  Support recovery of sparse signals from noisy measurements with orthogonal
matching pursuit (OMP) has been extensively studied. In this paper, we show
that for any $K$-sparse signal $\x$, if a sensing matrix $\A$ satisfies the
restricted isometry property (RIP) with restricted isometry constant (RIC)
$\delta_{K+1} < 1/\sqrt {K+1}$, then under some constraints on the minimum
magnitude of nonzero elements of $\x$, OMP exactly recovers the support of $\x$
from its measurements $\y=\A\x+\v$ in $K$ iterations, where $\v$ is a noise
vector that is $\ell_2$ or $\ell_{\infty}$ bounded. This sufficient condition
is sharp in terms of $\delta_{K+1}$ since for any given positive integer $K$
and any $1/\sqrt{K+1}\leq \delta<1$, there always exists a matrix $\A$
satisfying the RIP with $\delta_{K+1}=\delta$ for which OMP fails to recover a
$K$-sparse signal $\x$ in $K$ iterations. Also, our constraints on the minimum
magnitude of nonzero elements of $\x$ are weaker than existing ones. Moreover,
we propose worst-case necessary conditions for the exact support recovery of
$\x$, characterized by the minimum magnitude of the nonzero elements of $\x$.
