A Radio Propagation Model for Mixed Paths in Amazon Environments for the UHF Band

The present work proposes a radio propagation model for the Amazon region called Mixed Path. The techniques used for Mixed Path model are Geometrical Optics (GO) and the Uniform Theory of Diffraction (UTD). Only ten rays are considered the main contributors to calculate the total electric field. Increasing the number of rays does not improve the accuracy of Mixed Path model since the scenario is for receivers located in long distances. Then slope diffraction or multiple reflections means a low electric field that does not contribute significantly to the total electric field. The parameters of Mixed Path model such as electrical constants, antennas height, buildings height among others, are analyzed in order to know the influence of them in the received electric field. Measured data in the central frequency of 521 MHz of a Digital Television station in the city of Belem of Pará are used to validate Mixed Path model. This city is located in the Amazon region of Brazil and presents mixed routes formed by city, river, and forest. Because digital television has a wide coverage and reception flexibility, Mixed Path was designed for receivers at the user’s level for the service of Mobile Digital Television (M-DTV) and for fixed receivers on the rooftops of homes for Home digital television (H-DTV). Finally, the proposed model and other models in the literature are compared with the data measured for M-DTV, being Mixed Path the model with the lowest RMS error with a value of 3.15 dB for a receiver over the river behind the city and behind the forest.


List of Tables
Path loss for suburban areas : Electric field over the sea : Electric field on the land : Distance of Fraunhoffer : Wavelength : Number of radio of Fresnel 1 : Distance between the transmitter and the obstacles 2 : Distance between the obstacle and the receiver. : Fresnel Kirchoff diffraction parameter − : Phase shift along the ray path ( ): Spreading factor : Electric field for direct ray : Transmission Power. : Gain of transmitter antenna : Loss for connectors, cable, etc : Free space propagation constant Γ : Coefficient of reflection for Vertical polarization Γ : Coefficient of reflection for Horizontal polarization : Intrinsic impedance of media i : Conductivity in (S/m) : Permittivity (F/m) : Distance between the transmitter and the point of reflection : Distance between the receiver and the point of reflection τ : Coefficient of refraction/transmission for Vertical polarization : Coefficient of refraction/transmission for Horizontal polarization : Electric field of transmission : Distance between the transmitter and the point of transmission. : Electric field of reflection : Incidence point of transmission forest-air. : Distance inside the forest between and n, the point of incidence of refraction forest-air : Distance between n and the receiver : Transmission coefficient air-forest : Transmission coefficient forest-air : Phase constant : permeability : Attenuation constant : Angular frequency : Diffraction coefficient ′ : Incidence angle of diffraction : Angle between the 0 face and the diffracted ray : Distance between the fount (F) and the point of incidence of diffraction.

′:
Distance between point of incidence of diffraction and the observation point (O) 0 : Value to include the grazing effects in the surface 0. : Value to include the grazing effects in the surface n. : Diffracted electric field. : Point of diffraction : Distance between transmitter and b. : Critical Angle ℎ : Height of transmitter over ground ℎ : Height of receiver ℎ : Height of building over ground ℎ : Height of ground ℎ : Height of forest : Distance between transmitter and receiver over the ground : Direct distance between transmitter and receiver ′ : Width of the street considering (top view) Width of city with constructions ′ : Width of the building considering (top view). : Angle between transmitter-receiver and the street or building (top view) Fig Incidence point of transmission : Incidence point of diffraction : Grazing angle for reflection on a vertical surface ℎ : Grazing angle for reflection on a horizontal surface : Angle of direct ray : Incidence angle for transmission air-forest : Transmission angle for air-forest : Incidence angle for transmission forest-air : Transmission angle for forest-air : Point of transmitter placement : Point of the base of the building ℎ : Electric field for a reflection on a horizontal surface. : Electric field for a reflection on a vertical surface.

− :
Electric field for a diffraction on the left.

+ :
Electric field for a diffraction on the right.

− ℎ :
Electric field for a diffraction on the left, reflection on a horizontal surface.
Electric field for a diffraction on the right, reflection on a horizontal surface − : Electric field for a diffraction on the left, reflection on a vertical surface + : Electric field for a diffraction on the right, reflection on a vertical surface ℎ: Height of the obstacle for the calculation of the Fresnel parameter : Total electric field for MDTV in City cities [2]. The Brazilian System of Digital Terrestrial Television (SBTVD-T) uses the Japanese standard which is called Integrated System Digital Broadcasting (ISDB-T) [3]. Fig. 1 illustrates the capitals of the cities where the analogical signal was switched off until these days.

Motivation
Radio propagation models are important for planning and optimization of telecommunication systems. In Brazil, the most extended service of telecommunication is Television with coverage of 90%. Furthermore, there are not radio propagation models for the Amazon region formed by city-water and forest. In the literature can be found many papers about radio propagation for mobile digital TV (M-DTV) and fixed digital TV. In this study fixed digital TV is called Home digital TV (H-DTV) because the receiver antenna is over the rooftops of the houses. For M-DTV many scenarios have been studied including urban scenario [6], under ducts [7], mixed path land-sea [8] and indoor scenarios [9]. For the fixed DTV in [10] a study for different heights for the receiver, in different cities of Brazil for a suburban environment was presented while in [11] an indoor study to know the loss due to different materials of the constructions was analyzed.
The mixed paths currently studied are City-sea or City Forest. For City-sea most of the works are for low frequencies, the HF [12], [13], and MF bands [14], [15], [16] for vertical polarization. ITU-R Recommendation P.1546 [17], addresses the UHF band, but it does not differentiate if the first path is over land or sea, unlike the Okumura model [18], that establishes the correction for mixed path considering the first path is over water or land.
However, Okumura and ITU-R P.1546 do not consider obstacles in the transition zone from land to water.
There are radio propagation models that consider mixed City-Forest path for mobile services, in [7] and [19] loss caused by forest is calculated using knife diffraction.
Furthermore, the recommendation ITU-R P. 833 [20] gives an attenuation factor for forest of different countries. In the case of Brazil, the forest attenuation factor is only for Rio of Janeiro city. Furthermore, there are some studies in the Amazon Region as in [21] where were applying dyadic green functions using four layers for air, treetop, trunk, and land for vertical polarization. Another work uses parabolic equations to calculate the electric field but has limitations with large propagation angles [22].
A less study scenario for mixed path involves land-river as in [23], which analyses propagation over water with the presence of obstacles, like buildings and bridges; however, it does not study the interaction between diffraction from a city and reflection over water, and it presents results for distances larger than 1km. Earlier works for mixed path formed by citywater do not study the transition zone city-river.

