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The generalized orthogonal matching pursuit 
$\left( \text{gOMP} \right)$ algorithm has received much attention in recent years as a natural extension of the orthogonal matching pursuit 
$\left( \text{OMP} \right)$. It is used to recover sparse signals in compressive sensing. In this paper, a new bound is obtained for the exact reconstruction of every 
$K$-sparse signal via the 
$\text{gOMP}$ algorithm in the noiseless case. That is, if the restricted isometry constant 
$\left( \text{RIC} \right)$
${{\delta }_{NK+1}}$ of the sensing matrix 
$A$ satisfies
${{\delta }_{NK+1}}\,<\frac{1}{\sqrt{\frac{K}{N}+\,1}},$
then the $\text{gOMP}$ can perfectly recover every $K$-sparse signal 
$x$ from 
$y\,=\,Ax$. Furthermore, the bound is proved to be sharp. In the noisy case, the above bound on 
$\text{RIC}$ combining with an extra condition on the minimum magnitude of the nonzero components of $K$-sparse signals can guarantee that the $\text{gOMP}$ selects all of the support indices of the $K$-sparse signals.
