Nonuniform sparse array design for active sensing

Active sensing using multiple transmitting elements and independent waveforms has recently attracted much attention. Using M transmitting and N receiving elements, one can virtually simulate a physical array of MN elements by the sum co-array. Nonuniform sparse arrays can further be used in active sensing to produce the difference co-array of the given sum co-array with dramatically increased degree of freedom. However, current literature lacks an efficient design method for active sensing with nonuniform sparse arrays. In this paper, we address this problem and propose several systematic construction methods based on some classical results in number theory. By using these methods, we are able to construct active sensing sparse arrays, in which the difference co-array of the sum-co-array has aperture in the order of O(M^2N^2). Furthermore, it has no holes within this aperture. Several performance bounds on the maximum aperture of the sparse array are then provided. These can be used in the future to compare the performance of other suboptimal nonuniform sparse array geometries.