3.5 Clear-Air Multipath Propagation Modeling Techniques by Various Authors
3.5.21 Systematic Development of New Multivariable Techniques for Predicting the Distribution
In this approach, multipath fading data obtained from 47 terrestrial microwave line-of-sight links in France and the United Kingdom are analyzed to derive narrow-band prediction equations for the deep
Clear-Air Radioclimatological Modeling for Terrestrial Line of Sight Links in Southern Africa Odedina P.K. Aug , 2010 85
fading range of the cumulative distribution for the average worst month. A large number of possible predictor variables based on typical radio link parameters are investigated in their approach and equations are developed which reduce the standard error of prediction to less than half that of the previous techniques for this part of Europe.
Generalized Prediction Equation
In their analysis, the earlier power law equations [21] are generalized to:
i A /10
g i i
P K iF (x ) 10 (3.62)
where the Fi are appropriate functions of the variables xi. Linear multiple regressions [97] is employed to obtain the best values of the exponent i .For such calculations, it is more convenient to work with the logarithmic form:
o i i i i
A G a (a log[F (x )]) 10 log(P) (3.63)
Where ai 10 i and G + ao= 10 log (Kg). Here it is assumed that the ai‘s are fixed over the entire region of the database and that G is a geoclimatic factor that in general varies within this region. In some cases, F (x ) = xi i i are employed in the regression. In cases where functions Fi are used, they are obtained by a combination of physical intuition and trial and error, the least systematic aspect of the overall procedure, but perhaps one for which there is no alternative. Although it might be desirable to use functions of thexi that give physically reasonable behaviour as the xi!0, this has been found to not always be essential in order to obtain empirical equations that give good results down to fairly small values of xi. For some variables, or functions of variables, regressions are carried out both with
Wi log (F )i and Wi log(10 )Fi "Fias the regression variables.
Calculation of the Geoclimatic Factor
In order to obtain the geoclimatic factor G using a finite amount of fading data, it is assumed that this factor is a constant Gj within geographical zones smaller than the overall region (the index j denoting the zone), each zone containing several of the experimental links. The Gj and the regression coefficients ai are obtained by an “iterative multiple regression” on
Clear-Air Radioclimatological Modeling for Terrestrial Line of Sight Links in Southern Africa Odedina P.K. Aug , 2010 86
A0.01 Gj a0 a W (x )i i i (3.64)
In equation (3.64), A0.01 is the fade depth at 0.01% of the time. This is done by setting all G = 0j in (3.64) and performing the first regression. The mean prediction errors are then computed for all the links in each zone, and these values are given toGj. A new regression is done with A0.01 Gj as the dependent variable. For each additional regression the mean prediction errors for the links within each zone are added to the previous values ofGj. This is continued until the mean errors are negligible in each geoclimatic zone, allowing the final values of Gj to be interpreted as zonal geoclimatic factors.
Predictor Variable Selection
The initial selection of variables is based on both physical intuition and historical considerations (e.g., the use of d, f, and s). However, once the initial variables are selected for consideration, a systematic procedure involving partial regression and correlation on regression residuals is employed to determine the best combinations of these variables.
Procedure for the Selection of the Predictor Variables
(a) Distribution Slope
The prediction equations for the distribution tail are based on the assumption of “Rayleigh type” fading for small percentages of the time, with a distribution slope of 10 dB/decade. A mean distribution slope of 10.2 dB/decade was calculated for the 38 links for which both tails points were available. The estimated standard deviation is 1.5 dB, giving 95% confidence limits of ±0.5 dB/ decade (i.e. twice the standard error). The fairly large standard deviation, however, indicates quite a large variation in the sample distribution slopes at the lowest probabilities.
(b) Choice of Geoclimatic Zones
The links were grouped into eight geoclimatic zones with the number of links in each zone varying from four to eight. The four zones in the UK were chosen under the constraint of geographical proximity of links and are identical to those used by Doble [26]. Although other four-zone groupings of
Clear-Air Radioclimatological Modeling for Terrestrial Line of Sight Links in Southern Africa Odedina P.K. Aug , 2010 87
links in the UK were investigated, this was found to be the most reasonable grouping in terms of minimizing the errors for the prediction models considered.
(c) Correlation Analysis
From the correlation analysis, the results for nine single variables or group of variables are presented.
These variables are: path length, frequency, beamwidth, terrain profile roughness, clearance, path and terrain inclination, grazing angle, height above sea level, and relative reduction in combined antenna directivities in the direction of the specular reflection point. Some groups include several variables that are closely related. As seen in [21] the variables in general cannot be assumed to be statistically independent since several pairs have a large correlation coefficient. This includes path length, frequency and terrain roughness, not a completely surprising result since the experiments from which the data were obtained were not designed to minimize such correlation.
(d) Regression Analysis
Results were given in [21] for multiple regression models from one to six variables. In addition, the results of several standard statistical tests on each model are presented. The goodness of fit is best indicated by the standard error of regression . The overall statistical significance of each model is indicated by the values of the variance ratio statistic F and the multiple determination coefficient R2 (or multiple correlation coefficient R). The statistical significance of the individual variables in a model is indicated by the values of the Student’s t statistics. With a large enough sample in which the residual errors (predicted minus measured fade depths) are normally distributed, about 95% of measured fade depth will lie within ± 2 of the predicted fade depths. For a given number of variables, the higher the value of F in relation to the critical value of 95% confidence, the more statistically significant is the overall regression. For any number of variables, this is also true the closer R2 approaches 100%. For any variable in a model, the larger the value of | t | statistic in relation to the critical value | t |95% for 95%
confidence that its regression coefficient is not zero, the more statistically significant is that variable.