Objectives
General objective • Develop a radio propagation model for UHF band for a mixed path in the Amazon Region, using geometrical optics and the uniform theory of diffraction for H-DTV and M-DTV.

Specific objectives
• Study the transition zone city-water, for distances less than 1 km.
• Use only the principal rays to obtain the total electric field.
• Analyze the influence of the parameters of the proposed model in the calculation of electric field.
• Compare the results of the proposed model with measured data.
• Compare the attenuation caused by the city and for the forest.
• Determine electrical parameters of forest.
• Validate the model with measured data.

Technics used to develop the model
This work is focused on outdoor radio propagation for M-DTV and H-DTV in a mixed path formed by city-water-forest not studied before. Since for planning and optimizing radio communication services, radio propagation models are necessary.
The techniques to calculate the electrical field are Geometrical Optics (GO) and Uniform Theory of Diffraction (UTD) [24]. UTD considers obstacles with finite conductivity then it is more accurate than (Geometrical Theory of Diffraction) GTD according to [25].
Considering the electrical parameters allow characterizing each of the paths since all of them have different electrical parameters.
Earlier works implement ray reflection and diffraction for urban scenarios as the Ikegami, [26], and Walfish-Bertoni, [27] , models: the former considers grazing knife diffraction and the latter models buildings as several half-screens edges, the combination of the two being the COST231 Walfish-Ikegami model [28], duly assessed in Europe. Furthermore a correction factor is proposed to add the attenuation caused by the city when the electrical field is calculated over the water, the effect of the city was observed in [12] where a correction factor was proposed, however, this work was for low frequencies and does not consider obstacles on the border of the river. Additionally, using ray tracing as a deterministic model is possible to predict the electric field for various frequencies in the UHF band. Then, cities that have mixed paths formed by city-river-forest, the model can be used.

Contributions
This thesis presents a new radio propagation Model for a mixed path formed by City-

Organization of the thesis
The proposal of the thesis is divided in 7 chapters: Chapter 1: Describes the introduction, motivation and objectives and contribution of the present work.
Chapter 2: Presents earlier works about digital television and radio propagation models for mixed paths.  In [6], three transportation environments were studied, viaduct, road, and river, because Digital TV service, is used while traveling in signal covered regions. The purpose of this work is studying the influence of average noise power of Digital TV channel in a UHF band on public transportation. The average noise power in viaducts is the highest and the most scattered of the environments according to this study. Furthermore, the average noise power in the river was higher and more discrete than on the ground roads. In addition, ISDB-T standard with the option of using a long time interleave is the most robust to impulsive noise J. Yan and J. Bernhard, "Investigation of the Influence of Reflective Insulation on Indoor Reception in Rural Houses," IEEE Antennas and Propagation Letters.
In [11], studies how metal affects the signal reception of DTV inside the houses and the effectiveness of using directional antennas. Generally, indoor antennas have a small aperture.
Wireless Insite was used to simulate a typical house in North America, the points inside the house are distributed uniformly on a 30 cm grid. There were evaluated four transmitter positions with LOS and NLOS situations. Furthermore, the house was insulated and no insulated. From the simulations was observed that with LOS situation the reflective isolation is insignificant. However, in NLos situations, the reflective insulations can cause a significant signal fluctuation. In [12] the propagation of high-frequency surface radar, when the transmitter is on the coast of a Beach is studied. Earlier works used Millington for distances of several kilometers, in this study Millington was used for short distances. The values of the electrical parameters were chosen to give the best fit by eye to the measured data.
The measurements were carried out for dry sand, dry-wet sand and dry sand-wet sand and water. The paths were of 200 m to 300 m. All the three measurements were compared with predicted values using Millington, all of them have a good agreement, showing that the electrical parameters for the three different paths are correct.
Furthermore, a mixed-media propagation factor as an additional power loss is defined.
Since when the path is only water it does not present that attenuation. Furthermore, the recovery effect is evident when the first path is greater than 50 m. The PE showed coherence with measured data in a city-forest environment. It has a limitation with large propagation angles, however, the advantage is a less computational effort compared with similar techniques. To reduce the computational effort the system was reduced to a tridiagonal system. The method to solve the tridiagonal system is the implicit finite differences scheme of the Crank-Nicolson type.
After present some related works can be concluded that studies conducted on digital television do not contemplate mixed paths in the Amazon region. The mixed paths analyzed are city-forest or city-river; however, there are no studies about receivers over the river and behind the forest. In addition, the transition zone city-river was analyzed only for distances greater than 1 km. Therefore, a propagation model for the Amazon region considering cityriver-forest-river is necessary as well as the analysis of the transition zone for distances less than 1 km.

-Introduction
This chapter describes the radio propagation models existed in the literature for the UHF band in different scenarios as city, river, and forest. Furthermore, the technics used for the proposed model called Mixed Path, Geometrical Optics (GO) and the Uniform theory of diffraction (UTD) are also detailed.

-Radio propagation models
Radio propagation models predict the electric field that arrives at the receiver. They are divided into three groups: deterministic models, empirical models, and semi-deterministic models. Deterministic models use only theoretical methods, empirical models are based on measurements and the Semi-Deterministic models are a combination of measurements and theoretical methods. Next, a description of some radio propagation models.

