  This paper considers the exact recovery of $k$-sparse signals in the
noiseless setting and support recovery in the noisy case when some prior
information on the support of the signals is available. This prior support
consists of two parts. One part is a subset of the true support and another
part is outside of the true support. For $k$-sparse signals $\mathbf{x}$ with
the prior support which is composed of $g$ true indices and $b$ wrong indices,
we show that if the restricted isometry constant (RIC) $\delta_{k+b+1}$ of the
sensing matrix $\mathbf{A}$ satisfies \begin{eqnarray*}
\delta_{k+b+1}<\frac{1}{\sqrt{k-g+1}}, \end{eqnarray*} then orthogonal matching
pursuit (OMP) algorithm can perfectly recover the signals $\mathbf{x}$ from
$\mathbf{y}=\mathbf{Ax}$ in $k-g$ iterations. Moreover, we show the above
sufficient condition on the RIC is sharp. In the noisy case, we achieve the
exact recovery of the remainder support (the part of the true support outside
of the prior support) for the $k$-sparse signals $\mathbf{x}$ from
$\mathbf{y}=\mathbf{Ax}+\mathbf{v}$ under appropriate conditions. For the
remainder support recovery, we also obtain a necessary condition based on the
minimum magnitude of partial nonzero elements of the signals $\mathbf{x}$.
