MODELING AND ESTIMATION OF CELLULAR PATH LOSS …

TEL AVIV UNIVERSITY

The Iby and Aladar Fleischman Faculty of Engineering

The Zandman-Slaner School of Graduate Studies

MODELING AND ESTIMATION OF CELLULAR PATH

LOSS DISTRIBUTIONS

A thesis submitted toward the degree of

Master of Science in Electrical and Electronic Engineering

by

Yehonatan Broyde

August 9200

TEL AVIV UNIVERSITY The Iby and Aladar Fleischman Faculty of Engineering

The Zandman-Slaner School of Graduate Studies

MODELING AND ESTIMATION OF CELLULAR PATH

LOSS DISTRIBUTIONS

A thesis submitted toward the degree of

Master of Science in Electrical and Electronic Engineering

by

Yehonatan Broyde

This research was carried out in the Department of Electrical Engineering - Systems under the supervision of Prof. Hagit Messer-Yaron

August 2009

Acknowledgment

Warm thanks go to my supervisor Prof. Hagit Messer-Yaron, for her wise advices,

support and encouragement.

This research was initiated in cooperation with Schema Ltd. (http://schema.com), who

presented the issue of indoor/outdoor ratio estimation, and shared its parsing and

analysis tools, as well as raw cellular data. Fruitful discussions with Schema people

led to the possibility of using a mixed-Gaussian model for describing the path loss

distribution, in addition to some other useful ideas.

I therefore deeply thank Schema people, and especially to Dr. Michael Lifvschitz, Mr.

Shmuel Nowik and Mr. Roy Bass, for their ideas, knowledge and support.

Special thanks to my family and friends, who supported me throughout the long

journey of the thesis research and writing.

http://schema.com/

Abstract

Cellular network path losses are key factors in the behaviour and performance of the

network. Many models have been studied in the last few decades, for describing path

loss behaviour as a function of the distance between the transmitter and receiver, and

of other parameters such as antenna height, buildings and vegetation characteristics in

the signal path, and more.

In this thesis, "Sector-to-Users" path losses were studied, and a novel model

describing this path loss distribution is presented. First, the distribution is computed

theoretically, based on a few simple assumptions such as a power law signal decay,

uniform geographical distribution of cellular calls, etc. Next, a simple Gaussian model

is suggested, based on the theoretical computation with real-life parameters and on

analysis of empirical data. This thesis shows, both theoretically and empirically, that

the simple Gaussian model is valid for many real-life sectors, where the following

criteria are met: standard deviation of shadowing effect is above ~5dB, and mixing of

indoor and outdoor users is low. The complete model incorporates Indoor/Outdoor

mixing, and offers a Gaussian Mixture (GM) model for the path loss distribution.

Throughout the work, different algorithms are analyzed and used to estimate the path

loss distribution parameters: Gaussian Mixture estimation techniques for distributions

with known means and variances, and Expectation-Maximization (EM) algorithm for

distributions with unknown means and variances. The estimators were thoroughly

tested and analyzed based on empirical path loss data from a real UMTS cellular

network in Israel. The data was retrieved from various sectors, including sectors with

high and low Indoor/Outdoor mixing, sectors which use repeaters in their coverage

area and more.

Based on the model and estimators presented, two applications will be proposed in

this thesis, utilizing the fact that path loss data is already measured and logged

automatically in most cellular networks. The first application is a "coverage map" of

the cellular network composed by estimations of the sector's actual coverage area.

This can help locate coverage "blind spots" and overlaps in the network. The second

application is an estimation of the Indoor/Outdoor ratio, using the GM model and the

EM algorithm. Both applications can lead to significant improvements in network

design and optimization.

To conclude, a few directions for future research will be suggested. The most

prominent of these directions is calibration of the model studied in this thesis, which

can be done using standard "drive-tests" measurements.

Contents

1. Introduction

1.1. Background

1.2. Previous Work

1.2.1 General path loss models

1.2.2 Sector-to-users path loss models, and indoor/outdoor ratio

estimation

1.3. This thesis contribution

2. Cellular Path Loss Distribution – a Theoretical Computation

3. Gaussian Mixture Model

3.1. Why choose a GM model?

3.2. GM background

3.3. Estimators for distributions with known means and variances

3.4. EM Algorithm

4. The Monitored System

5. Results Analysis

5.1. Highly bi-modal sectors

5.2. Highly uni-modal sectors

5.3. Influence of repeaters

5.4. Dependence of initial values

5.5. EM results for pure Gaussian and theoretical model distributions

5.6. Sector coverage area estimation

6. Conclusions and Suggestions for Future Research

6.1. Conclusions

6.2. Suggestions for future research

6.2.1 Calibrating the models presented in this thesis

6.2.2 Studying the convergence rate of the EM algorithm and developing

techniques for its acceleration

6.2.3 Using the models as a new "measuring tool"

6.2.4 Combining location and path loss data

6.2.5 Mobility estimation, in-car path loss behavior and discrimination

6.2.6 Integrating additional information to the Gaussian mixture

estimation

Appendices

A. CRLB Computation

Bibliography

List of Acronyms

BSC – Base Station Controller

BTS – Base Transceiver Station

CDMA – Code Division Multiple Access

CPICH – Common Pilot Channel

CRLB – Cramer-Rao Lower Bound

EM – Expectation Maximization

GM – Gaussian Mixture

GPEH – General Performance Event Handler

GSM – Global System for Mobile communication

ITU – International Telecommunication Union

ML – Maximum Likelihood

MLC – Maximum Likelihood Classifier

MS – Mobile Station

MSC – Mobile Switching Center

MSE – Mean Square Error

OSS – Operations Support System

PDF – Probability Density Function

PPE – Prior Probability Estimator

RF – Radio Frequency

RNC – Radio Network Controller

RRM – Radio Resource Management

RSL – Received Signal Level

RSS – Received Signal Strength

UE – User Equipment

UMTS – Universal Mobile Telecommunication System

List of Figures

Figure 1.1 – Signal propagation in a wireless network --------------------------------- 2

Figure 2.1 – Cellular sector-to-users theoretical distribution - I --------------------- 8

Figure 2.2 – Cellular sector-to-users theoretical distribution - II --------------------- 9

Figure 2.3 – Theoretical Vs. Measured Path Loss Distribution ----------------------- 9

Figure 2.4 – Path loss mean Vs. Sector's maximal radius, for model values of

----------------------------------------------------------- [dB]and σ=9 3.7=10A

10

Figure 2.5 – Kullback-Leibler divergence between (2.9) and its Gaussian fit, Vs.

the standard deviation σ used in (2.9) ------------------------------------

11

Figure 3.1 – Measured Path Loss Distribution (EKG141952) Vs. Unimodal

Gaussian fit --------------------------------------------------------------------

13

Figure 3.2 – Measured Path Loss Distribution (EKG142181) Vs. Unimodal

Gaussian fit --------------------------------------------------------------------

13

Figure 3.3 – MLCunbiased Vs. Measured Path Loss Data, Sector EKG 141952 ------ 17

Figure 3.4 – MLCunbiased Vs. Measured Path Loss Data, Sector EKG 141952 ------ 18

Figure 3.5 – Estimators' MSE Vs. Distance between the Gaussians

(100 samples) -----------------------------------------------------------------

19

Figure 3.6 – Estimators' MSE Vs. Distance between the Gaussians

(10,000 samples) -------------------------------------------------------------

19

Figure 4.1 – Sector locations in the Beer-Sheva area ---------------------------------- 22

Figure 4.2 – Typical trisector Node B units --------------------------------------------- 23

Figure 5.1 – Google Earth photo of the central bus station, sectors

EKG153363 and EKG142181 ---------------------------------------------

25

Figure 5.2 – Measured Data Vs. EM Estimation of EKG153363 -------------------- 26


Figure 5.4 – Google Earth photo of the "BIG" shopping center, sector

EKG141952 ------------------------------------------------------------------

27


Figure 5.6 – Google Earth photo of an urban area, sector EKG143613 ------------- 28


Figure 5.8 – Google Earth photo of urban area, sector EKG159853 ----------------- 29


Figure 5.10 – Google Earth photo of the university area sectors and repeaters ---- 31

Figure 5.11 – Measured Data Vs. EM Estimation of EKG130711 ------------------- 31


Figure 5.13 – Google Earth photo of the area around EKG148911/2/3 ------------- 33




Figure 5.17 – EM estimation with different initial parameters values,

EKG141952 -----------------------------------------------------------------

36

Figure 5.18 – EM Algorithm steps of two different runs (right and left),

EKG141952 -----------------------------------------------------------------

36

Figure 5.19 – EM estimation with different initial parameters values,

EKG148912 -----------------------------------------------------------------

37

Figure 5.20 – EM estimation with different initial parameters values, pure

Gaussian data ---------------------------------------------------------------

38

Figure 5.21 – EM estimation with different initial parameters values, equation

(2.9) theoretical computation ----------------------------------------------

39

Figure 5.22 – Beer-Sheva city center coverage map estimation ---------------------- 40

Figure 5.23 – Coverage map estimation outside of Beer-Sheva ----------------------

40

Chapter 1

Introduction

1.1 Background

Cellular network path losses are one of the most significant factors influencing the

behavior of the network and its performance, and therefore have a great effect on

network design and optimization. Path loss models have been widely studied in the

last decades, though most of the study referred to "loss vs. distance" behavior and the

different factors influencing it. In this thesis we study "sector-to-users" path loss

distributions, present a model to describe these distributions, develop estimation

techniques and analyze empirical data from real cellular networks.

