Walfisch- Bertoni Model  


        Walfisch –Bertoni model, called also as diffracting screens model [1], is a semi -deterministic model valid for situations uniform building heights and spacing. The model approach the buildings as absorbing diffracting edges and finds the field strength at low grazing angles at what field is settled to a value. Further simplification is made by assuming an elevated fixed antenna achieved by using the local plane wave approximation to calculate the influence of buildings on field strength on the spherical wave radiation by elevated antenna.

            (1)
 
            The model firstly finds the amplitude Q(a) of the field at roof tops due to a plane wave of unit amplitude incident at the glancing angle a on a number of buildings. It is found that for large number of buildings, Q(a) settles to a constant value. Then, this value is multiplied by following factor (1) to find the field amplitude at the mobile.


 


 (2)
 
where ho is the average height of the buildings in meters , b is center to center distance between two successive buildings in meters  and g is given by following formula 2. and g and a are in radians.


 


Furthermore, formula 1 is simplified to 1/(g-a) by assuming that a is small compared to g. 


 


                                       Figure 1[1] Geometry for Walfisch-Bertoni Model

 

The geometry for the model is given in figure above. From the figure it is easily found that

               (3)
 
 



sina=H/R where H =hb-ho. For small a values, this expression gives a=H/R, by the inclusion of earth’s curvature effect yields

where Re is effective earth radius and Re=8.5x103 km.

 


Dependence of the field amplitude on parameter aÖ(b/l) is found by calculating the settled field amplitude for values of aÖ(b/l) from 0.01 to 1.0. The obtained figure is given below 2.

 


 


                                              Figure 2[1] Dependence of the settled field Q on the parameter aÖ(b/l) with a in radians

            (4)
 

Solid line is used as a fit to the settled field, thus with a in radians following formula is found for amplitude of field strength:

This is valid for 0.02<aÖ(b/l)<0.5.

By multiplying equation simplified version of 1 with 3 yields field strength value at mobile unit with respect to one unit field strength radiated by transmitter. Decibel expression of the multiplication yields excess path loss Lex formula proposed by the model.

 

              (5)
 

 


 


Summation of Lex and free space loss gives the total path loss Lo

(6)
 
 


               Lo=Lex+LFS

 

A comparison of the model with measurements is done in [1]. Figure 3 shows a comparison of the model with measurements carried out at Philadelphia for different H values at 820 MHz. In measurement data although range dependence is about 36.8 dB, model gives 38 dB range dependence, which is a sufficient result.   

 

 

 


 

 

 


Figure 3[1] Comparison of  sector-averaged signal strength for various transmitter sites, with theoretical predictions (solid lines). Signal level is in dBm, range in miles, H in meters.

 

Also, [1] gives a comparison of the model with okumura curves by the sight of a. Plot of excess attenuation as a function of the a  is given figure 4.


Figure 4 [1] Comparison of excess path loss found from theoretical model (solid curve) with measurement of Okumura plotted as a function of the a at f=922MHz and transmitter heights of 45 m and 140 m.

 


The plot shows that excess attenuation for both transmitting antenna heights agree with in 3 dB. This agreement means that excess path loss is a function of a rather than of R and H.

 

 

 

 

 

 

A comparison of the model with Hata model is given by the following figure taken by using Wireless Simulator output.

Terrain Parameters:

Average Width: 73.8 m

Average Building Height: 10.63m

Percentage of Buildings: 37%
 

Study Parameters:

Frequency: 900 MHz,

TX Height (hb)=50m

Mobile Height (hm)=1.5m

TX Gain:  13 dBi

City Size: Small/Medium

Area Type : Suburban

 

 
 
 



  

 


        Figure 5 Comparison of the Walfisch-Bertoni Model

                with Okumura –Hata Model.

 

In this study, average building height is taken to be 10.6 m, which is   a height of typical house  with 3 floor and a roof in a suburban. The path loss value at 4.96 km for the model is 120.78 dB whereas that value is 123.88 dB for Hata model. This study implies that propagation takes place over the buildings, with diffraction of the rooftop fields down to the mobile [2].

 

 

In the literature, comparison of the model with measurements carried out in different cities [3] are done and it is shown that model is not applicable for large values of parameter v= aÖ(b/l) greater than 0.5.  In [3], for v=1, error >3 dB and for v=2 error>6dB are observed. Actually, the amplitude of field strength Q(a) in 2.2.2.3 is approximated from curve in figure 2  and it is mentioned that the 3 is more accurate for v<0.5. 

 

 

     (7)
 
            This model can be considered as the limiting case of the flat edge model when the number of buildings is sufficient for the field to settle, i.e. n ³ ns [4] where


 

 

 


Use of Walfisch –Bertoni model is limited to case that large number of buildings are present and particularly grazing angle a is small. Also, it should be cared that model is valid for H>0 as seen in formula 4. That means model works when base station antenna height is above the average rooftop level.

 

[1] J.Walfisch and H.L. Bertoni, “ A Theoretical model of UHF propagation in urban environments,” IEEE Trans. Antennas Propagat., vol.36, 1988, pp.1788-1796

 

[2] H.L.Bertoni, “Radio Propagation for Modern Wireless Systems”,Prentice Hall,New Jersey, USA,200

 

[3] N.Cardona, P.Moller, and F.Alonso, “Applicability of Walfisch-type urban propagation models”, Electron.Letts., Vol.31,1995, pp.1971-1972

 

[4] Sounders, Simon.R., “ Antennas and Propagation for Wireless Communication Systems”,Wiley,New York,1999