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					                King’s College London

                 University of London



     MSc in Telecommunications by Research




An Outdoor-Indoor Interface Model for
      Radio Wave Propagation
       for 2.4, 5.2 and 60 GHz.



                          Prepared by:

                    Michael Döhler



                         Supervised by:

                 Prof. A. H. Aghvami




      A thesis submitted for the degree of MSc by Research.

                           1998-1999
                                 Dedication




To the particles of any race and colour,
      which have to obey their live long
                   Maxwell’s equations.




     ii
                                                                          Acknowledgements




Acknowledgments
        This first line is dedicated to my supervisor Monica Dell’Anna, who never tired of
supplying me with new work and a healthy portion of encouragement using her alluring
Italian smile. My special gratitude is for Prof. Aghvami who showed me what is really
important in the abstruse world of Digital Communications.
        I had the pleasure of working with Roger Cheung whose generous good-
naturedness and copiously interesting ideas helped me more than once out of a sheer
unsolvable situation. Without him this year would have been a tedious drudgery.
        Спасибо тебе, мама ты моя. Ты всегда давала мне правельные советы и
поддерживала меня в трудные моменты. Без тебя я бы этому никогда не достиг.
Благодорить я тоже хочу всем моим друзям в Москве: бабушке, Лиде, Зине, Ване,
Насте, Нин, Квадже, Оле, Максиму и Катарине за их откровенную любовь и
дружбу.
        Besonders lieblichen Dank meinem Bruder Eddie, welcher schlafend mein Leben
manchmal in einen stürmischen Ozean verwandelt hat. Auch meiner lieben Schwester
Anita, die ich einfach ungemein gern habe. Lächelnder Dank meinen Freunden Steffen,
André, Ilia und Friedi in Deutschland, jeder welcher in seiner Art skuril, witzig und eigen
meinen Weg begleitet hat. Leiser Dank auch Anja. Stiller Dank meinem Vater.
        Most gratitude to those who made London's gloomy days shiny. To my flat-mates
Yunis, Max, Ulrike, Yukako and Helena; to my lab-mates Victor, Vasileios, Patrick,
Giorgio, Nelly, Jean-Philippe and Julio; to the Chemist-mafia Eva, Marco, Piero, Alex
and Alberto; and not least to Victoria, Marta and Leo.
        Muchísimas gracias a mis amigos castellanos y catalanes, cuya sangre
mediterránea fue como una brisa fresca en mi vida.
        Gemmuli, deixa’m emprar aquesta llengua secreta per dir-te com t’estimo! Tu
m’has alliberat i m’has fet florir.




London, 9.9.1999                                                           Mischa




                                            iii
                                                                                                                                     Table of Contents




Table of Contents

ACKNOWLEDGMENTS ..........................................................................................................................III

TABLE OF CONTENTS ...........................................................................................................................IV

ABSTRACT.................................................................................................................................................VI

INTRODUCTION ........................................................................................................................................ 1

1.     AVAILABLE OUTDOOR-INDOOR MODELS ............................................................................... 8
     1.1.     INTRODUCTION ................................................................................................................................ 8
     1.2.     PATH-LOSS MODELS ....................................................................................................................... 8
       1.2.1         Linear Path-Loss Model.......................................................................................................... 8
       1.2.2         Angle dependent Path-Loss Model ......................................................................................... 9
       1.2.3.        COST 231 Keenan and Motley Model.................................................................................... 9
     1.3.     FIELD-STRENGTH PREDICTING METHODS ...................................................................................... 10
       1.3.1.        Ray tracing ............................................................................................................................ 10
       1.3.2.        Method of Moments (MoM) ................................................................................................. 10
     1.4.     PARAMETER DEPENDENCIES AND TENDENCIES ............................................................................. 11
       1.4.1.        Grazing Angle ....................................................................................................................... 11
       1.4.2.        Penetration Loss Model Parameter ....................................................................................... 11
       1.4.3.        Frequency dependent Loss.................................................................................................... 11
       1.4.4.        Receiver Height inside a Building ........................................................................................ 12
       1.4.5.        Moisture Effects.................................................................................................................... 12
       1.4.6.        Penetration Loss Statistics .................................................................................................... 12

2.     PROPAGATION ALLOTMENTS ................................................................................................... 13
     2.1.     INTRODUCTION .............................................................................................................................. 13
     2.2.     TRANSMISSION COEFFICIENTS ....................................................................................................... 16
     2.3.     NON-SPECULAR TRANSMISSION DUE INTERIOR PERIODIC STRUCTURES ........................................ 19
     2.4.     SCATTERING DUE TO SURFACE ROUGHNESS .................................................................................. 23
     2.5.     SCATTERING DUE TO WALL INTERIOR METALLIC LATTICES AND MESHES .................................... 24
     2.6.     DIFFRACTION ................................................................................................................................. 26

3.     THE PROPAGATION MODEL ....................................................................................................... 28
     3.1.     INTRODUCTION .............................................................................................................................. 28
     3.2.     DETERMINISTIC TRANSFORMATION ............................................................................................... 29
     3.3.     TRANSFORMATION OF THE PROBABILITY FUNCTIONS ................................................................... 31
     3.4.     OUTDOOR TRANSMITTER AND INDOOR RECEIVER ......................................................................... 36



                                                                             iv
                                                                                                                                    Table of Contents


4.     APPLICATION .................................................................................................................................. 38
     4.1.     INTRODUCTION .............................................................................................................................. 38
     4.2.     THE GENERIC CELL ....................................................................................................................... 38
     4.3.     THE MODIFIED COST 231 – MOTLEY MODEL ............................................................................... 42

5.     CONCLUSIONS ................................................................................................................................. 48
     5.1.     CONCLUSION ................................................................................................................................. 48
     5.2.     FURTHER OUTLOOK ....................................................................................................................... 49

6.     APPENDIX I (GRAPHICS)............................................................................................................... 50
     6.1.     INTRODUCTION .............................................................................................................................. 50
     6.2.     TRANSMISSION COEFFICIENTS ....................................................................................................... 51
     6.3.     NON-SPECULAR TRANSMISSION DUE INTERIOR PERIODIC STRUCTURES ........................................ 53
     6.4.     SCATTERING DUE TO SURFACE ROUGHNESS .................................................................................. 55
     6.5.     SCATTERING DUE TO WALL INTERIOR METALLIC LATTICES AND MESHES .................................... 55
     6.6.     DIFFRACTION ................................................................................................................................. 55
     6.7.     THE GENERIC CELL ....................................................................................................................... 57

7.     APPENDIX II (FORMULAS) ........................................................................................................... 59
     7.1.     INTRODUCTION .............................................................................................................................. 59
     7.2.     TRANSMISSION COEFFICIENTS ....................................................................................................... 59
     7.3.     NON-SPECULAR TRANSMISSION DUE INTERIOR PERIODIC STRUCTURES ........................................ 61
     7.4.     SCATTERING DUE TO SURFACE ROUGHNESS .................................................................................. 70
     7.5.     SCATTERING DUE TO WALL INTERIOR METALLIC LATTICES AND MESHES .................................... 70
     7.6.     DIFFRACTION ................................................................................................................................. 70
     7.7.     PROOF OF ABSENCE OF SIDE LOBES FOR THE CELL-PHILOSOPHY .................................................... 85

