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Table of ContentsEstimating Path Loss Exponent and Application with Log Normal Shadowing..................................2Abstract.................................................................................................................................................3

1Path Loss Models...........................................................................................................................41.1Free Space Path Loss Model..................................................................................................4

1.1.1Free Space Path Loss Equation:.....................................................................................41.1.2Received Power:.............................................................................................................41.1.3Propagation of a Radio Signal using Free Space Model...............................................5

1.2Log-Distance Path Loss Model..............................................................................................51.2.1Log-Distance Path Loss Equation..................................................................................51.2.2Propagation of a Radio Signal in Urban Area using Log Distance Model....................6

1.3Log-Normal Shadowing Path Loss Model.............................................................................71.3.1Log-Normal Shadowing Path Loss Equation.................................................................7

1.4Other Examples to Path Loss Models....................................................................................72Estimation of Path Loss Exponent.................................................................................................7

2.1Minimum Mean Square Error (MMSE) Function.................................................................72.2Applying MMSE to Log-Normal Shadowing Model............................................................8

2.2.1Equations........................................................................................................................82.2.2Cauchy Method..............................................................................................................82.2.3Application.....................................................................................................................9

2.3Frequency Extension to Log-Distance Model......................................................................122.3.1Application and Performance Analysis........................................................................122.3.2Comparasion of Estimated Model and Measured Values.............................................14

Summary.............................................................................................................................................16References..........................................................................................................................................17

Kocaeli University Computer Engineering Department – 2011 Spring – Mustafa KIYAR – Optimization Theory

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Estimating Path Loss Exponent and Application with Log Distance Path Loss Model

Kocaeli University Computer Engineering Department – 2011 Spring – Mustafa KIYAR – Optimization Theory

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AbstractWireless Communication is a big part of our life in this century and all calculations related to this field also effects our wireless experience quality from cell phones to bluetooth dongles. Signal propagation model's importance increasing day by day on behalf of quality of signal. There are some theoretically and practically proved models in this field. We will mention about some theoretical approaches and derive another one using an estimation technique. With this technique we will create an error function derived from measured values and minimize that error using Cauchy method to find optimized variables to fit our model.

Kocaeli University Computer Engineering Department – 2011 Spring – Mustafa KIYAR – Optimization Theory

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1 Path Loss Models

1.1 Free Space Path Loss Model

Free Space Path Loss Model is used when there is no obstacle, interference or any kind of factor that can cause weakening on the signal strength between two links ( these two links will be accepted as transmitter and receiver ). Working with satellite communication systems or line-of-sight microwave radio links satisfies similar conditions[1].

The main idea of the explanation of Free Space Path Loss Model here, providing basic understanding of the propagation characteristics of the radio signals in space or air.

PL = path loss, Pr = receiver power, Pt = transmitter power

Gt = transmitter antenna gain, Gr = receiver antenna gain

EIRP = Effective Isotropic Radiated Power calculated as “ Pt * Gt ”

d = distance

L = system loss factor (L>=1 but in free space we will accept as 1)

λ = wave length (in meters)

1.1.1 Free Space Path Loss Equation:

In Free Space Path Loss Model there is no factor weaken the signal strength other than distance as you can see from the equation below;

Equation 1

[2]

1.1.2 Received Power:

According to Free Space Path Loss Model received power can be calculated using the equation below ;

Equation 2: [3]

Kocaeli University Computer Engineering Department – 2011 Spring – Mustafa KIYAR – Optimization Theory

Lf dB=10∗log [

2

4∗2∗d2 ]

Pr dBm=EIRPLf GrLcable loss

5

1.1.3 Propagation of a Radio Signal using Free Space Model

The illustration below explains the propagation of a signal for Free Space Path Loss Model in log domain versus distance. Transmission power of the signal accepted as 10 Watt and frequency is 800 MHz .

1.2 Log-Distance Path Loss Model

Log-Distance Path Loss Model predicts the path loss over distance using known values (power measurement at an exact point/exact distance and path loss exponent).

1.2.1 Log-Distance Path Loss Equation

Equation 3:

[4]

In case of no fading zero-mean Gaussian random variable (Xσ ) is accepted as zero. Log-Normal Shadowing Path Loss Model is just a derivative of Log-Distance Path Loss Model in case of Xσ has Gaussian Distribution with standard deviation σ.