Okumura-Hata
A radio propagation model widely used is the empirical model proposed by Okumura who makes an extensive measurement campaign for 10 years in Japan [18]. In order to facilitate the application of this model, Hata proposed simple equations to calculate path loss [31]. This model was designed for Suburban, Urban, Rural and Mixed scenarios.
• ∶ Distance (km) is the correction factor for the receiver antenna due to the environment around it, if ℎ = 1.5 (ℎ ) = 0, in other cases: Small city: Large city: For suburban areas with small buildings and wide streets, the path loss formulation in dB is: For mixed path land-water, with d < 30 km: where: • 1 : Distance only over water.
• : Angle between the incidence wave and the street.

Recommendation ITU-R-P.1546
Recommendation ITU-R P.1546 [17] is for propagation point-zone in the band (30 For a mixed path is proposed a methodology that consists in first calculate all the path as land, second calculate all the path as water and finally and interpolation between these two curves. The equations for this methodology are: The interpolation factor: where: is a relation between the distance only over the water ( ) and the total distance of the link ( ).

= (21)
0 ( ) is defined analytically: The following expression is to calculate the value of with

-Plane wave
Wave is a function of both space and time. A plane wave is characterized by the vector of the electric or magnetic field, for its complex wave number and its direction of propagation.
In the far-field region, in a small portion of the sphere surface, the spherical waves can be approximated as a plane wave. The far field is known as the Fraunhofer region that is the distance bigger than the distance of Fraunhoffer, , given by the following expression [32]: where: D : is the largest dimension of the transmitter : Wave length   The radius of ℎ Fresnel zone circle, can be expressed in terms of , 1 and 2 :

-Fresnel zones
where: : Number of radio of Fresnel 1 : Distance between the transmitter and the obstacles 2 : Distance between the obstacle and the receiver.
The effect of shadowing is sensitive to the frequency as well as the location of obstructions with relation to the transmitter or receiver. Diffraction effects are neglected when the first Fresnel zone is free more or equal than 60%.
The difference between the direct path and the diffracted path is given for: One the most used parameters to determine diffraction is the dimensionless Fresnel Kirchoff diffraction parameter which is given by[33]: When ≥ −0.78, there is diffraction.

Geometrical Optics
The incorporation of such local plane wave behavior of the field allows reducing the electromagnetic wave equations to the simpler equations for the polarization, amplitude, phase and propagation path of the high frequency field. Geometrical Optics (GO) method allows determining the wave propagation for incident, reflected and refracted fields.
The transmission of the geometrical optics field for a general astigmatic ray tube is illustrated in Fig. 4. Source [24] In Fig. 4 the distance, , between the focal lines is called the astigmatic difference. The references constant phase is Ψ(0), and it has principal radii of curvature 1 and 2 measured on the central ray. The Ψ(s) surface has principal radii of curvature ( 1 +s) and ( 2 +s).
The expression describing the transmission of the geometrical optics (GO) field, for the general astigmatic ray tube shown in Fig. 4 is given by, where: • (0) : Amplitude, phase and polarization at the reference point ( = 0) • : Distance along the ray path from the reference point ( = 0) • − : Phase shift along the ray path.
• ( ) = √| 1 2 ( 1 + )( 2 + ) ⁄ |: Spreading factor which governs the amplitude variation of the GO field along the ray path • 1 2 : the principal radii of curvature of the wave front (which is a surface) at the reference point = 0. The sign convention is that a positive (negative) radius of curvature implies diverging (converging) rays in the corresponding principal plane.
• n (m): is the number of caustic lines crossed by the observer in moving from the references position = 0 to the given observation point in a direction of (opposite to that of) propagation.

Field of direct wave
Direct field is known as line of sight, because there is not an obstacle between the transmitter and the receiver. Electric field for direct ray, , decreases with the distance as is shown in the following equation [33]: where: • : Transmission Power.

Reflected field
When there are two media with different electrical parameters and an incident wave encounters the interface a fraction of the wave intensity will be reflected into medium 1, as can be seen in Fig. 5. Two grazing angles, , for the incident ray and for the reflected ray are also illustrated. Knowing the direction of the electric field, can be defined the wave polarization if the electric field is parallel to the incidence plane it is known as vertical polarization. When the electric field is perpendicular to the incidence plane, it is a horizontal polarization. a) E into the incidence plane b) E perpendicular to the incidence plane The formulation used to calculate the coefficient of reflection for these two polarizations are: Vertical polarization: Horizontal polarization where: : Intrinsic impedance of media i.
When the first media is the free space the coefficients are: For vertical polarization: For horizontal polarization: where: • : Conductivity in (S/m) The reflection coefficients illustrated in Fig. 6 are reproduced from [33]. The electric field for a reflective wave, , can be calculated as follows: where: • ( ): Electric field of incidence in, , the incidence point of reflection ( is illustrated in Fig. 16).
• : Distance between transmitter and .
• : Distance between and the receiver.

Refracted/Transmitted field
Refraction happens when a wave passes from medium one to medium two, and these two media have different electrical parameters. In Fig. 7  In order to calculate the refracted/transmitted electric field, for the present work are considered two transmissions/refractions. Then in the general formula for refraction is introducing one more refraction coefficient. There are not considering reflections inside the forest, the expression for refracted/transmitted field, , is: where: • ( ): Incidence electric field in , , the incidence point of transmission/refraction ( . 16) .
• : Distance between transmitter and .
• : Distance inside the forest between and n, the point of incidence of refraction forest-air, illustrated in Fig. 16.
• : Distance between n and the receiver.

Real angle of refraction/transmission
The incident and transmission/refraction angles are calculated using Snell's law, with Real transmission angles for Forest-Air: where: • , : Phase constant for forest and air [35].

