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Support recovery of sparse signals from noisy measurements with orthogonalmatching pursuit (OMP) has been extensively studied in the literature. In thispaper, we show that for any $K$-sparse signal $\x$, if the sensing matrix $\A$satisfies the restricted isometry property (RIP) of order $K + 1$ withrestricted isometry constant (RIC) $\delta_{K+1} < 1/\sqrt {K+1}$, then undersome constraint on the minimum magnitude of the nonzero elements of $\x$, theOMP algorithm exactly recovers the support of $\x$ from the measurements$\y=\A\x+\v$ in $K$ iterations, where $\v$ is the noise vector. This conditionis sharp in terms of $\delta_{K+1}$ since for any given positive integer $K\geq2$ and any $1/\sqrt{K+1}\leq t<1$, there always exist a $K$-sparse $\x$ and amatrix $\A$ satisfying $\delta_{K+1}=t$ for which OMP may fail to recover thesignal $\x$ in $K$ iterations. Moreover, the constraint on the minimummagnitude of the nonzero elements of $\x$ is weaker than existing results.

A Sharp Condition for Exact Support Recovery of Sparse Signals With Orthogonal Matching Pursuit