  We investigate the MIMO broadcast channel in the high SNR regime when linear
filtering is applied instead of dirty paper coding. Using a user-wise rate
duality where the streams of every single user are not treated as
self-interference as in the hitherto existing stream-wise rate dualities for
linear filtering, we solve the weighted sum rate maximization problem of the
broadcast channel in the dual multiple access channel. Thus, we can exactly
quantify the asymptotic rate loss of linear filtering compared to dirty paper
coding for any channel realization. Having converted the optimum covariance
matrices to the broadcast channel by means of the duality, we observe that the
optimal covariance matrices in the broadcast channel feature quite complicated
but still closed form expressions although the respective transmit covariance
matrices in the dual multiple access channel share a very simple structure. We
immediately come to the conclusion that block-diagonalization is the
asymptotically optimum transmit strategy in the broadcast channel. Out of the
set of block-diagonalizing precoders, we present the one which achieves the
largest sum rate and thus corresponds to the optimum solution found in the dual
multiple access channel. Additionally, we quantify the ergodic rate loss of
linear coding compared to dirty paper coding for Gaussian channels with
correlations at the mobiles.
