  We study a Compressed Sensing (CS) problem known as Multiple Measurement
Vectors (MMV) problem, which arises in joint estimation of multiple signal
realizations when the signal samples have a common (joint) sparse support over
a fixed known dictionary. Although there is a vast literature on the analysis
of MMV, it is not yet fully known how the number of signal samples and their
statistical correlations affects the performance of the joint estimation in
MMV. Moreover, in many instances of MMV the underlying sparsifying dictionary
may not be precisely known, and it is still an open problem to quantify how the
dictionary mismatch may affect the estimation performance.
  In this paper, we focus on $\ell_{2,1}$-norm regularized least squares
($\ell_{2,1}$-LS) as a well-known and widely-used MMV algorithm in the
literature. We prove an interesting decoupling property for $\ell_{2,1}$-LS,
where we show that it can be decomposed into two phases: i) use all the signal
samples to estimate the signal covariance matrix (coupled phase), ii) plug in
the resulting covariance estimate as the true covariance matrix into the
Minimum Mean Squared Error (MMSE) estimator to reconstruct each signal sample
individually (decoupled phase). As a consequence of this decomposition, we are
able to provide further insights on the performance of $\ell_{2,1}$-LS for MMV.
In particular, we address how the signal correlations and dictionary mismatch
affects its performance. Moreover, we show that by using the decoupling
property one can obtain a variety of MMV algorithms with performances even
better than that of $\ell_{2,1}$-LS. We also provide numerical simulations to
validate our theoretical results.
