Physics and mathematics of radio wave propagation in cellular wireless communications

The objective of this chapter is to illustrate that an electromagnetic macro model can accurately predict the dominant component of the propagation path loss for a cellular wireless communication. The reason a macro model can provide accurate results that agree with experiments is because the trees, buildings, and other man-made obstacles contribute second-order effects to the propagation path loss as the dominant component that affects propagation is the free-space propagation of the signal and the effect of the earth over which the signal is propagating. It is demonstrated using both measurements and an analytical theoretical model that the propagation path loss inside a cellular communication cell is first 30 dB per decade of distance, and later on, usually outside the cell, it is 40 dB per decade of the electrical distance between the transmitter and the receiver irrespective of their heights from the ground. This implies that the electric field decays first at a rate of ρ-1.5 inside the cell and later on, usually outside the cell, as ρ-2, where ρ stands for the distance between the transmitter and the receiver. This appears to be independent of the frequency of operation in the band of interest and the parameters of the ground. It is also illustrated that the so-called slow fading is due to the interference between the direct wave and the ground wave as introduced by Sommerfeld over 100 years ago. All these statements can be derived from the approximate integration of the Sommerfeld integrals using a modified path for the steepest descent method and also using a purely numerical methodology. Finally, an optical analog is described based on the image theory developed by Van der Pol to illustrate the mechanism of radio wave propagation in a cellular wireless communication system.