Geometric Analysis of the Doppler Frequency for General Non-Stationary 3D Mobile-to-Mobile Channels Based on Prolate Spheroidal Coordinates

Mobile-to-mobile channels often exhibit time-variant Doppler frequency shifts due to the movement of transmitter and receiver. An accurate description of the Doppler frequency turns out to be very difficult in Cartesian coordinates and any subsequent algebraic analysis of the Doppler frequency is intractable. In contrast to other approaches, we base our investigation on a geometric description of the Doppler frequency with the following three mathematical pillars: prolate spheroidal coordinate system, algebraic curve theory, and differential forms. The prolate spheroidal coordinate system is more appropriate to algebraically investigate the problem. After the transformation into the new coordinate system, the theory of algebraic curves is needed to resolve the ambiguities. Finally, the differential forms are required to derive the joint delay Doppler probability density function. This function is normalized by the equivalent ellipsoidal area of the scattering plane bounded by the delay ellipsoid. The results generalize in a natural way our previous model to a complete 3D description. Our solutions enable insight into the geometry of the Doppler frequency and we were able to derive a Doppler frequency that is dependent on the delay and the scattering plane. The presented theory allows describing any time-variant, single-bounce, mobile-to-mobile scattering channel.