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A coprime antenna array consists of two or more sparse subarrays featuringenhanced degrees of freedom (DOF) and reduced mutual coupling. This paperintroduces a new class of planar coprime arrays, based on the theory of ideallattices. In quadratic number fields, a splitting prime $p$ can be decomposedinto the product of two distinct prime ideals, which give rise to the twosparse subarrays. Their virtual difference coarray enjoys a quadratic gain inDOF, thanks to the generalized Chinese Remainder Theorem (CRT). To enlarge thecontiguous aperture of the coarray, we present hole-free symmetric CRT arrayswith simple closed-form expressions. The ring of Gaussian integers and the ringof Eisenstein integers are considered as examples to demonstrate the procedureof designing coprime arrays. With Eisenstein integers, our design yields adifference coarray that is a subset of the hexagonal lattice, offering asignificant gain in DOF over the rectangular lattice, given the same physicalareas.

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