  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.
