In this paper, we consider robust system identification of FIR systems when both sparse outliers and random noises are present. We reduce this problem of system identification to a sparse error correcting problem using a Toeplitz structured real-numbered coding matrix and prove the performance guarantee. Thresholds on the percentage of correctable errors for Toeplitz structured matrices are established. When both outliers and observation noise are present, we have shown that the estimation error goes to 0 asymptotically as long as the probability density function for observation noise is not "vanishing" around origin. No probabilistic assumptions are imposed on the outliers. (C) 2014 Elsevier Ltd. All rights reserved.