Knife Edge Diffraction—Deriving K and w

To simplify the analysis of diffraction, a novel averaging technique called “Chess Board filtering” is used to preferentially smooth the terrain (as described in the implementation section). Once this filter has been applied, the terrain, for the purposes of diffraction will appear to have only two elevation values, which alternate between each other as one travels from one 30m by 30m area to another. In other words the site will appear a chess board, with black areas at one elevation and white areas at another elevation.

           Each cell site is approximately a million square meters in area. Therefore, each 30m by 30m area that has a higher of the two elevation values can be assumed to be a “knife edge”. A wave may or may not experiences knife edge diffraction every time it is obstructed by this edge. Figure 3 represents one such diffraction (cite source). The ratio of the electric field at point B to the electric field at point A is given by equation 3 (cite source).

 

 

 

 

 

 

                                 Figure 3. Knife Edge Diffraction

 

 

 

 

 

 

           Equation 3. Relating E-fields (Knife Edge Diffraction)

 

Where v, the lower limit of the integral is given by equation X.

 

 

 

 

 

                      Equation 4. Lower limit of Integral

 

 

 

Under the conditions that d1, and d2 >> u and d1, d2 >> λ

 

When u is zero, by assumption 1, the power gained is by diffraction is zero (but the diffraction is not necessarily zero). By equation 4, the integral’s lower limit v is also zero. So let equation 3 evaluated at v = 0 be ‘Ia’ and corresponds to an index of 1 with respect to power gain. In other words, when v is zero, the power at point B will just be the power at point A multiplied by 1 (zero power gain (in dBm). When v is not zero (which is the case in our model) let’s say equation 3 evaluates to ‘Ib’. Therefore when v is not equal to zero, the power at point B will be the power at point A multiplied by (Ia/Ib)² . The ratio is squared because power is proportional to the square of the electric field. This multiplier is denoted as K.

 

=> K was derived to be (Ia/Ib)², characteristic of a given cell site.-> (i)

 

Note: d1 and d2 are equal in our case and they represent the distance between the centers of two adjacent 30m x 30m areas. (This area is an approximation. When the model is programmed however, the actual area is used)

 

The exponent ‘w’ is designed to be, for a point, the approximate number of diffractions a wave experiences between the antenna and the point. As mentioned earlier, a wave may or may not experience diffraction at every alternate square. Since there are about a million squares in total for each cell site, there would be no harm in approximating that the wave experiences knife edge diffraction once every 80 squares. This number was obtained by trial and error while modeling. Also a wave does not necessarily traverse perpendicular to the knife edge (i.e. it may not travel exactly parallel to the x or exactly parallel to the y axes). For our model however, diffraction is only considered for a wave perpendicular to the knife edge. Based on the above assumptions, w can be defined as

 

 w =( (# of squares in the x direction from source to point) + (# of squares in the y direction from source to point))/80                       ->(ii)