Half-Inverted Array Design Scheme for Large Hole-Free Fourth-Order Difference Co-Arrays
Yuan-Pon Chen, Chun-Lin Liu · IEEE Transactions on Signal Processing · 2023
Linear sparse arrays with fourth-order cumulant processing can resolve$\mathcal{O}(N^4)$directions-of-arrival (DOAs) using$N$physical sensors, provided that the fourth-order difference co-array$\Delta_4$contains a contiguous segment of size$\mathcal{O}(N^4)$. Furthermore, if$\Delta_4$has no holes, then the received data can be fully exploited in subspace-based DOA estimators. However, few existing arrays attain large hole-free$\Delta_4$. Many existing arrays designed for$\Delta_4$are constructed from two smaller arrays, called the basis arrays. Nevertheless, such arrays either restrict the basis arrays to certain types or have no guarantee of hole-free$\Delta_4$. This paper proposes the half-inverted (HI) arrays, parameterized by two basis arrays$\mathbb{S}^{(1)}$and$\mathbb{S}^{(2)}$, the shifting parameter$M$, and the scaling parameter$\sigma$. An HI array consists of$\mathbb{S}^{(1)}$and an inverted, scaled, and shifted version of$\mathbb{S}^{(2)}$. HI arrays are guaranteed with hole-free$\Delta_4$over a range of$(M,\sigma)$pairs. This property unifies several existing arrays with hole-free$\Delta_4$and admits an optimization problem over$(M,\sigma)$. The half-inverted general hole-free (HIGH) scheme is defined as the HI array with a closed-form and optimized$(M,\sigma)$pair determined by the second-order co-arrays of the basis arrays. The HIGH scheme enjoys a large hole-free$\Delta_4$. The shift-scale representation (SSR) is presented to study$\Delta_4$of HI arrays visually. From these results, the half-inverted array based on second-order optimization and extended shift (HI-SOES) is proposed. For a fixed$N$, HI-SOES synthesizes a hole-free$\Delta_4$larger than an existing array. Numerical examples demonstrate the DOA estimation performance of HI arrays and existing arrays.