  Frequency recovery/estimation from discrete samples of superimposed
sinusoidal signals is a classic yet important problem in statistical signal
processing. Its research has recently been advanced by atomic norm techniques
which exploit signal sparsity, work directly on continuous frequencies, and
completely resolve the grid mismatch problem of previous compressed sensing
methods. In this work we investigate the frequency recovery problem in the
presence of multiple measurement vectors (MMVs) which share the same frequency
components, termed as joint sparse frequency recovery and arising naturally
from array processing applications. To study the advantage of MMVs, we first
propose an $\ell_{2,0}$ norm like approach by exploiting joint sparsity and
show that the number of recoverable frequencies can be increased except in a
trivial case. While the resulting optimization problem is shown to be rank
minimization that cannot be practically solved, we then propose an MMV atomic
norm approach that is a convex relaxation and can be viewed as a continuous
counterpart of the $\ell_{2,1}$ norm method. We show that this MMV atomic norm
approach can be solved by semidefinite programming. We also provide theoretical
results showing that the frequencies can be exactly recovered under appropriate
conditions. The above results either extend the MMV compressed sensing results
from the discrete to the continuous setting or extend the recent
super-resolution and continuous compressed sensing framework from the single to
the multiple measurement vectors case. Extensive simulation results are
provided to validate our theoretical findings and they also imply that the
proposed MMV atomic norm approach can improve the performance in terms of
reduced number of required measurements and/or relaxed frequency separation
condition.
