  The sparse signal recovery in the standard compressed sensing (CS) problem
requires that the sensing matrix be known a priori. Such an ideal assumption
may not be met in practical applications where various errors and fluctuations
exist in the sensing instruments. This paper considers the problem of
compressed sensing subject to a structured perturbation in the sensing matrix.
Under mild conditions, it is shown that a sparse signal can be recovered by
$\ell_1$ minimization and the recovery error is at most proportional to the
measurement noise level, which is similar to the standard CS result. In the
special noise free case, the recovery is exact provided that the signal is
sufficiently sparse with respect to the perturbation level. The formulated
structured sensing matrix perturbation is applicable to the direction of
arrival estimation problem, so has practical relevance. Algorithms are proposed
to implement the $\ell_1$ minimization problem and numerical simulations are
carried out to verify the result obtained.
