Methodology for Calculating Path Loss Upper and Lower Bounds for WiMAX

Mohamed Ali Aboul- Dahab Arab Academy For Science And Technology

Cairo, Egypt Mdahab@aast.edu

Hossam Mohamed Kamel National Telecommunication Institute

Cairo, Egypt Hmka81@hotmail.com

Abstract: The WiMAX (IEEE 802.16) technology is widely deployed to provide users with wireless network connectivity in easy way. Propagation measurements at the frequency of 3.5 GHz for the WiMAX technology have been taken from different places in the world with different topologies. This paper deals with path loss calculation and analysis in order to fined the best methodology for calculating upper and lower bounds for the path loss using Least Square Approximation which help designers of the network to have clear vision of the losses. A comparison between path loss models also is presented here.

Keywords: WiMAX, path loss models, Propagation model, Least Square Approximation.

I. INTRODUCTION: Worldwide Interoperability for Microwave Access (WiMAX) is a wireless technology, which is mainly based on the IEEE standards 802.16-2004 and 802.16e-2005 for fixed and mobile applications, respectively [1]. The flexibility of wireless technology, combined with high throughput, scalability and long-range features of the IEEE 802.16 standard helps to fill the broadband coverage gaps and reach millions of new residential and business customers worldwide [2]. Based upon the filed measurements of signal strengths of different places, with different scenarios, path loss models have been devised. The widely used model namely Hata-Okumura model, is valid for the 500-1500 MHz frequency range and there exists an elaboration on this model that extends the frequency range up to 2000 MHz ( known with COST-231 Hata model). It was found that these models are not suitable for lower base station antenna heights, and hilly or moderate-to-heavy wooded terrain. To correct for these limitations, Erceg model covers three most common terrain categories. In Erceg model statistical path loss model derived from 1.9 GHz experimental data collected across the United States in 95 existing macro-cells using the least-squares linear regression on the measured path loss with distance [3].The model tuning is an important part of the wireless network design which involves propagation model adjustments[4]

The most common path loss model used in the WiMAX is Erceg model which used in conjunction with the SUI channel model. This model was modified in order to be used with different frequencies like in 3.5 GHz [5] and with different antenna heights.

All of the previously mentioned models are based upon finding a mathematical expression for the mean value of the path loss. This approach is appropriate if the variance of the filed measured is considerably small. However, if the variance is large, the model may not be suitable. In this paper, we present a methodology by which a mathematical expression for upper and lower bounds of the path loss are devised. The derivations of these bounds were based upon the analysis of some of the filed measurements of the path loss at different locations [6].The parameters of the derived bounds are obtained a numerical technique namely the least square approximation. Theses bounds could be suitable tools for the system designers for more precise planning of the cell coverage (e.g. WiMAX system)

II. THEORY Path loss can be defined as the ratio of the transmitted to received power, usually expressed in decibels. The equation for the Least Square (LS) regression analysis shows the path loss at distance d in the form [7]

Where, do is the reference point at 1 km and γ is known as the path loss exponent. The path loss values, L (.) are expressed in decibels. Note that for Free Space Loss (FSL) the path loss exponent is equal to two.

In the filed measurement the pass loss is measured using the following equation. [8]

Where Pt, Pr represents the transmitted and received power and Gt, Gr represents the transmitted and received gains and L represents the path loss.

L(d) = PL(do) + 10 γ log10 (d/ do ) (dB) (1)

Pr (dBm) = Pt (dBm) + Gt (dBi) + Gr (dBi) - L(dB) (2)

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Usually the pass loss is plotted versus distance to provide a profile of signal strength up to certain ranges. Actually, the measured points on the graph are rather scattered. In order to fined upper and lower bounds of these scattered points, we use suitable numerical technique. In this paper we use the Least Squares Approximation techniques.

A. Least Squares Approximation : In this case the path loss for each boundary can be descried as a straight line equation as following

y = a + k x (3)

Where” x” is value log (d), y is propagation path loss in (dB) and k is k= 10 γ. In other words:

L (dB) = a (dB) +10 γ log (d) (4)

The coefficient «a» is a function of frequency and other parameters (e.g. the transmitter antenna height, the receiver antenna height), It is worth mentioning that equation (4) is applied for both boundaries.

