  Support recovery of sparse signals from compressed linear measurements is a
fundamental problem in compressed sensing (CS). In this paper, we study the
orthogonal matching pursuit (OMP) algorithm for the recovery of support under
noise. We consider two signal-to-noise ratio (SNR) settings: i) the SNR depends
on the sparsity level $K$ of input signals, and ii) the SNR is an absolute
constant independent of $K$. For the first setting, we establish necessary and
sufficient conditions for the exact support recovery with OMP, expressed as
lower bounds on the SNR. Our results indicate that in order to ensure the exact
support recovery of all $K$-sparse signals with the OMP algorithm, the SNR must
at least scale linearly with the sparsity level $K$. In the second setting,
since the necessary condition on the SNR is not fulfilled, the exact support
recovery with OMP is impossible. However, our analysis shows that recovery with
an arbitrarily small but constant fraction of errors is possible with the OMP
algorithm. This result may be useful for some practical applications where
obtaining some large fraction of support positions is adequate.
