Time-Variant Doppler PDFs and Characteristic Functions for the Vehicle-to-Vehicle Channel

Non-stationarity of vehicle-to-vehicle channels is one of the key elements that has to be taken into account for accurate channel modeling. The time-variance and its dual-the frequency selectivity-lead to non-stationarity. These can be assessed by both the temporal autocorrelation function and the Doppler spectrum, respectively. For fixed-to-mobile channels closed-form solutions for autocorrelation functions and Doppler spectra are well known. For vehicle-to-vehicle channels closed-form solutions exist, if uncorrelated double-bounce scattering is assumed. For time-variant, delay-dependent, correlated single-bounce scattering expressions are yet to be found. This contribution addresses the mentioned problem. Specifically, the proportionality between the Doppler probability density function (pdf) and Doppler power spectral density in time-varying scenarios for non-stationary, uncorrelated scattering is demonstrated. The latter also implies a proportionality between the characteristic function and the corresponding autocorrelation function; these functions are the Fourier transforms of the Doppler pdf and Doppler power spectral density, respectively. It is shown that time-varying characteristic functions and Doppler pdfs for general vehicle-to-vehicle scenarios can be derived in prolate spheroidal coordinates. The investigation of the Doppler frequency in these coordinates allows us to derive expressions of the maximum and minimum frequencies of the Doppler pdf in the vicinity of line-of-sight. Several vehicular scenarios of interest are investigated and closed-form solutions for the Doppler pdf and characteristic function are presented. An analysis of the results shows that the obtained expressions generalize well the known closed-form results for stationary channels. This further permits deriving some time-variant statistical channel parameters like the mean Doppler and Doppler spread. These parameters are particularly important when designing a Wiener filter or estimating propagation channel characteristics for highly time-variant vehicle-to-vehicle channels.