Difference Coarrays of Rational Arrays

Rational arrays were recently proposed for direction of arrival (DOA) estimation, and some of their advantages were discussed. In this paper we discuss further advantages of rational arrays, by considering their difference coarrays. Although integer arrays such as nested arrays, and coprime arrays are well-known for their ability to identify ${{\mathcal{O}}}\left({{m^2}}\right)$ uncorrelated sources using m sensors through difference coarray domain, they can do so only when a large enough aperture is available. However, rational arrays can do so even when the aperture is constrained. We demonstrate that adding a few sensors at non-integer locations in an otherwise integer array can add a large number of fractional lags at which autocorrelation can be estimated. Appropriately designed sparse integer arrays can also be scaled to produce rational arrays that fit available aperture and have large uniform coarray segments at rational locations. Monte-Carlo simulations are provided to demonstrate the advantages, and practical issues associated with shrinking, such as increased mutual coupling, are discussed.