We consider the winding number of planar stationary Gaussian processes defined on a (increasing) time interval. Under mild conditions, we obtain the asymptotic variance and the Central Limit Theorem for the winding number as the time horizon tends to infinity. In the asymptotic regime, our discrete approach is equivalent to the continuous one studied previously in the literature and our main result extends the existing ones. Our model allows for a general dependence of the coordinates of the process and non-differentiability of one of them. Based on a joint work with Jean-Marc Azaïs (Toulouse III) and José R. León (Udelar, Uruguay).