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We shall show that if the restricted isometry constant (RIC)$\delta_{s+1}(A)$ of the measurement matrix $A$ satisfies $$ \delta_{s+1}(A) <\frac{1}{\sqrt{s + 1}}, $$ then the greedy algorithm Orthogonal MatchingPursuit(OMP) will succeed. That is, OMP can recover every $s$-sparse signal $x$in $s$ iterations from $b = Ax$. Moreover, we shall show the upper bound of RICis sharp in the following sense. For any given $s \in \N$, we shall construct amatrix $A$ with the RIC $$ \delta_{s+1}(A) = \frac{1}{\sqrt{s + 1}} $$ suchthat OMP may not recover some $s$-sparse signal $x$ in $s$ iterations.

A Sharp Restricted Isometry Constant Bound of Orthogonal Matching Pursuit