We proposed a weighted l1 minimization to recover a sparse signal vector and the corrupted noise vector from a linear measurement when the sensing matrix A is an m by n row i.i.d subgaussian matrix. We obtain both uniform and nonuniform recovery guarantees when the corrupted observations occupy a constant fraction of the total measurement, provided that the signal vector is sparse enough. In the uniform recovery guarantee, the upper-bound of the cardinality of the signal vector required in this paper is asymptotically optimal. While in the non-uniform recovery guarantee, we allow the proportion of corrupted measurements grows arbitrarily close to 1, and the upper-bound of the cardinality of the signal vector is better than those in a recent literature [1] by a ln(n) factor.