Path loss is usually described as the cumulative effect of three different phenomena

[1, 2, 3, 4, 5]–

1. Propagation loss1 –

This is the median attenuation of the signal power, usually described as a

power law relation between the signal power and the distance between the

transmitter and the receiver. Propagation loss is due to free space wavefront

spreading, and the median effect of the propagation environment, such as

buildings, vegetation, water sources etc.

2. Shadowing –

This is the low variations of the signal strength around the average curve

(Figure 1.1). The radio signal undergoes additional attenuation caused by local

obstacles (or masks) between the transmitter and the receiver. It is generally

modeled by a Gaussian distribution with zero mean and standard deviation σ

(when referring to the signal power in dB). A typical standard deviation for

rural environment is 6 dB and can be as high as 10-12 dB for urban

environments [6, 7].

3. Fast Fading –

This is the short-term fading, which is present in the signal in form of rapid

variations (Figure 1.1). It is due to multipath effects, and can cause variations

of up to 20 dB in the signal power, over distances in the order of magnitude of

the signal wavelength [8]. Since the variations happen over such a small scale

of distances, fast fading is usually averaged and not taken into account in large

scale optimization and design of cellular networks.

1 In the literature "path loss" is sometimes referred to only as the median large-scale loss. For the sake of clarity I chose to refer to the total signal loss from the transmitter to receiver as the "path loss", and refer to the median large-scale loss as "propagation

loss".

Figure 1.1 - Signal propagation in a wireless network (Drawing taken from [6])

1.2 Previous work

1.2.1 General path loss models

Path loss models have been widely studied in the last few decades. Many models have

been suggested for various environments, signal frequencies and distances, and these

models are widely used by network designers, operators and others in the cellular

industry.

The methods for modeling and predicting the path loss in a specific network can be

roughly divided into two main groups: statistical and computational.

I. Statistical methods.

These methods use measured path loss data, plus RF propagation models, to formulate

the path loss model. They are usually based on path loss measurements in specific

areas and cities, and give a mathematical description of the path loss behavior, usually

just for the median behavior (thus actually modeling the propagation loss). Besides

the radio frequency and the distance between the transmitter and receiver, methods

differ in the parameters they take into consideration. Such parameters can be the

height of the transmitter and receiver, the buildings' height, the building density etc.

The most widely used method for cellular path loss modeling is the Okumura-Hata

model, which is based on numerous measurements taken in and around Tokyo.

Okumura [9] suggested a method based on a series of curves describing the average

attenuation ( ),A frequency distance relative to the free-space loss in an urban

environment. Hata [10] developed a mathematical formula to describe Okumura's

curves, which takes into account the frequency of the signal, the distance between the

base station antenna and the mobile antenna, the height of the base station antenna

and of the mobile antenna, and the character of the area (urban – large and medium

size cities, suburban and rural).

Propagation Loss

Shadowing Fast

Fading

( )[ ] 069.55 26.16log( ) 13.82log( ) ( ) 44.9 6.55log( ) log( )dB T R TPL f h a h h r= + − − + −

For example, the median path loss for a medium-size city is modeled by:

(1.1)

Where 0f is the signal center frequency, r is the distance between the base station and

the mobile antenna, hT is the base station antenna height, hR is the mobile antenna

height, and

( ) ( )( ) ( )( )0[ ]1.1log 0.7 1.56log 0.8R R RdB

a h f h h= − − − (1.2)

0150 1500 , 30 200 , 1 10 , 1 20T Rf MHz h m h m r km (1.3)

Other statistical methods exist for path loss prediction, including COST-231,

Walfisch-Ikegami model, and Ibrahim-Parsons method [1]-[5].

II. Computational methods.

These methods are sometimes called "deterministic methods", and use wave

propagation models or ray-tracing techniques to compute the path loss in specific

radio links. They are usually more accurate than the statistical methods, but require

detailed and accurate description of the objects in the propagation space, plus

expensive computational resources.

This thesis is aimed to produce a general and easily applicable model for the path loss

distribution, without the need to rely on such detailed data (or expensive

computational resources). Therefore we focused on statistical methods for modeling

the path loss and did not go deep into computational methods.

A good review of computational methods can be found in [2, Section III].

1.2.2 Sector-to-users path loss models, and indoor/outdoor ratio estimation.

While most of the statistical methods provide formulas similar to (1.1), which relate

the path loss to the frequency, distance and environmental characteristics, we focus in

this thesis on the path loss distribution between the sector's antenna and the mobile

users. This sector-to-users path loss distribution is important because of three main

reasons:

i. It is more relevant for planning and optimizing cellular networks than the classic

path loss formulas.

ii. It can be extracted from actual measurements, which are usually performed

automatically by the cellular network.

iii. As we show later in this thesis, a general model can be built to describe the

sector-to-users path loss distribution. Using this model, we can extract

interesting network features from empirical path loss data.

One major difficulty which arises when we try to model the sector-to-users path loss

distribution is the indoor/outdoor path loss differences. It is accepted that for indoor

calls, penetration and in-building losses of 7-20dB [1, 7, 8] should be added to the

total path loss. Many of the path loss models mentioned in section 1.2.1, solve this

difficulty by dealing with the indoor and outdoor cases separately, and by adding a

penetration loss factor for the indoor case. In the "real world", the measurements

contain a mixture of indoor and outdoor calls, and we must take this into account.

Some previous work has been done on the subject of sector-to-users path loss models,

and on the indoor/outdoor issue:

Barucha and Haas (2008) [11] derived an analytic formula for sector-to-users path

loss distribution in the case of uniformly spatial calls distribution, without considering

the indoor/outdoor issue. We actually made an almost identical computation,

independently, which is presented in section 3 and compared to this paper's results.

Zhu and Durgin (2005) [12] studied Received Signal Strength (RSS) methods for

locating indoor and outdoor cellular calls. In their work, they calculated the RSSAN,

the average of the N strongest RSS measurements of a specific call. They assumed

that the RSSAN is log-normally distributed with different means and variances for

indoor and outdoor calls. In their measurement set at the Georgia Tech campus, they

measured an indoor mean RSSA6 of -97.8dBm with standard deviation of 14.1dB, and

outdoor mean RSSA6 of -85.5dBm with standard deviation of 9.7dBm. These

differences between the indoor and outdoor log-normal distributions allowed them to

achieve ~90% correct indoor/outdoor estimates with a simple threshold estimator.

Villebrun et al. (2006) [8] studied a few methods for discrimination of

indoor/outdoor/incar calls in a cellular network. They state the basic differences

between the three situations – penetration loss for indoor calls, and large RSS

fluctuations and cell change rates for incar calls. They then present a mobility

detection algorithm, based on fluctuations of the RSS. Finally an indoor/outdoor

discrimination algorithm is presented, based on a Neyman-Pearson criterion, where

the different indoor and outdoor a-priori distributions are calculated using a network

planning software simulation. Results are given only for a small number of

measurements, using special mobile equipment that records the power control

measurements it takes and sends over the network. Even though their methods are

designed to use data from standard wireless networks, no such examples are

presented.

Alaya-Feki and Le Cornec (2007) [6] suggest methods for intelligent real-time

analysis of RRM (Radio Resource Management) measurements, for improving RRM

processes. They too rely on RSS measurements taken by the mobile user, and use

smoothing and regression techniques to estimate the different attenuation components

– propagation loss, shadowing and fast fading. They use these estimations to

discriminate unmoving, pedestrian and incar mobile users, according to differences in

the fluctuations of the shadowing and propagation loss components. Indoor situations

are ignored, and the authors do not state what kind of fluctuations is expected in

indoor measurements. Another application they present is an amelioration of the

handover process, by better choosing the next server cell, relying on the slope of the

neighboring cells RSS Vs. time measurements, in addition to the RSS values

themselves.

1.3 This thesis contribution

In this thesis, we introduce new methods for measuring and modeling the path-loss

statistics in cellular networks.

In section 2 we develop a theoretical model for the cellular "sector-to-user" path loss

distribution. We show that under real-life conditions, this model is close to Gaussian,

and use empirical data to show that the model corresponds well to sectors which have

high rates of either indoor or outdoor calls.

In section 3 we deal with the issue of indoor/outdoor mixing, and suggest a Gaussian

mixture (GM) model for the path loss distribution. We study different estimators for

Gaussian mixture distributions. Three different estimators are compared, for

distributions with known means and variances, and we introduce an Expectation-

Maximization (EM) algorithm for distributions with unknown means, variances and

mixing probability.

Empirical path loss distributions are presented in section 5, from a real UMTS cellular

network in Israel. Results are analyzed and examples are given for two different

applications – estimating sectors' physical coverage area, and estimation of the

sectors' indoor/outdoor ratio.

