Two-Tx Precoding Codebooks for Variable Spatial Correlation

We design precoding codebooks for the situation when the spatial correlation is not known at the transmitter, but its distribution is. The optimum codebooks combine features of Grassmannian precoding, optimum in the limit of spatially i.i.d. channels, and equatorial precoding, where all antennas transmit the same power. Parametrizable families of codebooks interpolating between these two are considered. The optimum choice of parameter depends on the distribution of spatial correlations. In the case of two transmit antennas, the interpolation is between sphere packings in 2D and sphere packings in ID, and the codebooks can be understood as the vertices of deformed Platonic solids.