Six-rays model

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 approach 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.[1]
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 formula_1, 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 formula_2. 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 formula_3.
Being located in the center of the street the distance between the antennas formula_4 and formula_5, the buildings and the width of the streets are equal in both sides so that formula_6, defining thus a single distance formula_7.
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 formula_8 that separates the two antennas, is equal to the first direct ray formula_9 or line of sight (LOS), that is:
formula_10
For the ray reflected under formula_9 applies the theorem of Pythagoras, in the right triangle that forms between the reflection of formula_9 as the hypotenuse and the direct ray obtaining:
formula_13
For formula_14 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 formula_7 and the diagonal distance to the wall:
formula_16
For formula_14 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 formula_18 is the half of the distance of formula_14 and these must be the half of the line of sight distance formula_8:
formula_21
For formula_14 y formula_23 the deduction and the distances are equals, therefore:
formula_24
formula_25
Antennas of heights equal located in any point of the street.
As the direct ray LOS does not vary and has not angular variation between the rays, the distance of the first two rays formula_9 and formula_27 of model does not vary and deduced according to the mathematic model for two rays. For the other four rays it applies the next mathematical process:
formula_14 is obtained through a geometric analysis of the top view for the model and it applies the Pythagorean Theorem triangles, taking into account the distance between the wall and the antennas formula_29, formula_30, formula_31, formula_32 are different:
formula_33
For likeness of triangles in the top view for model is determined the equation formula_14:
formula_35
formula_36
formula_37
For formula_14 and formula_23 the deduction and the distances are equal then:
formula_24
formula_25
Antennas of heights different located in the center of the street.
For antennas of different heights with rays that rebound in the wall, it is noted that the wall is the half point, where the two transmitted rays they fall on such wall. This wall has half the height between the height of the formula_4 and formula_5, it means smaller than the transmitter and higher than the receiver and this high is where the two rays impact in the point, then rebound to the receiver. The ray reflected leaves two reflections, one that it has the same high of the wall and the other the receiver, and the ray of the line of sight maintains the same direction between the formula_4 and the formula_5. The diagonal distance "d´" that separates the two antennas divides in two distances through of the wall, one is called formula_46 and the other formula_47"."
Antennas of heights different located in any point of the street.
For the mathematical model of six-ray propagation for antennas of different heights located at any point in the street, formula_48, there is a direct distance formula_8 that separates the two antennas, the first ray is formed by applying The Pythagorean theorem from the difference of heights of the antennas with respect to the line of sight:
formula_50
The second ray or reflected ray is calculated as the first ray but the heights of the antennas are added to form the right triangle.
formula_51
For deducing the third ray it is calculated the angle between the direct distance formula_8 and the distance of line of sight formula_9
formula_54
Now deducing the height that subtraction of the wall with respect the height of the receiver called formula_55 by similarity the triangles:
formula_56 
formula_57
By similarity of triangles it can deduce the distance where the ray hits the wall until the perpendicular of the receiver called "a" achieved:
formula_58 
formula_59
formula_60
By similarity of the triangles can be deduced the equation of the fourth ray:
formula_61
For formula_14 y formula_23 the deduction and the distances are equal, therefore:
formula_24
formula_25
Free-space path loss on the model.
Consider a transmitted signal in the free space a receptor located a distance "d" of the transmitter. One may add rays bouncing off each side of a street in an urban corridor, leading to a six-rays model, with rays formula_9, formula_14 and formula_23 each one having a direct and a ground bouncing ray.
An important assumption must be made to simplify the model: formula_69 is small compared to the symbol length of the useful information, that is formula_70. For the rays rebound outside the earth and on each side of the street, this assumption is fairly safe, but in general will have remembered that these assumptions mean the dispersion of delays (diffusion of the values formula_69) is smaller than symbols speed of transmission.
Free-space path loss of six rays model is defined as:
formula_72
formula_73
formula_74
formula_75
formula_76 formula_77 is the wavelength.
formula_78 Is the time difference between the two paths.
formula_79 Is the coefficient of ground reflection.
formula_80 Gain of the transmitter.
formula_81 Receiver gain.