8.     APPENDIX III (MATLAB) ............................................................................................................... 87
     8.1.     INTRODUCTION .............................................................................................................................. 87
     8.2.     TRANSMISSION COEFFICIENTS ....................................................................................................... 88
     8.3.     NON-SPECULAR TRANSMISSION DUE INTERIOR PERIODIC STRUCTURES ........................................ 89
     8.4.     SCATTERING DUE TO SURFACE ROUGHNESS .................................................................................. 91
     8.5.     SCATTERING DUE TO WALL INTERIOR METALLIC LATTICES AND MESHES .................................... 91
     8.6.     DIFFRACTION ................................................................................................................................. 91

TABLE OF FIGURES................................................................................................................................ 95

BIBLIOGRAPHY....................................................................................................................................... 97

INDEX ..................................................................................................................................................... MM




                                                                              v
                                                                                   Abstract




Abstract
The study presented in this thesis has been undertaken as part of the Radio Environment

work area of the UK’s Mobile VCE, whose Core Program of research involves seven UK

universities and more than twenty industrial organizations. The main objective of this

work is to provide a model for electromagnetic wave propagation through a windowed

wall, possibly with an internal periodic structure. The model operates as an interface

between outdoor and indoor propagation models at frequencies of 2.4GHz, 5.2GHz and

60GHz. An approximated deterministic approach has been chosen to be able to match

existing semi-empirical, deterministic and stochastic outdoor models to the appropriate

indoor models. The model embraces all participating propagation phenomena like

specular & non-specular transmission, scattering and diffraction. The extracted

approaches are then utilized to ease site-specific calculations. One approach considers a

generic wall as several sufficiently large cells embedding typical window-wall

constellations. The formulas elaborated in this thesis can be applied to such cells to give

tabled field and power distributions, where the cell shares should be added to give an

overall prediction. Another approach extends the COST 231 - Motley outdoor-indoor

model justified by strong influences of diffraction in primary penetrated rooms.

Furthermore, to make use of existing statistical indoor models, the given outdoor pdf's are

transformed into indoor pdf's using the well-known transformation of multi-dimensional

random variables. Thus the model developed allows one to predict the transformed

indoor field parameters from the known outdoor field-state, including their pdf’s. This

method is a trade-off between calculation time, accuracy and the ability to transform

pdf's.


                                            vi
                                                                                         Introduction




Introduction
                                                      Suddenly it appears to be so obvious why the
                                                      evolution brought us out of the water to the land,
                                                      since it’s quite impossible to use a Mobile Phone,
                                                      what for some among us seems to be the crème de la
                                                      crème of the evolutionary ladder, under water.



        From its very beginnings humankind seems to have been ruled by the desire to

commune. Wasn’t it this aspiration that made us speak? Wasn’t it this craving that forced

our brains to develop, not to talk nonsense the whole day long? Humans started to talk, to

express themselves and not last, to communicate. First they merely used their vocal

chords. When the distances increased they started to utilize tools like drums to make

noise over vast wilds. But all these methods were disadvantageous. They were annoying,

unreliable and everybody could listen and intercept the ongoing conversation. For those

reasons the Mobile Phone was just a question of time. In fact the only thing humanity

always had plenty of. A couple of milleniums have had to pass before a bunch of

celebrities like Kirchhoff, Maxwell, Sommerfeld or Shannon were born to make this

dream come true. None of them initially had a clue about what kind of ball they set

rolling. With the aid of them and thousands of famous and fameless scientists, many of

them not even capable of holding a normal conversation, we juggle around with

acronyms like UMTS, WLAN or CDMA and actually do know what they are supposed to

mean.

        Not all the work is done, though. The main path is hinted at, what remains is to

walk it. The edge to the second Millennium is characterized by the endeavour to merge

all telecommunication services into UPT (Universal Personal Telecommunications). It

should offer access to all kinds of services at a reasonable expense at any place and any


                                            1
                                                                                  Introduction


time. Certain steps have already been mastered as the elaboration of the 3rd generation

mobile phones standardized in UMTS (Universal Mobile Telecommunications System)

via GSM (Global System for Mobile Communications), DECT (Digital European

Cordless Telecommunications), TETRA (Trans-European Trunked Radio) and ERMES

(European Radio Message System). UMTS gives way to the accelerating and rising

demand for ISDN (Integrated Services Digital Network) and B-ISDN (Broadband-

ISDN), emphasizing high-bit-rate services, like multimedia, as well as voice and low-bit-

rate services. Restrictions of the transmitted bit rate are due to the limits of physical

resources such as the electromagnetic spectrum or the available power. Therefore highly

sophisticated coding, modulation and transmission techniques have to be investigated and

applied. The engineers’ struggle to achieve dBm coding or modulation gain certainly

reaches its limits. Yet there is still some hope left to ameliorate existing technologies or

even to find new ones. For example, I believe, quantum electrodynamics could be

exploited for novel transmission techniques. The work undertaken in this Master attaches

more importance to the approach mentioned first. The aim of the following is to give a

profound and complete background of the theory applied, namely classical

electrodynamics, which is then being used to extract a new model for outdoor-indoor

wave propagation.

       As    mentioned     above,   both    emerging     and   well-established    Personal

Communications Services (PCS) require an accurate prediction of the wave propagation

mechanisms for development of new techniques as well as system deployment. The

limited bandwidth available and the tremendously increasing number of users does

compel cell-network operators to a highly terse frequency re-use pattern. The objective is




                                             2
                                                                                 Introduction


to minimize interference with maximum range depending on cell type and power control

mechanisms. Coverage and power control algorithms are less crucial for cells with base

stations located above rooftop level, i.e. Macro, Large, Small or Umbrella Cells, since

those cells cover more or less geometrically manageable areas. Problems arise lowering

the base station below rooftop level, as it is for Micro and Pico cells, where the path

between transmitter and receiver is usually more randomly obstructed. Frontiers of this

cell type are uppermost fractal and naturally follow externally imposed architectural

patterns, e.g. street alignments or a certain room distribution on a floor. Interferences

become more predominant since cell isolation is an arduous task to achieve. Until

recently Micro and Pico Cells were exclusively restricted to either outdoor or indoor

environment. Soon the question arose whether an outdoor base station was able to cover a

given indoor area and vice versa or whether the loss through building walls is high

enough to reliably separate indoor and outdoor cells. Therefore, Micro and Pico cells

demand a meticulous knowledge of the radio wave behaviour through a building’s wall.

       Assuming the channel characteristics to be known in a certain area of interest,

accurate prognosis about Base Station and Mobile Station coverage-area, power-drop

probability, inter-cell and intra-cell interferences, etc. can be made. Various models exist

which predict both outdoor and indoor propagation parameters such as field strength,

polarization, (spatial) angle of arrival, time of arrival and their pdf’s. All these models

assume that the mobile and base stations are located in a similar environment, i.e. either

outdoor or indoor. They can be classified into Models for Field Prediction and Radio

Channel Models [1] that is circumstantiated below.




                                             3
                                                                                Introduction


       Models for Field Prediction give as one output the main transmission paths

including the losses along them. These models can be subclassified into Empirical,

Abstract-structure-based, Semi-empirical and Deterministic Models [2]. Empirical

Models are extracted from measurement data by means of regression methods. Typical

representatives are the legendary Okumura, Hata, COST-231-Hata and RACE dual-slope

Models. Abstract-structure-based Models, as the Walfish&Bertoni and Ikegami Models,

analytically provide propagation loss assuming a facile terrain structure. In the Semi-

empirical Models the given parameters of the Abstract-structure-based Models are

empirically corrected, as was done in the COST-231-Walfisch-Ikegami Model. All

models that use electrodynamic field integral equations or ray tracing/launching

techniques form the intricate, but not necessarily superior, class of Deterministic Models.

