On the Foundation of Sparsity Constrained Sensing—Part II: Diophantine Sampling With Arbitrary Temporal and Spatial Sparsity

In the second part of the series papers, we set out to study the algorithmic efficiency of sparsity-constrained sensing. Stemmed from co-prime sampling/array, we propose a generalized framework, termed Diophantine sensing, which utilizes generic Diophantine equation theory and higher-order sparse ruler to strengthen the sampling time (delay), the degree of freedom (DoF), and the sampling sparsity, simultaneously. It is well known that co-prime sensing can reconstruct the autocorrelation of a sequence with significantly more lags based on Bézout theorem. However, Bézout theorem also puts two practical constraints in this framework. For frequency estimation, co-prime sampling needs sampling time proportional to the product of down-sampling rates; As for Direction-of-arrival (DoA) estimation, the array cannot be arbitrarily sparse, where the least sensor inter spacing needs to be less than a half of wavelength. Resorting to higher-moment statistics, the proposed Diophantine framework presents two fundamental improvements. First, on frequency estimation we prove that given arbitrarily large down-sampling rates, there exist sampling schemes where the number of samples needed is only proportional to the sum of DoF and the least number of snapshots required for each lag, which implies a linear sampling time. Second, on DoA estimation, we propose two generic array constructions such that given $N$ sensors, the minimal spacing among sensors can be as large as a polynomial of $N$, which indicates that an arbitrarily sparse array (with arbitrarily small mutual coupling) exists given sufficiently many sensors. In addition, the proposed array configurations produce the best known asymptotic DoF bound.