Ambiguity Analysis and Design of Sparse Arrays via Generalized Vandermonde Rank Conditions

Abstract

Sparse linear arrays obtained by thinning a uniform linear array (ULA) achieve large effective apertures with a reduced number of physical sensors and have become a key enabling technology across radar, sonar, communications, and integrated sensing and communications. The price of thinning, however, is the emergence of ambiguities in the array manifold: distinct sets of directions of arrival that produce identical sensor measurements, precluding unique identification of multiple sources. Conventional sparse-array design criteria, based on beampattern shaping or estimation-performance optimization, do not fully capture how multiple steering vectors interact jointly to produce such ambiguities. This paper develops a scalable algebraic framework for the multi-source identifiability analysis of thinned ULAs. By relating the rank deficiency of the generalized Vandermonde matrix associated with the sparse steering matrix to that of a thinned Toeplitz matrix, and further to a rank condition on an augmented full-ULA steering matrix with prescribed generators, we obtain a systematic characterization of the ambiguity sets in large sparse arrays together with constructive design guidelines for ambiguity-free geometries. Algebraic and numerical examples demonstrate that the proposed framework characterizes ambiguity sets at scales well beyond the practical reach of previous sparse-array design and synthesis methods