Kocaeli University Computer Engineering Department – 2011 Spring – Mustafa KIYAR – Optimization Theory

Illustration 1:

P L=PtxdBm−P rxdBm=PL010∗n∗log10 dd0

X

6

Different path loss exponents can be seen below for different environments;

Environment Path Loss Exp. (n)

Free Space 2

Urban Area 2.7 to 3.5 (for cellular radio)

Shadowed Urban Area 3 to 5 (for cellular radio)[5]

1.2.2 Propagation of a Radio Signal in Urban Area using Log Distance Model

The illustration below explains the propagation of a signal for Log Distance Model and Urban Area path loss exponent in log domain versus distance. Transmission power of the signal accepted as 10 Watt and frequency is 800 MHz . (No fading)

• We can find received power at 100 meter using free space model and then use Log-Distance Model;

Lf dB=10∗log [

2

4∗2∗d2 ] =

10∗log [0.14062

4∗2∗1002

]= - 69,588 dB

Pr dBm=EIRPLf GrLcable loss = 50−69,5883−1 = -17,588 dBm

• Then we can apply the known power and distance to Log-Distance Model for Urban Area;

P idBm=P0−P Li=P0−10∗n∗log10 d i

d 0

= −17,588−10∗3∗log10 d n

100m

Kocaeli University Computer Engineering Department – 2011 Spring – Mustafa KIYAR – Optimization Theory

Illustration 2:

7

1.3 Log-Normal Shadowing Path Loss Model

Log-Normal Shadowing Path Loss Model has Xσ zero-mean Gaussian random variable. We will use this method in chapter 2.

1.3.1 Log-Normal Shadowing Path Loss Equation

Equation 4:

Note: Xσ can be generated using matlab randn function. Return value of randn function must be multiply with standard deviation (σ) to have Xσ.

1.4 Other Examples to Path Loss Models

• Hata Urban Path Loss Model• Hata SubUrban Path Loss Model• Hata Rural Path Loss Model• PCS Extension to Hata Model (COST-231) [6]

2 Estimation of Path Loss Exponent

2.1 Minimum Mean Square Error (MMSE) Function

Basically MMSE means total value of square errors as seen below;

Equation 5:

Pi=estimated power , Pi=known power

Kocaeli University Computer Engineering Department – 2011 Spring – Mustafa KIYAR – Optimization Theory

P Ld dB=P L dodB10∗n∗log10 ddo

Xσ dB

f n=∑ P i− Pi2

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2.2 Applying MMSE to Log-Normal Shadowing Model

2.2.1 Equations

Error function represented as f , dependent to path loss exponent. To find the optimum path loss exponent, error must be minimized.

Equation 6:

& Equation 7:

Equation 8:

2.2.2 Cauchy Method

To minimize a given f(x) function starting from x0 point we should iterate xk follows;

xk1=xk∗∇ f xk

when

∇ f xk1=∇ f =0 then xk is accepted as optimum point for f(x) function. [7]

Kocaeli University Computer Engineering Department – 2011 Spring – Mustafa KIYAR – Optimization Theory

P idBm=P0−10∗n∗log10 d i

d0

f n=∑ P i− Pi2

f n=∑ P i−P010∗n∗log10d i

d 0

2

9

2.2.3 Application

Assuming that we have a power measurement set as below;

Power (dBm) Distance (meter)

-15 100

-45 1000

-60 2500

-65 4000

-70 5000

We generated the values in the table using the values at 100 meter calculated by free path loss model and then generated new power measurements for different distances, because characteristic of log distance model converges to real values. [8]

Kocaeli University Computer Engineering Department – 2011 Spring – Mustafa KIYAR – Optimization Theory

Illustration 3:

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• Error function for measurements table;

Equation 9:

• Expansion of f(n) function for the measurement table above;

f n=−451510∗n∗log10 1000 /1002

−601510∗n∗log102500 /1002

−651510∗n∗log10 4000 /1002

−701510∗n∗log10 5000 /1002

Error function found as;

f n=840.7∗n2−5329∗n8450

• Applying Cauchy Method

Gradient of error function;

∇ f n=1681,4∗n−5329

Randomly selected n0 value is 100;

first step;

nk1=nk∗∇ f nk

n1=n00∗∇ f n0=1000∗1681.4∗100−5329

∇ f n1= f ' 0=1681.4∗1000∗1681.4∗100−5329−5329=0

from the equation above α0 found as -5.9474 * 10-4

n1=n00∗∇ f n0=100−5,9474∗10−4∗168140−5329=3,1694

second step;

n2=n11∗∇ f n1

n2=n11∗∇ f n1=3.16941∗1681.4∗3.1694−5329

∇ f n2= f ' 1=1681.4∗3.16941∗0.029160−5329=0

Kocaeli University Computer Engineering Department – 2011 Spring – Mustafa KIYAR – Optimization Theory

f n=∑ P i−P010∗n∗log10 d i

d 0

2

11

from the equation above α1 found as -5.9474 * 10-4

n2=n11∗∇ f n1=3.1694−5.9474∗10−4∗1681.4∗3.1694−5329=3.1694

No incrementation in variable n means that we already reached to optimum point. For n = 3.1694 , lets find f(n)

f n=841.4∗n2−5329∗n8450=5.1799

2=∣

f n5

∣=1.036 => =1.0178

We can get Xσ variable using matlab function randn multiplying with σ value for each calculation to draw propagation graph using equation 4;

Measured Power (dBm) Frequency (MHz) Distance (Meter)

-15 800 100

-50 900 1000

-65 1600 2500

-80 1900 4000

Kocaeli University Computer Engineering Department – 2011 Spring – Mustafa KIYAR – Optimization Theory

Illustration 4:

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2.3 Frequency Extension to Log-Distance Model

We added frequency dependent variable to path loss exponent and created a new model. Now we can find optimum values for n1 and n2 using field measurements.

