  In this paper we study the recovery conditions of weighted $l_{1}$
minimization for signal reconstruction from incomplete linear measurements when
partial prior support information is available. We obtain that a high order RIP
condition can guarantee stable and robust recovery of signals in bounded
$l_{2}$ and Dantzig selector noise settings. Meanwhile, we not only prove that
the sufficient recovery condition of weighted $l_{1}$ minimization method is
weaker than that of standard $l_{1}$ minimization method, but also prove that
weighted $l_{1}$ minimization method provides better upper bounds on the
reconstruction error in terms of the measurement noise and the compressibility
of the signal, provided that the accuracy of prior support estimate is at least
$50\%$. Furthermore, the condition is proved sharp.
