The calculation of time derivatives has a state-space representation of the form of a perturbed linear system. This description enables the application of the so-called robust filtering differentiator, i.e. a non-linear state observer with extended state and homogeneous input injection terms. Therefore, given the practical importance of having an accurate discretization of such differentiator, this paper presents the design of two discrete-time implementation schemes. The first discrete-time realization is explicit, while the second one is implicit. The implicit allows reducing the numerical chattering phenomenon caused by the explicit discretization of discontinuous terms. Numerical comparisons between the presented scheme and an existing discrete-time representation show that the performance of the proposed explicit implementation scheme is similar to the most recent results from the literature. Finally, the proposed implicit discrete-time realization presents better accuracy, especially when considering large sampling periods.