  In this paper, we consider the problem of recovering an unknown sparse signal
$\xv_0 \in \mathbb{R}^n$ from noisy linear measurements $\yv = \Hm \xv_0+ \zv
\in \mathbb{R}^m$. A popular approach is to solve the $\ell_1$-norm regularized
least squares problem which is known as the LASSO. In many practical
situations, the measurement matrix $\Hm$ is not perfectely known and we only
have a noisy version of it. We assume that the entries of the measurement
matrix $\Hm$ and of the noise vector $\zv$ are iid Gaussian with zero mean and
variances $1/n$ and $\sigma_{\zv}^2$. In this work, an imperfect measurement
matrix is considered under which we precisely characterize the limiting
behavior of the mean squared error and the probability of support recovery of
the LASSO. The analysis is performed when the problem dimensions grow
simultaneously to infinity at fixed rates. Numerical simulations validate the
theoretical predictions derived in this paper.
