Joint Delay Doppler Probability Density Functions for Air-to-Air Channels

Recent channel measurements indicate that the wide sense stationary uncorrelated scattering assumption is not valid for air-to-air channels. Therefore, purely stochastic channel models cannot be used. In order to cope with the nonstationarity a geometric component is included. In this paper we extend a previously presented two-dimensional geometric stochastic model originally developed for vehicle-to-vehicle communication to a three-dimensional air-to-air channel model. Novel joint time-variant delay Doppler probability density functions are presented. The probability density functions are derived by using vector calculus and parametric equations of the delay ellipses. This allows us to obtain closed form mathematical expressions for the probability density functions, which can then be calculated for any delay and Doppler frequency at arbitrary times numerically.