Equation 10:

In section 1.4 we mentioned about other propagation models. These models can fail in real world in some conditions or in some fields, such as tunnels or bridges. More specific models can be created for specific fields. [9]

2.3.1 Application and Performance Analysis

f n=−501510∗n1∗log10 1000/10010∗n2∗log10 900/8002

−651510∗n1∗log10 2500 /10010∗n2∗log101600/8002

−801510∗n1∗log10 4000 /100 10∗n2∗log10 1900 /8002

f n=23.373∗n22214.76∗n1∗n2552.8∗n1

2−825.20∗n2−4180.6∗n17950

∇ f n=214.76∗n21105.6∗n1−4180.646.746∗n2214.76∗n1−825.2

first step:

∇ f 55=2421.2482.33

n1=n0

−∗∇ f n0 =55−∗2421.2

482.33=5−2421.2∗5−482.33∗

f ' =−542523.910185−482.33∗−2780463.91085−2421.2∗1.0520087436000001E+7

=8.7147e-04

n1=5−2421.2∗

5−482.33∗=2.89004.5797

Kocaeli University Computer Engineering Department – 2011 Spring – Mustafa KIYAR – Optimization Theory

f (n)=∑ (P i−P0+ 10∗n1∗log10 (d i

d 0

)+ 10∗n2∗log10(f i

f 0

))2

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second step:

∇ f 2.89004.5797=−1.8796

9.5391 n2

=n1−∗∇ f n1

=2.89004.5797−∗−1.8796

9.5391 =2.89001.8796∗4.5797−9.5391∗

f ' =29.468644000000041.8796∗2.89−42.25187260000001 4.5797−9.5391∗13.80

=0.20620

n2=2.89001.8796∗

4.5797−9.5391∗=3.27762.6128

Estimated Values:Measured Power (dBm) Frequency (MHz) Distance (Meter)

-49.11 900 1000

-68.68 1600 2500

-77.32 1900 4000

third step;

∇ f 3.27762.6128= 4.239488

0.8353248

n3=n2

−∗∇ f n2=3.2776

2.6128−∗ 4.2394880.8353248= 3.27764.239488∗

2.61280.8353248∗f ' =4866.57224.239488∗3.2776949.5205 0.8353248∗2.6128−18412.9135

=0.0008715

n3= 3.27764.239488∗

2.61280.8353248∗=3.28129462.613528

Estimated Values:Measured Power (dBm) Frequency (MHz) Distance (Meter)

-49.15 900 1000

-68.74 1600 2500

-77.39 1900 4000

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2.3.2 Comparasion of Estimated Model and Measured Values

1. Estimated Model Comparasion vs distance at 900 Mhz ;

2. Estimated Model Comparasion vs distance at 1200 Mhz ;

Kocaeli University Computer Engineering Department – 2011 Spring – Mustafa KIYAR – Optimization Theory

Illustration 5:

Illustration 6:

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3. Estimated Model Comparasion vs distance at 1600 Mhz ;

4. Estimated Model Comparasion vs distance at 1900 Mhz ;

Kocaeli University Computer Engineering Department – 2011 Spring – Mustafa KIYAR – Optimization Theory

Illustration 7:

Illustration 8:

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5. Estimated Model Comparasion vs distance at 2200 Mhz ;

SummaryIn this paper we studied mostly the optimization of error functions. We used this technique to make Log-Distance Model estimated to measured values using n matrix. Frequency Extension to Log-Distance Model has been added and by this way we created a new path loss model. We saw that frequency dependent model estimated real values. According to all these datas we can say that new path loss models can be derived from Log-Distance Model using error estimation minimization and make the path loss model to other variables such as antenna lengths.

Kocaeli University Computer Engineering Department – 2011 Spring – Mustafa KIYAR – Optimization Theory

Illustration 9:

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References[1], [2], [3], [4], [5], [6] T.S Rappaport, Wireless communications –Principles and practice, 2nd

Edition , Prentice Hall, 2001, pp. 100-125

[7] Edwin K.P Chong & Stanislaw H. Zak, An Introduction to Optimization, 2nd Edition, Wiley, 2001, pp. 115-118

[8] Purnima K. Sharma, Comparative Analysis of Propagation Path loss Models with Field Measured Data , International Journal of Engineering Science and Technology Vol. 2(6), 2010, 2008-2013

[9] Yan Wu, Min Lin, Ian Wassell , Path Loss Estimation in 3D Environments Using a Modified 2D Finite Difference Time-Domain Technique

Kocaeli University Computer Engineering Department – 2011 Spring – Mustafa KIYAR – Optimization Theory