International Journal of Systems Science and Applied Mathematics 2017; 2(2): 51-56 http://www.sciencepublishinggroup.com/j/ijssam doi: 10.11648/j.ijssam.20170202.11 Empirical Validation Of Effective Earth Radius Adjustment Factors For Earth Bulge and Diffraction Loss Parameters Computation Eduediuyai Dan, Constance Kalu, Ogungbemi Emmanuel Oluropo Department of Electrical/Electronic and Computer Engineering, University of Uyo, Uyo, Nigeria Email address: [email protected] (C. Kalu) To cite this article: Eduediuyai Dan, Constance Kalu, Ogungbemi Emmanuel Oluropo. Empirical Validation Of Effective Earth Radius Adjustment Factors For Earth Bulge and Diffraction Loss Parameters Computation. International Journal of Systems Science and Applied Mathematics. Vol. 2, No. 2, 2017, pp. 51-56. doi: 10.11648/j.ijssam.20170202.11 Received: January 9, 2017; Accepted: February 25, 2017; Published: March 25, 2017 Abstract: In this paper, the effect of effective earth radius adjustment factors (k-adjustment factors) on various parameters associated with single knife edge diffraction loss is studied. The parameters considered are, the earth bulge, Fresnel-Kirchoff diffraction parameter and the number of Fresnel zones that are partially or fully blocked by obstruction in the signal path. The k-adjustment factors analytical expressions are derived and then validated using empirical elevation profile data for line-of-sight (LOS) communication link between Eket and Akwa Ibom state University. Also, k-factors considered in this paper are k1 = 0.5, k2 = 0.9 and k3= 1.333. In all, the results show that when the value of any of the three parameters is known at a given k-factor, k1, then the value of that parameter can be determined at any other k-factor, k2 by adding the k1-to-tk2 adjustment factor of that parameter to the value of the parameter at k1. The result is essential is evaluating the influence of variations in effective earth radius factor on the parameters associated with single knife edge diffraction loss. Keywords: K-Factor, Diffraction Parameter, Diffraction Loss, Effective Earth Radius, K-Adjustment Factors 1. Introduction In the communication industry, Fresnel geometry is usually density with altitude [11-12]. In order to ensure clear line of used in the analysis of line-of-sight (LOS) communication sight the curvature of the earth and atmospheric refraction link design. The analysis utilizes the terrain elevation profile, effect must be considered. In LOS communication link design, earth bulge profile and other network and terrain parameters to the effective earth radius K-factor takes into account the determine the suitable transmitter and receiver antenna mast curvature of the earth and atmospheric refractivity which height that will ensure clear line of sight between the bends the beam either up or down [13-15]. Effective earth transmitter and the receiver [1-6]. radius is the radius of a hypothetical spherical Earth, without Further, the Fresnel geometry is also used in the analysis of atmosphere, for which propagation paths follow straight lines, the knife edge diffraction loss for such LOS link in situations the heights and ground distances being the same as for the where obstructions intrude into the key Fresnel zones of the actual Earth in an atmosphere with a constant vertical gradient signal path. In that case, Fresnel diffraction parameter is of refractivity [15]. K-factor is the ration of effective earth determined from the knowledge of the signal frequency, the radius and true earth radius [14-16]. In effect, the bending of obstruction clearance height and the distance of the the beam either up or down makes it appear as though the obstruction from the transmitter and the receiver. radius of the earth is less than or greater than the true radius. A Importantly, the atmosphere is not homogeneous and K-factor of >1.0 means the beam is bent towards the earth. usually refracts or bends radio waves passing through it [7-10]. Consequently, for LOS link design, the effective earth radius Atmospheric refraction is simply the deviation of light or factor (k- factor) must be set carefully to optimize its other electromagnetic waves from a straight line as it performance. transverses the atmosphere as a result of the variation of air Particularly, in Fresnel geometry analysis of LOS links, 52 Eduediuyai Dan et al.: Empirical Validation Of Effective Earth Radius Adjustment Factors For Earth Bulge and Diffraction Loss Parameters Computation k-factor directly affects the earth bulge and therefore affects bulge. other link parameters that include the earth bulge. Among these parameters are the obstruction clearance height, the 2.2. The Path Elevation Profile Fresnel diffraction parameter, knife