Hole Identification and Filling in k-Times Extended Co-Prime Arrays for Highly Efficient DOA Estimation

Among various sparse array techniques, co-prime array is found to be more attractive because of its higher DoF with a smaller number of sensing elements. Generally, a co-prime array with O(N) sensors can offer O(N^{2})number of DoFs by exploiting the difference co-array that can be obtained from the second-order statistics of received signals. However, the number of achievable DoFs of co-prime arrays is significantly smaller than expected due to the existence of 'holes' in the difference co-array. In this paper, we make three major contributions to co-prime arrays. We first introduce a k-times extended co-prime configuration that can achieve lager number of DoFs and higher flexibility in array configurations, and the later helps better meet different application needs. Second, based on our k-times extended geometry, we uncover the mysterious veil of the holes in the difference co-arrays. We find several general rules of the locations of holes and derive the close-form expressions of the exact locations of all holes in the difference co-arrays of different co-prime array structures. Finally, we propose a specific array structure called complementary sub-array that can fill all of holes existing in the difference co-array. Compared with the traditional hole-existing co-prime arrays, our k-times complementary co-prime array has either similar number of sensors and DoFs with much smaller array aperture required, or much higher DoFs with the same aperture.