  In this paper, we consider robust system identification under sparse outliers
and random noises. In our problem, system parameters are observed through a
Toeplitz matrix. All observations are subject to random noises and a few are
corrupted with outliers. We reduce this problem of system identification to a
sparse error correcting problem using a Toeplitz structured real-numbered
coding matrix. We prove the performance guarantee of Toeplitz structured matrix
in sparse error correction. Thresholds on the percentage of correctable errors
for Toeplitz structured matrices are also 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 0.
