Sparse Symmetric Linear Arrays with Low Redundancy and a Contiguous Sum Co-Array

Sparse arrays can resolve significantly more scatterers or sources than sensor by utilizing the co-array - a virtual array structure consisting of pairwise differences or sums of sensor positions. Although several sparse array configurations have been developed for passive sensing applications, far fewer active array designs exist. In active sensing, the sum co-array is typically more relevant than the difference co-array, especially when the scatterers are fully coherent. This paper proposes a general symmetric array configuration suitable for both active and passive sensing. We first derive necessary and sufficient conditions for the sum and difference co-array of this array to be contiguous. We then study two specific instances based on the Nested array and the Kl{\o}ve-Mossige basis, respectively. In particular, we establish the relationship between the minimum-redundancy solutions of the two resulting symmetric array configurations, and the previously proposed Concatenated Nested Array (CNA) and Kl{\o}ve Array (KA). Both the CNA and KA have closed-form expressions for the sensor positions, which means that they can be easily generated for any desired array size. The two array structures also achieve low redundancy, and a contiguous sum and difference co-array, which allows resolving vastly more scatterers or sources than sensors