Scattered signal distributions, parametric uncertainties, and Bayesian sequential updating

Abstract

A variety of probability density functions (pdfs) have been proposed for scattered signals, which have varying analytical advantages and ranges of physical applicability. We discuss here modeling with a compound pdf, in which a basic pdf, describing the underlying scattering process, has uncertain parameters or is modulated by variability in the environment. The parameters of the modulating pdf are termed hyperparameters. Some previous examples of compound formulations include the K-distribution, for which strong scattering (exponential pdf) is modulated by a gamma pdf for the mean signal power, and scattering by intermittent turbulence, for which strong scattering is modulated by a log-normal pdf for the structure-function parameter. We describe some alternative formulations, including strong scattering modulated by a gamma pdf for the inverse mean power, and Rytov (log-normal) scattering modulated by a normal pdf for the log-mean of the signal. These lead to relatively simple marginalized signal power distributions (Lomax and log-normal, respectively). Furthermore, the basic scattered signal pdf may be viewed as a likelihood function in which the modulating pdf is the Bayesian conjugate prior. Hence the hyperparameters of the modulating process can be refined by sequential Bayesian updating as additional transmission data become available.