Half-Inverted Array Design Scheme for Large Hole-Free Fourth-Order Difference Co-Arrays

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.