The more accurate output of these models is taken from a detailed terrain database.

       Radio Channel Models provide a more comprehensive description of the

propagation phenomenon providing the full characterization of the CIR (channel impulse

response) in a mobile radio environment. It has been distinguished between Stored,

Deterministic and Stochastic Channel Models. The first retain gathered site specific

channel impulse responses, the second make use of field calculations, whereas the latter

gives its parameters as realizations of random processes.

       The few existing outdoor-indoor studies and models [3-13] belong to the Models

for Field Prediction. In fact it makes little sense to extract a channel impulse response

exclusively for the interface. It is common to attach the outdoor-indoor model at least to

the indoor environment where predictions are required. The reason is that the large

number of different interface constellations would give raise to a enormous amount of




                                            4
                                                                                           Introduction


CIR’s or a too broad spread for the statistical parameters. It is perspicuous that the wave

penetration loss through a building is a function of parameters such as construction

materials of the buildings, distribution and size of the windows, existence of frames, fire

escapes, air conditioners, internal wall-reinforcement, foliage, the nature of the

surrounding buildings, the structure of the rooms, and the position of transmitter and

receiver. Beside this the direction of arrival has a predominant leverage, which is

therefore the main input parameter for all available interface models. Due to this broad

gamut of partially non-comparable parameters it became customary to distinguish

between several parameter classes. The first differentiation is made with the sites of

concern, such as Urban (typical, bad or dense), Suburban (hilly or non-hilly) and Rural.

The distinction is necessary to use the appropriate models to predict the field-strength at

the exterior wall with a given outside base station. The second differentiation classifies

various building types. In literature [3] it became normal to distinguish between eight

classes given as     (1)   Residential houses in suburban areas;   (2)   Residential houses in urban

areas; (3) Office buildings in suburban areas; (4) Office buildings in urban areas; (5) Factory

buildings with heavy machinery;           (6)   Other factory buildings, sports halls, exhibition

centres;   (7)   Open environment, e.g. railway stations or airports;         (8)   Underground. The

building type supplies information about the choice of the interface model and the values

of the parameters to be inserted into these deterministic models. A third differentiation

sorts the availability of certain more specific building details, e.g. frames, coating,

reinforcement, influential external scatters, percentage window-wall, etc. They determine

the magnitude of certain parameters and their spread. Further differentiation embraces

the frequency range and the bandwidth with respect to the coherence bandwidth of the




                                                    5
                                                                               Introduction


given channel. Concerning the frequency most elaborated outdoor-indoor models are

designed for frequencies around 1.8GHz, the operation frequency of PCN (Personal

Communication Networks). Regarding the bandwidth, it must be said that, although the

behaviour of the channel varies heavily over a vast range from 2.4GHz to 60GHz, the

channel might be regarded as wideband. This can be explained with the broad coherence

bandwidth of the interface compared with the current maximum available data rate. In the

literature distinguished wideband and narrowband measurements are taken with respect

to the outdoor and indoor environments.

       The challenge of this work is to expunge the disadvantages of the already

developed outdoor-indoor models, which use either time consuming site-specific

calculations or very simple (semi-) empirical propagation formulas. The former are

unable to transform given outdoor pdf’s into their indoor counterparts, whereas the latter

neglect strong diffracted waves acting severely in the primary base station facing rooms.

Within the core research, an important output of the Radio Environment work area is the

development of such an interface model for outdoor-indoor propagation. An

approximated deterministic approach has been chosen here to be able to match existing

semi-empirical, deterministic and stochastic outdoor models to the appropriate indoor

models. The model developed allows one to predict the transformed indoor field

parameters from the known outdoor field-state, including their pdf’s. This method is a

trade-off between calculation time, accuracy and the ability to transform pdf's. Another

enhancement pertains to the frequency range that is extended to 2.4GHz, 5.2GHz and

60GHz. The frequency around 2GHz is exploited by PCN. HIPERLAN captures the

frequency around 5.2GHz, although it might be extended to 17GHz. The 60GHz




                                            6
                                                                                 Introduction


frequency band is restricted to indoor applications yet, but an exhaustive study might

allow outside base stations to cover larger indoor areas. The physical picture drawn at

this frequency is quite different from the lower frequencies since it approaches the optical

boundary, and penetration losses increase enormously. Therefore the model developed

applies mainly to frequencies below 10GHz, where some comments for the 60GHz case

are added.



       Chapter 1 gives a broad survey about already existing outdoor-indoor models

treating in depth parameters of interest. In Chapter 2 all the propagation allotments are

handled and summarized to give the necessary basis for the actual outdoor-indoor model,

which is exhaustively treated in Chapter 3. In Chapter 4 this model is then applied to two

completely different signal-strength predicting models. Finally conclusions and further

vistas are drawn in Chapter 5. Appendix I (Graphics) includes all relevant graphics,

which would otherwise have overloaded the actual thesis, and in Appendix II (Formulas)

all needed formulas are derived. To enable the reader to follow and prove the given

formulas and graphics, Appendix III (Matlab) embodies the Matlab source-code used

throughout this work.




                                             7
                                                                                  Chapter 1




1. Available Outdoor-Indoor Models

1.1. Introduction
Network Operators require a thorough knowledge of the channel characteristic as power

distribution, its statistics, e.g. Rayleigh or log-normal, the maximum fading margins and

several probabilities of occurrence, e.g. Angle of Arrival (AOA), Time of Arrival (TOA)

or polarization. Furthermore general tendencies are of interest, e.g. power behaviour with

varying floors or the influence of open windows, moisture walls, etc. The fading statistics

are confirmed to be Rayleigh or Rice for the small-scale fading and log-normal for the

large-scale fading [9]. The fading margins are given either implicitly through the

magnitude or spread of the parameters, as in the COST 231-Motley Model [13], or

explicitly as a result of precise site-specific computations [12]. The latter method is

unnecessarily accurate since the precise location of the power drop is not required, but

rather its existence and extent. Thus they should be used to verify more approximated

models. Other parameters of interest are discussed in the following.

1.2. Path-Loss Models
1.2.1    Linear Path-Loss Model
The most frugal model with regard to the set of parameters is the Linear Path-Loss Model

proposed by Horikoshi [4]. It assumes the excess penetration loss in dB to be

approximately linearly dependent on the angle of incidence. The absolute loss in terms of

the angle of incident φ and perpendicular loss L0° is given as:

                                                     field at φ ° incidence
        L = L 0o − 20 ⋅ log(Γ(φ )) , where Γ(φ ) =                          .         (1.1)
                                                     field at 0° incidence

The model gives best predictions for normal incidence but fails for grazing angles.


