Algorithm for Computing N Knife Edge Diffraction Loss Using Epstein-Peterson Method

In this paper, algorithm for computing N knife edge diffraction loss using Epstein-Peterson method and International Telecommunication Union (ITU) knife edge diffraction loss approximation model is presented. Requisite mathematical expressions for the computations are first presented before the algorithm is presented. Then sample 10 knife edge obstructions are used to demonstrate the application of the algorithm for L-band 1 GHz microwave signal. The results showed that for the 10 knife edge obstructions spread over a path length of 36 km the maximum virtual hop single knife edge diffraction loss is 8.054711 dB and it occurred in virtual hop j =10 which has the highest diffraction parameter of 0.233333. However, the virtual hop j =10 has line of site (LOS) clearance height of 2.333333 m whereas the highest LOS clearance is 3.454545 m and it occurred in virtual hop j =6. The minimum virtual hop single knife edge diffraction loss is 6.109884 dB and it occurred in virtual hop j =3 which has the lowest diffraction parameter of 0.008909 as well as the lowest LOS clearance height of 0.142857 m. The algorithm is useful for development of automated multiple knife edge diffraction loss system based on Epstein-Peterson method and ITU knife edge diffraction loss approximation model.


Introduction
Propagation paths of line of sight microwave signals are in many cases obstructed by obstacles [1,2]. When a single isolated obstruction is considered, such as hill or building, such obstruction is approximated as knife edge obstruction [3,4]. In that case, single knife edge obstruction approach is used to estimate the diffraction loss that signals may suffer as they encounter the knife edge obstruction.
In reality, there are always two or more obtrusions that are often located in the signal path. In such case, multiple knife edge diffraction methods are used to determine the expected diffraction loss [5,6]. Studies have shown that computation of multiple knife edge diffraction loss is complex and it complexity increases especially as the number of knife edge obstructions considered increases [7,8]. As such, available studies limit the computation to two or three obstructions. In this paper, an algorithm that can be used to determine the diffraction loss of any number of knife edge obstruction is presented. The multiple knife edge computation is based on the Epstein-Peterson method [9][10][11][12][13] whereas the International Telecommunication Union Recommendations (ITU-R) P 526-13 knife edge diffraction loss approximation model [14][15][16][17] is used to determine the diffraction loss for any diffraction parameter obtained for each of the knife edge obstruction. Sample 10 knife edge obstructions are used to demonstrate the applicability of the algorithm. designated as N+1. Each of the N obstructions blocks the line of sight and consitutes an edge that will cause diffraction loss and also introduces a virtual hop in the multiple edge diffraction loss analysis. Each virtual hop has one knife edge that causes diffraction and either one or two other knife edge obstructions that serve as either the virtual transmitter or the virtual receiver for a given virtaul hop. In figure 1   In figure 1, H is the height of the obstruction from the sea level. Idealy, H takes into account the earth bulge, the elevation and the obstruction height measured from the ground level. Again, j =0 referes to the receiver whereas j =N+1 referes to the transmitter. J = 1 to J = N referes to the obstructions 1, 2, 3,…N respectively. According to Epstein-Peterson multiple knife edge diffraction loss method, for any given hop j, the clearance height to its LOS is given The knife-edge diffraction parameter for any hop j is given For any given diffraction parameter, v the knife-edge diffraction loss, A according to ITU-R 526 is given as; where A is in dB Then, in respect of knife-edge diffraction loss for any hop j with diffraction parameter, v " , the knife-edge diffraction loss is denoted as A " , where ITU approximation model for A " is given as; where A " is in dB According to the Epstein-Peterson multiple diffraction loss method, the effective diffraction loss for all the m hops is given as;

The Procedure For Computing N Knife Edge Diffraction Loss Using Epstein-Peterson Method
The Procedure for computing N knife edge diffraction loss using Epstein-Peterson method and the ITU knife edge diffraction loss approximation model is as follows: Step 1: For j = 0 to N +1 obtain height H (j) of obstruction, where j includes the transmitter with j=0, the receiver with j =N +1and the N obstructions with j =1 to N.
Step 2: For j=1 To N +1 obtain the distance d (j) of obstruction (j) from obstruction (j-1) Step 3: For j = 1 to N compute the LOS clearance heights Step 5: For j = 1 to N compute the knife-edge diffraction loss (A ) for each v (Use 4) Step 8: A = A + A + A + ⋯ + A B + A B (Use Eq 5)

Numerical Example and Discussion of Results
The numerical example is for 10 knife edge obstructions located in the path of 1 GHz microwave signal. In this case, N = 10. The height, H (j) of the obstructions for j = 0 to j = N +1 are given in Table 1 while Table 2 shows the distance d (j) of obstruction (j) from obstruction (j-1) for j=1 to j= N+1. The results of the computations are presented according to the steps given in the algorithm. In all, for the given 10 obstructions, the total diffraction loss is 67.35065 dB. The path length is the sum of all the d (j) for j=1 to N+1. From Table 2, the path length is 36 km.
Result for Step 1: The height H (j) of obstruction for j = 0 to N +1, where j includes the transmitter with j=0, the receiver with j =N +1and the N obstructions with j =1 to N. Result for Step 2: The distance d (j) of obstruction (j) from obstruction (j-1) for j=1 to N+1.

Conclusion
Algorithm for computing N knife edge diffraction loss using Epstein-Peterson method and ITU knife edge diffraction loss approximation model is presented. The mathematical expressions required for the computations are first presented before the algorithm. Then sample 10 knife edge obstructions are used to demonstrate the application of the algorithm for L-band 1 GHz microwave signal.