Electromagnetic Radiation

In order to define a suitable transmitter and receiver for an indoor model it is useful to examine theoretically ``ideal'' models. The source that is best suited to simulation is one that radiates equally in all directions; whose transmitted power decreases with distance from the receiver; but for a fixed distance has constant power irrespective of transmitter orientation. In this context, the radiated power is given by the Poynting vector [83, p. 389]. This is the ideal omnidirectional antenna, or point source. If we define some power at a radial distance from the transmitter, we can write an equation for the power at distance as

Proakis[3, p 508] defines as

This equation is valid for the far field of the transmitter, that is for , but when the receiver moves into the near field the equation no longer holds. At some distance, , the received power as defined by (3.1) may exceed the transmitted power. Clearly this does not describe the real situation correctly, and it is also clear that the equation has a singularity, called a caustic, at . On reviewing the definition, this is a perfectly understandable result as a finite power has been defined for all spheres concentric on the transmitter. As the radius of the sphere decreases, the power density on the sphere increases giving rise to infinite power at an infinitesimally small radius.

The theoretical point source model is also deficient in another respect; electromagnetic theory shows that such a source of radiation is impossible[84]. For an electric and magnetic field to be generated-and hence radiated-there must be a change in charge, that is a current. With a point source, or point charge, such a change or flow of charge is not possible. However, it is possible to define a dipole of an infinitely small size with some alignment. For a dipole aligned on the -axis at the origin (Figure 3-1) with current defined within the length , an expression for , the current on the -axis can be defined by

where is the current moment. This results in the magnetic field being defined, in spherical coordinates, as

where . That is is a vector that rotates around the -axis. The term is the angular frequency of oscillation, the permittivity of the medium, and the permeability of the medium. The electric field is given by

, and are unit vectors in the directions of , and respectively where , and define a point in spherical coordinates. is the angle made between the vector from the origin to the point of interest and the plane with respect to the -plane, and is the angle between the -axis and the point of interest (Figure 3-1). Thus, the Poynting vector, which denotes the power that is being radiated, is defined by

where , and are unit vectors in the radial and polar angular directions, and denotes the complex conjugate of .

For points distant from the transmitter, the terms and higher powers can be neglected, resulting in a plane wave where

and

Thus, the power, , at a given point in space, assuming that it is far from the transmitter, can be expressed as

where denotes the complex conjugate of , and is the intrinsic impedance of the medium given by

which simplifies to when . Using this, an equation for can be derived as

Thus the relationship in (3.1) can be expressed as

The implication for constructing a software model for the channel is that if the reflecting and diffracting surfaces are sufficiently far from the radiating source, the radiation can be treated as a plane wave, and calculations involving only one field, namely the field, need to be performed. This creates a significant reduction in computational complexity.