Mathematical Model and Algorithm for Determination of Minimum Antenna Mast Height for Terrestrial Line of Sight Microwave Link with Zero Path Inclination

In this paper, mathematical model and algorithm for determination of minimum antenna mast height for terrestrial line of sight microwave link with zero path inclination is presented. The mathematical expressions developed are used for computing relevant link parameters while the algorithm gives the procedure for using the mathematical expressions for computing the minimum antenna mast heights. Sample 40km 10GHz Ku-band microwave link is used to demonstrate the application of the model and algorithm; in this case, the transmitter is located at longitude 7.711747 and latitude 5.178536 and the receiver is located at longitude 8.039903and latitude 5.055223. The link is required to make a minimum of 100% clearance with respect to the Fresnel zone 1. The results show that the transmitter and receiver antenna are at the same line of sight height of 158.7 m whereas the transmitter antenna mast height is 68.8 m while the receiver antenna mast height is 109.7m. Also, the maximum obstruction height of 128.58m occurred at a distance of 14306.98m from the transmitter with percentage clearance of 100% in respect of Fresnel zone 1. The result shows that the model can be used to ensure that the specified percentage clearance is achieved through the application of the models presented in this paper.


Introduction
Microwave signals travel in a straight line [1,2]. Consequently, they are used for Line Of Site (LOS) communication. Researches however, have found that microwave signal do not travel in a pencil tip straight line. Rather, the signal spread from the transmitter and terminate at the receiver forming an ellipsoidal shape that is described by Fresnel and modeled as Fresnel zones [3,4,5,6]. Also, secondary radioclimatic parameter called effective earth radius factor (k-factor) affect the signal path [8][9][10]. At different values of k-factor the signal can bend away from the earth, towards the earth or it can travel in a straight line with respect to the earth surface. Particularly, the effective earth radius factor is dependent on the three primary radioclimatic parameters, namely, temperature, pressure and relative humidity. The effect of the effective earth radius factor is modeled using the concept of earth bulge [11]. Importantly, in LOS communication system, the transmitter antenna and the receiver antenna are installed at a height that will ensure that no obstacle obstruct the LOS between the (straight line joining) transmitter and the receiver. In order to ensure that the required LOS clearance is achieved, during the design of the LOS communication link the Fresnel Zone clearance and the earth bulge are taken into consideration along with the elevation profile of the terrain. Also, the path inclination, which is the angle subtended by the LOS with the horizontal is considered in the determination of the antenna mast height. In the case where the path inclination is zero, it means that the transmitter and the receiver antenna heights are equal. However, that does not mean that the transmitter mast and the receiver antenna mast heights are equal.
In any case, there are several approaches that can be used to determine the minimum transmitter mast and the receiver antenna mast heights when the path inclination is zero. In this paper, one of such methods is presented based on a given Fresnel geometry for LOS link. The method assumes that the required LOS percentage clearance with respect to Fresnel zone 1 is specified along with the expected obstruction height. Also, the path elevation profile is required as input data for the model. The requisite mathematical expressions and algorithm for the computation of the minimum transmitter mast and the receiver antenna mast heights are presented.

Fresnel Geometry for the Line of Sight Link
The Fresnel geometry for the line of sight link is given in figure 1. In the model, it is assumed that the transmitter has the lower antenna height, especially, when the two antenna heights are not equal. However, if the transmitter antenna height is higher than the receiver antenna height, then the analysis still holds by swapping the receiver subscripts with that of the transmitter.

The Elevation Profile
The model make use of the link elevation profile which is the set of data on the elevation at various points between the transmitter and the receiver. The terrain elevation profile provides a number of elevation points and their distance from the transmitter and the receiver. Each elevation point consist of the elevation height and the distance of the point from the transmitter or distance from the receiver. Let be the number of elevation points taken from the transmitter location to the receiver location. Also, let ( ) be the elevation at point x, where x = 1,2,3,…, ; let ( ) be the distance of location x from the transmitter; let ( ) be the distance of location x from the receiver, where x = 1,2,3,…, and let be the distance (in meters) between the transmitter and the receiver. Then, Since the transmitter is taken as the reference point for the measurements, then, the transmitter is located at x = 0 and the receiver is located at x = . Therefore, is the elevation at the transmitter location where x = 0, hence, = ( ) is the elevation at the receiver location where x = , hence, = Height for Terrestrial Line of Sight Microwave Link with Zero Path Inclination

