Coprime Sensing via Chinese Remaindering over Quadratic Fields, Part I: Array Designs

TL;DR

This paper introduces a novel class of planar coprime arrays based on ideal lattices in quadratic number fields, enhancing degrees of freedom and aperture size for array design using the generalized Chinese Remainder Theorem.

Contribution

It proposes a new array design methodology utilizing quadratic fields and ideal lattices, expanding array capabilities with closed-form hole-free symmetric arrays.

Findings

Quadratic gain in degrees of freedom (DOF) achieved.

Difference coarray can be a subset of hexagonal lattice.

Designs outperform rectangular lattice in DOF for same area.

Abstract

A coprime antenna array consists of two or more sparse subarrays featuring enhanced degrees of freedom (DOF) and reduced mutual coupling. This paper introduces a new class of planar coprime arrays, based on the theory of ideal lattices. In quadratic number fields, a splitting prime $p$ can be decomposed into the product of two distinct prime ideals, which give rise to the two sparse subarrays. Their virtual difference coarray enjoys a quadratic gain in DOF, thanks to the generalized Chinese Remainder Theorem (CRT). To enlarge the contiguous aperture of the coarray, we present hole-free symmetric CRT arrays with simple closed-form expressions. The ring of Gaussian integers and the ring of Eisenstein integers are considered as examples to demonstrate the procedure of designing coprime arrays. With Eisenstein integers, our design yields a difference coarray that is a subset of the hexagonal lattice, offering a significant gain in DOF over the rectangular lattice, given the same physical areas.