edge diffraction loss and The path elevation profile is represented by and other parameters that depend on diffraction parameter. In this where; () () paper, the focus is to use terrain elevation profile of a given is the elevation taken at point x, where x = 1,2,3,…, LOS link to validate the analytical expressions for evaluating () is the number of elevation points in the topographic ! the variation of the LOS link and knife edge diffraction profile;N parameters on the effective earth radius k-factor. Particularly, the effective earth radius adjustment factors (k-adjustment 2.3. The Effective Obstruction Height, factors) for the following parameters are examined; Earth +(,) bulge, obstruction height, Fresnel-Kirchoff diffraction The effective obstruction clearance height, is the parameter and the Fresnel zones in which the tip of the single height (in meters) from the tip of the obstruction at- ()location x knife edge obstruction lies. to a point on the line of sight at location x, where x is between the transmitter and the receiver. is given as; -() 2. Methodology (6) -() -.() + () + () − /01() 2.1. The Earth Bulge Where is the height of obstruction x from the The earth bulge at a distance from the transmitter and ground -.() is the overall height (in meters) of a point on the line distance from the receiver is() given as [17, 18, 19]: of /sight01() at location x between the transmitter and the receiver () where point x is a distance of from the transmitter. The for x = 1,2,3,…, . (1) ()() equation for the line of sight that() passes through the point () .∗ N Where ( , ) is given as: earth's curvature at the point x between the () /01() (7) transmitter() and the receiver (m) 2 32 is effective earth radius factor /01() ) * () + / Where; is the number of elevation points in the topographic is the effective transmitter antenna heights which is also profileN the overall height (in meters) of the transmitter antenna, = distance between the point and the transmitter (km) / including the elevation measured from the sea level and the dwhere() x = 1,2,3,…, . earth bulge = distance between N the point and the receiver (km) is the effective receiver antenna heights which is also where() x = 1,2,3,…, . the/ overall height (in meters) of the receiver antenna, is the distance (in N meters) between the transmitter and including the elevation measured from the sea level and the the receiver. earth bulge and are given as follow; (2) / / () + () = + + (8) (3) / - () − () = + + (9) The transmitter is located at x = 0. Hence, = . / - Also, the receiver is located at x = . Therefore, =() = Where; d. At the transmitter, = 0,! = d, hence, (") is the height (in meters) of the transmitter antenna mast measured- from the ground . At the receiver,() () = d, = 0, hence,() ()() is the height (in meters) of the receiver antenna mast () () .∗ measured from the ground ()() . So, . - is the elevation at the transmitter location. ()The earth .∗ bulge for two ()effective () earth radius factors, () and are related as follows; is the elevation at the receiver location. is the() earth bulge at the transmitter. (4) E5 is the earth bulge at the receiver. #$(,&') ( EThe56 effective obstruction height, can be expressed #$(,&() ' Hence, with respect to location, x and effective-() earth radius factor, k as follows; (5) ( = (10) (,') (,() )'* -(,7) -() -.() + () + (,7) − /01() ( is the effective earth radius scaling factor for earth The effective obstruction height, for two effective )'* -() International Journal of Systems Science and Applied Mathematics 2017; 2(2): 51-56 53 earth radius factors, and are related as follows; ' R (17) OPQ ) * (11) ( Furthermore, can be expressed in terms of distance, x (,' ,( ,7 - - )' 1* and the effectiveO earthPQ radius factor, k as . Then, for Hence, is the effective earth radius ( two k-factor, and , the are relatedOPQ,7 as follows; , adjustment factor for)' the 1*effective obstruction height. OPQ,7 ' (18) 2.4. The Fresnel-Kirchoff Diffraction Parameter S,&(< ST&(,&' OPQ,' The Fresnel-Kirchoff diffraction parameter ( ) at any ' S,&(ST&(,&'<ST&(,&' (19) given location x between the transmitter and the receiver is ' ( 9 OPQ, OPQ, U V given as [17, 18, 19]: ' is the k-factor adjustment S,&(ST&(,&'<ST&(,&' W X factor which can be represented as . Hence; < (12) OPQK(,' 9 - :; = > (20) Where OPQ,' OPQ,( OPQK(,' is effective obstruction height which is the height (in 3. The Results and Discussions meters)- from the tip of the obstruction at location x to a point on the line of sight at location x, where x is between the Elevation data for a LOS link between Eket and Akwa Ibom transmitter and the receiver. state University are used along with mathematical expressions λ is the wavelength of the