In order to find the value of «a» and «γ» for the upper and lower bounds, we use least squares approximation approach making use of the measured data. In this approach, we seek the best-fit line of the measured data by minimizing the sum of the squares of the errors from this line. A line in slope-intercept form looks like a + k x= y where «a» is the y -intercept and «k» is the slope. We want to find «a» and «k» such that is a + k x = y true for all our data points. For example if we take these data samples from the upper bound as flowing :

a+ 0.1006 k = 99.78

(5)

a + 0.1033 k = 101.1 a + 0.1035 k = 101.1

.................. ……………….

a + 0.9862 k = 145.2 a + 0.9969 k = 145.5 a + 0.9109 k = 143.2 a + 0.9137 k = 144.4

We have more equations than the unknowns. There is a system of equations called the normal equations that can be used to find least squares solution to systems with more equations than unknowns as we have different measurements of the path loss with different distances [8]. We know that there may not exist «a» and «k» that fit all these equations, so we try to find the best fit. We can write these equations in the form Xc = y (these are just new letters for our familiar equation Ax = b ) where

X1 0.10061 0.10331 0.1035 1 0.9862 1 0.9969 1 0.9109 1 0.9137

c ak y99.78101.1101.1145.2145.5143.2144.4

(6)

As in [8] X be an m by n matrix such that XTX is invertible, then the solution to the normal equations, XTXc = XTy, is the least squares approximation to c in Xc = y.

By solving this equation XTXc = XTy we calculate the augmented matrix and use Gauss-Jordan elimination to find the solution of the normal equations. This solution will be the coefficients of the line which give the best fit in the least squares sense. Where the final solution of the augmented matrix AG is:

AG 1 00 1 97.906152.7003 (7)

97.906152.7003 (8)

So a = 97.9061 and k = 10 γ = 52.7003, which is the requirements in the equation (4).

IV. DATA EXTRACTION The data can be extracted from measurements results using software package FindGraph version 1.871. This package has been used to get the data of the path loss given in [6]. A comparison between the measured and extracted data is shown in figure 1.a and b. It is clear that the extracted data are very close to the measured data.

Figure 1.a Urban Propagation Measurements at 3.5 GHz [6]

Figure 1.b Extracted data form the from measurements results

It is clear that the maximum distance of these measurements is 1 km, However after we derive the upper and lower bounds equations we will extend the distance up to 5 km.

III. THE UPPER AND LOWER BOUNDS

A. The upper bound using the Lest Squire Approximation In this section we will take the data of the upper most points of figure 1.b which are shown in figure 2. We use these data to derive the upper bound equation and the path loss exponent. (Sample of these data is given in table 1).

Figure 2. The upper bound from the measured data

TABLE 1. SAMPLE OF THE UPPER BOUND MEASURED DATA

Distance (km) Path Loss (dB) 0.1006 99.78 0.1033 101.1 0.1035 101.1 …… ……… …… ……..

0.9862 145.2 0.9969 145.5 0.9109 143.2 0.9137 144.4

The upper bound is plotted in figure 2, it is clear that it is close to straight line where

a = 97.9061 , γ = 5.27 (9)

So the path loss equation for upper bound can be expressed as

L(dB)= 97.9061+10*5.27*log (d) (10)

B. The lower bound using the Lest Squire Approximation Following the same approach given in pervious subsection but take the data of the lower most points of figure 1.b which are shown in figure 3. We use these data derive the lower bound equation and the path loss exponent. (Sample of these data is given in table 2).

Figure 3. The lower bound from the measured data

TABLE 2. SAMPLE OF THE LOWER BOUND MEASURED DATA

Distance (km) Path Loss (dB) 0.1017 84.02 0.1029 83.69 0.1041 83.36 …… ……… …… ……..

0.9673 115.7 0.9743 115.1 0.9813 114.4

Approximation to derive the roots of equation (4) we obtain the following

a = 78.9939, γ = 3.93113 (11)

So the path loss equation for lower bound can be expressed as

L(dB)= 78.99+10*3.93*log (d) (12)

C. Final upper and lower bounds Now we will use the upper and lower equations after least square approximation to calculate the upper and lower bounds for long distances like 1000 m and 5000 m.

Figure 4 upper and lower bounds using least square approximation for

1 km

As shown in figure 4, it is clear that the difference between the upper and lower bounds at the small distances is small, about few dB and the difference increase at large distances to be tens of dB that is because the value of coefficient «a» in equation (4) is function of frequency and other conditions of the measured path loss values also the difference between the measured data values at the same distances which affect the value of the pass loss exponent γ.

Figure 5 upper and lower bounds compared with SUI model for 5 km

When we extend the distance from 1 km to 5 km as in figure 5 the difference still large and it is follow the behaviour of the most common WiMAX model that is SUI as in figure 5 [10] Where the SUI model equation 10 log

(13)

Where 20 log 4 , (14)

Where hBTS is the base station antenna height

TABLE 3. PATH LOSS EXPONENT PARAMETERS OF SUI MODEL

The results we get in figure 5 for our bounders compared to SUI model show how close the predication of our curves with the used models but here we present a new approach by using the bounds which contain the measured data between it is bounds. In this case we use terrain A which is close to the environment we were working on. It is clear as we see in figure 5 that the SUI curve located between the upper and lower bounds and it is more close to the lower bound this result show how our new approach can help designers .We apply these bounds for another measurements data as in the next section.

