  This paper considers efficient sampling of simultaneously sparse and
correlated (S$\&$C) signals. Such signals arise in various applications in
array processing. We propose an implementable sampling architecture for the
acquisition of S$\&$C at a sub-Nyquist rate. We prove a sampling theorem
showing exact and stable reconstruction of the acquired signals even when the
sampling rate is smaller than the Nyquist rate by orders of magnitude.
Quantitatively, our results state that an ensemble $M$ signals, composed of
a-priori unknown latent $R$ signals, each bandlimited to $W/2$ but only
$S$-sparse in the Fourier domain, can be reconstructed exactly from compressive
sampling only at a rate $RS\log^{\alpha} W$ samples per second. When $R \ll M$,
and $S\ll W$, this amounts to a significant reduction in sampling rate compared
to the Nyquist rate of $MW$ samples per second. This is the first result that
presents an implementable sampling architecture, and a sampling theorem for the
compressive acquisition of S$\&$C signals. The signal reconstruction from
sub-Nyquist rate boils down to a sparse and low-rank (S$\&$L) matrix recovery
from a few linear measurements. The conventional convex penalties for S$\&$L
matrices are provably not optimal in the number of measurements. We resort to a
two-step algorithm to recover S$\&$L matrix from a near-optimal number of
measurements. This result then translates into a signal reconstruction
algorithm from a sub-Nyquist sampling rate.