Uniform Theory of Diffraction
Diffraction is a local phenomenon which depends on two things: 1. The geometry of the object where the incident ray for diffraction arrives, 2. The characteristics of the incident field at the point of diffraction such as: amplitude, phase and polarization. In Fig.8 a diffracted ray is showed for an arbitrary wedge with flat faces, Φ ′ is the incidence angle of diffraction, Φ is the angle formed between the face 0 and face n of the wedge. Sproj is the distances of diffraction and S'proj is the incidence distance.
Previously diffraction formulation is studied for the General Theory of Diffraction (GTD), however, it is inaccurate in the vicinity of the shadow boundaries [24]. The Uniform Theory of Diffraction proposes an additional term for the diffraction coefficient, known as Fresnel transition function, to solution the inaccuracy of GTD formulation. The Diffraction coefficient for 2D is [24]: The values of + and − , are related with the incident shadow boundary (ISB) and the reflection shadow boundary (RSB), as functions of ± ′ and [24]. where: • : Used to calculated the angle of the obstacle, this work uses = 1.5.
• : Angle formed between the 0 face and the diffracted ray The formulation for diffraction is the following: where: • ( ): Electric field at the point of incidence b.
• : Incidence distance between transmitter and b.
• : Diffraction distance between b and receiver.
• : Diffraction coefficient In order to prove that formulation is executed with accuracy, the Fresnel transition function is reproduced from [24] and can be observed in Fig. 9. Source [24] Furthermore, in Fig. 10 the scattered field from a wedge with plane incidence using UTD diffraction coefficients [24] is shown. The parameters for calculation are the amplitude of the electric field which is 1, with n=1.778, = 40°, the incidence angle of diffraction ′ = 55 degrees, the distance of incidence is 1m, the frequency of 3 GHz, horizontal polarization.
The total electric field, is calculated according with the region I, II or III, which are separated for the shadow boundaries RSB and ISB. The equation is as follows where , is the electric field given by the direct ray, is the reflected ray and finally is the diffracted ray. a) Scattered field b) Reproduced scattered field Source: [24]

Image Method
The present study implements the image theory in order to calculate the reflected field trajectories. In Fig. 11, the image method is illustrated.
The surfaces are smooth and plane, the image method is efficient to calculate the reflected rays. The virtual source in Fig. 10 is T'. Knowing the reception point R, the reflected ray trajectory is perfectly defined. Then distances T-R' is the same distance of Reflected ray T-R.
The expression two find T-R' is: where: • ℎ : height of the transmitter • : distance between transmitter and receiver Fig. 11. Image method is used to find the difference between direct ray and reflected ray Source [33]

Lateral Wave
Tamir in 1967 presented a study about the lateral wave that is produced for a critical angle when a ray passes from a dense medium to a less dense medium. The lateral wave contributes to the total electric field that arrives at a receiver. Fig. 12 illustrates a transmitter inside the forest which originates a refracted ray forest-air, this ray forms a critical angle which originates a lateral wave. The critical angle according with [38] is given by the following expression: In order to calculate the critical angle only the real part is considered The proposed Model Mixed Path pretended to use a lateral wave in the total electric field, however the geometry does not produce the value needed for the critical angle. Using (74 ) with the electrical parameters for a dense forest from [19], the value for the critical angle is 61.29 degrees. Fig. 13 illustrates that the critical angle for the scenario of Mixed Path depends on the incidence angle of refraction air forest which depends on the reflection angle over the water. Then the angles formed instead the critical angle are around 88.41 and 88.68 degrees, which do not produce a lateral wave.

Final considerations
Mixed Path model uses some of the existing theory of literature, for example, reflection and diffraction coefficients. Furthermore, to ensure that existing formulation was implemented correctly some applications were reproduced from books. Ray tracing and the method of the images was also explained since these methods are used in the proposed model.

Scenario
The Mixed Path model is designed for both M-DTV and H-DTV. In Fig. 14 Table I. In Fig. 16 are illustrated the ten rays used for Mixed Path model in mixed City-Water1-Forest1-Water2-Forest2 path, however in each section the number of rays is different.
For example for M-DTV in the City path the maximum number of rays is eight which are the principal contributors to the total electric field.   Angle formed by the straight line between transmitter-receiver and the street or building (top view) Width of the street from top view Fig. 15 Width of the building from top view Fig. 15 Incidence point of reflection.  Table II and illustrated in Fig. 16.

Formulation for City Path for M-DTV
Mixed Path model uses maximum eight rays to calculate the total electrical field for a receiver on the City. Furthermore, the value of the Fresnel parameter is calculated to know if the first Fresnel Ellipsoid is unobstructed or not. In the City, two Fresnel parameters are considered, the first one for line of sight ray and the second one for reflected ray on the building wall.
The Fresnel parameter for line of sight, , is given by the following expressions,

Symbol Description
Line of sight. The Fresnel parameter for line of sight, , is as follows, Furthermore, to know if the reflected ray will be added in the total electric field, first, ℎ is calculated, it is the height between the incidence point of reflection and g, illustrated in Fig. 16.
Then, if ℎ < ℎ , the reflected ray can be added to the total electric field After calculating the Fresnel parameters, the total electrical field for City is calculated considering four cases: 1) When and are lower than -0.78 and ℎ < ℎ , the total electric field for City is: 2) When and are lower than -0.78 and ℎ > ℎ or is lower than -0.78 and is higher than -0.78, the total electrified for City is: 3) When and are higher than -0.78 or is higher than -0.78 and is lower than -0.78 and ℎ > ℎ the total electrified for City is:

4)
When is higher than -0.78 and is lower than -0.78 and ℎ < ℎ , the total electrified for City is:

