We study compressive sensing in the spatial domain for target localization in terms of direction of arrival (DOA), using multiple-input multiple-output (MIMO) radar. A sparse localization framework is proposed for a MIMO array in which transmit/receive elements are placed at random. This allows to dramatically reduce the number of elements, while still attaining performance comparable to that of a filled (Nyquist) array. Leveraging properties of a (structured) random measurement matrix, we develop a novel bound on the coherence of the measurement matrix, and we obtain conditions under which the measurement matrix satisfies the so-called isotropy property. The coherence and isotropy concepts are used to establish respectively uniform and non-uniform recovery guarantees for target localization using spatial compressive sensing. In particular, non-uniform recovery is guaranteed if the number of degrees of freedom (the product of the number of transmit and receive elements MN) scales with K (log G)~2, where K is the number of targets, and G is proportional to the array aperture and determines the angle resolution. The significance of the logarithmic dependence in G is that the proposed framework enables high resolution with a small number of MIMO radar elements. This is in contrast with a filled virtual MIMO array where the product MN scales linearly with G.
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