Table of Contents. Kocaeli University Computer Engineering Department 2011 Spring Mustafa KIYAR Optimization Theory

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1 1 Table of Contents Estimating Path Loss Exponent and Application with Log Normal Shadowing...2 Abstract...3 1Path Loss Models Free Space Path Loss Model Free Space Path Loss Equation: Received Power: Propagation of a Radio Signal using Free Space Model Log-Distance Path Loss Model Log-Distance Path Loss Equation Propagation of a Radio Signal in Urban Area using Log Distance Model Log-Normal Shadowing Path Loss Model Log-Normal Shadowing Path Loss Equation Other Examples to Path Loss Models...7 2Estimation of Path Loss Exponent Minimum Mean Square Error (MMSE) Function Applying MMSE to Log-Normal Shadowing Model Equations Cauchy Method Application Frequency Extension to Log-Distance Model Application and Performance Analysis Comparasion of Estimated Model and Measured Values...14 Summary...16 References...17

2 2 Estimating Path Loss Exponent and Application with Log Distance Path Loss Model

3 3 Abstract Wireless 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.

4 4 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. P L = 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) 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; 2 Lf db =10 log[ 4 2 d ] 2 Equation 1 [2] Received Power: According to Free Space Path Loss Model received power can be calculated using the equation below ; Pr dbm =EIRP Lf Gr L cable loss Equation 2: [3]

5 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. Illustration 1: 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) Log-Distance Path Loss Equation P L = P txdbm P rxdbm = P L0 10 n log10 d d 0 X Equation 3: 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. [4]

6 6 Different path loss exponents can be seen below for different environments; Environment Free Space 2 Urban Area Shadowed Urban Area [5] Path Loss Exp. (n) 2.7 to 3.5 (for cellular radio) 3 to 5 (for cellular radio) 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; 2 Lf db =10 log[ 4 2 d ] 2 = log[ ] 2 = - 69,588 db Pr dbm =EIRP Lf Gr L cable loss = 50 69, = -17,588 dbm Then we can apply the known power and distance to Log-Distance Model for Urban Area; P i dbm =P 0 P Li =P 0 10 n log10 d i = 17, log10 d 0 100m d n Illustration 2:

7 7 1.3 Log-Normal Shadowing Path Loss Model Log-Normal Shadowing Path Loss Model has X σ zero-mean Gaussian random variable. We will use this method in chapter Log-Normal Shadowing Path Loss Equation P L d db=p L do db 10 n log10 d Xσ db do 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; f n = P i P i 2 Equation 5: Pi=estimated power, Pi=known power

8 8 2.2 Applying MMSE to Log-Normal Shadowing Model Equations Error function represented as f, dependent to path loss exponent. To find the optimum path loss exponent, error must be minimized. P i dbm =P 0 10 n log10 d i & f d 0 n = P i P i 2 Equation 7: Equation 6: f n = P i P 0 10 n log10 d 2 i d 0 Equation 8: Cauchy Method To minimize a given f(x) function starting from x 0 point we should iterate x k follows; x k 1 =x k f x k when f x k 1 = f =0 then x k is accepted as optimum point for f(x) function. [7]

9 Application Assuming that we have a power measurement set as below; Power (dbm) Distance (meter) 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] Illustration 3:

10 10 Error function for measurements table; f n = P i P 0 10 n log10 d 2 i d 0 Equation 9: Expansion of f(n) function for the measurement table above; f n = n log / n log / n log / n log /100 2 Error function found as; f n =840.7 n n 8450 Applying Cauchy Method Gradient of error function; f n =1681,4 n 5329 Randomly selected n 0 value is 100; first step; n k 1 =n k f n k n 1 =n 0 0 f n 0 = f n 1 = f ' 0 = =0 from the equation above α 0 found as * 10-4 n 1 =n 0 0 f n 0 =100 5, =3,1694 second step; n 2 =n 1 1 f n 1 n 2 =n 1 1 f n 1 = f n 2 = f ' 1 = =0

11 11 from the equation above α 1 found as * 10-4 n 2 =n 1 1 f n 1 = = No incrementation in variable n means that we already reached to optimum point. For n = , lets find f(n) f n =841.4 n n 8450= = f n =1.036 => = We can get Xσ variable using matlab function randn multiplying with σ value for each calculation to draw propagation graph using equation 4; Illustration 4: Measured Power (dbm) Frequency (MHz) Distance (Meter)

12 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. f (n)= (P i P n 1 log10 ( d i )+ 10 n d 2 log10( f 2 i )) 0 f 0 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] Application and Performance Analysis f n = n 1 log / n 2 log10 900/ n 1 log / n 2 log / n 1 log / n 2 log /800 2 f n = n n 1 n n n n f n = n n n n first step: f 5 5 = n 1 =n 0 f n 0 = = f ' = E+7 n 1 = = =8.7147e-04

13 13 second step: f = n 2 =n 1 f n 1 = = f ' = n 2 = = = Estimated Values: Measured Power (dbm) Frequency (MHz) Distance (Meter) third step; f = n 3 =n 2 f n 2 = = f ' = n 3 = = = Estimated Values: Measured Power (dbm) Frequency (MHz) Distance (Meter)

14 Comparasion of Estimated Model and Measured Values 1. Estimated Model Comparasion vs distance at 900 Mhz ; Illustration 5: 2. Estimated Model Comparasion vs distance at 1200 Mhz ; Illustration 6:

15 15 3. Estimated Model Comparasion vs distance at 1600 Mhz ; Illustration 7: 4. Estimated Model Comparasion vs distance at 1900 Mhz ; Illustration 8:

16 16 5. Estimated Model Comparasion vs distance at 2200 Mhz ; Illustration 9: Summary In 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.

17 17 References [1], [2], [3], [4], [5], [6] T.S Rappaport, Wireless communications Principles and practice, 2 nd Edition, Prentice Hall, 2001, pp [7] Edwin K.P Chong & Stanislaw H. Zak, An Introduction to Optimization, 2 nd Edition, Wiley, 2001, pp [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, [9] Yan Wu, Min Lin, Ian Wassell, Path Loss Estimation in 3D Environments Using a Modified 2D Finite Difference Time-Domain Technique