Conclusions and suggestions for future research are presented in section 6.

The uniqueness and innovation of this thesis consist of three main aspects:

1. Development of a theoretical model for "sector-to-user" path loss distributions

plus a quantitative range of parameters (which are met in real-life conditions), in

which the theoretical model is equivalent to a simple unimodal Gaussian model.

2. Introduction of new applications – monitoring each sector's path loss

performances, physical coverage area and indoor/outdoor ratios. The monitoring

can be done almost in real-time, and further improve cellular network design and

optimization tools.

3. Development of the model and applications based on readily available

measurements from the cellular network, which are measured and logged

automatically by network operators.

Chapter 2

Cellular Path Loss Distribution – a theoretical computation

In this section we present a theoretical computation of the expected sector-to-users

path loss distribution, based on a few simple assumptions. The aim of the computation

is to give a "first order" analysis of the nature of the path loss distribution, and to

better understand the factors which dominate it.

The main assumptions of the computation are:

I. The geographic distribution of the cellular calls is uniform –

As we have no prior information about the geographical distribution of the calls,

the most reasonable assumption is to assume the distribution is uniform. We

believe that in urban areas this assumption is usually valid, on the average,

although in each sector the geographic distribution of the calls might be

clustered in specific areas, such as office buildings, malls, main junctions etc.

II. The signal power median decays according to a power law –

Since we want a general model, we turned to the group of statistical methods,

like COST-231 and the Okumura-Hata model. Both models (plus some other

statistical models) give a power law decays for signal power medians in cellular

frequency bands. Each model provides different constants and factors for

different parameters (such as the height of the antenna, the height of the

buildings in the signal space etc.). But for examining the general nature of the

distribution, as we shall see ahead, it was sufficient to consider the path loss

model as a power law, e.g. sent receivedP X Ar P

= (2.1),

where X represents a slow-fading zero-mean log-normal distribution.

III. The geographic area in which mobile phones communicate with the sector

antenna is a slice with an opening angle θ and radius R –

In addition this is actually an assumption of a distinct division of the cellular

network into separate cells. In reality this division is not distinct nor has clear

borders, and the spatial form of the cells does not have a well defined shape.

Nonetheless, it is a good enough assumption for the geographical area of a

general cell.

IV. The measurements are statistically independent –

This assumption validity depends on the way the measurements are taken. We

expect the measurements to be statistically independent if, for example, each

measurement is taken for a different call, and the call positions are independent

too. In this thesis the measurements are taken in multiple times during a call (at

each handover), and so the independency is weaker.

V. Shadowing effect add an independent log-normal factor, with constant mean and

variance in the sector area –

Since we develop a statistical model, the independency is reasonable. The

constant mean and variance of the shadowing effect is a common assumption in

2( ) 2 , 0r

rf r r R

R=

the literature, though non-constant values can be added to the model, requiring

numerical computation.

Under these assumptions, we compute the distribution of the path loss:

We first compute the distribution of the path loss using the power law formula, and

then consider the X log-normal factor. Relying on the statistical independency of

the two phenomena, taking the log-normal factor into account is done by convoluting

the two distribution functions.

We define PL to be a random variable representing the path loss.

According to assumption II –

[ ] [ ] 10log( ) 10 log( ) (0, )ratio dBPL X Ar PL A r N

= = + + (2.2)

According to assumptions I and III, we can compute the distribution of r, the random

variable representing the distance between the sector's antenna and the mobile user –

(R represents the maximal radius of the sector's slice) (2.3)

Ignoring the log-normal factor (for now), we can compute the distribution of PL as a

function of r –

10( ) 10log( ) 10 log( ) '( )

ln(10)g r A r g r

r

+ = (2.4)

[ ]

2

[ ] 2 2

( ) 2 ln(10) ln(10)( )

'( ) 10 5dB

rPL dB

f r r rf PL r

g r R R = = = (2.5)

According to (2.2), ignoring the independent log-normal factor – [ ]1

10[ ] 10log( ) 10 log( ) 10

dBPL

dBPL A r r A −

= + = (2.6)

Therefore [ ]

[ ]

2

5[ ] 2

ln(10)( ) 10

5

dB

dB

PL

PL dBf PL AR

−

= [ ] 10log( ) 10 log( )dBPL A R− + (2.7)

And we get an exponential distribution for PL. As a sanity check we validate that

[ ] [ ] [ ]( ) 1dBPL dB dBf PL dPL

+

−

= (2.8)

Taking the shadowing log-normal factor into account, we convolute (2.7) with the

log-normal distribution to get: 2[ ][ ]

2

[ ]

2

5 2[ ] 2 2

ln(10) 1( ) 10

5 2

dBdB

dB

PLPL

PL dBf PL A eR

−−

= (2.9)

Figure 2.1 shows the path loss distribution according to equation (2.9), for values of

α=3, A=103.7 (parameters specified for UMTS path loss model by ITU, [7]),

R=500[meters] and σ = 6[dB] and 12[dB]. The red line represents the propagation loss

without the log-normal shadowing factor, and the blue lines represent the total path

loss.

Figure 2.2 shows the path loss distribution according to equation (2.9) for the same

parameters, but with two different R values of 500 and 1000 meters.

Figure 2.3 shows a real sector path loss distribution (black), compared to equation

(2.9) (blue) and a unimodal Gaussian estimation (dashed purple). In this figure and

A were specified as in figures 2.1, 2.2, by ITU [7], while R and were set manually

to create a good fit.

Figure 2.1 – Cellular sector-to-users theoretical distribution - I

Red line – Propagation loss without shadowing

Blue narrow distribution – Total path loss with shadowing, σ = 6[dB]

Blue wide distribution – Total path loss with shadowing, σ = 12[dB]

Figure 2.2 – Cellular sector-to-users theoretical distribution - II

Blue narrow distribution – R = 500[meters], σ = 6[dB]

Blue wide distribution – R = 500[meters], σ = 12[dB]

Purple narrow distribution – R = 1000[meters], σ = 6[dB]

Purple wide distribution – R = 1000[meters], σ = 12[dB]

Figure 2.3 – Theoretical Vs. Measured Path Loss Distribution

Black line – Measured path loss distribution from sector EKG142183

Blue line – Equation (2.9), with values manually chosen for a good fit

Purple dashed line – A unimodal Gaussian fit

Two important conclusions arise from the above computation and from equation (2.9):

1. As might be expected, the maximal propagation loss (without the shadowing

effect) depends logarithmically on R (eq. (2.7)). Taking the shadowing factor into

account, we still get a clear dependence of the maximum likelihood path loss (the

maximum of the path loss PDF), and of the path loss mean on the sector's radius.

Figure 2.4 shows this dependency for α=3, A=103.7 [7] and σ=9[dB] (the value of σ

is less relevant, since a 6-12[dB] range of values changes the maximum likelihood

and mean path loss by only ~1dB). As we present in section 5.6, this dependency

can be used reversely, to estimate the coverage area of cellular sectors by their

path loss distribution, and thus help cellular operators to better monitor and

optimize their network.

2. Looking at figures 2.1-2.3, a question rises as to how different the suggested

model is from a simple Gaussian model. Though we get a clear exponential

distribution for the propagation loss, we see that with real propagation loss and

shadowing parameters, the path loss distribution is quite similar to Gaussian.

Figure 2.5 shows the Kullback-Leibler divergence between equation (2.9) and a

Gaussian fit, for fixed α, A, and R values and different standard deviations. We see

that for real-life cellular path loss parameters (i.e. standard deviations above

~5[dB]), a simple Gaussian model describe the path loss distribution well, and in

most cases the complicated expression (2.9) is not needed.

Figure 2.4 – Path loss mean Vs. Sector's maximal radius, for model values of A=103.7

and σ=9[dB]

Figure 2.5 – Kullback-Leibler divergence between (2.9) and its Gaussian fit, Vs.

the standard deviation σ used in (2.9)

As mentioned in section 1.2.2, Barucha and Haas [11] made an almost identical

computation (though they didn't relate to the conclusions presented above). Their goal

was to develop an analytic formula for the distribution of path losses for uniformly

distributed nodes in a circle. This is the same case as ours, though they didn't target

their computation specifically for the cellular case, and indeed it is relevant for any

uniformly scattered wireless network, with a distance power law path loss behavior.

They used different symbolization but came to the same result as (2.9). They then

continued the analytical computation of the convolution, to get:

2 2[ ]

2

[ ]

2 ln(10)( ) 2 (ln(10))

[ ] 2

ln(10)( ) 1 ( )

dB

dB

b PL a

bPL dBf PL e erf D

bR

− +

= − (2.10)

Where: 2 2

[ ] log( ) 2 ln(10)10log( ) 10

2

dBPL b ab b Ra A b D

b

− − += = = (2.11)

Barucha and Haas also compared the computation to Monte-Carlo simulations of the

path loss distribution, and as expected the computation and simulations were in good

agreement. Good agreement was also found between the computation and simulations

done for hexagonal sectors, which are slightly more similar to the real shape of

cellular sectors.