Formulation for Water1 Path for M-DTV
A receiver located over Water 1 can have maximum 4 rays. In order to know if direct ray and the reflected ray over the water can be added in the total electric field are calculated Fresnel parameters for direct ray, 1 , and reflected ray, 1 . The equations to calculate the Fresnel parameter for direct ray, 1 , are: Fresnel parameter for reflected ray, 1 , is: Additionally, to be added in the total electric field the reflected field over the Water1, the following condition has to be accomplished, > , where, is the distance between z and , depicted in Fig.16.
Finally, for the total electric field in Water1, an environmental correction factor is added. This factor introduces the attenuation caused for City before that signal arrives at the receiver over Water1. The correction factor, 1 , is given by de following expression: where: ( ): Total electric field for City at a distance equal to the width of City with constructions.
( ): Electric field for Line of sight at a distance equal to the width of City.
Then total electric field over Water1 is calculated. Considering Fresnel parameters, the conditions for reflection and the correction factor, there are four cases: 1) When 1 and 1 are lower than -0.78 and > the total electric field for Water1 is: 2) When 1 and 1 are lower than -0.78 and < or 1 is lower than -0.78 and 1 is higher than -0.78, the total electric field for Water1 is: 3) When 1 and 1 are higher than -0.78 or 1 is higher than -0.78 and 1 is lower than -0.78 and < , the total electric field for Water1 is: 4) When 1 is higher than -0.78 and 1 is lower than -0.78 and > , the total electric field for City is:

Formulation for Water2 Path for M-DTV
In Water 2 the maximum number of rays which arrive to the receiver is nine rays. For the scenario because of the frequency a slab represents the forest. Generally, the receiver is far away from the transmitter, and then diffraction on the left of the receiver can be calculated as grazing diffraction. On the other hand diffraction on the right of the forest will be slope diffraction, it means a huge attenuation of the signal, and therefore, this ray will not contribute to the total electrical field.
First of all the Fresnel parameters are calculated for direct, reflected and diffracted ray on the right. The Fresnel parameter for direct ray in Water2 Path, 2 , is calculated as follows: Fresnel parameter for reflected ray in Water2 path, 2 , is calculated as follows: The second condition for the reflected ray to be added at the total electric field over Water2 is: > 1 , where is the distance between and , illustrated in Fig. 13 for Water2.
The Fresnel parameter for ray diffracted on the right is used to know if there is slope diffraction. When the incidence ray of diffraction is diffracted on the right then, the signal is attenuated for double diffraction and it does not contribute significantly to the total electric field. The equations for 2 + the Fresnel parameter for diffracted ray on the right are: Another parameter to be calculated is , the distance of grazing diffraction. Grazing diffraction means that the incidence angle of diffraction is zero degrees, therefore the incidence ray of diffraction is grazing the 0 surface. In the case of Water2 as forest is not a uniform height, first is calculated, ∆ℎ , the difference height of forest. Then when , is bigger than 1, the incidence diffraction ray is grazing the forest. For example, if dg is 50 m, the incidence diffracted ray is grazing the surface of diffraction (forest) 50 m before it reaches the point of diffraction. The equations to calculate the grazing distance are: where: ℎ : Maximum forest height ℎ : Minimum forest height ′ : Incidence angle of diffraction The incidence angle of diffraction is near zero degrees in Water 2 because the receiver is far away from the transmitter. When the parameter , , distance of grazing diffraction is higher than 1, it means the incidence ray of diffraction is grazing the incidence surface of the obstacle. Furthermore in [26], when the angle is near zero it is considered a grazing diffraction.
Finally, for Water2 there is an environmental correction factor, to introduce the attenuations produced by city and Water1. The equation for a correction factor, 2 ( + 1 ), is: The expressions for total electrical field in Water2, are divided into six cases based on Fresnel parameters and condition of the reflected ray and are detailed next, 1) When 2 , 2 and 2 + are lower than -0.78 and > 1 the total electric field for Water1 is: 2) When 2 , 2 and 2 + are lower than -0.78 and < 1 or 2 and 2 + are lower than -0.78 and 2 is higher than -0.78, the total electric field for Water1 is: 3) When 2 , 2 are lower than -0.78 and 2 + higher than -0.78 or > 1 and > 1 the total electric field for Water1 is: 4) When 2 , 2 are lower than -0.78, 2 + higher than -0.78 or > 1 and < 1 or 2 is lower than -0.78 and 2 and 2 + are higher than -0.78, the total electric field for Water2 is: 5) When 2 and 2 are higher than -0.78 and 2 + lower than -0.78 or 2 is higher than -0.78 and 2 and 2 + are lower than -0.78 and < 1 , the total electric field for Water2 is: 6) When 2 and 2 are higher than -0.78 and 2 + higher than -0.78 or > 1, or 2 is higher than -0.78 and 2 is lower than -0.78 and < 1 and 2 + higher than -0.78 or > 1 the total electric field for Water2 is:

Formulation for City Path for H-DTV
In the case of H-DTV the receiver is on the houses' rooftop then, is considered to have line of sight between the transmitter and the receiver. For the City path the maximum number of rays is three, the total electric field, , considers two cases to be calculated: 1) If the distance, , illustrated in Fig. 13, between the incidence point of reflection over the rooftop, , and j is lower than ′ /2, then reflected ray is added to, , 2) If > ′ /2 , the total electric field for City Path is,

Formulation for Forest1 Path for H-DTV
In Forest 1 and Forest 2 are used maximum 4 rays. For Forest1, the Fresnel parameter is used to know if the reflected ray on water is being diffracted for the house on the border of the river. It is calculated using the following expressions: After, calculating the Fresnel parameter, 1 , then two cases are possible, as detailed next, 1) If 1 < −0.78, total electric field in Forest1, 1 , is: where: 2) If 1 > −0.78, total electric field in Forest1, 1 , is:

Formulation for Forest2 Path for H-DTV
For Forest2, the Fresnel parameter determines if the house on the border of the river is diffracting the reflected ray. The expressions to calculate it are as follows, For Forest2 there are two cases for the total electric field, 2 , described as follows, 1) If 2 < −0.78, where: • ℎ 2 ( ): Reflected field over Water2

Summary of Mixed Path model formulation for M-DTV
This summary has all the equations and conditions to implement Mixed Path model. In the case of the City, as described before, four cases are considered in order to calculate the total electric field. The first case evaluates if the Fresnel parameter is lower than -0.78, then Direct ray and reflected ray are not obstructed and all the eight rays are added to the total electric field. In the second case even when the Fresnel parameter for the reflected ray is lower than -0.78, ℎ is higher than ℎ , and there is not surface of reflection, then in the total electric field the reflected ray is not added. The third case is when the Fresnel parameters are higher than -0.78, then the total electric field is the addition only of diffracted rays. In the fourth case, the Fresnel parameter of direct ray is higher than -0.78 and the Fresnel parameter of reflected ray is lower than -0.78 and ℎ is lower than ℎ , as a consequence the total electric field does not consider only the direct ray.
For Water 1 and Water 2, as the same as in City, the Fresnel parameters are evaluated in order to add or not the direct ray or reflected ray or both in the total electric field.

