  This paper considers the implicit Euler discretization of Levant's arbitrary
order robust exact differentiator in presence of sampled measurements. Existing
implicit discretizations of that differentiator are shown to exhibit either
unbounded bias errors or, surprisingly, discretization chattering despite the
use of the implicit discretization. A new, proper implicit discretization that
exhibits neither of these two detrimental effects is proposed by computing the
differentiator's outputs as appropriately designed linear combinations of its
state variables. A numerical differentiator implementation is discussed and
closed-form stability conditions for arbitrary differentiation orders are
given. The influence of bounded measurement noise and numerical approximation
errors is formally analyzed. Numerical simulations confirm the obtained
results.
