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Weyl-Heisenberg type representations occur in several areas of communication theory. This talk considers examples from signal theory in doubly dispersive channels and the transmission of classical information through (bosonic) i.i.d. quantum Gaussian channel. Both of them can be formulated in a joint framework with similar performance characterizations based on quantities called channel fidelities. The first case covers the problem of the approximate eigenstructure of time-varying channels which is important for an information-theoretic treatment of communication. Furthermore, pulse shaping with respect to the scattering function of wide sense stationary uncorrelated scattering (WSSUS) mobile communication channels is discussed. Interestingly, this topic is intimately connected to the problem of quantum state design for optimal hypothesis testing in Gaussian channels. A formulation based on the notion of Weyl--Heisenberg maps is presented and further investigated in terms of scaling behavior and upper bounds.