  Compressive sensing is a signal acquisition framework based on the revelation
that a small collection of linear projections of a sparse signal contains
enough information for stable recovery. In this paper we introduce a new theory
for distributed compressive sensing (DCS) that enables new distributed coding
algorithms for multi-signal ensembles that exploit both intra- and inter-signal
correlation structures. The DCS theory rests on a new concept that we term the
joint sparsity of a signal ensemble. Our theoretical contribution is to
characterize the fundamental performance limits of DCS recovery for jointly
sparse signal ensembles in the noiseless measurement setting; our result
connects single-signal, joint, and distributed (multi-encoder) compressive
sensing. To demonstrate the efficacy of our framework and to show that
additional challenges such as computational tractability can be addressed, we
study in detail three example models for jointly sparse signals. For these
models, we develop practical algorithms for joint recovery of multiple signals
from incoherent projections. In two of our three models, the results are
asymptotically best-possible, meaning that both the upper and lower bounds
match the performance of our practical algorithms. Moreover, simulations
indicate that the asymptotics take effect with just a moderate number of
signals. DCS is immediately applicable to a range of problems in sensor arrays
and networks.
