  The problem of dirty paper coding (DPC) over the (multi-antenna) fading dirty
paper channel (FDPC) Y = H(X + S) + Z is considered when there is imperfect
knowledge of the channel state information H at the transmitter (CSIT). The
case of FDPC with positive definite (p.d.) input covariance matrix was studied
by the authors in a recent paper, and here the more general case of positive
semi-definite (p.s.d.) input covariance is dealt with. Towards this end, the
choice of auxiliary random variable is modified. The algorithms for
determination of inflation factor proposed in the p.d. case are then
generalized to the case of p.s.d. input covariance. Subsequently, the largest
DPC-achievable high-SNR (signal-to-noise ratio) scaling factor over the no-CSIT
FDPC with p.s.d. input covariance matrix is derived. This scaling factor is
seen to be a non-trivial generalization of the one achieved for the p.d. case.
Next, in the limit of low SNR, it is proved that the choice of all-zero
inflation factor (thus treating interference as noise) is optimal in the
'ratio' sense, regardless of the covariance matrix used. Further, in the p.d.
covariance case, the inflation factor optimal at high SNR is obtained when the
number of transmit antennas is greater than the number of receive antennas,
with the other case having been already considered in the earlier paper.
Finally, the problem of joint optimization of the input covariance matrix and
the inflation factor is dealt with, and an iterative numerical algorithm is
developed.
