  Optimal link adaption to the scattering function of wide sense stationary
uncorrelated scattering (WSSUS) mobile communication channels is still an
unsolved problem despite its importance for next-generation system design. In
multicarrier transmission such link adaption is performed by pulse shaping
which in turn is equivalent to precoding with respect to the second order
channel statistics. In the present framework a translation of the precoder
optimization problem into an optimization problem over trace class operators is
used. This problem which is also well-known in the context of quantum
information theory is unsolved in general due to its non-convex nature. However
in very low dimension the problem formulation reveals an additional analytic
structure which again admits the solution to the optimal precoder and
multiplexing scheme. Hence, in this contribution the analytic solution of the
problem for the 2x2 doubly--dispersive WSSUS channel is presented.
