  Let $\mu(n)$ be the M\"{o}bius function and $e(\alpha)=e^{2\pi i\alpha}$. In
this paper, we study upper bounds of the classical sum
$$S(x,\alpha):=\sum_{1\leq n\leq x}\mu(n)e(\alpha n).$$ We can improve some
classical results of Baker and Harman \cite{BH}.
