  Exact recovery of $K$-sparse signals $x \in \mathbb{R}^{n}$ from linear
measurements $y=Ax$, where $A\in \mathbb{R}^{m\times n}$ is a sensing matrix,
arises from many applications. The orthogonal matching pursuit (OMP) algorithm
is widely used for reconstructing $x$. A fundamental question in the
performance analysis of OMP is the characterizations of the probability of
exact recovery of $x$ for random matrix $A$ and the minimal $m$ to guarantee a
target recovery performance. In many practical applications, in addition to
sparsity, $x$ also has some additional properties. This paper shows that these
properties can be used to refine the answer to the above question. In this
paper, we first show that the prior information of the nonzero entries of $x$
can be used to provide an upper bound on $\|x\|_1^2/\|x\|_2^2$. Then, we use
this upper bound to develop a lower bound on the probability of exact recovery
of $x$ using OMP in $K$ iterations. Furthermore, we develop a lower bound on
the number of measurements $m$ to guarantee that the exact recovery probability
using $K$ iterations of OMP is no smaller than a given target probability.
Finally, we show that when $K=O(\sqrt{\ln n})$, as both $n$ and $K$ go to
infinity, for any $0<\zeta\leq 1/\sqrt{\pi}$, $m=2K\ln (n/\zeta)$ measurements
are sufficient to ensure that the probability of exact recovering any
$K$-sparse $x$ is no lower than $1-\zeta$ with $K$ iterations of OMP. For
$K$-sparse $\alpha$-strongly decaying signals and for $K$-sparse $x$ whose
nonzero entries independently and identically follow the Gaussian distribution,
the number of measurements sufficient for exact recovery with probability no
lower than $1-\zeta$ reduces further to
$m=(\sqrt{K}+4\sqrt{\frac{\alpha+1}{\alpha-1}\ln(n/\zeta)})^2$ and
asymptotically $m\approx 1.9K\ln (n/\zeta)$, respectively.
