  In this paper we revisit the sparse multiple measurement vector (MMV) problem
where the aim is to recover a set of jointly sparse multichannel vectors from
incomplete measurements. This problem has received increasing interest as an
extension of the single channel sparse recovery problem which lies at the heart
of the emerging field of compressed sensing. However the sparse approximation
problem has origins which include links to the field of array signal processing
where we find the inspiration for a new family of MMV algorithms based on the
MUSIC algorithm. We highlight the role of the rank of the coefficient matrix X
in determining the difficulty of the recovery problem. We derive the necessary
and sufficient conditions for the uniqueness of the sparse MMV solution, which
indicates that the larger the rank of X the less sparse X needs to be to ensure
uniqueness. We also show that the larger the rank of X the less the
computational effort required to solve the MMV problem through a combinatorial
search. In the second part of the paper we consider practical suboptimal
algorithms for solving the sparse MMV problem. We examine the rank awareness of
popular algorithms such as SOMP and mixed norm minimization techniques and show
them to be rank blind in terms of worst case analysis. We then consider a
family of greedy algorithms that are rank aware. The simplest such algorithm is
a discrete version of MUSIC and is guaranteed to recover the sparse vectors in
the full rank MMV case under mild conditions. We extend this idea to develop a
rank aware pursuit algorithm that naturally reduces to Order Recursive Matching
Pursuit (ORMP) in the single measurement case and also provides guaranteed
recovery in the full rank multi-measurement case. Numerical simulations
demonstrate that the rank aware algorithms are significantly better than
existing algorithms in dealing with multiple measurements.
