Physics of propagation in a cellular wireless communication environment

The objective of this paper is to illustrate from a physics standpoint the nature of the fields that carry energy from a transmitter to a receiver located over an imperfectly conducting Earth. In an accompanying paper [1], it was shown from both the theoretical (developed from the classical Sommerfeld attenuation function) and experimental points of view that the fields inside a cellular-communication cell decay as 30 dB per decade, following a slow fading region. The 30 dB per decade region is followed by a region (in the fringe area) where the rate of decay is 40 dB per decade. This means that the fields decay as 1/ρ^1.5 (where ρ is the radial distance between the transmitter and the receiver) and 1/ρ², respectively inside a cell. The first goal is to illustrate that Sommerfeld had no error in sign in his classic 1909 paper. However, while computing his asymptotic development of the branch-cut integral, Sommerfeld did not notice that beside the space wave, this contains the surface wave with a negative sign, which would cancel the residue of the pole. In other words, no surface wave can be generated from a pole that is a part of the original Sommerfeld solution for radiation from a dipole operating over an imperfectly conducting ground. Other type of waves, e.g., Norton's surface wave and a radiation field that decays as 1/ρ¹⁵, are part of the ground wave. They can be derived from the same Sommerfeld attenuation function, but using different types of approximations. As already mentioned, Norton's surface wave and the radiation field (decaying as 1/ρ^1.5) are a part of the solution. These points were illustrated in an accompanying paper [1]: here, we focus on the physics of these waves. We also explain what the mathematical characteristics of a surface wave are, as there is a lot of confusion in the usage of the same word for different types of waves. The goal of this paper is hence to develop an understanding of the physics associated with radiating fields that decay with a rate of 30 dB per decade with distance, and with an emphasis on the following phenomena: 1. To illustrate that radiation fields emanating from a two-dimensional source and associated with the launching of a true surface wave are responsible for producing a field that decays as 30 dB per decade. This type of field can also be generated from a line source located over an imperfectly conducting ground. 2. To illustrate that radiation fields in the form of Norton's surface wave contribute only in the fringe areas of the cells, where the fields decay as 40 dB per decade. 3. To illustrate that the error in Sommerfeld's classical paper of 1909 had nothing to do with an error in sign. He did not consider the effect of the occurrence of a pole near the saddle point in his asymptotic expansion. In other words, the poles were not located on the proper Riemann sheet. The popular belief that he made an error in sign in his paper is indeed a myth! 4. To illustrate that the electromagnetic field produced in a cellular communication system (where the dipole constituting the base-station antenna is placed at a given height over the imperfectly conducting surface of the Earth) is due to an infinite set of images starting at the location of the image of the first point of the transmitting antenna and extending to negative infinity, resulting in a line source. This will be illustrated by both theoretical and experimental data. This further illustrates that ray tracing is not the appropriate tool to use in predicting the fields near the ground. 5. Finally, to illustrate that all the experimental data indeed clearly show that the path-loss exponent in a cellular environment is three (at moderate distances from the transmitting antenna) or four (at larger distances), in spite of buildings and trees being present in the measurements. Hence, one of the surprising conclusions is that one can get by with macro modeling, and micro modeling of the environment is overkill!