                                                8
                                                                                     Chapter 1


1.2.2     Angle dependent Path-Loss Model
The European COST 231 project [5] extrapolated an empirical formula out of numerous

measurement campaigns varying the building type, the distance and the angle of

incidence. The all embracing formula yields the loss between the transmitter location and

a reference point just inside the building:

         L = 32.4 + 20 ⋅ log(f /GHz ) + 20 ⋅ log(d ) + We + WGe (1 − sin (θ )) .      (1.2)
                                                                             2




θ is now the grazing angle of the impinging wave. d is the physical distance between

transmitter and external building wall just outside the reference point, where free-space

and LOS conditions are assumed. We is the loss for the perpendicularly illuminated outer

wall and is gauged to be 4-10dB. WGe , in order of magnitude of about 20dB, is an

additional loss for perfectly grazing angles (see 1.4.). Additional terms accounting for the

floor-gain inside the building are omitted here.

1.2.3.    COST 231 Keenan and Motley Model
Within the framework of COST 231 the Keenan and Motley outdoor-indoor propagation

model was amended and floor-gain corrections were added to give the following formula:

                                         I                   J
         L = L 0 + 10 ⋅ n ⋅ log(d ) + å k f ,i ⋅ L f ,i + å k w, j ⋅ Lw, j            (1.3)
                                        i =1                 j =1



n is given as the power decay index, d is the distance, L0 is the path loss at 1m

distance, L f ,i and Lw, j are the loss for floor i and wall j , respectively. I and J are the

number of penetrated floors and walls.




                                                         9
                                                                                    Chapter 1



1.3. Field-strength predicting Methods

1.3.1.   Ray tracing
Ray tracing and cognate methods are powerful in the sense that they can predict

arbitrarily accurate with increasing site precision. One obvious drawback is that the static

character of this approach does not solve the entire field distribution at one instant in the

given space. Rather, it follows certain optical paths, which have to be truncated

depending on the accuracy required. Huge sites and high precision usually result in

formidable simulation times. A second severe drawback is the arduousness of including

diffracted rays, even though a whole theory, the Geometrical Theory of Diffraction, was

developed. Ray tracing naturally leads to penetration loss overestimation for grazing

angles and underestimation for angles > 60° . The reason for the former is the missing

diffraction term, whereas the latter is due to the neglected finite wall-thickness. Incoming

rays sense the window gaps to be wider than they actually are. And finally the ray tracing

philosophy is restricted to high frequencies, thus small wavelengths with regard to the

obstacles.

1.3.2.   Method of Moments (MoM)
A breakthrough was recently achieved by the research-group around B. De Backer [12]

who were able to combat the MoM’s most striking disadvantage: inefficient evaluation

techniques and towering memory consumption. This allows one to handle even large sites

in a passable computation time. The idea behind this approach is to solve the whole field-

distribution numerically. First of all the site is split into a set of linear segments along

which the field components are expanded into a series of pulse and overlapping triangle

functions. Galerkin testing is applied to achieve a system of linear equations. They can be



                                             10
                                                                                  Chapter 1


dramatically eased using some neglecting approximations and transforming it into a

sparse matrix. Finally well known linear algebra is applied to solve the set of equations.

Unlike ray tracing the MoM is not hampered by any high frequency limitation.



1.4. Parameter Dependencies and Tendencies
1.4.1.     Grazing Angle
It is remarkable to note that the penetration loss for perfectly grazing angle does not

approach infinity as Fresnel’s Theory postulates, but reaches a constant level of around

27.5 dB almost independent of frequency [5]. This can be attributed to diffraction and

scattering by any external scatters like window edges.

1.4.2.     Penetration Loss Model Parameter
The explicitly given penetration losses for normal or grazing incident angles in equation

(1.1), (1.2)   and (1.3) are not real physical losses, but are extracted from empirical

measurements and embrace all participating propagation effects including multiple

reflection and diffraction.

1.4.3.     Frequency dependent Loss
A general agreement on penetration loss in dependency of the chosen frequency has not

been found yet. The reason is that buildings, the material and general premises differ too

much from one measurement campaign to another. It has been reported [9] that as

frequency increases the penetration loss does not necessarily also increase, but might

succumb to serious fluctuations. Therefore, for increasing frequencies additional path loss

can be compensated by decreasing building penetration.




                                            11
                                                                                     Chapter 1


1.4.4.     Receiver Height inside a Building
Most of the papers cited make clear statements concerning the influence of the signal

strength in dependency of the receiver height inside the building. Most influential are the

shadowing clutters in front of the building. Trees let the receiver field strength decrease

from tree-height to the first floor. Above the obstruction height, when LOS conditions are

given, the power lever gradient is expected to level off. Measurements reported in [9] and

[10] show an augmentation of around 2dB/floor up to the surrounding hindrance-height

after which it remains nearly constant.

1.4.5.     Moisture Effects
Network operators in regions with frequent or torrential rainfalls should incorporate an

additional penetration loss coefficient for a sufficient confidence level of the operating

system. It could be shown [7] that due to higher reflection the penetration losses of a wet

wall are raised by 10 percent compared with a dry wall.

1.4.6.     Penetration Loss Statistics
Also in [7] the fading margin inside a building is evaluated. For it the outdoor log-normal

and the penetration loss statistic is considered. The assumed large number of independent

random processes that determine the penetration loss, e.g. material, permittivity,

moisture, incident angle, thickness, scatters, etc., presume one to expect the statistic to be

roughly log-normal, too. Since both effects arise independently the overall statistics is

log-normal with zero mean and a standard deviation given by:

         σ = σ outdoor + σ 2
               2
                           penetration .                                              (1.4)

The fade margin γ at the cell boundaries with given outage probability is gained through:

         poutage = Q(γ / σ ) .                                                        (1.5)


                                             12
                                                                                     Chapter 2




2. Propagation Allotments

2.1. Introduction
To find a tractable approach to the outdoor-indoor propagation model, first the objective

has to be clarified and then the solution methods scrutinized. The aim of the Radio

Environment work area of the Mobile VCE is to develop a Simulation Platform for real-

time and real-site studies [1]. Its backbone is a meticulous database including site specific

details from building to window and door positions. Any arbitrary number of users can be

assigned to a certain number of base stations using all imaginable transmission features,

e.g. different modulations, power control, hand-over etc. The claim for real-time ability

coping with the mountainous amount of computations makes it impossible to use either

straightforward solutions of Maxwell’s equations or the methods mentioned in the

previous chapter, e.g. MoM or ray-tracing. Since measurements are available, empirical

models ought to be constituted in the first stage. The parameters needed are extracted

from fitted regression curves and inserted into the above-given formulas. The detriments

are apparent since the parameters are confined to the measurement-site, probabilities are

not transformable and the whole physical picture behind the penetration is basically not

understood. Therefore, before a practical model is elaborated in the second stage, all

eventual propagation components are examined to form an approximated deterministic

model, which overcomes the aforementioned disadvantages.

       Using a versatile outdoor wave propagation model for three-dimensional terrain

[14] the field strength, polarization state and angle of arrival, as well as their pdf’s, at a

given wall surface can be predicted. Assuming the field states are known over the

windowed-wall surface being considered, it is possible to decompose the field into TE


                                             13
                                                                                          Chapter 2


and TM components with respect to the plane of incidence. The TE wave is linearly

polarized with the electric field vector perpendicular to the plane of incidence, whereas

the TM wave has its electric field vector in the plane of incidence. The impinging wave is

partly reflected, partly absorbed and partly transmitted (Table 1). The specular and non-

specular reflected components as well as the scattered and backward diffracted ones

contribute to the total outdoor-model. The specular and non-specular transmitted and

forward-diffracted components are captured by the outdoor-indoor-model (Figure 1). All

these propagation processes arise from the same physical principles obeying Maxwell’s

equations. Some of them, e.g. scattering and propagation through periodic structures,

show similar physical behavior, but are listed separately to make use of theory already

developed.

