  The problem of Dirty Paper Coding (DPC) over the Fading Dirty Paper Channel
(FDPC) Y = H(X + S)+Z, a more general version of Costa's channel, is studied
for the case in which there is partial and perfect knowledge of the fading
process H at the transmitter (CSIT) and the receiver (CSIR), respectively. A
key step in this problem is to determine the optimal inflation factor (under
Costa's choice of auxiliary random variable) when there is only partial CSIT.
Towards this end, two iterative numerical algorithms are proposed. Both of
these algorithms are seen to yield a good choice for the inflation factor.
Finally, the high-SNR (signal-to-noise ratio) behavior of the achievable rate
over the FDPC is dealt with. It is proved that FDPC (with t transmit and r
receive antennas) achieves the largest possible scaling factor of min(t,r) log
SNR even with no CSIT. Furthermore, in the high SNR regime, the optimality of
Costa's choice of auxiliary random variable is established even when there is
partial (or no) CSIT in the special case of FDPC with t <= r. Using the
high-SNR scaling-law result of the FDPC (mentioned before), it is shown that a
DPC-based multi-user transmission strategy, unlike other beamforming-based
multi-user strategies, can achieve a single-user sum-rate scaling factor over
the multiple-input multiple-output Gaussian Broadcast Channel with partial (or
no) CSIT.
