Cascaded scattering functions for sonar signal processing

This paper derives the total scattering function of a channel as a cascaded convolution of propagation and other scattering functions in a general structure with the narrowband (time-frequency) and wideband (time-scale) details presented as special cases. To do this requires the assumption that the probing signal is sufficiently rich so that the total spreading function can be represented by an inverse transform of the received signal. The derivation of this cascade of scattering functions then exploits properties of reproducing kernel Hilbert spaces (RKHS). Since scattering functions behave similarly to ambiguity functions (there is an increase in ambiguity as the signal propagates through the medium), a convolution of scattering functions can be viewed as propagation of ambiguities through a time-varying multipath medium. The incorporation of cascaded scattering functions into a detection processor then yields an improved technique for detecting signals in more complex time-varying environments.