                          §   Specular reflection (Snell’s Law)
                          §   Non-specular reflection due to periodic structures (Floquet’s
    Reflected part            Theorem)
                          §   Scattering (Gaussian Scattering Matrix)
                          §   Backward diffraction (UTD)

                          §   Due to the structures conductivity, which results in a complex
    Absorbed part             permittivity ε = ε real − j ⋅ σ / ω assuming a time-harmonic
                              electromagnetic field

                          §   Specular transmission (Snell’s Law)
                          §   Non-spec. transmission due to periodic structures (Floquet’s
                              Theorem)
  Transmitted part
                          §   Scattering due to exterior wall periodicity and internal wall lattices
                          §   Diffraction (UTD)

       Table 1: Synopsis of the propagation effects


In order to thoroughly understand the outdoor-indoor propagation mechanism a

characteristic window-wall configuration was chosen (Figure 2 and Figure 3). Figure 2

depicts the top-view of a horizontal cross-section of such wall, where the left and the




                                                  14
                                                                                                                                                         Chapter 2


right part of the picture correspond to a solid and an internally periodic wall, respectively.

The periodic wall typically consists of a thin outer and inner wall and reinforcing

concrete cross girder. Figure 3 clarifies the notation for the angles used throughout the

following work. The main transmission phenomena, including specular and non-specular

propagation, scattering and diffraction, are now examined separately.

                                                                        Outdoor Channel




        Reflection at the                        Reflection at the                        Scattering at the exterior                Backward-Diffraction at
          exterior wall                       internally periodic walls                                  wall                       the exterior wall-edges




                                     Field-state at the wall-surface:                                E = ETE + E TM



                                              Transmission through                                                                   Forward-Diffraction at
     Transmission through                                                                       Transmitted scattered
                                              the internally periodic                                                                   the window-wall-
                                                                                                   components
     the window and wall                                   walls                                                                           configuration




                                                                          OUTDOOR-INDOOR
                                                                              CHANNEL



                                                                             Indoor Channel


              Figure 1: Decomposition of the propagation allotments


           Outer-Wall                                                                                         Mode n=-1
                                                       Specular impinging wave
                                                                                                                          Mode n=0, specular reflection
                    Exterior edge                                                                   φ
                                                                        Window aligned
 y             x                                                        wall face
                                                                                                                            Mode n=1
                                       Window (sing/ dbl)
                                                                                                                            h1…Outer Wall (εW)
 z                                                                                                                          2h2…Periodic Structure ( εWd1 ,εa d2)
                                                w
                                                                                                                            h1…Inner Wall (εW)


                                                                                                                               Mode n=1
                   Interior edge 1                                Interior edge 2

                                                                                                                          Mode n=0, specular transm.
                                    Window Frame                                                     Mode n=-1



              Figure 2:             Horizontal cross-section of a typical Wall-Window configuration showing both solid
                                    and periodic wall structures on the left and right part of the picture, respectively




                                                                                     15
                                                                                                Chapter 2




                                                                               θ … Elevation angle with
                                               y                               respect to the x-z-plane
                                x                      k
Normal to the                                      θ                           φ … Azimuthal angle with
                                           α
wall surface                                                z                  respect to the x-z-plane
                                               φ
                                                                               α … Azimuthal angle with
                                                                               respect to the plane of
                                                                               incidence or impinging angle
                                                                               with respect to the normal z



       Figure 3: Elevation and azimuthal decomposition for window and wall




2.2. Transmission Coefficients
The transmission coefficients for a single and double glazed window and a lossy wall are

easily obtained using the transmission line model. For the sake of computational ease the

generic case of a 5-layered dielectric is assumed as depicted in Figure 4, where the

thickness and permittivity ε of the individual layers is set according to the penetrated

object mentioned above.

                                    α                            Air, semi-infinite thickness


                                                                 Dielectric, thickness t




                 Air, thickness d



                 Dielectric, thickness t


                Air, semi-infinite thickness


       Figure 4: Generic 5-layer structure used for derivation of the propagation formulas.




                                                       16
                                                                                    Chapter 2


In the case of single glazing d is put equal to zero, t to half the actual pane-thickness and

ε = 19 − j ⋅ 0.1 . The double-glazing requires d to be put to the pane-distance and t to the

pane-thickness. Similarly, for the lossy wall d is put to zero, t to the half of the wall-

thickness and ε = 3.5 − j ⋅ 0.9 . Both permittivities are given in [15] for 1.8GHz and have

to be corrected for higher frequencies through measurements. The derivation of the

transmission formula can be gleaned from Appendix II; merely some important impacts

are discussed herein.

       It is remarkable to note that, in the case of single and double-glazing, the mutual

cancellation of the multiple reflected waves leads to severe fluctuations. The single

glazing case is shown in Figure 5 with the normalized transmitted TE power vs. the

impinging angle in parametrical dependency of the pane thickness 2t for all frequencies

concerned. The curves are shifted by 10dB and 20dB for f=5.2GHz and f=60GHz,

respectively. It can be seen already that an increase of the pane-thickness of 1mm can

result in a power drop of 5dB. For double-glazing a pane-thickness of 1.5mm and normal

incidence were assumed. Figure 6 depicts that case with the normalized transmitted TE

power in dependency of the pane-separation. The wall attenuates the electromagnetic

wave of approximately 1.1dB/cm, 2.2dB/cm and 26.6dB/cm for 2.4GHz, 5.2GHz and

60GHz, respectively. The wall-attenuation for 60GHz is so vigorous that the penetration

scenario severely depends on fluctuating short-time effects like open or closed windows,

moving people inside, etc. Additionally, the free-space propagation ‘loss’ becomes

intolerably high, limiting the range of application to a few hundred meters.




                                             17
                                                                                        Chapter 2




Figure 5: TE transmitted power vs. impinging angle for single glazing; f=2.4GHz (not shifted),
          f=5.2GHz (shifted by 10dB), f=60GHz (shifted by 20dB) pane-thickness 2t: 0.5mm,
          1.0mm, 1.5mm, 2.0mm




Figure 6: TE transmitted power vs. pane separation for double glazing; f=2.4GHz (not shifted),
          f=5.2GHz (shifted by 10dB), f=60GHz (shifted by 20dB) pane-thickness 2t: 1.5mm




                                           18
                                                                                                Chapter 2



2.3. Non-specular Transmission due Interior Periodic Structures
Buildings typically have walls constructed from concrete blocks or bricks. Reinforcing

grids or slabs form the periodicity in concrete walls. Bricks, possibly hollow, and cement

mould a much subtler periodic structure, which becomes relevant for the propagation

mechanisms at higher frequencies. Both wall types show general interior periodic

structures as depicted in Figure 7 and Figure 2, right. This leads to additional

transmission in directions other than the specular one, due to the excitation of higher-

order space harmonics, which can carry a significant amount of power depending on the

angle of incidence. The main impact is that ray-tracing and plain point-to-point path-loss

models lose their applicability. They have to be completed by the additional rays

emanating from the inner wall surface.

       To deal with this effect, the generic structure in Figure 7 has been proposed [15],

what is mathematically examined in Appendix II.


