  We consider the problem of detecting the locations of targets in the far
field by sending probing signals from an antenna array and recording the
reflected echoes. Drawing on key concepts from the area of compressive sensing,
we use an $\ell_1$-based regularization approach to solve this, in general
ill-posed, inverse scattering problem. As common in compressed sensing, we
exploit randomness, which in this context comes from choosing the antenna
locations at random. With $n$ antennas we obtain $n^2$ measurements of a vector
$x \in \C^{N}$ representing the target locations and reflectivities on a
discretized grid. It is common to assume that the scene $x$ is sparse due to a
limited number of targets. Under a natural condition on the mesh size of the
grid, we show that an $s$-sparse scene can be recovered via
$\ell_1$-minimization with high probability if $n^2 \geq C s \log^2(N)$. The
reconstruction is stable under noise and under passing from sparse to
approximately sparse vectors. Our theoretical findings are confirmed by
numerical simulations.
