  The purpose of this paper is to introduce a communication model for Large
Intelligent Surfaces (LIS). A LIS is modelled as a collection of tiny closely
spaced antenna elements. Due to the proximity of the elements, mutual coupling
arises. An optimal transmitter design depends on the mutual coupling matrix.
For single user communication, the optimal transmitter uses the inverse of the
mutual coupling matrix in a filter matched to the channel vector. We give the
expression of the mutual coupling for two types of planar arrays. The
conditioning number of the mutual coupling matrix is unbounded as the antenna
element density increases, so only the dominant values can be inverted within
reasonable computation. The directivity is partial but still significant
compared to the conventional gain. When the spacing between elements becomes
small (smaller than half a wavelength), the directivity surpasses the
conventional directivity equal to the number of antennas, as well as the gain
obtained when modelling the surface as continuous. The gain is theoretically
unbounded as the element density increases for a constant aperture.
