  We consider the pulse design problem in multicarrier transmission where the
pulse shapes are adapted to the second order statistics of the WSSUS channel.
Even though the problem has been addressed by many authors analytical insights
are rather limited. First we show that the problem is equivalent to the pure
state channel fidelity in quantum information theory. Next we present a new
approach where the original optimization functional is related to an eigenvalue
problem for a pseudo differential operator by utilizing unitary representations
of the Weyl--Heisenberg group.A local approximation of the operator for
underspread channels is derived which implicitly covers the concepts of pulse
scaling and optimal phase space displacement. The problem is reformulated as a
differential equation and the optimal pulses occur as eigenstates of the
harmonic oscillator Hamiltonian. Furthermore this operator--algebraic approach
is extended to provide exact solutions for different classes of scattering
environments.
