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== Free-Space Wave Propagation ==

In a free-space line-of-sight (LOS) communication system, the signal propagates directly from the transmitter to the receiver without encountering any obstacles (scatterers). Electromagnetic waves propagate in the form of spherical waves with a functional dependence of e^j(&omega;^t-k₀R)/R, where R is the distance between the transmitter and receiver, <math>\omega = 2\pi f</math>, f is the signal frequency, <math>k_0 = \frac{\omega}{c} = \frac{2\pi}{\lambda}</math>, c is the speed of light, and &lambda;₀ is the free-space wavelength at the operational frequency. By the time the signal arrives at the location of the receiver, it undergoes two changes. It is attenuated and its power drops by a factor of 1/R², and additionally, it experiences a phase shift of <math>\frac{2\pi R}{\lambda_0}</math>, which is equivalent to a time delay of R/c. The signal attenuation from the transmitter to the receiver is usually quantified by '''Path Loss''' defined as the ratio of the received signal power (P_R) to the transmitted signal power (P_T). Assuming isotropic transmitting and receiving radiators (<i>i.e.</i> radiating uniformly in all directions), the Path Loss in a free-space line-of-sight communication system is given by Friis’ formula:

where '''u_T''' and '''u_R''' are the unit polarization vectors of the transmitting and receiving antennas, and G_T and G_R are their gains, respectively.

== Basic Wave Interaction Mechanisms ==

== Ray Reflection & Transmission ==

The incident, reflected and transmitted rays are each characterized by a triplet of unit vectors:

* <math>( \mathbf{ \hat{u}_{\|}^{\prime}, \hat{u}_{\perp}', \hat{k}' } )</math> representing the reflected parallel polarization vector, reflected perpendicular polarization vector and reflected propagation vector, respectively.

* <math>( \mathbf{ \hat{u}_{\|}^{\prime\prime}, \hat{u}_{\perp}^{\prime\prime}, \hat{k}^{\prime\prime} } )</math> representing the transmitted parallel polarization vector, transmitted perpendicular polarization vector and transmitted propagation vector, respectively.

The reflected ray is assumed to originate from a virtual image source point. The three triplets constitute three orthonormal basis systems. Below, it is assumed that the two dielectric media have permittivities &epsilon;₁ and &epsilon;₂, and permeabilities &mu;₁ and &mu;₂, respectively. A lossy medium with a conductivity &sigma; can be modeled by a complex permittivity &epsilon;_r = &epsilon;'_r –j&sigma;/&epsilon;₀. Assuming '''n''' to be the unit normal to the interface plane between the two media, and Z₀ = 120&Omega; , the incident polarization vectors as well as all the reflected and transmitted vectors are found as:

In &quot;Thin Wall Approximation&quot;, we assume that an incident ray gives rise to two rays, one is reflected at the specular point, and the other is transmitted almost in the same direction as the incident ray. The reflected ray is assumed to originate from a virtual image source point. Similar to the case of reflection and transmission at the interface between two dielectric media, here too we have three triplets of unit vectors, which all form orthonormal basis systems.

The transmission coefficients are calculated for the two parallel and perpendicular polarizations as:

* <math>\mathbf{(\hat{u}_f, \hat{u}_b, \hat{t})}</math> representing the incident forward polarization vector, incident backward polarization vector and incident propagation vector, respectively.

* <math>\mathbf{(\hat{u}_f', \hat{u}_b', \hat{t}')}</math> representing the diffracted forward polarization vector, diffracted backward polarization vector and diffracted propagation vector, respectively.

The three triplets constitute three orthonormal basis systems. The propagation vector '''k'''' of the diffracted ray has to be constructed based on the diffraction cone as follows: