Capacity loss for multilayer codebook precoding in MIMO systems

In limited feedback multiple-input multiple-output (MIMO) systems, it is important that the codebooks used provide a good approximation to the channel. Any mismatch between the codebook entries and the actual channel creates a non zero chordal distance (or subspace angle) between the codebook element and the corresponding columns of the right singular matrix of the channel. As a result, the system exhibits a capacity loss. We evaluate and quantify the amount of this capacity loss with long term evolution (LTE) and Grassmannian codebooks in the case of Layer 1 (L1) and Layer 2 (L2) transmission using linear receivers in three different channel models. We evaluate the distribution of the minimum distance for LTE and Grassmannian codebooks via simulations. We also derive capacity loss approximations for zero forcing (ZF), minimum mean-squared error (MMSE) and singular value decomposition (SVD) receivers. We find that the capacity loss approximations for L1 transmission are accurate for all receivers and lead to an extremely simple relationship between capacity loss and minimum distance.