Chapter 3

Gaussian Mixture Model

3.1 Why choose a GM model?

In section 2 we derived a theoretical expression of the sector-to-user path loss

distribution. As was demonstrated in figure 2.3, there are some real cellular sectors

whose path loss distribution is in agreement with that expression. Another simple

model which might be suggested (As was shown in figures 2.3, 2.5) is a unimodal

Gaussian distribution. However, it turns out there are some sectors with completely

different path loss behavior. Figures 3.1 and 3.2 show two such sectors. A unimodal

Gaussian estimator is presented (The purple dashed line), but it is clear that neither

the unimodal Gaussian model nor the expression derived in section 2 can describe

these kinds of distributions.

The main factor which was not referred to in section 2 is the indoor penetration loss,

usually taken to be 7-20dB [1, 7, 8]. Taking the indoor penetration loss into account,

we propose a bimodal Gaussian mixture (GM) model for the path loss distribution:

2 2

1 1 2 2( ) ( , ) (1 ) ( , )PLf pl p N p N = + − (3.1)

Where: 2

1 1( , )N – Represents a Gaussian model for outdoor calls path loss distribution.

2

2 2( , )N – Represents a Gaussian model for indoor calls path loss distribution.

p – Represents the mixing probability between indoor and outdoor calls in

the sector coverage area.

We note that in Israel there is a mandatory use of hands free cell phones in cars, and

therefore the effect of in-car penetration loss is expected to be small. Since in-car calls

are transmitted through an antenna outside the vehicles, these calls are treated as

outdoor calls.

In section 3.2 we give a short background about the Gaussian Mixture estimation

problem. In section 3.3, we present some estimation methods for cases in which the

means and variances of the two Gaussians are known or can be estimated separately.

In addition a comparison is presented between the different estimators and the

Cramer-Rao Lower Bound, which is computed in appendix A. In section 3.4, an

Expectation-Maximization (EM) algorithm is presented, for estimating all five

parameters ( 1 1 2 2, , , ,p ) with no prior knowledge other than the model of

equation 3.1.

Figure 3.1 – Measured Path Loss Distribution (EKG141952) Vs. Unimodal Gaussian fit



Figure 3.2 – Measured Path Loss Distribution (EKG142181) Vs. Unimodal Gaussian fit



3.2 GM background

Parameter estimation of Gaussian mixtures has been widely studied in the past, with

studies on many different aspects of the problem – univariate and multivariate

densities, estimating parameters when some a-priori information is given, lower

bounds for different cases etc (general background on Gaussian distributions and

estimations can be found in [13, 14]). The problem of estimating the five parameters

( 1 1 2 2, , , ,p ) in a mixture of two univariate normal distributions was considered

as long ago as 1894 (by Karl Pearson, [15]).

Studies which are more relevant to this thesis include those made by –

B.S. Everitt (1984) [16], who presents a comparison between different Maximum

Likelihood (ML) algorithms for estimating the five parameters in a mixture of two

univariate normal distributions (Expectation-Maximization algorithm, Newton's

method, Fletcher-Reeves algorithm and Nelder-Mead Simplex algorithm).

Dick and Bowden (1973) [17], who studied the estimation of the five parameters,

when independent sample information is available from one of the populations.

G.R. Dattatreya, who studied a few aspects of the problem –

• Estimating parameters of a mixture of M Gaussian densities, with known and

distinct means and unknown (possibly different) variances (Dattatreya 2002,

[18]).

• Estimating the prior probabilities in a mixture of M classes with known class

conditional distribution (Dattatreya & Kanal 1990, [19]). This study deals with

general distributions, though it's applicable to Gaussian distributions as well,

and even presents an example regarding the estimation of the prior probability

of a Gaussian mixture.

• Estimating prior probabilities in a mixture of M multivariate Gaussians with

known means and unknown common covariance matrix (Dattatreya & Fang

2003, [20]).

The most suitable estimation algorithm for our problem is probably the Expectation-

Maximization algorithm, which is presented in section 3.4.

3.3 Estimators for distributions with known means and variances

We begin our study of Gaussian mixture estimators by looking at a simple Maximum-

Likelihood estimator, for distributions with known means and variances.

Given a bi-modal Gaussian mixture model – 2 2

1 1 2 2( ) ( , ) (1 ) ( , )Yf y p N p N = + − (3.2)

and a vector of N statistically independent measurements y , the total distribution

function of y is:

1

( ) ( )N

Y Y n

n

f y f y=

= (3.3)

The total likelihood and log-likelihood functions are, respectively:

( ) ( ) ( )| |

1

| |N

Y p Y p n

n

L p f y p f y p=

= = (3.4)

(3.5)

Maximizing the total likelihood function is a difficult task, which will be dealt with in

section 3.4.

A simpler task will be maximizing the likelihood function for each measurement

separately. This estimator thus classifies each measurement to one of the two

Gaussians, and after classifying the N measurements, we define a "Maximum-

Likelihood Classifier" (MLC) estimator by:

1 1( , ) _

_

#ˆ

#

N classifications

MLC

Total Samples

p

= (3.6)

We denote

2121

0

( )

2 0 1

2 2

1 1

1 1

22 2

x

x

xA e dx erfc

− − − = =

(3.7)

And

22022

( )

2 0 2

2 2

2 2

1 11

22 2

xxx

B e dx erf

−−

−

− = = +

(3.8)

For 0x such that

2 20 1 0 2

2 21 2

( ) ( )

2 2

2 2

1 2

1 1

2 2

x x

e e

− −− −

= (3.9)

Notice that when 0p = ˆ1

MLC

BE p

B=

−, when 1p =

1ˆ

MLC

AE p

A

−= and in

general:

( ) 1 2

1 1ˆ 1

1 1 1MLC

A B A B BE p p p p C p C

A B A B B

− − = + − = − + = +

− − − (3.10)

1 2

1,

1 1

A B BC C

A B B

− = − =

− − (3.11)

We see that ˆMLCp is a biased estimator, though an unbiased version of it can be easily

defined: 1_

2

ˆˆ MLC

MLC unbiased

p Cp

C

−= (3.12)

( 1C and 2C are computed for each set of 1 1 2 2, , , .)

( ) ( )( ) ( )( )|

1

ln ln |N

Y p n

n

l p L p f y p=

= =

Dattatreya and Kanal [19] developed a different estimator, for the general case of

estimating mixing probabilities in multiclass finite mixtures, when the class

conditional density functions are known:

Consider a random variable X with a multiclass distribution function:

1

( ) ( | )M

X i X i

i

f x p f x =

= (3.13)

Where

M is the number of classes.

, 1,...,ip i M= are the unknown prior probabilities

( | ), 1,...,X if x i M = are the known class conditional density functions of X.

Now consider ( )h X , a vector of M functions of the mixture random variable X :

1

1 1

( ) ( ) ( ) ( ) ( | )

( ) |

M

i i X i j X j

j

M M

j i j j ij

j j

E h X h x f x dx h x p f x dx

p E h x p h p

=− −

= =

= = =

= =

H

(3.14)

Since ( | ), 1,...,X if x i M = are known, ( ) |i j ijE h x h are known too.

( )iE h X can be estimated by averaging 1

1( ) ( )

n

i i k

k

h n h xn =

= , and the resulting

estimator is: 1 ˆˆ ( ) ( )PPEp n h n−= H (3.15)

The design of a Prior Probability Estimator (PPE) then boils down to finding M

functions ( )ih X such that the matrix H is invertible. Such functions exist if and only

if ip are identifiable [19, p. 152].

The ( )ih X functions presented by Dattatreya and Kanal for the Gaussian mixture

problem (and for some other problems) are:

( ) ( | )i X ih x f x = (3.16)

And the estimator is:

( )

( )

1| 1

1

1

|

|1

ˆ ( )

|M

X kn

PPE

k

X k M

f x

p nn

f x

−

=

=

E (3.17)

When E is a matrix whose elements ije are defined by:

( | | )ij X i je E f x (3.18)

And for a bi-modal Gaussian mixture: 2

2 2

( )

2( )

2 2

1

2 ( )

i j

i j

ij

i j

e e

−−

+=

+ (3.19)

Figure 3.3 shows the estimation of p for the data of sector EKG141952. We estimate

the means of the two Gaussians as the local maxima of the distribution, and estimate

the variances of the Gaussians by taking into account only the data samples that falls

in the 'outer' side of the two Gaussians. Figure 3.4 shows the estimation of p by the

same algorithm, but with a slight shift outwards (2%) in the Gaussians mean values.

As expected, figure 3.4's estimation is better than figure 3.3's, and simple estimation

algorithms can be developed by trying few Gaussian mean values or by iterative

methods.

Figure 3.3 – MLCunbiased Vs. Measured Path Loss Data, Sector EKG 141952

Black line – Measured Path Loss Distribution

Blue line – MLCunbiased estimation

Red line – The two separate Gaussians

Figure 3.4 – MLCunbiased Vs. Measured Path Loss Data, Sector EKG 141952


Blue line – MLCunbiased estimation

Red line – The two separate Gaussians

Figures 3.5, 3.6 show the mean square error (MSE) of the two unbiased estimators,

compared with the Cramer-Rao lower bound (for CRLB computation, see appendix

A). The MSE was computed versus the distance between the Gaussians' means, with

standard deviation of 1, and taking the mean value of the MSE over p values between

0 and 1. Sample size of 100 samples was used in figure 3.5, and 10,000 samples were

used in figure 3.6.