                                                         Mode n=-1
           Specular impinging wave
                                                                      Mode n=0, specular reflection
                                                  φ
                                                                          Mode n=1




                        d                                               Mode n=1

                                                                    Mode n=0, specular transmission

                                                      Mode n=-1


       Figure 7: Top view of a horizontal cross-section of an internally periodic wall.




                                                   19
                                                                                      Chapter 2


A. Concrete wall:

Assuming a commonly available 6″ thick concrete block with d=15cm, it can be shown

that, depending on the frequency and the angle of incidence, a different number of

excited space harmonics is coupled to the air and propagates into non-specular directions:

                                      n
       φ n = arcsin(sin ϕ 0 + 2π ⋅        )                                            (2.1)
                                     d ⋅k

                                              φn    … angle of the non-specular components
                                              ϕ0    … angle of the impinging wave
                                              n     … number of coupled space harmonics
                                              d     … periodicity of the structure
                                              k     … wave number
For a fixed impinging angle ϕ = 30 ° and a frequency 5.2GHz, the following propagation

directions can be calculated:

                         § φ 2 = imaginary, hence evanescent
                         § φ1 = 62.2°

                         § φ 0 = 30.0°

                         § φ −1 = 6.6°

                         § φ −2 = -15.6°

                         § φ −3 = -40.8°

                         § φ −4 = imaginary, hence evanescent
Those angles can also be obtained using Figure 8, which depicts the case for 5.2GHz. It is

the graphical realization of formula (2.1) . The vertical line at φ 0 = 30.0° crosses the

family of lines corresponding to the appropriate coupled space harmonics at the

propagating outbound angles.




                                                   20
                                                                                         Chapter 2




              Figure 8: The Outbound angles of the coupled Space Harmonics for f=5.2GHz in
                        dependency of the Inbound angles.



Using Floquet’s Theorem and some basic propagation formulas the power carried by the

individual specular and non-specular components can easily be calculated. Figure 9

shows the transmitted power (red line) for each n and the relative transmitted power with

respect to the transmitted specular component (black line) for the 5.2GHz case. It is

important to note that the graph below depicts the power carried by the space harmonics

as a function of the inbound angle and does not say anything about the actual propagation

direction of the space harmonics. The numbers on the right hand side of Figure 9 indicate

the coupled space-harmonics. The black reference lines clearly indicate that for

impinging angles between 20˚ and 60˚ the non-specular components carry, beside the

high absolute value, up to 30dB more power than the specular one.


                                              21
                                                                                            Chapter 2




                                                                                                n = +5


                                                                                                n = +4


                                                                                                n = +3


                                                                                                n = +2


                                                                                                n = +1


                                                                                                n=0

                                                                                                n = -1


                                                                                                n = -2


                                                                                                n = -3


                                                                                                n = -4


                                                                                                n = -5




       Figure 9: Relative radiated Power of the accordant Space Harmonics depending on the inbound
                 angle




B. Bricked wall:

It can be said that the non-specular components can be neglected for a bricked wall with a

frequency less than 20GHz, whereas for frequencies above 20GHz the non-specular

components start to carry a notable amount of energy. This fact is important for

predicting indoor propagation models for indoor communications at 60GHz.




                                                22
                                                                                    Chapter 2



2.4. Scattering due to Surface Roughness
The wavelength for 2.4GHz, 5.2GHz and 60GHz are λ =12.5cm, λ =5.7cm and

λ =0.5cm, respectively. Applying the Frauenhofer-criterion for flat surfaces yields:


                                                                α
                      ∆h          π
                  4π ⋅   ⋅ cosα >
                      λ           8
                             λ
                  cosα >                                                                 ∆h
                         32 ⋅ ∆h



  A rough estimation for bricked and concrete walls gives: ∆ h=2 ± 1mm. Therefore, the

  angle α has to be:

                         >0 degrees (arbitrary) for f=2.4GHz,

                         >26° degrees for f=5.2GHz and

                         >86° degrees for f=60GHz

  the surface to be considered flat. Are those conditions violated, the wall radiates:

                         (1) coherent scattering for highly correlated surfaces

                         (2) diffuse scattering for random surfaces.

Since no scattering is expected for the 2.4GHz case and some scattering for angles α <26°

for a frequency of 5.2GHz, which can hardly be handled itself, forward and backward

scattering can be neglected. The windowless 60GHz case is quite unsuitable for outdoor-

indoor propagation due to high attenuation losses. The existence of windows makes

propagation possible for a realistic power budget, however scattering then plays a

secondary role.




                                               23
                                                                                             Chapter 2




2.5. Scattering due to Wall Interior Metallic Lattices and Meshes
Most European buildings are reinforced with consecutively arranged metallic lattices

(Figure 10, left). Furthermore, plasterboard or wall covering plaster is often reinforced

with thin wire mesh (Figure 10, right). This results in attenuation and non-specular

transmission. The periodic nature of the structure allows decomposition of the scattered

field into a two-dimensional series including a specular component and a double sum of

grating lobes. For a one-layered mesh the same approach as for the internally periodic

wall is taken (see section 2.3.). The formulas differ only in the dielectric constant of the

periodic medium, which is highly conducting in this case.




                                                                                      h≈w




                                                                          w

               d                w


       Figure 10: Common metallic lattice (left) and Common reinforcing wire mesh (right).

As a consequence of some parametric calculations for metallic lattices it can be said that

the dominant propagation mode is the specular one. Assuming a rod diameter of 1cm, a

lattice periodicity of w=10cm and normal incidence for the TE-case the attenuation

amounts to 2dB for both frequencies 2.4GHz and 5.2GHz (Figure 11). Again, the case for

60GHz is omitted here due to strong general wall losses. Figure 12 depicts the

dependency of the transmitted TE power vs. the lattice-periodicity w for the specular and

three non-specular space-harmonics for the frequencies concerned. For w>10cm the

strongest component is constantly attenuated by 2dB.


                                                 24
                                                                                       Chapter 2




Figure 11: Transmitted Power of the induced space-harmonics for a metallic mesh at 2.4GHz
           (circle) and 5.2GHz (star)




Figure 12: Transmitted Power of the induced space-harmonics vs. lattice-periodicity at 2.4GHz
           (dashed) and 5.2GHz (solid)




                                          25
                                                                                   Chapter 2



2.6. Diffraction
Diffraction mainly occurs at window-wall transitions. The geometrical optics solution

leads to discontinuities across the shadow and reflection boundaries. The uniform theory

of diffraction eliminates the discontinuity of the electric and magnetic fields across these

boundaries. Unfortunately, closed simple solutions exist merely for the perfectly

conducting wedge case, whereas the conducting case is sufficiently solved only for non-

oblique incidence. To meet calculation time limits approximations have to be done.


   1.   The diffracted part is concentrated around the specular and non-specular

        transmitted and reflected directions.

   2.   We assume that half wall-thickness is about one wavelength.

   3.   The closed window and the lossy wall weaken the diffracted rays depending on

        the impinging angle and the wall and window type.


Due to the first assumption, remote diffraction can be neglected. For a fixed position of

the receiver in the room, only the diffraction around the adjacent two or three optical

boundaries is taken into account. To back up this approach it is essential to know that due

to interior multiple reflections and power leakage through the wall, the power level in

even the remote parts of the room never drop below a certain threshold. Therefore,

diffraction is not used to account for the power level in the shadow regions, rather to

compensate the discontinuities. The interference between the diffracted waves of the

adjacent optical boundaries gives a realistic picture about power drops due to

reciprocative cancellation.




