From Co-prime to the Diophantine Equation Based Sparse Sensing

With a careful design of sample spacings either in temporal and spatial domain, co-prime sensing can reconstruct the autocorrelation at a significantly denser set of points based on Bazout theorem. However, still restricted from Bazout theorem, it is required O(M1 + M2) samples to estimate frequencies in the case of co-prime sampling, where M1 and M2 are co-prime down-sampling rates. Besides, for Direction-of-arrival (DOA) estimation, the sensors can not be arbitrarily sparse in co-prime arrays. In this letter, we restrain our focus on complex waveforms and present a framework under multiple samplers/sensors for both frequency and DOA estimation based on Diophantine equation, which is essentially to estimate the autocorrelation with higher order statistics instead of the second order one. We prove that, given arbitrarily high down-sampling rates, there exist sampling schemes with samples to estimate autocorrelation only proportional to the sum of degrees of freedom (DOF) and the number of snapshots required. In the scenario of DOA estimation, we show there exist arrays of N sensors with O(N^3) DOF and O(N) minimal distance between sensors.