  The performance of estimating the common support for jointly sparse signals
based on their projections onto lower-dimensional space is analyzed. Support
recovery is formulated as a multiple-hypothesis testing problem. Both upper and
lower bounds on the probability of error are derived for general measurement
matrices, by using the Chernoff bound and Fano's inequality, respectively. The
upper bound shows that the performance is determined by a quantity measuring
the measurement matrix incoherence, while the lower bound reveals the
importance of the total measurement gain. The lower bound is applied to derive
the minimal number of samples needed for accurate direction-of-arrival (DOA)
estimation for a sparse representation based algorithm. When applied to
Gaussian measurement ensembles, these bounds give necessary and sufficient
conditions for a vanishing probability of error for majority realizations of
the measurement matrix. Our results offer surprising insights into sparse
signal recovery. For example, as far as support recovery is concerned, the
well-known bound in Compressive Sensing with the Gaussian measurement matrix is
generally not sufficient unless the noise level is low. Our study provides an
alternative performance measure, one that is natural and important in practice,
for signal recovery in Compressive Sensing and other application areas
exploiting signal sparsity.
