Novel joint delay Doppler probability density functions for vehicle-to-vehicle communications channels are introduced. Prior measurements of vehicle-to-vehicle channels have unveiled their nonstationarity; thus, the wide-sense stationary and also the uncorrelated scattering assumption for such channels is often violated, which makes their modeling challenging. In this work it is proposed to exploit geometry-based stochastic modeling to cope with the nonstationarity of vehicle-to-vehicle channels. To this end, delay-dependent Doppler pdfs are derived for arbitrary times. It is assumed that scatterers are randomly distributed on an ellipse with two moving vehicles being in its foci. The proposed approach allows reducing the dimensionality of the resulting problem. This in turn leads to a significantly simplified derivation of the delay-dependent Doppler pdfs for general vehicle-to-vehicle propagation environments; moreover, the resulting computations can be performed almost fully analytically. By combining the calculated Doppler pdf with a delay pdf, the joint pdf of delay and Doppler is obtained. The joint pdf then can be put into relation with the generalized local scattering function. The presented modeling approach is simple yet very scalable and accurate, which allows its application in different vehicular scenarios. The obtained modeling results correspond very well with measurement data reported in prior works.
