geometries—Part II: Robustness and Mutual Coupling
Zixiang Yang, Qing Shen, Wei Liu, Yonina C. Eldar, Wei Cui
Abstract
Array robustness and mutual coupling are two important factors affecting the direction finding performance of various array structures. These properties are not typically analyzed in DOA estimation based on <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-arrays. In the first part of this work, we proposed fractal arrays for exploiting 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). In this second part, we analyze robustness and mutual coupling leakage of 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-O-Fractal framework. We introduce a robustness analysis tool, and derive a fragility upper bound 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-O-Fractal as a function of the generator array fragility and the fractal order. Our results show that the robustness of <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 can be improved by designing a robust generator or increasing the fractal order. The mutual coupling leakage of <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 is also shown to be upper bounded by that of its generator array. Simulation results demonstrate that the proposed <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 has superior performance over existing structures exploiting 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, in the presence of random sensor damage and mutual coupling effect.