Terrain Type C

Terrain Type B

Terrain Type A

Model parameter

3.6 4 4.6 a 0.005 0.0065 0.0075 b

20 17.1 12.6 c

D. Applying the results to other area measurements When we apply the upper and lower bounds for other measurements as in figure 7[11] namely (Empirical Propagation Model for WIMAX AT 3.5 GHz in an Urban Environment) we found that our bounds are suitable for the path loss measured in this case.

Figure 6 Path loss measurements for urban environment [11] The measured path loss data in figure 6 are extracted using the software package FindGraph; we apply the derived upper and lower bounds to the extracted data as in figure 8.

Figure 7 upper and lower bounds compared with the mean path loss as in [11]

As in figure 7 it is clear that the bounds derived follow the pattern of the measured data. It can therefore be used to give an estimate of the data beyond the maximum distance of the measured data. The mean path loss from [11] is 10 log

(15)

Where A= 105.45 dB , do= 200 m From figure 8 we find that the mean of the path loss in equation (15) located between the upper and lower bounds we have derived and it is close to the upper bound than the lower bound this due to the difference of the measurements parameters (e.g. antenna heights , transmit power )

V. CONCLUSION The path loss models available in literature are based upon formulating mathematical expressions that describe the mean value of the path loss for different propagation environments.

However, the mean value will not indicate the profile of the path loss behavior at different distances especially when the variance of the data is considerably large. Depicting the upper and lower bounds of the path loss shall be useful in such case. In this paper, mathematical expression for the path loss bounds (upper and lower) have been predicted. The parameters included in these expressions are derived from the measured data using the least squares approximation of the first order which is numerical technique. The measured data are extracted from some filed measurements available in literature by using FindGraph package, whereas the numerical calculations are carried out using MATLAB 2008.

The case of urban area is considered where the maximum distance varies between 1-5 km with an operating frequency 3.5 GHz. The bounds obtained using the least squares approximation of the first order match with the measured data up to the maximum distance of measurements. It is obvious that the parameters predicted in the mathematical expression for both techniques will be different if operating frequency and/or the environment of the measurements varies (suburban, rural, etc.). The proposed methodology can be used to give more clear vision of the path loss profile in WiMAX system especially if the variance of measured data is large. The upper and lower bounds are not the exact boundaries; they are rather boundaries for the path loss so that it can be used as boundaries to estimate the possible maximum and minimum values of path loss.

REFERENCES: [1] A. Rial, H. Krauss, J. Hauck, and Others “EMPIRICAL PROPAGATION MODEL FOR WIMAX AT 3.5 GHz IN AN URBAN ENVIRONMENT”, IEEE MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 50, No. 2, February 2008 [2] D. Luca, F. Fiano, F. Mazzenga, and Others “Outdoor Path loss models for IEEE 802.16 in suburban and campus-like environments”, IEEE ICC 2007 [3] Vinko Erceg, , Larry J. Greenstein, Others “An Empirically Based Path Loss Model for Wireless Channels in Suburban Environments” IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 17, NO. 7, JULY 1999 [4] Zerihun Abate “WiMAX RF System Engineering”, Artech House, 2009 [5] T.S. Chu, Larry J. Greenstein, Others “A Quantification of Link Budget Differences Between the Cellular and PCS Bands ” IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 48, NO. 1, JANUARY 1999 [6] Marcus C. Walden, Frank J. Rowsell, “Urban Propagation Measurements and Statistical Path Loss Model at 3.5 GHz”, 2005 [7] V.S. Abhayawardhana, I.J. Wassell, D. Crosby and Others “Comparison of Empirical Propagation Path Loss Models for Fixed Wireless Access Systems” ,2005 [8] Winton Afric1, Branka Zovko Cihlar, Sonja Grgic “Methodology of Path Loss Calculation using Measurement Results”, IEEE 2007 [9] Tamara A. Carter, Richard A. Tapia and Anne Papakonstantinou “Linear Algebra-An Introduction to Linear Algebra for Pre-Calculus Students ", May 1995, Rice University. [10] Mohamed Ouwais Kabaou, others “Path Loss Models Comparison in Radio Mobile Communication”, Medwell Journals 2008 [11] A. Valcarce Rial1, Harald Krauss, Joachim Hauck, and others “EMPIRICAL PROPAGATION MODEL FOR WIMAX AT 3.5 GHz IN AN URBAN ENVIRONMENT”, 2008