Coprime Sensing via Chinese Remaindering Over Quadratic Fields—Part II: Generalizations and Applications

Abstract

The practical application of a new class of coprime arrays based on the Chinese remainder theorem (CRT) over quadratic fields is presented in this paper. The proposed CRT arrays are constructed by ideal lattices embedded from coprime quadratic integers with B1 and B2 being their matrix representations respectively, whereby the degrees of freedom (DOF)The geometrical constructions and theoretical foundations were discussed in the accompanying paper in great detail, while this paper focuses on aspects of the application of the proposed arrays in two-dimensional (2D) remote sensing. A generalization of CRT arrays based on two or more pairwise coprime ideal lattices is proposed with closed-form expressions on sensor locations, the total number of sensors and the achievable DOF. The issues pertaining to the coprimality of any two quadratic integers are also addressed to explore all possible ideal lattices. Exploiting the symmetry of lattices, sensor reduction methods are discussed with the coarray remaining intact for economic maximization. In order to extend conventional angle estimation techniques based on uniformly distributed arrays to the method that can exploit any coarray configurations based on lattices, this paper introduces a hexagon-to-rectangular transformation to 2D spatial smoothing, providing the possibility of finding more compact sensor arrays. Examples are provided to verify the feasibility of the proposed methods.

Exact and Robust Reconstructions of Integer Vectors Based on Multidimensional Chinese Remainder Theorem (MD-CRT)

“…we have M = LK 11 and N = LK 12 , where K i j for 1 ≤ i, j ≤ 2 are all integer matrices due to the unimodularity of V. Therefore, L is a cld of M and N. Then, we demonstrate that L is actually a gcld of M and N. For any other cld T of M and N, i.e., M = TA and N = TB for some integer matrices A and B, we have, from (11), T (AV 11 + BV 21 ) = L, which means that T is a left divisor of L. Therefore, L is a gcld of M and N, and is given by…”

“…In particular, if M and N are left coprime, their gcld L must be unimodular. We right-multiply L −1 on both sides of (11), and can further get…”

“…To date, there has been a surge in work on applying the CRT for partitioning a large task into a number of smaller but independent subtasks, which can be performed in parallel. For example, the CRT has been intensively utilized in the signal processing community in the context of cyclic convolution [4], [5], fast Fourier transform [6], [7], coprime sensor arrays [8]- [11], to name a few. It also finds applications in various other fields, such as computer arithmetic based on modulo operations (e.g., multiplication of very large numbers), coding theory (e.g., This work was supported in part by NIH under Grants R01GM109068 and R01MH104680, and in part by NSF under Grant 1539067.…”

Multi-Channel Modulo Samplers Constructed From Gaussian Integers

“…For two Gaussian integers x 0 and x 1 , they are relatively prime (or co-prime) if and only if there exist two Gaussian integers α and β so that the Bézout's identity holds, i.e., x 0 α + x 1 β = 1. Alternatively, x 0 = p 0 + iq 0 and x 1 = p 1 + iq 1 are co-prime if and only if their greatest common divisor (gcd) satisfy [17] gcd (N(x 0 ), N(x 1 ), p 0 p 1 + q 0 q 1 ) = 1.…”

Sequential Estimation of Frequency and Direction-of-Arrival Based on the Relaxed Coprime Array for Multiple Targets

“…Specifically, the parameterization in coprime undersampling is guided by Chinese Remainder Theorem (CRT) [14]- [16], which is a powerful approach to achieve the parameter estimation under the subsampling condition. Authors in [17]- [19] presented the CRT-based estimation methods to achieve DOA estimation with high accuracy. In [20], [21], some methods utilizing CRT were proposed to acquire the frequency estimates.…”