Small-Scale Spatial-Temporal Correlation and Degrees of Freedom for Reconfigurable Intelligent Surfaces

Abstract: The reconfigurable intelligent surface (RIS) is an emerging promising candidate technology for future wireless networks, where the element spacing is usually of sub-wavelength. Only limited knowledge, however, has been gained about the spatial-temporal correlation behavior among the elements in an RIS. In this paper, we investigate the spatial-temporal correlation for an RIS-enabled wireless communication system. Specifically, a joint small-scale spatial-temporal correlation model is derived under isotropic scattering, which can be represented by a four-dimensional sinc function. Furthermore, based upon the spatial-only correlation at a certain time instant, an essential RIS property -- the spatial degrees of freedom (DoF) -- is revisited, and an analytical expression is propounded to characterize the spatial DoF for RISs with realistic hence non-infinitesimal element spacing and finite aperture sizes. The results are vital to the accurate evaluation of various system performance metrics.