On the Foundation of Sparse Sensing (Part II): Diophantine Sampling and Array Configuration

In the second part of the series papers, we set out to study the algorithmic efficiency of sparse sensing. Stemmed from co-prime sensing, we propose a generalized framework, termed Diophantine sensing, which utilizes generic Diophantine equation theory and higher-order sparse ruler to strengthen the sampling time, the degree of freedom (DoF), and the sampling sparsity, simultaneously. 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 number of snapshots required, which implies a linear sampling time. Second, on Direction-of-arrival (DoA) estimation, we propose two generic array constructions such that given N sensors, the minimal distance between sensors can be as large as a polynomial of N, O(N^q), which indicates that an arbitrarily sparse array (with arbitrarily small mutual coupling) exists given sufficiently many sensors. In addition, asymptotically, the proposed array configurations produce the best known DoF bound compared to existing sparse array designs.