Figure 3.5 – Estimators' MSE Vs. Distance between the Gaussians (100 samples)

Black dotted line – _ˆ

MLC unbiasedp MSE

Purple line - ˆPPEp MSE

Blue line – Cramer-Rao Lower Bound

Figure 3.6 – Estimators' MSE Vs. Distance between the Gaussians (10,000 samples)

Black dotted line – _ˆ

MLC unbiasedp MSE

Purple line - ˆPPEp MSE

Blue line – Cramer-Rao Lower Bound

3.4 The EM Algorithm

In section 3.3 we developed GM estimators for cases in which the means and

variances of the Gaussians are known. Other estimators exist for other types of a-

priori information (some of them are mentioned in section 3.2). Theoretically, even

without known means and variances, we could have used separate estimators to

estimate the means and variances of the two Gaussians (as was exampled in figure 3.3

and 3.4). These estimators could use additional information, such as geographical

information (distances to the mobile users, distances from other sectors, buildings

heights in the sector's area etc.), information about the expected indoor attenuation,

information from drive tests and more.

Since we searched for good estimation without relying on additional information, we

chose to study estimators which estimate all five parameters together. In section 6 we

relate to the issue of integrating additional information and/or constraints to our

estimator and present some ideas for future research in this area.

Among the different estimators groups for the GM case (most of them are reviewed in

[16]), the EM estimator is the most widely used and is well suited to our problem.

The algorithm is thoroughly described in the literature ([21] – [26]) and we will give

here only a short overview of it.

The EM (Expectation-Maximization) algorithm was introduced in 1977, in a classic

paper by Dempster, Laird and Rubin [21]. Though the authors themselves state that

the method had been "proposed many times in special circumstances" by other

authors, the 1977 paper generalized the method and developed the theory behind it.

The algorithm is a general method of performing ML parameter estimation of an

underlying distribution from a given data set, when the data is incomplete or has

missing values. It is also used in many cases when optimizing the likelihood function

is analytically intractable, but the likelihood function can be simplified by assuming

the existence of missing or hidden parameters. The latter is the case with our problem

of Gaussian mixtures estimation.

The EM algorithm is an iterative algorithm, composed of two steps (as hinted in its

name) – The Expectation step and the Maximization step.

Let's assume we have a data set X, and we want to estimate the parameter set Θ. The

classic ML estimator will try to maximize the likelihood function:

( ) ( | ) ( | )XL L X f X = = (3.20)

As mentioned earlier, there are many cases in which maximizing L(Θ) is analytically

intractable. In some of these cases, we can add another "hidden" data set Y and define

a new "complete" data set Z={X, Y}.

The new "complete-data" likelihood function will be:

,( ) ( | ) ( | ) ( , | )complete complete Z X YL L Z f Z f X Y = = = (3.21)

And we denote the likelihood function of equation (3.20) by ( )incompleteL .

The new "hidden" data set Y is unknown, and it is actually a random variable (or

vector). Thus the complete data likelihood function is not deterministic, and can be

regarded as a function ,( | , ) ( )complete XL X Y h Y = (3.22) of a random variable Y

where X and Θ are constants. Now we can assume initial parameters 0 and

compute the expectation of the complete-data likelihood function with respect to

random variable Y. We define

( ) ( ) ( )1 1 1

, |, | , | , , | | ,i

i i i i i i

Y complete Y X YQ E L X Y X E f X Y X− − −

=

(3.23)

And respectively the log-likelihood function:

( ) ( )( )

( )( )

( )( ) ( )

1 1

1

, |

1

, |

, log | , | ,

log , | | ,

log , | | ,

i

i

i i i i

Y complete

i i

Y X Y

i i

YX Y

y Y

q E L X Y X

E f X Y X

f x y f y X dy

− −

−

−

=

= =

=

(3.24)

In many cases, as in the Gaussian mixture case, although it is impossible to find an

analytical solution for maximizing the incomplete likelihood (or log-likelihood)

function, computing ( )1,i iq − and maximizing it with respect to i is analytically

possible.

Fixing i as ( )( )1max ,i iArg q −

, it can be proven that:

( ) ( )1i i

incomplete incompleteL L − (3.25)

And the process converges to a local maximum of the likelihood function.

We wish to state two important remarks on the issue of local maximum:

1. Notice that in the Gaussian mixture case (and others), the likelihood function is

not bounded and has several singularity points. One such singularity point, for

example, is when the mean of one of the Gaussians is equal to one of the samples,

and the Gaussian's variance tends to zero.

2. When the two Gaussians are too close to each other (the Rayleigh criterion, as a

rule of thumb), there might be many close local maxima, and the estimation result

might be very dependant on the initial values of the process. We encounter this

problem in some of the sectors' estimates, and we'll discuss this issue more

thoroughly in section 5.4.

Chapter 4

The monitored system

The work presented in this thesis has been implemented on data records from an

actual UMTS cellular network in Israel. The network is comprised of 256 sectors, in

and around Beer-Sheva, a medium-size city in the south of Israel (population of

~185,000). Almost all of the sectors lie in the city area of around ~100km2, a few lie

near main rural roads, and others in some small towns or villages. Among the sectors

which lie inside Beer-Sheva, some are located densely near the city center, and the

density becomes smaller towards the outskirts of the city (see Figure 4.1).

Figure 4.1 – Sector locations in the Beer-Sheva area1

In general, cellular networks are quite complex, and contain many levels,

relationships, hierarchies, and interconnections between the different network

modules. To make things worse, different terminology is used to describe different

types of cellular networks, e.g. GSM, UMTS, CDMA etc.

Since the focus of this thesis is on path loss models, which should be similar and

easily applicable to all kinds of cellular network protocols, we define a simplified

model of a cellular network, which is suitable to our needs and can be easily

implemented to any other cellular network protocol.

1 This figure is based on data from the cellular operator, which was analyzed using Schema Ltd. Ultima

Mentor software and presented using Google Earth software.

The network model consists of three main levels:

1. The User Equipment (UE) –

This is the mobile user end equipment, usually a hand-held cellular phone.

While the appropriate term in 3rd generation systems (like UMTS) is UE, GSM

systems use the term Mobile Station (MS) with similar meaning.

2. The cellular Sector –

We use this term to describe the basic communication module which

communicates with the UEs. Usually in the literature, this is referred to as a cell

or a sector. Though there are sectors which cover 360o, in most cases 2-3

sectors are located at the same place, each covering an 180o or 1200 slice

respectively, thus creating a full 360o cover. A network device called Node B

facilitates the wireless communication in the sectors. Usually each Node B is

responsible of a group of 1-3 sectors, all positioned in the same place. The

common term for Node B in GSM networks is the BTS (Base Transceiver

Station).

3. The radio network controller (RNC) –

This is the governing element in the network. UMTS networks use the term

RNC, and it is responsible for control of the Node Bs that are connected to it. It

also carries out radio resource management, some of the mobility / handover2

management functions and it's the point where encryption is done before user

data is sent to and from the mobile. GSM networks use slightly different

network governing mechanism, and use the term BSC (Base Station Controller).

Usually the UEs' and sectors' radio measurements are logged in the RNCs.

Figure 4.2 – Typical trisector Node B units [27, 28]

As this thesis was meant to be easily applicable, no special measurements were taken,

and only data which is regularly collected by the network was used.

We used GPEH (General Performance Event Handler) data, which is the format used

by Ericsson's UMTS equipment to monitor the network parameters and performances.

GPEH records are in fact messages which are sent over the network, and contain

numerous types of measurements. These messages are sent between different modules

in the network (UE's, Node B's, RNC's etc.), and the cellular operator can decide

which messages will be saved for later analysis. Log files are created every 15

minutes, containing all the relevant messages from that time period.

As can be understood from its name, GPEH messages are event-evoked, meaning that

a message will be sent (and saved if defined so) only when a specific event happens.

For example, messages containing radio signal power measurements are sent

whenever a handover process happens. We used these messages to measure the path

loss, by looking at the power of a pilot signal which is transmitted by the sectors at

known power levels (The power levels change from sector to sector, but these

changes are accounted for in the analysis process).

2 A UMTS handover event occurs when a UE gains or looses connection to one of the sectors in its

vicinity. A hard handover is the event of changing the UE's "best-server" sector, while in a soft

handover event the "best-server" sector stays the same, but the list of the UE's available sectors

changes.

Chapter 5

Result Analysis

5.1 Highly bi-modal sectors

We begin our result analysis by looking at sectors which show a highly bi-modal

behavior. Unsurprisingly, these sectors are located in places with large outdoor and

indoor areas. Such places were found near the city's central bus station, in large

shopping complexes with large outdoor parking areas and in the city's university area.

