Multiple signal direction finding with thinned linear arrays

The evaluation of the direction-finding capabilities of non-uniform arrays is approached via algorithm-independent lower bounds on achievable angle estimation errors. Two classes of bounds are considered. The first, known was the Cramer-Rao bound, applies only to unbiased estimates. Compact analytical expressions for these bounds are developed which are applicable to very general direction-finding problems, including an arbitrary number of emitters. It is well-known that Cramer-Rao bounds are overly optimistic at low signal-to-noise ratios. As this ratio decreases, a point is reached at which estimation accuracy decreases abruptly. Another class of bounds, known as Ziv-Zakai bounds, provide information about the location of this threshold point. Their study suggests that poor direction-finding performance occurs in situations where the emitter direction vectors are part of a set which is nearly linearly dependent. Such linear dependence does not occur in the case of uniformly spaced linear arrays (without getting lobes). However, it does occur when elements are removed from such arrays. A systematic test is developed to test a given array geometry for this condition.