Analysis of Non-Stationary 3D Air-to-Air Channels Using the Theory of Algebraic Curves

Non-stationary channel models play a crucial role in today's communication systems. Mobile-to-mobile channels are known to exhibit non-stationary behavior caused by the movement of transmitter and receiver. Non-stationarity can be addressed by introducing time-variant stochastic functions such as the time-variant instantaneous Doppler probability density function or time-variant instantaneous characteristic function. An algebraic analysis of the time-variant Doppler probability density function (pdf) in a classical Cartesian coordinate system is only numerically tractable due to trigonometric functions in the resulting expressions. In contrast, it has been shown that by using prolate spheroidal coordinates for 2D vehicle-to-vehicle channels the algebraic analysis becomes analytically tractable. In this paper, the analysis is extended to air-to-air channels. It is shown that the time-variant Doppler pdf can be represented without trigonometric functions. The description of the Doppler frequency in prolate spheroidal coordinates allows describing it as an algebraic curve. This permits the use of algebraic methods to analyze the Doppler frequency and derive the boundaries of the resulting Doppler pdf. Using the developed tools, we have investigated exemplary air-to-air scenarios. However, the methodology can be extended to any air-to-air configuration.