A detailed derivation of arrays free of higher rank ambiguities

Since linear dependence of steering vectors would lead to ambiguous direction-of-arrival (DOA) estimates, it is crucial to employ array with linearly independent steering vectors in DOA estimation applications. In fact, it is well known that in order to determine uniquely the DOAs of ν uncorrelated signals, one requires an array that is free of up to rank-ν ambiguities (i.e., an array whose every (ν+1) steering vectors with distinct DOAs are linearly independent). However, the question as to the sensor arrangement that gives rise to an array that is free of up to rank-ν ambiguities, where ν∈{3,...}, remains unanswered. We construct a class of cross arrays that are free of up to rank-ν ambiguities, where ν∈{1,...}. Through this study, we derive a theorem for characterizing inherent ambiguities associated with cross arrays. We also provide counterexamples to the conjecture for characterizing higher rank ambiguity proposed by Lo and Marple (see ibid., vol.40., p.2641-50, 1992) and that proposed by Wang et al. (see ICASSP´9, vol.IV, p.509-12, 1994)