Final considerations
In this section was presented Mixed Path model formulation. City path uses eigth rays, Water1 uses four rays and a correction factor because of the attenuation caused by City, Water2 uses nine rays and the correction factor because of attenuation caused by City and Water1. The first Fresnel zone has to be unobstructed to add the direct and/or reflected ray in the total electric field. Finally, the Mixed Path model formulation predicts the electric field for mixed City-river-forest-path.

Introduction
The formulation of Mixed Path uses parameters such as the height of antennas, the height of buildings, permittivity and conductivity of the materials, the width of the city, etc.
To know the sensibility of the model some parameters are modified and evaluated.
For Mixed Path are of special interest parameters as the level of the river, electrical constants of forest and transition zone. Since these parameters are part of the contributions of this work for the mixed city-river-forest path not studied before.

Analysis of Mixed Path model parameters
In • ℎ Height of building over ground.
• Width of City.
• ℎ Height of transmitter over ground.
• Incidence angle for air-forest.
First of all, are established the values for calculation of electric field with Mixed Path model, these values are in Table III. Next, the parameters mentioned before are evaluated giving different values.

1) Width of street.
The values of the width of the street are between 6 and 18 meters. In Fig. 17 can be observed the results for a receiver located at a distance of 3km in City. field are lower than for a street of 10 m because the diffraction is stronger for narrow streets than for wide streets. For a street between 10 and 18 meters, the electric field has a soft variation. Then for wide streets, the Electric field is higher than for narrow streets.

2) Angle
This parameter considers a receiver on the center of the street. Then the variation of the angle indicates the position of the receiver in relation with, , the point of diffraction as illustrated in Fig.18 from a top view. The decreasing of the angle indicates that the distance between transmitter and receiver increases and the receiver is getting far from the diffraction point.
Source: Author   Fig. 19. Electric field for the variation of alpha when the receiver is in the same street. For a constant distance between transmitter and receiver, while α has different values, it means that the receiver has different positions, as Fig. 20 illustrates.
In Fig. 21, for angles up 45 degrees different rays reach the receiver, therefore, the signal presents significant increasing and attenuation. However, for angles higher than 45 degrees the signal shows a soft decreasing because only diffracted rays reach the receiver.

1) Height of building over the ground
Three different distances are used to evaluate the influence of the building's height in the received electric field, 3km for a receiver in the city, 4 km for a receiver over Water1 and 8.95 km for a receiver over Water2. Fig. 22 illustrates that the signal decreases as the height of the buildings increases for the three different receivers. The receiver in the City path is surrounded by buildings; therefore, the signal is lower than the signal received in Water1, which considers buildings only on the left of the receiver. This parameter is the total length of the city in the mixed city-river-forest path. Results for the receivers over Water 1 and over Water 2 show that the values of electric field decrease with the width of the City. For Water 1 the receiver is fixed at 4 km from the transmitter, therefore when city width increases it means city is getting near to the receiver located over water, and then the receiver enters in the shadow zone. In the case of Water2 when the City width increases the value of the correction factor increases and electric field is lower than for Water 1. However, when the direct ray exists is not advisable to increase the transmitter height because the distance between the transmitter and receiver increases, therefore, the electric field decreases.
For a receiver in the City in Fig.24, when the transmitter is higher than 300 m, a reflected ray is the principal contributor of the total electric field. The reflected ray for different transmitter heights has similar values because the distances are similar. In some occasions the addition is not in phase, for a transmitter of 400 m, 500 m and 550 m which have lower values. For the receiver of Water1 and Water2, diffracted rays are the principal contributors to the total electric field and for a transmitter of 400m, 500m and 550m the rays are added in opposite phases.

