Resulting Models

In [19], the data for LOS (Line-of-Sight) and OBS (Obstructed) floors were analysed separately.

The model to be fitted to m^p' for LOSfloors, was of the form

where Po';]p'''';]?are the parameters of the model to be estimated from the data.

Variables d, h_brel and h_mrel have units of metres; A^fhas units ofm² xI 0^3; a_i has

57

units of degrees; andN_r andg are unitless.

The question of which subset of the variables mentioned above best models the data was considered. A statistic method was applied, using the Cpstatisticwhere the subscriptp refers to the number of independent variables in the subset model under study. To have a brief explanation of how the C_p statistic is defined, see the Appendix 1. The Cp statistic was calculated for all the possible subsets for the full model given by the equation and the models with smallest C^pand those with

C_p

=

^{pwereselected.}

After doing this, it was found that relative base station antenna height, relative mobile antenna height, and floor area are the least significant variables. The exclusion of this variables does not significantly increase the RMSE and the resulting model has a C^p

=

^p. The inverse power law for path length is near two, the same as that for the free-space equation. Combining this information with the relative insensitivity of the best model fitted to the data to either the base station or mobile antenna heights suggested that the free-space equation may be a suitable model for m^p for S floors, and it is shown below

(3.2)

where

fi

_f ^{is the} free-space clutter factorfor LOS floors and Lf is the path loss calculated using the free-spaceequation given by

L_f =32.44+2010gd+2010gf (3.3)

wheredis expressed in km,fis expressed in MHz, and isotropic transmitting and receiving antennas are assumed.

fi

_f was calculated for each floor and these values were then regressed against A_f , N_r ^, g,and a_{i .}The resulting equation for

fi

_f ^is

fi

_f

=

^17.6+3.55A_f ^+0.387N_r ⁺6.1g - 0.202a_i (3.4)

This equation states that

fi

_f increases with increasing floor area and increasing number of rooms and decreases with increasing angle of illumination, i.e. fewer obstructions. The presence of mirror glass windows increases

fif'

The same variables and the same equation to be fit were used in the case of DESfloors. Nevertheless, the analysis produced inconsistent results because of insufficient randomisation of the data with respect to the regressor variables. Due to this, only those variables that did not exhibit anomalous results and that were known to have a significant effect on path loss were included. This analysis only included path length and relative mobile antenna height.

58

Penetration and Transmiuion qllfHF Radio Waves into/through Buiidings-a Literature Review

The result model was

m_p

=

42.9+2.56 x 10logd - 0.126h_mrel (3.5) The distance dependence was closed to the d²relationship predicted by the free-space equation; due to this, the use of the same free-space based model as that for LOS floor was proposed, where

/ll

was given by

/ll =

27.5 - 0.14h_mrel (3.6)

The model was significant using theF-test at all standard a levels and had an RMSE of3.9 dB for the case of LOS floors and a value of7.2 dB for the case ofOBS floors.

The number of predictor variables used in [20] with the regression technique should not be greater than one quarter of the sample size. Taking into account that it can be known from [4] and [5], that the sample size was equal to 30, the regression model should no have more than seven independent variables at anyone time.

In total over 60 models have been considered: some of which failed

completely, while others show a reasonable degree of acceptance. After investigating the relationships between all the variables, the best of all results was obtained when three variables were present in the regression equations: the logarithm of the distance, d, the logarithm of the floor area,AI' and the number of buildings sides seen by the transmitter on each floor of the building housing the receiver, 8^Q'

The resulting models for the path loss at 900, 1800 and 2300 MHz, were found to be:

1';,900 = -37.7+40.0 10glOd +17.6log10 AI - 27.58_Q

1';,1800 = -27.9+40.0 log,o d +_23.3loglO AI - 20.98_Q

Yi,2300= -7.9+_40.0loglO d +_{16. 110glO}AI - 27.38_Q

(3.7)

with the root mean square errors (RMSE's) equal to 2.4, 2.2 and 1.7 dB respectively.

Itmust be pointed out, taking the results above into account, that different conditions of signal transmission and reception are more important in detennining the values of each model coefficient than the variation of the frequency.

3.3.2.Limits to Proposed Models

The model proposed by Barry and Williamson in [19], is a reliable model for m^{p ,}but the variables AI' N_r,g and a_iincluded in this model are unavailable from current terrain databases or from other sources. If these variables were excluded from the model then

/ll

would be described by its mean and standard deviation (which is

59

70l.-- ~-~--~-~-__.

equal to the RMSE of the model). The mean and standard deviation calculated from the data in the case of LOS floors are 18.3

dB and 6.8 dB, respectively. Taking into the account the values ofRMSE from the previous section, it can be seen that the inclusion of variables describing the floors environment can significantly reduce the RMSE of the model.

The path loss predicted by the reduced free-space model for m^ptogether with the data points, the path loss

predicted by the best-fit inverse power Figure 15:Measured and Predicted Median Path

law (which has an exponent,n,equal to Lossfor LOS Floor in [l9].

2.22), and the path loss predicted by the free-space equation, for LOS floors, is shown in Figure 15.Itcan be seen the large scatter of the data about the reduced free-space model for mp as suggested by the relatively large RMSE for this model.

The RMSE for the model described in equation (3.5) is 7.35 dB, while the value for the model in equation (3.6) is 7.2 dB. The same values plotted in Figure 15

are now plotted in Figure 16 but for LOS floors. The figure shows the large

variability ifthe data points about the regression line as suggested by the large RMSE for the model.

From the point of view of [20], it is interested to observe that the Barry and Williamson model yielded slightly worse

70\----~---~-- values ofRMSE's: 3.9 dB for the line of

0' 02 0 ~,slonC^•.~m5 ^{07 10} sight case and 7.2 dB for the obstructed

path case. Thus, predicting propagation

Figure 16:Measured and Predicted Median Path

Lossfor OBS Floors in[19]. path loss using the models in [20] should produce more precise results.

The penetration loss for the proposed model, for Williamson model and the measured values are shown in Figure 17 for 2300 MHz, along with the measured and predicted signal strength. There is a good agreement between the measured and predicted values. The same results for analysis carried out for 1800 and 900 MHz can be found in the same Reference.

Itis clear then, from the results of the two articles, that better predictions are possible in buildings where the features and transmission conditions are well known because, in that case, particular models can be applied yielding, in consequence, smaller errors.

60

Penetration and Tran£wiuioll o,/UHFRadio Waves into/through Building£-a Literature Review

15

III

j10 R

.~ ⁶ D

e

_i ^{i D 6}

k ~ • ⁶D ⁶D ^D

09'd 2nd 4th 61h 8th

1l00r c

/::).Proposed model XWilliamson model DMeasured values

a measured and predicted signal strength b CDF of residuals (dB)

c penetration loss results

Figure 17: Into-Bui/dings Measurements at 2300 MHz in [20].

61

CHAPTER 4

In document Penetration and transmission of UHF radio waves into/through buildings a literature review (Page 49-54)