A New Restriction on Low-Redundancy Restricted Array and Its Good Solutions

In array signal processing, a fundamental problem is to design a sensor array with low-redundancy and reduced mutual coupling, which are the main features to improve the performance of direction-of-arrival (DOA) estimation. For a $N$-sensor array with aperture $L$, it is called low-redundancy (LR) if the ratio $R=N(N-1)/(2L)$ is approaching the Leech's bound $1.217\leq R_{opt}\leq 1.674$ for $N\rightarrow\infty$; and the mutual coupling is often reduced by decreasing the numbers of sensor pairs with the first three smallest inter-spacings, denoted as $\omega(a)$ with $a\in\{1,2,3\}$. Many works have been done to construct large LRAs, whose spacing structures all coincide with a common pattern ${\mathbb D}=\{a_1,a_2,\ldots,a_{s_1},c^\ell,b_1,b_2,\ldots,b_{s_2}\}$ with the restriction $s_1+s_2=c-1$. Here $a_i,b_j,c$ denote the spacing between adjacent sensors, and $c$ is the largest one. The objective of this paper is to find some new arrays with lower redundancy ratio or lower mutual coupling compared with known arrays. In order to do this, we give a new restriction for ${\mathbb D}$ to be $s_1+s_2=c$ , and obtain 2 classes of $(4r+3)$-type arrays, 2 classes of $(4r+1)$-type arrays, and 1 class of $(4r)$-type arrays for any $N\geq18$. Here the $(4r+i)$-Type means that $c\equiv i\pmod4$. Notably, compared with known arrays with the same type, one of our new $(4r+1)$-type array and the new $(4r)$-type array all achieves the lowest mutual coupling, and their uDOFs are at most 4 less for any $N\geq18$; compared with SNA and MISC arrays, the new $(4r)$-type array has a significant reduction in both redundancy ratio and mutual coupling. We should emphasize that the new $(4r)$-type array in this paper is the first class of arrays achieving $R<1.5$ and $\omega(1)=1$ for any $N\geq18$.