  The "Writing onto Fast Fading Dirt" (WFFD) channel is investigated to study
the effects of partial channel knowledge on the capacity of the "writing on
dirty paper" channel. The WFFD channel is the Gel'fand-Pinsker channel in which
the output is obtained as the sum of the input, white Gaussian noise and a
fading-times-state term. The fading-times-state term is equal to the
element-wise product of the channel state sequence, known only at the
transmitter, and a fast fading process, known only at the receiver. We consider
the case of Gaussian distributed channel states and derive an approximate
characterization of capacity for different classes of fading distributions,
both continuous and discrete. In particular, we prove that if the fading
distribution concentrates in a sufficiently small interval, then capacity is
approximately equal to the AWGN capacity times the probability of this
interval. We also show that there exists a class of fading distributions for
which having the transmitter treat the fading-times-state term as additional
noise closely approaches capacity. Although a closed-form expression of the
capacity of the general WFFD channel remains unknown, our results show that the
presence of fading can severely reduce the usefulness of channel state
knowledge at the transmitter.
