Mobile communication channels are often modeled as linear
time-varying filters or, equivalently, as time-frequency integral
operators with finite support in time and frequency. Such a
characterization inherently assumes the signals are narrowband and
may not be appropriate for wideband signals. In this paper
time-scale characterizations are examined that are useful in
wideband time-varying channels, for which a time-scale integral
operator is physically justifiable. A review of these time-frequency
and time-scale characterizations is presented. Both the
time-frequency and time-scale integral operators have a
two-dimensional discrete characterization which motivates the design
of time-frequency or time-scale rake receivers. These receivers have
taps for both time and frequency (or time and scale) shifts of the
transmitted signal. A general theory of these characterizations
which generates, as specific cases, the discrete time-frequency and
time-scale models is presented here. The interpretation of these
models, namely, that they can be seen to arise from processing
assumptions on the transmit and receive waveforms is discussed. Out
of this discussion a third model arises: a frequency-scale
continuous channel model with an associated discrete frequency-scale
characterization.