In the following figures we present empirical path-loss distributions (black), EM

estimation results (dashed blue), estimated parameters and the Kullback-Leibler

divergence between the modeled and empirical probability density functions.

The red lines represent the relevant sectors. Each "V" is located where the sector

antenna is located, directed at the same direction, and has a 60° opening (similar to

the 55°~70° horizontal widths of most of the urban sectors' antennas). The length of

the red lines does not represent the actual size of the coverage area (see section 5.6 for

more on sector coverage area estimation).

Figure 5.1 – Google Earth photo of the central bus station, sectors EKG153363 and

EKG142181

Figure 5.2 – Measured Data Vs. EM Estimation of EKG153363


Blue dashed line – Mixed Gaussian path loss distribution with 1 1 2 2( , , , , )p =

estimated by the EM algorithm





estimated parameters:

2

1 1

2

2 2

ˆ ˆ101.2 103.7

ˆ ˆ120.1 102.3

ˆ 0.556p

= =

= =

=

Kullback divergence = 0.00073


2

1 1

2

2 2

ˆ ˆ96.8 75.8

ˆ ˆ118.0 100.7

0

ˆ 0.391p

= =

= =

=


Figure 5.4 – Google Earth photo of the "BIG" shopping center, sector EKG141952






2

1 1

2

2 2

ˆ ˆ107.7 61.9

ˆ ˆ126.1 61.4

ˆ 0.516p

= =

= =

=


We wish to highlight a few interesting remarks:

1. The difference between the two estimated indoor Vs. outdoor Gaussian means

is ~18-20dB, as expected from penetration loss.

2. The variances of the Gaussians are similar to those mentioned in the literature

[6, 7], expressing a standard deviation of ~7-11dB.

3. The path loss distribution of figures 5.2 and 5.3, near the central bus station,

looks a bit like a tri-modal Gaussian mixture. This might be caused by an

indoor/outdoor/in-car discrepancy (Despite Israel mandatory hands free cell

phone usage, cellular calls from within buses can be considered as 'regular' in-

car calls). Notice that near the shopping center (figure 5.5) the distribution

shows a much more explicit bi-modal behavior.

5.2 Highly uni-modal sectors

Our analysis found highly uni-modal sectors in most of the city's dense urban area.

Figures 5.6-5.9 show the location, distribution and estimation of some of these

sectors. Notice that due to a relatively high shadowing effect in dense urban areas, the

measured standard deviation of these distributions is around ~9-10dB. As was

presented in section 2, with these values of standard deviations, we expect to see

distributions very similar to Gaussian, and indeed our EM estimator produces very

high p̂ values.

Figure 5.6 – Google Earth photo of an urban area, sector EKG143613





(The small peak around 147dB (marked with a red circle in figure 5.7) is due to the

measurement equipment minimal power threshold, leading to a maximum possible

path loss measurement)

Figure 5.8 – Google Earth photo of urban area, sector EKG159853


2

1 1

2

2 2

ˆ ˆ106.2 64.6

ˆ ˆ123.8 71.6

ˆ 0.055p

= =

= =

=






5.3 Influence of repeaters

In areas which contain cellular repeaters, we should expect to see similar bi-modality

(or multi-modality of higher order) in the path loss distribution.

In UMTS, the cellular repeaters do not communicate with a specific sector, but rather

act as amplifiers for the UEs, and connect to the 'strongest' sectors in the area, much

like the UEs themselves. Therefore we can't examine a specific sector, and we need to

examine sectors which look towards the repeater location.

Figures 5.10-5.16 show path loss distributions in two areas which contain repeaters.

One area (figure 5.10) is in the city university campus. Since it contains large outdoor

and indoor areas as well, we should expect to see a more complex path loss behavior.


2

1 1

2

2 2

ˆ ˆ98.1 9.4

ˆ ˆ1

0

21.5 81.9

ˆ 0.002p

= =

= =

=


Figure 5.10 – Google Earth photo of the university area sectors and repeaters






2

1 1

2

2 2

ˆ ˆ100.9 64.6

ˆ ˆ119.1 100.7

ˆ 0.304p

= =

= =

=






Figures 5.13-5.16 show an industrial area in the outskirts of the city. We look at the

three sectors of EKG148911/2/3, and see a clear difference in the distributions of the

sectors looking toward the repeater. Regarding these sectors (EKG148912 and

EKG148913), the EM estimation results were indistinct and dependent on the

estimation initial values. All EM runs estimated a Gaussian with ~90% mixing

probability and a mean around ~114-117dB (EKG148912) and ~117-120dB

(EKG148913). However, for some initial values the second Gaussian was estimated

to be around 110dB (EKG148912) and 112dB (EKG148913), and for others the

second Gaussian was around 134dB (EKG148912) and 136dB (EKG148913). This

might be caused by a multi-modality of a higher order in the distributions (a tri-modal

of indoor/outdoor/repeater might be expected in such sectors), and can be solved by

integrating additional information (as the expected indoor/outdoor path loss means),

by calibrating the model, by regarding the Kullback-Leibler divergence or by other

estimation techniques, for distributions with multi-modality of a higher order (see

section 6.2).


2

1 1

2

2 2

ˆ ˆ106.9 62.0

ˆ ˆ123.6 57.5

ˆ 0.502p

= =

= =

=


Figure 5.13 – Google Earth photo of the area around EKG148911/2/3





1

3 2


2

1 1

2

2 2

ˆ ˆ110.8 111.7

ˆ ˆ123.2 90.2

ˆ 0.039p

= =

= =

=





estimated by the EM algorithm (first option)

Red dashed line – Mixed Gaussian path loss distribution with 1 1 2 2( , , , , )p =

estimated by the EM algorithm (second option)

2

1 1

2

2 2

ˆ ˆ114.4 110.0

ˆ ˆ134.3 37.3

ˆ 0.905p

= =

= =

=


estimated parameters (blue line):

estimated parameters (red line):

2

1 1

2

2 2

ˆ ˆ110.0 22.0

ˆ ˆ117.1 145.7

ˆ 0.11p

= =

= =

=





estimated by the EM algorithm (first option)

Red dashed line – Mixed Gaussian path loss distribution with 1 1 2 2( , , , , )p =

estimated by the EM algorithm (second option)

5.4 Dependence of initial values

As mentioned in sections 3.4, the EM algorithm gets initial values as an input for the

iterative estimation process. Therefore we must validate that our estimation is

independent of these initial values. In fact we want to validate that the likelihood

function has a clear global maximum, and in cases with different local maxima we

want to be able to choose the right local maximum, describing the actual path loss

distribution.

Figures 5.17 and 5.19 show different estimation results from 100 different initial

means and variances (chosen randomly), for the path loss distributions of sector

EKG141952 (see figure 5.5) and EKG148912 (see figure 5.15). The blue and red dots

represent the log-likelihood value and Kullback-Leibler divergence, respectively,

achieved in the last step of the EM algorithm. In the background we plot the

histogram of p̂ , the estimated indoor/outdoor mixing probability.

2

1 1

2

2 2

ˆ ˆ117.3 100.4

ˆ ˆ135.6 34.4

ˆ 0.90p

= =

= =

=


estimated parameters (blue line):

estimated parameters (red line):

2

1 1

2

2 2

ˆ ˆ111.7 18.4

ˆ ˆ119.9 128.1

ˆ 0.10p

= =

= =

=


Figure 5.17 – EM estimation with different initial parameters values, EKG141952

Blue dots – Log Likelihood final value of each algorithm run

Red dots – Kullback-Leilbler Divergence of algorithm run

Background blue histogram – p̂ estimation histogram

The two 'groups' of estimated values marked by the black arrows, represents the same

global maximum of the likelihood function. They are due to stopping of the algorithm

run coming from different 'directions', as shown in figure 5.18.

Figure 5.18 – EM Algorithm steps of two different runs (right and left), EKG141952

Blue dots – Log Likelihood value of each step

Red dots – Kullback-Leilbler Divergence of each step

left side estimated parameters: 2

1 1

2

2 2

ˆ ˆ107.5 60.3

ˆ ˆ125.8 63.0

ˆ 0.50304p

= =

= =

=

right side estimated parameters: 2

1 1

2

2 2

ˆ ˆ107.9 63.3

ˆ ˆ126.3 59.9

ˆ 0.52712p

= =

= =

=

Figure 5.19 – EM estimation with different initial parameters values, EKG148912

Blue dots – Log Likelihood function of each algorithm run


Background blue histogram – p̂ estimation histogram

In figure 5.19, we see two relevant estimations (i.e. two relevant local maxima of the

likelihood function): one estimates p̂ = ~0.1, 1̂ = ~110dB, 2̂ = ~117dB, and the

other estimates p̂ = ~0.9, 1̂ = ~114dB, 2̂ = ~134dB (see estimated PDFs in figure

5.15). While the first estimation yields a smaller Kullback-Leibler divergence, it

doesn't have a higher log-likelihood value. In such cases we should combine other

information in order to decide which estimation to choose. This information can be

the expected outdoor path loss mean, the expected difference between the indoor and

outdoor Gaussian means etc.