                                                26
                                                                                 Chapter 2


       The second assumption allows the wall to be considered as a half-plane. The exact

or asymptotic solution of the diffraction problem for an impedance wall takes into

account its lossy character and coupling between the wall-faces. In fact the wall can be

regarded as a perfectly conducting half-plane, since the wave inside the wall decays as

heavily as it does for the perfectly conducting half-plane. For the same reason, the

coupling-effect between the wall-faces is neglected here. It should be noted that both the

perfectly conducting half-plane and the perfectly conducting edge retrieve similar results

for the inbound and outbound angles concerned. A further interesting fact reveals that the

entire set of diffraction problems leads to the modified Fresnel Integral, which has a

tractable asymptotic solution. This has been used throughout this work to give a simple

denouement of the window-wall diffraction.

       The third assumption demands the unsteadiness between the fields transmitted

through the window and wall be compensated for. Consequently, the diffraction

coefficient has to be multiplied by the difference of both real transmission coefficients.

The diffraction coefficient is obtained via the method mentioned above, using the

approximated Fresnel Integral.

       The mathematical background can be found in [16] and [17], the formulas and a

more profound treatment in Section 3-D and Appendix II (Diffraction). The uniform

diffraction solution for the non-specular propagation occurring for internally periodic

walls is discarded here due to the cumbersome theory. Therefore, only the specular

propagating component is corrected with the uniform solution.




                                           27
                                                                                   Chapter 3




3. The Propagation Model

3.1. Introduction
All the participating components given in Chapter 2 have now to be combined to give an

overall prediction of the wave propagation from outdoor to indoor. Before this is done

lets recapitulate the allotments to be included for the frequencies of interest.

   f=2.4GHz

               •   Specular transmitted component through window and wall with
                   appropriate losses.

               •   5 non-specular transmitted modes if applicable.

               •   2dB loss per mesh-layer if applicable.

               •   Diffraction correction for the specular transmitted window and wall
                   components.

   f=5.2GHz

               •   Specular transmitted component through window and wall with
                   appropriate losses.

               •   11 non-specular transmitted modes if applicable.

               •   2dB loss per mesh-layer if applicable.

               •   Diffraction correction for the specular transmitted window and wall
                   components.

   f=60GHz

               •   Specular transmitted component through the window.

               •   Diffraction correction for the specular transmitted window component.




                                              28
                                                                                                                  Chapter 3



3.2. Deterministic Transformation
The linearity of the electromagnetic field allows one to decompose it into its TE and TM

components with respect to the wall-surface and to treat them separately. The generic

propagation formula is now given through:

        E indoor (φin ,θ in ) = Ψ (φin ,θ in ,φ out ,θ out ) ⋅ E outdoor (φ out ,θ out )                           (3.1)
                            æ E TE            ö
where   E indoor ,outdoor = ç TM ,outdoor ÷ and
                                indoor
                            çE                ÷
                            è indoor ,outdoor ø
                                                           æ Ψ TE              0 ö
                                                         Ψ=ç
                                                           ç 0
                                                                                   ÷
                                                           è                  Ψ TM ÷
                                                                                   ø
                                                                                                 K transmission
                                                                                æ T TE ,TM   ö
                                                                                ç            ÷
                                                         Ψ TE ,TM   = (1 n 1) ⋅ ç L          ÷   K lattice
                                                                                ç TE ,TM     ÷   K diffraction
                                                                                èD           ø

The mean power loss is then easily obtained: <PowerLoss> = 10 ⋅ log 1 2 ΨΨ ∗ . The                       (         )
previous section provided the individual figures to be put into equation (3.1) , which has

been summarized below. It should be borne in mind that aforementioned calculations

assume a non-oblique plane of incidence to ease the calculations for the periodicity,

hence θ≈90°, α≈φ.

A. Specular Transmission

        T TE ,TM (φ in ,θ in ) = T TE ,TM ⋅ δ (φ in − φ out ,θ in − θ out )                                        (3.2)

        φ in ,θ in              … refer to the variable indoor angle

        φ out , θ out           … refer to the fixed impinging outdoor angle

        δ (•,•)                 … is exclusively one for both arguments equal to zero

         T TE ,TM               … transmission coefficient for window or wall




                                                                    29
                                                                                                   Chapter 3


B. Non-Specular Transmission

       T TE ,TM (φ in ,θ in ) = å TnTE ,TM (φ out ,θ out ) ⋅ δ (φ in − φ in ,n ,θ in − θ in ,n )    (3.3)

C. Lattice or Mesh

       LTE ,TM (φ in ,θ in ) = n ⋅ 2.0dB ⋅ δ (φ in − φ in ,n ,θ in − θ in ,n )                      (3.4)

       n           … number of consecutive lattices or meshes (n=0 for 60GHz)

D. Diffraction

        D TE ,TM = d in ( ρ in , φin ) m d out ( ρ in , φin )                                       (3.5)
                                            (              )           (
       d in ,out ( ρ in , φ in ) = − sgn a in ,out ⋅ K − a in ,out ⋅ k ⋅ ρ in ⋅ e − j⋅k ⋅ρ in)
                          2
                                (
       a in ,out = 2 cos 1 ⋅ (φ in m φ out )                       )
              2π
       k=
               λ
                                (      (            )
                             − j arctan x 2 +1.5 x +1 −π       )
                  1 e               4
        K − (x ) ≈ ⋅                                                       for all x ≥ 0 .          (3.6)
                  2   π ⋅ x2 + x +1
        K − ( x ) represents the approximated modified Fresnel integral, which can be
                   simplified for x > 3.5 into the first term of its asymptotic expansion:
                      1     1
        K − (x ) ≈      ⋅        .                                                                  (3.7)
                      2 x ⋅ j ⋅π

The most frequently [17] utilised diffraction term is given through equation (3.7 ) , which

is exclusively valid in the remote shadow regions. It can be applied to outdoor

propagation since part of the signal reaches the shadowed receiver via hilltop or roof

diffraction. Unlike the outdoor environment, the indoor environment always provides

enough signal strength to neglect just these terms, whereas the discontinuities have to be

smoothed since the receiver can be in their region. Equation (3.6) should be used.



                                                                             30
                                                                                                        Chapter 3



3.3. Transformation of the Probability Functions
The approach developed allows one to transform, at least numerically, the expected

outdoor power distribution and the pdf’s of the angles of arrival into the indoor power

and angular distributions. In the following the index ‘o’ refers to outdoor variables and ‘i’

to indoor variables. It should be noted that these statistics do not refer to the signal itself,

rather to its describing parameters.

The generic global power delay-azimuthal-elevation spectrum can be expressed as


                                                            {                   }
        P (τ , φ ,θ ) ∝ ò EE * (t ,τ , φ ,θ ) ⋅ dt = E EE* | τ , φ ,θ ⋅ f (τ , φ ,θ ) .                  (3.8)

f (τ , φ ,θ ) is the joint probability function of the delay, azimuth and co-elevation and

  {              }
E EE * | τ , φ ,θ the expected power conditioned on the delay, azimuth and co-elevation.

