Correlation Subspaces: Generalizations and Connection to Difference Coarrays

Direction-of-arrival (DOA) estimation finds applications in many areas of science and engineering. In these applications, sparse arrays such as minimum redundancy arrays, nested arrays, and coprime arrays can be exploited to resolve <inline-formula><tex-math notation="LaTeX">$O(N^2)$</tex-math></inline-formula> uncorrelated sources using <inline-formula><tex-math notation="LaTeX">$N$</tex-math></inline-formula> physical sensors. Recently, it has been shown that <italic>correlation subspaces</italic>, which reveal the structure of the covariance matrix, help to improve some existing DOA estimators. However, the bases, the dimension, and other theoretical properties of correlation subspaces remain to be investigated. This paper proposes <italic>generalized correlation subspaces</italic> in one and multiple dimensions. This leads to new insights into correlation subspaces and DOA estimation with prior knowledge. First, it is shown that the bases and the dimension of correlation subspaces are fundamentally related to <italic>difference coarrays</italic>, which were previously found to be important in the study of sparse arrays. Furthermore, generalized correlation subspaces can handle certain forms of prior knowledge about source directions. These results allow one to derive a broad class of DOA estimators with improved performance. It is demonstrated through examples that using sparse arrays and generalized correlation subspaces, DOA estimators with source priors exhibit better estimation performance than those without priors, in extreme cases like low SNR and limited snapshots.