Although some of the sectors exhibited behaviors similar to that presented in figure

5.19, most analyzed cases had a clear global maximum and yielded a clear estimation,

independent of the EM algorithm initial parameters values.

5.5 EM results for pure Gaussian and theoretical model distributions

As a sanity check, we want to test our EM estimator with a pure Gaussian distribution

and with distributions given by the theoretical computation of section 2 (equation

(2.9)).

Figure 5.20 shows results of the EM estimation for a pure Gaussian distribution. We

see that the EM algorithm does not, supposedly, converge to the "right" solution (i.e.

p̂ close to zero or unity). However, further analysis of the results shows that the EM

algorithm would have eventually converged to zero or unity results if we let it

continue running. Because the tested distribution is pure Gaussian, the estimation fits

the data well for many different p̂ values, and the algorithm stopping conditions are

met at a relative early stage. Other than that reason, we see some abnormal results,

marked with black arrows, which we believe are caused by the finiteness of the data

and the excellent fit that can be achieved for a unimodal Gaussian, by a combination

of two other close Gaussians.

Figure 5.20 – EM estimation with different initial parameters values, pure Gaussian

data



Figure 5.21 shows results of the EM estimation for the theoretical distribution of

equation (2.9), with standard variation of 9dB. We see that although the distribution is

similar to pure Gaussian, we don't see the phenomena of figure 5.20, and the EM

algorithm estimates similar p values for different initial values. The fact that the

algorithm estimates a ~80% indoor ratio even for the "purely indoor" (or outdoor)

theoretical distribution of equation (2.9), places a limit on the validity of the

indoor/outdoor estimation to clear bi-modal distributions. Calibration of the model

(see section 6.2.1) might lead to better estimation of the indoor/outdoor ratio, in

general and specifically for distributions with low bi-modality.

Figure 5.21 – EM estimation with different initial parameters values, equation (2.9)

theoretical computation



5.6 Sector coverage area estimation

Using the theoretical computation of the path loss distribution, which was presented

in section 2 (figures 2.2, 2.4, and equation (2.9)), we can estimate the sector's

maximal radius. Combined with the antenna's aperture and location, we can

empirically estimate the actual geographical area covered by the sector. As mentioned

in section 2, real-life sectors usually do not have clear and "nice" shapes or borders.

However, the coverage area estimation presented here enables a preliminary

estimation, which is useful when no other data (such as "drive-tests") is available, or

when easily accessible or frequent coverage area estimation is needed.

Figure 5.22 shows a Google Earth photo of central Beer-Sheva coverage map

estimation. The semi-transparent slices represent the sectors' coverage area. Each slice

is positioned according to the original sector's antenna position and aperture. The

mean and variance of the "outdoor" Gaussian, estimated by the EM algorithm, are

used to determine the sector's maximal radius.

Figure 5.22 – Beer-Sheva city center coverage map estimation

Figure 5.23 shows a Google Earth photo and coverage map estimation of Omer, a

small village outside of Beer-Sheva, and of the roads near it. As expected, we see

larger sectors in rural areas, with less sector overlaps (notice that both figures are

identically scaled).

Figure 5.23 – Coverage map estimation outside of Beer-Sheva

The estimation results presented here are not calibrated and the model constants were

taken from the UMTS model of ITU [7]. We relate to the calibration issue in section

6.2.1, and believe that calibrating the model parameters will lead to much more

accurate results.

Chapter 6

Conclusions and suggestions for future research

6.1 Conclusions

In this thesis we presented a new model for "sector-to-users" path loss distributions,

based on a theoretical computation and empirical data.

The computation assumed uniform spatial distribution of the users, a power law decay

of the radio signal and a "pie slice" shaped sector.

We then compared the computation to a Gaussian model, and showed both

theoretically and empirically that the Gaussian model is valid in most cases, where the

shadowing effect standard deviation is above ~5dB and the indoor/outdoor mixing is

low.

The complete model incorporates the indoor/outdoor differences, and the effect of

indoor penetration losses on the path loss distribution. A Gaussian mixture (GM)

model was suggested, and the problem of estimating GM parameters was studied. We

developed two simple estimators for estimating the GM mixing probability with

known means and variances, and compared them with a third estimator developed by

Dattatreya and Kanal [19].

Since usually the means and variances of the two Gaussians are unknown, we studied

the Expectation-Maximization (EM) algorithm for estimating the distribution's

parameters. The EM estimator was used with empirical data from an actual UMTS

cellular network in Israel, and results were analyzed for different sectors. We

presented examples of sectors with high bi-modality behavior, located in areas with

high indoor/outdoor mixing, and sectors with high uni-modal behavior, located in

dense urban areas with high indoor rate.

Two main applications have resulted from the research:

1. Estimation of the sector's physical coverage area. Since inter-sector

interferences are one of the key factors influencing network capacity, knowing

the sectors' real physical area and boundaries is a highly important goal. Our

novel method enables the estimation of the physical boundaries without

relying on positioning data (based on time of arrival, received signal strength

etc.), which is sometimes not available or hard to implement in cellular

networks.

2. Estimation of a sector's indoor/outdoor call ratio from path loss data which is

automatically measured and logged by cellular operators. The indoor/outdoor

ratio is important both in design of cellular networks (e.g. installation of

indoor sectors or repeaters in certain places), and in optimization of network

parameters.

In addition, this thesis presented a theoretical background with quantitative validity

boundaries (shadowing effect > ~5dB), for describing "sector-to-users" path loss

distributions as Gaussian.

6.2 Suggestions for future research

6.2.1 Calibrating the models presented in this thesis

Calibration of the models is important mainly for estimating the sectors' boundaries

and the indoor/outdoor ratio. It can be done in several ways, though two ways seem

more practical than others:

I. "Drive-tests" – "Drive-test" data can be used to estimate the outdoor path loss

distribution in a given sector, with a uniform or other spatial distribution of the

calls. "Drive-test" data can also be used to measure the physical boundaries of

sectors, at least for outdoor calls, and thus calibrate the sectors' boundary

estimation.

II. Geographic location data – Integrating geographic location data of the calls can be

used to calibrate the models for both the indoor/outdoor and sectors' boundary

applications, in addition to other results which are discussed in 6.2.4.

6.2.2 Studying the convergence rate of the EM algorithm and developing techniques

for its acceleration

The issue of the EM algorithm convergence rate hasn't been deeply dealt with in this

thesis, though it is widely studied in the literature (for example, see [26]). We did

notice that the EM convergence rate was quite slow for many of the sectors,

especially those with low bi-modality. Practical applications which use

indoor/outdoor ratio estimation will need to be based on a faster version of the EM

estimator. This might be achieved by using standard EM acceleration techniques, or

by using methods specifically "tailored" for the indoor/outdoor case (such as choosing

good initial values, relying on known penetration losses etc.).

6.2.3 Using the models as a new "measuring tool"

The models presented in this thesis constitute a basis for future analysis of the "sector-

to-users" path loss behavior. An interesting research might be analyzing the path loss

distribution, sector's size or indoor/outdoor ratio in different hours of the day, on

different days of the week, seasons etc. The path loss distribution will be effected also

from non-routinely events, such as large gatherings ("hot spots") or even

environmental phenomena such as heavy rainfall, snow etc (for more on

environmental monitoring by cellular networks, see [29]).

6.2.4 Combining location and path loss data

As cellular localization becomes widespread in recent years, as well as the use of GPS

receivers in cellular mobile phones, combining the geographic location of the call

with path loss and other power control data will lead to promising results. Model

calibration, more accurate indoor/outdoor ratio estimation and further improvements

in the power management processes in cellular networks can be achieved. In our

opinion, the lack of easily accessible data is the current bottleneck for this kind of

researches. However, we believe that this bottleneck will be removed in the coming

few years and a whole field of research opportunities will become available.

6.2.5 Mobility estimation, in-car path loss behavior and discrimination

Another future research direction might be expanding the current indoor/outdoor ratio

estimation to indoor/outdoor/in-car discrimination. Using localization data, path loss

or other power control data (see Villebrun et al., [8]), a better indoor/outdoor/in-car

discrimination might be achieved, statistically or per single call. This in turn can lead

to further improvements in the efficiency of the network.

6.2.6 Integrating additional information to the Gaussian mixture estimation

In section 3.3 we discussed estimators for Gaussian mixture distributions with known

means and variances. In section 3.4 we presented the EM algorithm for distributions

with unknown means, variances and mixing probability. As was mentioned in section

3.4, additional information can be used to improve these estimations. Some

constraints can be suggested, like a certain difference between the means of the indoor

and outdoor Gaussians, limits or even specific values for the variances of the

Gaussians etc. Information about the physical size of the sectors' area (from separate

data, such as geographic location of the calls or the network sectors), can help

estimate the Gaussian means separately, and thus help the EM algorithm or even

estimate the indoor/outdoor mixing probability using simpler estimators.

Appendix A

CRLB Computation

In section 3.3, we examined three estimators for estimating the prior probability p for

two Gaussians mixture with known means and variances ( ˆMLCp – equation 3.6,

_ˆ

MLC unbiasedp – equation 3.11 and ˆPPEp – equation 3.16).

