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With a careful design of sample spacings either in temporal and spatialdomain, co-prime sensing can reconstruct the autocorrelation at a significantlydenser set of points based on Bazout theorem. However, still restricted fromBazout theorem, it is required O(M1 + M2) samples to estimate frequencies inthe case of co-prime sampling, where M1 and M2 are co-prime down-samplingrates. Besides, for Direction-of-arrival (DOA) estimation, the sensors can notbe arbitrarily sparse in co-prime arrays. In this letter, we restrain our focuson complex waveforms and present a framework under multiple samplers/sensorsfor both frequency and DOA estimation based on Diophantine equation, which isessentially to estimate the autocorrelation with higher order statisticsinstead of the second order one. We prove that, given arbitrarily highdown-sampling rates, there exist sampling schemes with samples to estimateautocorrelation only proportional to the sum of degrees of freedom (DOF) andthe number of snapshots required. In the scenario of DOA estimation, we showthere exist arrays of N sensors with O(N^3) DOF and O(N) minimal distancebetween sensors.

From Co-prime to the Diophantine Equation Based Sparse Sensing