Time-frequency and time-scale representations of doubly spread channels

We study the correspondence between time-frequency and time-scale integral operators and determine the mapping between time-scale kernels and time-frequency kernels, which are both used to model communication channels. Time-scale models, which have a physical interpretation in wireless communication, are often approximated as time-frequency models for narrowband communication. In determining this mapping we show that the causal time-scale channel is time-invariant and that the causal time-forward time-frequency channel is time-invariant. A time-forward channel is defined as a channel for which the Hardy space and its orthogonal complement are invariant. We derive the form of the equivalent lowpass characterization of the time-varying time-frequency channel and show that the equivalent lowpass characterization only exists for time-forward channels. We show that the mapping between the narrowband channel description based on a time-frequency kernel with constrained support and the wideband channel description based on a time-scale kernel with constrained support does not exist for the channels encountered in realistic time-varying communication scenarios. In light of this result, we develop a canonical time-scale channel model for wideband communication analogous to the canonical time-frequency channel model proposed by Sayeed and Aazhang.