The six-rays model is applied in an urban or indoor environment where a radio signal transmitted will encounter some objects that produce reflected, refracted or scattered copies of the transmitted signal. These are called multipath signal components; they are attenuated, delayed and shifted from the original signal (LOS) due to a finite number of reflectors with known location and dielectric properties, LOS and multipath signal are summed at the receiver. This model approaches the propagation of electromagnetic waves by representing wavefront as simple particles. Thus reflection, refraction and scattering effects are approximated using simple geometric equation instead Maxwell's wave equations. The simplest model is two-rays which predicts signal variation resulting from a ground reflection interfering with the loss path. This model is applicable in isolated areas with some reflectors, such as rural roads or hallway. The above two-rays approach can easily be extended to add as many rays as required. We may add rays bouncing off each side of a street in an urban corridor, leading to a six-rays model. The deduction of the six-rays model is presented below.
Mathematical deduction
Antennas of heights equal located in the center of the street
For the analysis of antennas with equal heights then h t = h r = h {\displaystyle h_{t}=h_{r}=h} , determining that for the following two rays that are reflected once in the wall, the point in which they collide is equal to said height h {\displaystyle h} . Also for each ray that is reflected in the wall, there is another ray that is reflected in the ground in a number equal to the reflections in the wall plus one, in these rays there are diagonal distances for each reflection and the sum of these distances is denominated d ′ {\displaystyle d'} . Being located in the center of the street the distance between the antennas T X {\displaystyle T_{X}} and R X {\displaystyle R_{X}} , the buildings and the width of the streets are equal in both sides so that w t 1 = w r 1 = w t 2 = w r 2 {\displaystyle w_{t1}=w_{r1}=w_{t2}=w_{r2}} , defining thus a single distance w {\displaystyle w} . The mathematical model of propagation of six rays is based on the model of two rays, to find the equations of each ray involved. The distance d {\displaystyle d} that separates the two antennas, is equal to the first direct ray R 0 {\displaystyle R_{0}} or line of sight (LOS), that is:
R 0 = d {\displaystyle R_{0}=d}
For the ray reflected under R 0 {\displaystyle R_{0}} applies the theorem of Pythagoras, in the right triangle that forms between the reflection of R 0 {\displaystyle R_{0}} as the hypotenuse and the direct ray obtaining:
R 0 ′ = d 2 + ( 2 ∗ h ) 2 {\displaystyle R_{0}'={\sqrt {d^{2}+(2*h)^{2}}}}
For R 1 {\displaystyle R_{1}} the Pythagorean theorem is reapplied, knowing that one of the hinges is double the distances between the transmitter and the building due to the reflection of w {\displaystyle w} and the diagonal distance to the wall:
R 1 = d 2 + ( 2 ∗ w ) 2 {\displaystyle R_{1}={\sqrt {d^{2}+(2*w)^{2}}}}
For R 1 {\displaystyle R_{1}} the second ray is multiplied twice but it is taken into account that the distance is half of the third ray to form the equivalent triangle considering that d 1 {\displaystyle d_{1}} is the half of the distance of R 1 {\displaystyle R_{1}} and these must be the half of the line of sight distance d {\displaystyle d} :
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