  This paper details a class of metal-based space-time metasurfaces for
application in wireless communications scenarios. Concretely, we describe
space-time metasurfaces that periodically alternate their properties in time
between three spatial states: "air", "conductor" and "grating". We analyze the
physics of these metastructures via a computationally-efficient analytical
technique based on the use of Floquet-Bloch series, integral equations and
circuit models. By doing so, we reveal important features of these
spatiotemporal metasurfaces: scattering parameters, field profiles, diffraction
angles and nature of the space-time harmonics. The results, corroborated with a
self-implemented numerical FDTD approach, show the potential application of
these space-time metasurfaces as beamformers acting in reflection, in
transmission or both. The amplitude and direction of the diffracted orders can
be electronically controlled with the paramaters of the metasurface. Moreover,
the intrinsic ability of time-modulated diffractive metasurfaces to mix and
multiply frequencies is tested. We show how two different modulations can lead
to the same diffraction angle but with different mixed output frequencies.