4) Height of receiver antenna
The first receiver is in the city, the second receiver is over Water1 and the third receiver is over Water2. In Fig. 25 for higher receivers is illustrated higher electric field, because of the direct ray. The receiver height values are from 2 to 17 meters, and the height of the buildings is 18 m since the receiver in this model is considered to be lower than the constructions. Some values of the electric field are lower even when the receiver is higher because of the adding in opposite phase of the rays. A receiver over Water1 has higher values than a receiver on the City for some receiver's height because; the diffraction is only on the left of the receiver. According to ABNT 1506, the minimum value of the electric field for Digital television is 54.54 dBuV/m and the maximum value is 111 dBμV/m or more. Then, considering the river level the best receiver antenna height is calculated using an optimization tool.
Furthermore, in the calculation of the total electric field is included the cable loss.
In order to know the best height of the receiver antenna to obtain the highest electric field was used a Genetic Algorithm function from Matlab. The objective function is also implemented in Matlab and uses equations (123) and (124) for total electric field in Forest 1.  First is calculated the best height of the receiver antenna for each river level and 13 results are obtained, after from these results are calculated the best height of the receiver antenna for all the rivers levels. In Fig. 26 the better antennas heights are shown for a fixed receiver at 5 km. The best height of the receiver antenna is 11.07 m. Then a receiver antenna of 11 m has a minimum value of 115 dbuV/m and the cable loss is 3 dB.
Other receiver antennas heights were evaluated, from 1 m to 5 m, we can conclude from For a situation where the receiver is in line of sight with the transmitter, for different distances the height of the antenna is 1 m, as can be observed in Fig. 28. Source: Author 6) Distance over the water A receiver is evaluated moving away from the city. In Fig. 29 the blue line illustrates that at all reception points is added the diffracted ray. In the case of the red line the diffracted ray is not added for distances greater than 0.4 km.
In addition, the arrow in Fig. 29 indicates that for distances greater than 0.8 km the third zone of Fresnel is unobstructed. Therefore, the diffracted ray is not more a significant contributor and the values represented with the blue and the red lines are similar. In other words, after the transition zone, a direct ray and a reflected ray are enough to represent the signal over the river. the variation of the signal is more significant because it is close to the city. In the three remaining cases, the electric field value is almost constant for different river level values.
In Fig. 31, are illustrated the results for a receiver of 10 m height. The electric field values in Fig. 30 are greater than the values in Fig. 29 because of the receiving antenna height. However, only for a distance of 2.08 km when the river decreases between 2.3 m and 3 m, the signal attenuates up to 8 dB. At the distance of 1.78 km, the variation of the electric field strength is of a maximum of 5 dB. The variation in these two distances can be attributed to the fact that they are close to the city. In the other cases, the signal remains constant with the variation of the river level.
Comparing the results of Fig. 27 and Fig. 30, the water level influences the signal variation in up 28 dB for H-DTV, unlike for M-DTV where the maximum variation is 8 dB. In Fig. 32 is illustrated the received signal for a receiver of 4 m height and different values of incidence angle. The width of the city is 1km, the river width is 1km and the width of the forest is 1.980 km. Then the electric field is low for high values on incidence angles.
Furthermore, from Table IV for higher incidence angles lower transmitted angles. Then for an incidence angle close to 90 degrees, the transmitted ray reaches receivers for greater distances. For lower incidences angles the transmitted ray reaches receivers near the forest.

9) Electrical parameters at the forest
In [19] the forest electrical parameters are: for a sparse forest = 0.03 / and = 1.01, for a medium forest = 0.1 / and = 1.1 and for a dense forest = 0.3 / and = 1.3. In Fig. 33 the results are for a receiver located behind the forest and over Water2. For distances up to 9.1 km, the received electric field value is higher for the sparse forest than for the medium and the dense forest. In Figs. 34-36 for the different types of the forest is presented the value of the amplitude of each ray that contributes in the total electric field. Fig. 33. Electrical parameters for forest vs Electric field.

Source: Author
In Fig. 34, the transmitted field, , exists for distances lower than 9.1 km, the amplitude of this ray is 73 dBuV/m, then the total electric field is higher in the first points of reception over water 2 for a sparse forest. After 9.1 km only diffracted rays exist, the diffracted ray, − , and diffracted reflected ray on the water, − ℎ , have significant values however the diffracted reflected ray on the forest, − , has low values that do not contribute to the total electric field. The diffracted ray and diffracted reflected ray on the water, have values that are increasing with the distances because, the values of the diffracted coefficients are higher when − ′ produce values near , it means is near the shadow boundaries. In this case, the incidence angle of diffraction, ′ , has low values around 1°, and has values a little higher than 180°.
For a medium forest in Fig. 35, the , is only in the three first points of reception with lower values than for a sparse forest, 64 dBuV/m. Furthermore, the first points for − and − ℎ , have lower values than for sparse forest, when distances increase the values of the electric field are similar to a sparse forest but still lower. The transmitted electrical field arrives only to one point of reception for a dense forest in Fig. 36. The values of − , are higher than for a medium forest and for distances lower than 9.1 km the addition with the other rays is in phase, then the electric field value is higher than for a medium forest. The values of − ℎ and − , are similar to the values of a medium forest and are the principal contributors to the total electric field after 9.1 km of distance.  Table   V. In Fig. 37 the results are for a receiver located in the city at a distance of 3 km and another receiver located on Water1 at a distance of 4km. The electric field on the city has lower values than electric field over the water because in the city the receiver is surrounded by constructions while the receiver over the water has constructions only in the transition zone.
The difference between heavy concrete and light concrete is 8 dB in the city and 7 dB over water. Then, the electric field is higher when the permittivity has higher values, as the cases of the street and heavy concrete from Table V.   Table VI are analyzed. The receiver is 4m height and it is at a distance of 4km. In Fig. 37, the electric field is practically the same for all types of water with permittivity 80. For ice, the electric field is different in 0.1 dB.

12) Radiation pattern of transmitter and receiver antennas
For Digital television, the polarization of the antennas most of the times is horizontal.
At homes, the common external antenna is a log periodic. In Figs. 39 and 40 are exhibited the radiation patterns of a transmitter antenna and a log periodic antenna respectively.

Results for M-DTV City
First of all, for a receiver located in the city are calculated the results. Fig. 48 illustrates the amplitude of each ray which contributes to the total electric field for Mixed Path model. The lowest electric field is given by Diffracted reflected ray on the street. Then for this case of study the rays which are the principal contributors are the diffracted on the right. The parameters for the calculations are in Table III. Furthermore, measured data were organized in annuli of 200 m to have a uniform scenario.

Radial 1
The transmitter is in the city, and the receiver is over Water1 and over Water2. The parameters used for the calculations are in Table III After the calculation of electric field using Mixed Path model, in Fig. 51, the results are compared with measured data and two models of radio propagation, Okumura Hata and ITU-R P.1546-5. In Water1 for distances up to 2.5 km measured data shows lower values than for bigger distances, this is because it is the transition zone (city-water) and is affected by constructions near the river border.

Radial 2
First of all the amplitude of each ray that contributes in the total electric field are illustrated in Fig. 52.   Fig. 52. Electric field amplitude of each ray for total electric field calculated for Radial 2 over Water1 and Water2.