We now compute the Cramer-Rao lower bound of the problem.

For the CRLB computation we make the following definitions:

y – A random variable which represents the path loss in a given measurement.

y – A random vector of N path loss measurements.

f1 – The path loss PDF for outdoor measurements.

f2 – The path loss PDF for indoor measurements.

fy – The total path loss PDF.

Thus

( ) ( )

( )2

1

212

12

1

1|

2Y

y

Yf y f y outdoor e

− − = = (A.1)

( ) ( )

( )2

2

222

22

2

1|

2Y

y

Yf y f y indoor e

− − = = (A.2)

And

( ) ( ) ( ) ( )

( )

( )

( )2 2

1 2

2 21 22 2

1 22 2

1 2

1 11 1

2 2Y Y

y y

Yf y pf y p f y p e p e

− − − − = + − = + − (A.3)

We assume the measurements are statistically independent, and therefore –

( ) ( )1

N

Y Y n

n

f y f y=

= (A.4)

The likelihood and log-likelihood functions are, respectively:

( ) ( ) ( )| |

1

| |N

Y p Y p n

n

L p f y p f y p=

= = (A.5)

( ) ( )( ) ( )( )| |

1

ln | ln |N

Y p Y p n

n

l p f y p f y p=

= = (A.6)

( )( )( ) ( )( )

( ) ( )

( )

( )

( )

( ) ( )

2 2

1 2

2 21 2

2 2

1 2

2 21 2

2 2

1 2

21 2

||1

1

2 2

1 2

12 2

1 2

2 2

1 2

ln |ln |

1 1

2 2

1 11

2 2

1 1

2 2

n n

n n

n n

N

Y p n NY p nn

n

y y

N

y yn

y y

f y pf y pl p

p p p

e e

p e p e

e e

=

=

− − − −

− − = − −

− − − −

= = =

−

= =

+ −

−

=

( ) ( ) ( )

2

2 2 2

1 2 2

2 2 21 2 2

12 2 2

1 2 2

1

1 1 1

2 2 2

n n n

N

y y yn

Nn

n n n

p e e e

A

pA B

− − −= − − −

=

=

− +

=+

The first derivative:

(A.7)

When An and Bn are defined to be:

( ) ( ) ( )2 2 2

1 2 2

2 2 21 2 22 2 2

1 2 2

1 1 1,

2 2 2

n n ny y y

n nA e e B e

− − − − − − − (A.8)

Thus the second derivative will be:

( )

( )

2 2

221

Nn

n n n

l p A

p pA B=

= −

+ (A.9)

And now we can compute Fisher's information:

( )( )

( )

( ) ( )

2 2

221

2 2

2 21

Nn

n n n

Nn

n n n

l p AJ p E E

p pA B

A AE N E

pA B pA B

=

=

= − = =

+

= =

+ +

(A.10)

For

( ) ( ) ( )2 2 2

1 2 2

2 2 21 2 22 2 2

1 2 2

1 1 1,

2 2 2

y y y

A e e B e

− − − − − − − (A.11)

And the Cramer-Rao lower bound:

( ) ( )1CRLB p

J p= (A.12)

Though we didn't get a 'nice' analytic expression for the CRLB, it can be easily

computed for any set of parameters.

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http://en.wikipedia.org/wiki/File:CellPhoneTower_OR.jpg

http://www.mbs.ie/images/antenna3.jpg

תקציר

מהווים גורם חשוב ביותר בהתנהגות הרשת ובביצועיה, ומודלים רבים תהפסדי ערוץ ברשתות סלולאריו

נבנו בעשורים האחרונים לתיאור התופעה. הרוב המכריע של המודלים מתייחס לתלות של הפסד הערוץ

וצמחייה במסלול במרחק בין המשדר למקלט, ובפרמטרים אחרים כגון גובה האנטנות, תכונות בניינים

האות ועוד. בעבודה זו, נבחנה התפלגות הפסדי הערוץ שבין אנטנת התא הסלולארי לבין משתמשי הקצה,

ומוצג מודל חדשני לתיאור התפלגות זו.

בשלב ראשון אנו מבצעים חישוב אנליטי של ההתפלגות. חישוב זה מבוסס על מספר הנחות יסוד, כגון

חק ע"י חוק חזקה, התפלגות מרחבית אחידה של השיחות הסלולאריות תאור דעיכת האות כתלות במר

יועוד. בהמשך, אנו מציעים מודל גאוסי פשוט לתיאור ההתפלגות, אשר מתבסס על החישוב התיאורט

בשילוב נתונים סטנדרטיים מהספרות. אנו מראים, באופן תיאורטי וכן אמפירית, שהמודל הגאוסי מתאים

, המקיימים את שני התנאים הבאים: לתאים סלולאריים רבים

. 5dBסטיית התקן של אפקט ההצללה בתא גדולה מ~ .א

(, outdoor)( או לחילופין מחוץ למבנים indoorרוב גדול מהשיחות בתא נעשות מתוך מבנים ) .ב

אך ללא ערבוב גדול ביניהם.

התא, ומציע המודל השלם המוצג בעבודה זו מכליל עירוב של שיחות מתוך ומחוץ למבנים באותו

על מנת לתאר את התפלגות הפסדי הערוץ. (Gaussian Mixture)להתבסס על עירוב גאוסיאנים

במהלך העבודה נלמדו ופותחו מספר אלגוריתמים לשערוך הפרמטרים של התפלגות הפסדי הערוץ:

עות מראש, שיטות שונות לשערוך פרמטרים עבור התפלגות עירוב גאוסיאנים עם תוחלות וסטיות תקן ידו

עבור התפלגות עירוב גאוסיאנים עם תוחלות וסטיות EM (Expectation-Maximization)ואלגוריתם

תקן אשר אינן ידועות מראש. אלגוריתמי שערוך אלה נותחו ונבחנו בצורה נרחבת, עבור נתונים

ממספר של אחת מחברות הסלולאר בישראל. הנתונים נמדדו UMTSאמפיריים של הפסדי ערוץ מרשת

רב של תאים, ביניהם תאים עם עירוב גבוה ונמוך של שיחות מתוך ומחוץ למבנים, תאים אשר משתמשים

בממסרים לשיפור הכיסוי שלהם, ועוד.

בהתבסס על המודל והמשערכים אשר מוצגים בעבודה זו, ובהתבסס על העובדה שנתוני הפסד הערוץ

של הרשתות הסלולאריות, אנו מציעים שתי אפליקציות נמדדים ונשמרים בצורה אוטומאטית ברוב הגדול

עיקריות לשיפור תכנון וטיוב רשתות סלולאריות. האפליקציה הראשונה היא שערוך אמפירי של "מפות

הכיסוי" של התאים ברשת הסלולארית. "מפות כיסוי" אלו יאפשרו זיהוי של חורים בכיסוי, וכן חפיפות

פליקציה השנייה היא שיערוך היחס בין מספר השיחות אשר מבוצעות אשר פוגעות ביעילות הכיסוי. הא

מתוך ומחוץ למבנים. יחס זה חשוב למתכנני רשתות סלולאריות, וכן תורם לטיוב הפעלת רשתות קיימות.

לסיכום, מוצעים מספר כיוונים למחקרי המשך. העיקרי מביניהם הוא כיול של המודל המוצג בעבודה זו,

)ביצוע מדידות בשטח הכיסוי של התאים ע"י "drive-tests"ע ע"י שימוש במדידות אשר יכול להתבצ

ציוד מדידה ייעודי(. כיווני מחקר נוספים יכולים להיות שילוב של מידע נוסף במודל כגון מידע על מיקומי

השיחות, שילוב המודל באלגוריתמים קיימים לתכנון וטיוב רשתות סלולאריות וכן ניטור פרמטרים

סביבתיים בהתבסס על המודל המוצע בעבודה זו ובשילוב מחקרים אחרים בתחום הניטור הסביבתי.

אביב -אוניברסיטת תל הפקולטה להנדסה ע"ש איבי ואלדר פליישמן

סליינר-בית הספר לתארים מתקדמים ע"ש זנדמן

מידול ושיערוך התפלגות הפסדי ערוץ ייםראסלול יםבתא

ת חשמל בהנדס" מוסמך אוניברסיטה"חיבור זה הוגש כעבודת גמר לקראת התואר ואלקטרוניקה

ידי -על

יהונתן ברוידא

תשס"טאב

אביב -אוניברסיטת תל הפקולטה להנדסה ע"ש איבי ואלדר פליישמן

סליינר-בית הספר לתארים מתקדמים ע"ש זנדמן

מידול ושיערוך התפלגות הפסדי ערוץ םים סלולארייבתא

חשמל בהנדסה ה" מוסמך אוניברסיט" חיבור זה הוגש כעבודת גמר לקראת התואר

ואלקטרוניקה ידי -על

יהונתן ברוידא

מערכות –הנדסת חשמל העבודה נעשתה במחלקה ל

ירון-חגית מסרפרופ' הנחיתב

תשס"טאב

MODELING AND ESTIMATION OF CELLULAR PATH LOSS …

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