  We study compressive sensing in the spatial domain to achieve target
localization, specifically direction of arrival (DOA), using multiple-input
multiple-output (MIMO) radar. A sparse localization framework is proposed for a
MIMO array in which transmit and receive elements are placed at random. This
allows for a dramatic reduction in the number of elements needed, while still
attaining performance comparable to that of a filled (Nyquist) array. By
leveraging properties of structured random matrices, we develop a bound on the
coherence of the resulting measurement matrix, and obtain conditions under
which the measurement matrix satisfies the so-called isotropy property. The
coherence and isotropy concepts are used to establish uniform and non-uniform
recovery guarantees within the proposed spatial compressive sensing framework.
In particular, we show that non-uniform recovery is guaranteed if the product
of the number of transmit and receive elements, MN (which is also the number of
degrees of freedom), 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. In contrast with a filled virtual MIMO array where the product MN
scales linearly with G, the logarithmic dependence on G in the proposed
framework supports the high-resolution provided by the virtual array aperture
while using a small number of MIMO radar elements. In the numerical results we
show that, in the proposed framework, compressive sensing recovery algorithms
are capable of better performance than classical methods, such as beamforming
and MUSIC.
