In this paper, we investigate the weighted H∞ Luenberger observation and the weighted H8 filtering. The motivation is the design of signal differentiators. The standard Luenberger observer structure is modified in order to formulate the design problem as a convex optimization problem involving Linear Matrix Inequality (LMI) constraints. Furthermore, we reveal the relationship between the Luenberger observer and the filter and we discuss the interest of the first one versus the other one. These results are then applied to the signal differentiator design problem. A numerical example, simulation and real time results are finally presented.