Source:Author
In contrast with Radial 1 over Water1, Fig. 52 illustrates that the values of − , have a different amplitude in the first points of reception. On the other hand, the values of − ℎ , are higher than − , due to the value of the angles of diffraction. For Water2 only one point of reception receives , then the value of − and − ℎ are the principal contributors, however, − ℎ , is the principal contributor in the first points of reception over Water2.
After showing the amplitude of the principal rays for Mixed Path model, Fig. 53 presents the total electric field . Furthermore, Okumura Hata, ITU-R P. 1546-5 and Mixed Path model are compared. The transition zone is different from Radial 1, then the interaction of the phase of the rays that arrive at the receiver is important. In contrast with Radial 1, the width of the city for Radial 2 is l.4 km. In addition, in Fig. 52

Radial 3
For Radial 3 the distances over Water1 and over Water2 are lower than for Radial 1 and Radial 2. Source:Author Furthermore, the forest is the shortest with 1. 079 km of length, but the trees are taller than for Radial 1 and Radial 2. From Fig. 54, can be observed that the principal rays are and ℎ . For distances up to 2.5 km consider as the transition zone, diffracted rays influence the total electric field. For Water2, the behavior of each ray is similar to Radial 1 and Radial 2, there is only one point of reception for , the − is the principal contributor in the first points of reception. After the addition of − , − ℎ and − forms the total electric field.

Annuli of City-Water1
In order to have a uniform scenario Annuli was used for distances of 200 meters as shown in Fig. 56. The annuli were not made for Water2 because the width of the forest was different for each radial. In Fig. 56   In order to know the attenuation caused by forest, Table XI compares the values of electric field calculated with Mixed Path for the first point behind the forest and the electric field calculated in the city in similar distances. The received electric field behind the forest is around 20 dB lower than electric field received in the city in similar distances.
The value of the electric field is equal to the electric field in free space, for greater distances than 3 the electric field is always smaller than the electric field in free space.
To calculate the break point distance, the receiver in the city for H-DTV is 18 m high and 6 m in Forest1 and Forest2. Therefore, using (129) the classical breakpoint distances are 14.2 km for the city and 4.7 km for Forest1 and Forest2. We can deduce that the city is in the interference zone and Forest1 and Forest2 are no longer in the interference zone. On the other hand, using (130) the break point distance for Forest1 e Forest2 is 10 km and with (131) the break distance point is 15 km. Therefore, they would be in the interference zone. Another explanation for the values greater than free space is the contribution of the diffracted beam on the corner of the building, the same one that is being added constructively to the direct beam.
In Figs. 57 and 58, the first point on Forest1 and Forest2 has a higher value due to the contribution of the reflected ray. In the case of ITU-R P.1546 the values on Forest1 and Forest2 are low and different from Mixed Path, because the recommendation considers grazing diffraction, this explains the low values.
In Fig. 59  Source: Author  In order to incorporate the attenuation caused by City over Water1, in the total electric field is added an environmental correction factor. Measurement data in a mixed path in Belém-Pará validates the environmental correction factor.

Comparison between H-DTV and M-DTV
The transition zone City-Water1 for M-DTV presents attenuation that depends on the height of the buildings in the city. The interaction of diffracted rays from the city and reflected rays over water are significant in the transition zone. After the transition zone, the main rays are the direct ray and reflected ray from the water.
The predicted values for H-DTV show higher values than the estimated values for M-DTV. In the City, the difference is up 19 dB, because H-DTV is calculated considering line of sight.
The electric field calculated for different electrical parameters of forest, show that for the reception points near the forest a sparse forest has higher electric field because of the transmitted ray that reaches the receiver, for medium forest the transmitted ray reaches maximum three reception points and for a dense forest transmitted ray reaches only one reception point. For points that are far from forest, the three types of forests have similar values, for a sparse forest the diffracted reflected signal does not contribute to total electrical field, in the case of medium forest and dense forest diffracted reflected rays from forest contributes to the total electrical field specially in the first points of reception.
The Angle is calculated from a top view to know the location of the receiver, the electric field decreases with the increase of the angle. For angles higher than 45 degrees with a constant distance the variation of the signal is around 4dB.
The increasing of the city width means the decreasing of the signal for a receiver in a constant distance since when the city is closer to the receiver, the city attenuates the signal.
The analysis of receiver and transmitter antennas height shows that signal has a tendency of increasing when the antennas height increases. Only when receiver is in line of sight with the transmitter, increasing the height of the transmitter antenna decreases the strength of the signal because the distance between transmitter and receiver increases.
For incidence angle air-forest, the electric field decreases with the increase of the incident angle. However, not all the incidence angles air-forest produce a ray transmitted from forest to air.
The level of the river affects the signal for a fixed receiver on the border of the Forest1 at a distance of 5 km between transmitter and receiver. The best height of the receiver antenna is 11 m, at this height antenna receives the highest electric field even when the level of the river changes. On the other hand, for a receiver at 5 km of distance, the signal is attenuated more than 30 dB for 5 m height antenna when the level of the river decreases 1.6 m.
Furthermore, for different distances between 1km and 5 km, the signal is stable for an antenna of 1 m height, with a minimum electric field around 109 db V/m. The value of 109 db V/m or greater is the maximum electric field for Digital TV according to ABNT 1506. Walfish-Ikegami (10.6 dB). However, Walfish Ikegami was evaluated only in the city.
Finally, the Mixed Path model has the best agreement for a mixed path in the Amazon region.
For future works, in order to validate the Mixed Path model for H-DTV, different measurement campaigns will be carried out in Belém of Pará considering the different positions for the reception antenna over the rooftop of the buildings. Furthermore, knowing that Belem has at the center of the city high buildings between 20 and 30 floors, in the total electric field for H-DTV a diffracted ray can be added.
Additionally, Mixed Path model can add more rays in order to predict the electric field for higher frequencies, for example, 3.5 GHz. The goal is to calculate the dense forest attenuation since 3.5 GHz is the frequency for the fifth generation of mobile communications for outdoor scenarios. Then a tool to predict the electric field is fundamental in order to look for solutions to reduce attenuation and have good coverage. Then measurement campaigns near a dense forest in large and small scale will be carried out in Belem of Pará.