A sufficient condition reported very recently for perfect recovery of a K-sparse vector via orthogonal matching pursuit (OMP) in K iterations is that the restricted isometry constant of the sensing matrix satisfies delta_K+1<1/(sqrt(delta_K+1)+1). By exploiting an approximate orthogonality condition characterized via the achievable angles between two orthogonal sparse vectors upon compression, this paper shows that the upper bound on delta can be further relaxed to delta_K+1<(sqrt(1+4*delta_K+1)-1)/(2K).This result thus narrows the gap between the so far best known bound and the ultimate performance guarantee delta_K+1<1/(sqrt(delta_K+1)) that is conjectured by Dai and Milenkovic in 2009. The proposed approximate orthogonality condition is also exploited to derive less restricted sufficient conditions for signal reconstruction in several compressive sensing problems, including signal recovery via OMP in a noisy environment, compressive domain interference cancellation, and support identification via the subspace pursuit algorithm.

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