This paper deals with the Tobit Kalman filtering (TKF) process when the one-dimensional measurements are censored and the noises of the state-space model are coloured. Two improvements of the standard TKF process are proposed. Firstly, the exact moments of the censored measurements are calculated via the moment generating function of the censored measurements. Secondly, coloured noises are considered in the proposed method in order to tackle real-life problems, where the white noises are not common. The designed process is evaluated using two experiments-simulations. The results show that the proposed method outperforms other methods in minimizing the Root Mean Square Error (RMSE) in both experiments.