On the Approximate Eigenstructure of Time-Varying Channels

In this article we consider the approximate description of doubly–dispersive channels by its symbol. We focus on channel operators with compactly supported spreading, which are widely used to represent fast fading multipath communication channels. The concept of approximate eigenstructure is introduced, which measures the accuracy Ep of the approximation of the channel operation as a pure multiplication in a given Lp–norm. Two variants of such an approximate Weyl symbol calculus are studied, which have important applications in several models for time–varying mobile channels. Typically, such channels have random spreading functions (inverse Weyl transform) defined on a common support U of finite non–zero size such that approximate eigenstructure has to be measured with respect to certain norms of the spreading process. We derive several explicit relations to the size |U | of the support. We show that the characterization of the ratio of Ep to some Lq–norm of the spreading function is related to weighted norms of ambiguity and Wigner functions. We present the connection to localization operators and give new bounds on the ability of localization of ambiguity functions and Wigner functions in U . Our analysis generalizes and improves recent results for the case p = 2 and q = 1.