This article deals with the boundary observability and controllability properties of a space finite-differences semidiscretization of the clamped beam equation. We make a detailed spectral analysis of the system and, by combining numerical estimates with asymptotic expansions, we localize all the eigenvalues of the corresponding discrete operator depending on the mesh size h. Then, an Ingham's type inequality and a discrete multiplier method allow us to deduce that the uniform (with respect to h) observability property holds if and only if the eigenfrequencies are filtered out in the range O (1/h(4)).