  Compressed sensing (CS) demonstrates that sparse signals can be estimated
from under-determined linear systems. Distributed CS (DCS) further reduces the
number of measurements by considering joint sparsity within signal ensembles.
DCS with jointly sparse signals has applications in multi-sensor acoustic
sensing, magnetic resonance imaging with multiple coils, remote sensing, and
array signal processing. Multi-measurement vector (MMV) problems consider the
estimation of jointly sparse signals under the DCS framework. Two related MMV
settings are studied. In the first setting, each signal vector is measured by a
different independent and identically distributed (i.i.d.) measurement matrix,
while in the second setting, all signal vectors are measured by the same i.i.d.
matrix. Replica analysis is performed for these two MMV settings, and the
minimum mean squared error (MMSE), which turns out to be identical for both
settings, is obtained as a function of the noise variance and number of
measurements. To showcase the application of MMV models, the MMSE's of complex
CS problems with both real and complex measurement matrices are also analyzed.
Multiple performance regions for MMV are identified where the MMSE behaves
differently as a function of the noise variance and the number of measurements.
  Belief propagation (BP) is a CS signal estimation framework that often
achieves the MMSE asymptotically. A phase transition for BP is identified. This
phase transition, verified by numerical results, separates the regions where BP
achieves the MMSE and where it is suboptimal. Numerical results also illustrate
that more signal vectors in the jointly sparse signal ensemble lead to a better
phase transition.
