An Analysis of Probability Distribution of Doppler Shift in Three-Dimensional Mobile Radio Environments

The Doppler shift (DS) distribution of the signal that is received by a mobile station in 3D mobile radio environments is closely related to the signal's power spectral density (PSD). From the relationship between the arriving angles and the DS, general equations are derived, bridging the probability density function (pdf) of the elevation angle with the pdf of the DS, its variance, and the characteristic function. These equations are then used to analyze the statistic distribution of the DS based on the semispheroid model, which has been recently proposed by Janaswamy. For this particular model, analytical expressions for the pdf and the variance of the DS have been derived, and a definite-integral expression is provided for the characteristic function. When the average strengths of the waves that are distributed on different equal-Doppler-shift surfaces are assumed to be equal, the derived pdf of the DS is equivalent to the signal's PSD, and the derived characteristic function of the DS is equivalent to the signal's autocorrelation function.