Shibuya Method and Modified ITU Knife Edge Diffraction Loss Model for Computing N Knife Edge Diffraction Loss

In this paper, algorithm for applying Shibuya multiple knife edge diffraction method and modified ITU-R P 526-13 knife edge diffraction loss approximation model are presented. Particularly, in this paper, algorithm for using the two models for computing N knife edge diffraction loss 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 C-band 6 GHz microwave link. The results showed that for the 10 knife edge obstructions spread over a path the maximum virtual hop single knife edge diffraction loss is 14.97452dB and it occurred in virtual hop j =6 which has the highest diffraction parameter of 1.027072 and the highest line of site (LOS) clearance height of 8.480769m. The minimum virtual hop single knife edge diffraction loss is 7.881902 dB and it occurred in virtual hop j =9 which has the lowest diffraction parameter of 0.114761 as well as the lowest LOS clearance height of 0.628571m. The algorithm is useful for development of automated multiple knife edge diffraction loss system based on Shibuya method and the modified ITU-R P 526-13 knife edge diffraction loss approximation model.


Introduction
Diffraction loss is one of the key components of pathloss that is udsed in link budget for line of sight (LOS) microwave link [1][2][3][4][5]. Diffraction occurs when wireless signal encouter obstacle in its path [7][8][9][10][11]. In such case, the signal bend and hence move round the obstacle to the receiver. The diffracted signal experiences loss in signal strenght which is reffered to as diffraction loss.
Huygens-Fresnel principle is used to explain the diffraction concept [11][12][13]. Particularly, in order to simplify the analysis of diffraction loss, an isolated obstruction like hill or building can be considered as a knife edge obstruction [14][15][16]. When there are two or more of such knife edge obstructions, then multiple knife edge diffraction loss methods can be employed to determine the effective diffraction loss of all the knife edge obstructions [17].
Available studies show that computation of multiple knife edge diffraction is quite complex [18][19][20]. The complexity increases with increasing number of obstructions considered. As such, most studies limit the multiple knife edge computation to three obstructions. In this paper, algorithm is presented which can be used to compute diffraction loss for any number of knife edge obstructions. The algorithm is based on the use of Shibuya multiple knife edge diffraction method and the modified ITU-R P 526-13 knife edge diffraction loss approximation model are presented. Sample 10 knife edge obstructions are used to demonstrate the applicability of the algorithm.

Methodology
Present studies on multiple knife edge diffraction loss computation limit the number of obstructions considered to a maximum of three. This is due to the fact that complexity of the computation increases so much as the number of obstructions increases. This paper focuses on presenting a method computing multiple knife edge diffraction loss where as muany as ten obstructions are considered. The computation is based on the Shibuya Multiple knife edge diffraction loss method. The mathematical expressions are presented for N-knife edge obstruction. The N knife edge obstructions with n =1,2,3,…,N-1,N is shown in Figure 1. The transmiter is denoted with N = 0 and the receiver is designated as N+1. In the computation, each of the N obstructions gave rise to a virtual hop which resulted in a knife edge diffraction loss. The overall diffraction loss, according to the Shibuya method is the sum of the diffraction loss computed for each of the N virtual hops. Accordingly, 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.
Shibuya method relies on the assumption that the ray grazing the obstacles at edge H and H generates a fictitious transmitter E [19][20][21]. The procedure for determination of the attenuation due to the diffraction by multiple knife edges is the same as in the Epstein-Peterson method with the difference however that the transmitter E is replaced here by a fictitious transmitter (Shibuya 1983). According to Shibuya multiple knife edge diffraction loss method, for any given hop j, the clearance height to its LOS is given as h where [19][20][21]; The transmitter height in hop j can be denoted H & , where; The knife-edge diffraction parameter for any hop j is given as v where [19][20][21] For any given diffraction parameter, v the knife-edge diffraction loss, A according to ITU-R P 526-13 model is given as [22]; 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; According to the Shibuya multiple diffraction loss method, the effective diffraction loss for all the m hops is given as; The original ITU-R P 526-13 knife edge diffraction loss approximation model is modified by replacing it with equivalent piecewise model that consists of linear function and linear -log functions without radical terms. The modified ITU knife edge diffraction loss approximation model is given as; Where U and W are constants and A (7) is a function of diffraction parameter, v. The values of U, W and function A(7) are given in 1. Again, for Shibuya method the effective multiple knife edge diffraction loss, A(0, 7) is given as; Let _ be the number of knife edges with diffraction parameter (7 ) values in the range −0.57 < 7 < 0. Let _ be the number of knife edges with diffraction parameter (7 ) values in the range 0 ≤ 7 < 1.414214 . Let _` be the number of knife edges with diffraction parameter (7 ) values in the range 1.414214 ≤ 7 < 2.828427 . Let _ a be the number of knife edges with diffraction parameter (7 ) values in the range 7 ≥ 2.828427 Where For all the _ knife edge obstructions in the range −0.57 < 7 < 0 , the total diffraction loss is denoted as A (0, 7) where; A (0, 7) = _ (Z ) + Y 5∑ 5A 7 9 Similarly, for all the _ knife edge obstructions in the range 0 ≤ 7 < 1.414214, the total diffraction loss is denoted as A (0, 7) A (0, 7) = _ (Z ) + Y 5∑ 5A 7 9 C^ C 9 (16) For all the _` knife edge obstructions in the range 1.414214 ≤ 7 < 2.828427, the total diffraction loss is denoted as A`(0, 7) For all the _ a knife edge obstructions in the range 7 ≥ 2.828427 , the total diffraction loss is denoted as A a (0, 7) A a (0, 7) = _ a (Z a ) + Y a 5∑ 5A a 7 9 A a (0, 7) = _ a (Z a ) + Y a mLN ∏ 7 C^a C n (24) Therefore, the effective diffraction loss by the multiple knife edeg diffracting obstructions 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 modified 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.

Numerical Example and Discussion of Results
Ten (10) knife edge obstructions located in a 6 GHz C-band microwave link is used for the numerical example. In this case, N = 10. The height, H(j) of the obstructions for j = 0 to j = N +1 are given in Table 2 while Table 3 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 92.15261 dB.
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.  6  d6  6  7  d7  5  8  d8  4  9  d9  3  10  d10  2  11  d11  1  d  36 Result for Step 3: The LOS clearance heights h = h &s tu ^( ) for 1 to N. The results are given in Table 4.

Result for
Step 4: For j = 1 to N compute the knife-edge diffraction parameter (v ) for each h .The results are given in Table 5.

Result for
Step 5: For j = 1 to N compute the knife-edge diffraction loss (A ) for each v .The results are given in Table  6. The minimum virtual hop single knife edge diffraction loss is 7.881902 dB and it occurred in virtual hop j =9 which has the lowest diffraction parameter of 0.114761 as well as the lowest LOS clearance height of 0.628571m.

Conclusion
Algorithm for computing N knife edge diffraction loss using Shibuya method and modified ITU-R P 526-13 knife edge diffraction loss approximation model is presented. The mathematical expressions required for the computations are first presented before the algorithm. Then 10 knife edge obstructions located in a 6 GHz C-band microwave link is used to demonstrate the application of the algorithm.