The Szeg\"o-Asymptotics for Doubly-Dispersive Gaussian Channels

Peter Jung

We consider the time-continuous doubly--dispersive channel with additive Gaussian noise and establish a capacity formula for the case where the channel operator is represented by a symbol which is periodic in time and fulfills some further integrability, smoothness and oscillation conditions. More precisely, we apply the well-known Holsinger-Gallager model for translating a time-continuous channel for a sequence of time--intervals of increasing length $\alpha\rightarrow\infty$ to a series of equivalent sets of discrete, parallel channels, known at the transmitter. We quantify conditions when this procedure converges. Finally, under periodicity assumptions this result can indeed be justified as the channel capacity in the sense Shannon. The key to this is result is a new Szeg\"o formula for certain pseudo--differential operators with real-valued symbol. The Szeg\"o limit holds if the symbol belongs to the homogeneous Besov space $\dot{B}^1_{\infty,1}$ with respect to its time-dependency, characterizing the oscillatory behavior in time. Finally, the formula justifies the water-filling principle in time and frequency as general technique independent of a sampling scheme.

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