Changes

== 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:

:<math> \mathbf{ \hat{u}_b' = \hat{k}' \times \hat{u}_f' } </math>

The diffraction coefficients are calculated in the following way:

:<math> D_h = \frac{-e^{-j\pi/4}}{2n \sqrt{2\pi k} \sin\beta_0'} \left\lbrace \begin{align} & \cot \left(\frac{\pi + (\phi-\phi')}{2n}\right) F[kLa^+(\phi-\phi')] + \cot \left(\frac{\pi - (\phi-\phi')}{2n}\right) F[kLa^-(\phi-\phi')] + \\ & R_{0 \|} \cot \left(\frac{\pi - (\phi+\phi')}{2n}\right) F[kLa^-(\phi+\phi')] + R_{n \|} \cot \left(\frac{\pi + (\phi+\phi')}{2n}\right) F[kLa^+(\phi+\phi')] \end{align} \right\rbrace </math>

where ''F(x)'' is the Fresnel Transition function:

:<math> F(x) = 2j \sqrt{x} e^{jx} \int_{\sqrt{x}}^{\infty} e^{-j\tau^2} \, d\tau </math>

In the above equations, we have

where <math>N^{\pm}</math> are the integers which most closely satisfy the equations <math> 2n\pi N^{\pm} - \nu = \pm \pi </math>.

[[Image:Top_icon.png|48px]] '''[[#Free-Space Wave Propagation | Back to the Top of the Page]]'''