The Earth Bulge
Earth bulge is the height an obstruction is raised higher in elevation (into the path) owing to earth curvature. Earth bulge is given as [12]; Where ( ) is the height (in meters) of the earth bulge at location x between the transmitter and the receiver; is the height (in meters) of the earth bulge at the transmitter mast location; is the height (in meters) of the earth bulge at the receiver mast location; ( ) and ( ) are as defined earlier. For Line of Sight (LOS) point-to-point links design K-factor of 4/3 is often used. At the transmitter, ( ) = 0, hence, Similarly, at the receiver, ( ) = 0, hence, In essence, at the transmitter and at the receiver, the earth bulge is zero.

Radius of the Fresnel Zones
In many cases obstructions do exist in the signal path. For acceptable clear line of sight, at least 60% clearance is required in the first Fresnel zone. So, obstructions in the signal path must maintain a clearance height that is at least 60% of the radius of the first Fresnel none. Theoretically, there are infinite number of Fresnel zones in any LOS link. Let λ be the wavelength of the radio wave; let c be the speed of the radio wave (where c = 3x10 ) */, and let f be the frequency of the radio wave in Hz, then, the radius of the nth Fresnel zone (-(.,0) ) at location x is given as ] [13][14][15]; ; for n =1,2,3,… and ( ) >> λ in metres is given as;

Obstruction Location and Height
In the model, the reference line for measuring the earth bulge is the chord (line segment) that joins the sea level at the transmitter and the sea level at the receiver. In the model of

Determination of Minimum LOS Height at Location x
For the specified percentage clearance : ;(.) , the minimum LOS height at location x is given as H HIJ(D) , where; for x = S TUVWX

Determination of the Minimum Antenna Height When the Path Inclination Is Zero, (_`= _ a )
Let be the overall height (in meters) of the transmitter antenna, including the elevation measured from the sea level and the earth bulge at the transmitter. Also, let be the overall height (in meters) of the receiver antenna, including the elevation measured from the sea level and the earth bulge at the receiver. Let path inclination be denoted as b c where; b c = |e fe | (15) When the path inclination is zero, then, = The transmitter and receiver antenna mast heights are selected so that the resultant overall heights of the antennas, namely, and will ensure clear line of sight between the transmitter and the receiver antennas.

Results and Discussion
The path profile used is obtained using online Geocontext-Profiler -an online topographic software available at http://www.geocontext.org/publ/2010/04/profiler/en/.
The path profile data are taken for a transmitter located at longitude 7.711747 and latitude 5.178536 and the receiver located at longitude 8.039903and latitude 5.055223. The elevation profile in Table 1 is plotted in figure 2. In Table 1 and figure 2, there is a total of 512 elevation data points and the link is 40,000 km long. The link's maximum elevation of 98.3 meters occurred at 11,415.1 meters from the transmitter and the link's minimum elevation of 10.2 meters occurred at a distance of 33,788.8 meters from the transmitter.  In Table 2 the maximum obstruction height of 128.58 m occurred at a distance of 14306.98m from the transmitter. The Percentage Clearance of 100% is achieved at the maximum obstruction point. This is equal to the 100% clearance given in the link design specification. In Table 3, the transmitter and receiver antenna are at the same line of sight height of 158.7 m. However, the transmitter antenna mast height is 68.8 m while the receiver antenna mast height is 109.7m. Maximum earth bulge of 22.2 occurred at the middle of the like at a distance of 19405.74 m from the transmitter.

Conclusion
Mathematical model and algorithm for determination of minimum antenna mast height for terrestrial line of sight microwave link with zero path inclination is presented. The mathematical expressions developed are used for computing relevant link parameters while the algorithm gives the procedure for using the mathematical expressions for computing the minimum antenna mast heights. Sample Ku-band microwave link is used to demonstrate the application of the model and algorithm. The result shows that the model can be used to ensure that the specified percentage clearance is achieved through the application of the models presented in this paper.