Phase differences estimation for diagonal ULAs within a fully filled rectangular array based on degenerated spatial ARMA process

A generalized estimation method of diagonal phase differences of external sources incident upon a fully-filled rectangular array (FFRA) is proposed based on degenerate spatial ARMA process. Various diagonal uniform linear arrays (ULAs) within FFRAs are first classified by function forms of diagonal phase differences, the ULAs sharing the same diagonal phase difference belong to one category. The modified Yule–Walker (MYW) system of linear equations and the root-finding polynomial are first derived for FFRA ULAs. Owing to diagonal interspacings larger than half of carrier wavelength, ambiguity problem of diagonal phase differences has arisen in estimation. Utilizing the explicit linear-combination relationships satisfied by diagonal and axial phase differences, a simple and effective elimination scheme of estimate ambiguity of diagonal phase differences is proposed in which actual intervals of no ambiguity are deduced by making use of the estimates of axial phase differences. With different FFRA diagonal ULAs on $X-Y$ sensor plane, it is numerically manifested by Monte-Carlo trials that the proposed method is effective for both independent and coherent sources and the Root mean square errors (RMSEs) are slowly convergent to the corresponding Cramer–Rao bounds (CRBs) after estimate ambiguities are eliminated. The consistency of estimation performance for diagonal ULAs belonging to one category is exhibited by their identical RMSEs. Because of the ability to exploit axial and diagonal ULAs, the proposed estimation method provides the basis of two-dimensional direction of arrival (2-D DoA) estimation with ULA combinations of FFRAs.