## 4 Transmission Channel Analysis
### Fresnel Zone  
The Fresnel zone is an infinite series of confocal prolate ellipsoidal three-dimensional regions enveloping the direct LoS path between the transmitting and the receiving antennas. It is employed in propagation theory to evaluate losses due to reflection and diffraction between the endpoints of the wireless link. Such zones are usually called \(F_{i}\) with \(i\in\mathbb{N}^{+}\) where \(F_{1}\) is the smallest one. Regularly,  
Figure 1: Gains, in blue, and losses, in yellow, of a communication channel to be accounted within link budget equation. a slight abuse of notation is committed since also the maximum radius relative to the \(i\)th zone is marked as \(F_{i}\). In order to grasp the practical meaning of the Fresnel zone, it has to be recalled that radio frequency waves propagate not only directly along the LoS path, but also in an off-axis fashion. This phenomenon entails reflections which in turn implies a phase variation on the reflected signal. Unfortunately though, if such unwanted and often unpredictable changes of phase amount to \((2k+1)\pi\) with \(k\in\mathbb{N}\), then the direct and reflected signals cancel out at the receiver side since they have opposite phases. On the other hand, if the change of phase is equal to \(2k\pi\) with \(k\in\mathbb{N}\) then the two signals will enhance each other at the receiver side due to the fact that they are perfectly in phase. However, reflections should be avoided since phase changing ordinarily leads to destructive interference and signal weakening. A more rigorous definition is that the \(n\)th Fresnel zone is the locus of points belonging to the three-dimensional space such that a two-segment transmitted reflected signal, that deflects off a point on such a surface, will have a phase change equal to \((i+1)\pi\) with respect to the signal propagated along the direct LoS path. Those zones are ellipsoids whose foci are the transmitter and the receiver antennas. In other words, even numbered Fresnel zones cause a phase change of \(\pi\), thus damaging wave propagation, whilst odd numbered ones produce a phase change of \(2\pi\), wherefore intensifying the broadcast. Hence, despite an intuitive belief, LoS between transmitter and receiver is not theoretically sufficient to assure adequate wireless transmissions. Indeed, due to the complex nature of radio propagation, obstructions within the first Fresnel zone should be shunned since they can cause significant signal weakening consequently vanishing the enhancing effect of such a zone. Sadly though, it is almost unfeasible to obtain a completely empty \(F_{1}\) so at least 60% clearance should be achieved [34]. Figure 2 reports an example of the first Fresnel zone: \(D\) is the distance between the transmitter and the receiver (i.e., the direct LoS path) while \(F_{1}\) is the maximum radius of the zone which occurs at \(\frac{D}{2}\). There exist formulas that evaluate \(F_{1}\) depending on the units of measure in which its terms are expressed. If \(D\) is specified in km and the transmission frequency \(f\) is defined in GHz, then \(F_{1}\) is calculated in m as  
\[F_{1}=8.656\sqrt{\frac{D}{f}}. \tag{5}\]  
So far it was pointed out that achieving the direct LoS from the transmitter and the receiver along with a significant clearance of the first Fresnel zone is considerably beneficial to increase the chances of adequate wireless transmissions. For what matters the direct LoS, it is obtained by checking whether or not the receiving antenna is beyond the horizon with respect to the transmitting one. Fresnel zone clearance has to be assured not only by avoiding obstacles like buildings or mountains, but also by accounting for the Earth curvature bulge especially for long transmission distances. Both of these issues may be faced by overhead installing the antennas. The direct LoS assessment considering the distance of the horizon from the transmitting antenna may be simply performed by making use of the Pythagorean theorem.