The MUSIC method is widely used in the field of DoA estimation using an array of M sensors, and is known to perform well as long as the number of available samples N is much larger than M. Nevertheless , in the scenario where N is of the same order of magnitude than M , its performance degrades, essentially because the sample covariance matrix (SCM) is no more a good estimator. A classical improvement, known as "rectification", consists in forcing the SCM to have a Toeplitz structure. In this paper, we analyze this method, by considering the asymptotic regime where M, N both converge to infinity at the same rate, and by studying consistency and asymptotic normality of the related DoA estimates.