  The problem of estimating an unknown deterministic parameter vector from sign
measurements with a perturbed sensing matrix is studied in this paper. We
analyze the best achievable mean square error (MSE) performance by exploring
the corresponding Cram\'{e}r-Rao Lower Bound (CRLB). To estimate the parameter,
the maximum likelihood (ML) estimator is utilized and its consistency is
proved. We show that the perturbation on the sensing matrix exacerbates the
performance of ML estimator in most cases. However, suitable perturbation may
improve the performance in some special cases. Then we reformulate the original
ML estimation problem as a convex optimization problem, which can be solved
efficiently. Furthermore, theoretical analysis implies that the
perturbation-ignored estimation is a scaled version with the same direction of
the ML estimation. Finally, numerical simulations are performed to validate our
theoretical analysis.