To get the appropriate power dependencies one merely has to integrate:


        Pτ (τ ) = òò P(τ ,φ ,θ ) ⋅ dφ ⋅ dθ

        P φ (φ ) = òò P(τ , φ ,θ ) ⋅ dτ ⋅ dθ                                                             (3.9)
        Pθ (θ ) = òò P(τ ,φ ,θ ) ⋅ dφ ⋅ dτ


Outdoor measurements have shown [18] that the processes arise quite independently

though a certain dependency cannot be denied. Using this approximation, one obtains for

the outdoor case,

        Po (τ o , φ o ,θ o ) ∝ Poτ o (τ o ) ⋅ Poφo (φ o ) ⋅ Poθ o (θ o )
        f o (τ o , φ o ,θ o ) = f oτ o (τ o ) ⋅ f oφo (φ o ) ⋅ f oθ o (θ )                               (3.10)
          {  o                    }
        E EE * | τ o , φ o ,θ o ∝ E EE * | τ o ⋅ E EE * | φ o ⋅ E EE * | θ o
                                       o   {          o  } {         o         } {         }
The expression of the single functions in                                    (3.10)   was obtained through many

measurements, i.e. [18], and is given below.


                                                                31
                                                                                                                                            Chapter 3


A. Delay Spectrum:

        Poτ o (τ o ) ∝ e                         , where σ τ o equals the Delay Spread (1K3 µs )
                              −τ o / σ τ o




                                                                                                               (                            )
                                       '
                              −τ o / σ τ
        f oτ o (τ o ) ∝ e                    o
                                                 , where σ τ' o equals the standard deviation σ τ' o ≈ 1.17σ τ o

                                                                                                                                  −τ o (1 / σ τ o −1 / σ τ )
                                                                                                                                                         '


                                                            {         }                        E{ o | τ o } e
                                                                                                   *
From (3.8) the expected power E EE | τ o is obtained as                                         EE         ∝                                            o
                                                                *
                                                                o                                                                                              .

B. Azimuthal Spectrum:


                                                     , where σ φo equals the Azimuthal Spread (5K10°)
                         − 2 φ o / σ φo
      Poφo (φo ) ∝ e

                          (                      )
                                                                                                                   (                            )
                                    '            2
                         − φ o / 2σ φo
      f oφ o (φo ) ∝ e                               , where σ φ' o equals the standard deviation σ φ' o ≈ 1.38σ φo

                                                                                      (φ / (            )) −
                                                                       {        }
                                                                                                  '      2
                                                                                               2σ φ o          2 φo / σ φ o
Again, the expected power is obtained as E EE* | φ o ∝ e
                                             o
                                                                                        o
                                                                                                                              .


C. Elevation Spectrum:

No measurements are available for this case. Even a long distance between the outdoor

Base Station and the wall surface cannot assure that both, the azimuthal and elevation

distribution, resemble. The reason is that in this case, i.e. Micro cells or larger, most of

the energy propagates via roof-top diffraction, where the last roof provides the strongest

component to the street-canyon or the wall-surface. This diffracted wave is being

reflected not often enough to guarantee a Gaussian distribution. Therefore, in case of

NLOS and Micro-Cells or larger the statistic is expected to resemble the tail of the

diffraction term given in (3.7 ) . The LOS-case would give a peak with a fringe similar to

the shifted (3.7 ) . The Pico-Cell statistic is expected to change from case to case but is

more likely to resemble a Gaussian distribution due to the large number of scatters in the

vicinity of transmitter and receiver.




                                                                          32
                                                                                                                    Chapter 3


Since the outdoor statistics are now defined, a transformation rule from outdoor to indoor

for the assumed independent components has to be found.

A. Delay Spectrum:

      τ i = τ o + δτ , where we assume δτ << τ o .

This results in

       f iτ i (τ i ) = f oτ o (τ o )
        {               } {
      E EE* | τ i = E EE* | τ o               }
what gives the indoor power spectrum Piτ i (τ i ) ≈ Poτ o (τ o ) ∝ e
                                                                                       −τ o / σ τ o




B. Azimuthal Spectrum:

First, let’s transform the angular pdf f(φ) assuming an internally periodic wall and no

diffraction. Then the angular dependency can be expressed as sin φi = sin φo + 2π ⋅ n / (kd ) ,

where k is the wave-vector and d represents the structure’s periodicity. The case for a

plain wall can be obtained by equaling n to zero. An impinging wave under φ o causes

several indoor waves under φi( n ) . Thus, the random variable φ o leads to several random

variables φi( n ) , whose distribution is deduced now. The well-known formulas for the

transformation of a random variable yields

                                                         ∂g −1
            φi
          f i φi ( ) = f (φ
                  (n)
                            o
                             φo
                                  o
                                         −1
                                       = g φi ( ))(n)
                                                        ⋅ (n) with φi( n ) = g (φo ) = arcsin (sin (φo ) + 2π ⋅ n / (kd ))
                                                         ∂φi

The differentiation gives ultimately the desired indoor pdf:


                 ( )              (           ( ( )
          f iφ i φi( n ) = f oφ o arcsin sin φi( n ) − 2π ⋅ n / (kd ) ⋅     ))                  ( )
                                                                                              cos φi(n )
                                                                                                                 (3.11)
                                                                                 1 − (sin (φ ) − 2π ⋅ n / (kd ))
                                                                                            ( )    n            2
                                                                                               i




                                                                    33
                                                                                                                                       Chapter 3


The next step is to transform the conditioned outdoor power E EE* | φ o , which might be
                                                                o                                            {             }
written in another form:

                                        (φ / (             )) −
          {            } ( )
                                                            2

                                                                                                                                         (3.12)
                                                     '
                                                  2σ φ o          2 φo / σ φ o
        E EE* | φ o = A φoA ⋅ e
            o
                                          o
                                                                                 .

  ( )
A φ oA is the maximum impinging power and φ oA the corresponding azimuthal angle. To

obtain E{EE* | φi( n ) } several steps have to be performed. First, φ o is substituted by φ i(n )
           i



through φi(n) = arcsin(sin(φo ) + 2π ⋅ n /(kd )) in equation (3.12) . Second, A φ oA is multiplied                         ( )
by the transmission coefficient for the nth coupled harmonic assuming an impinging angle

of φ oA . Third, some assumptions about the indoor spread and deviation have to be done.

For the case assumed above of no diffraction, these coefficients remain constant.

Diffraction, however, leads to a broadening of the indoor wave, which comes along with

an increase of both the spread and deviation. The increase severely depends on the

frequencies, where higher frequencies cause less spread. The figures to be put have to be

estimated to give best agreement with measurements.

The indoor power spectrum is now calculated merging (3.8) , (3.11) and (3.12) :


                         ( )         ( )
        Piφi (φi( n ) ) = A φoA ⋅ T ( n) φoA ⋅ e
                                              2                    ( ( )                  )
                                                    − 2 arcsin sin φi( n ) −2π ⋅n / ( kd ) / σ φi
                                                                                                    ⋅
                                                                                                                        ( )
                                                                                                                   cos φi(n )
                                                                                                        1 − (sin(φ ( ) ) − 2π ⋅ n / (kd ))
                                                                                                                       n                 2
                                                                                                                   i




C. Elevation Spectrum:

As soon as the outdoor elevation statistic is given the same approach as for the azimuthal

spectrum can be taken. Usually there is no periodicity for this case, what allows one to

put n to zero in the aforementioned formulas.




                                                                        34
                                                                                            Chapter 3


Figure 13 compares the indoor power spectra with the outdoor power spectra. Since the

interface introduces negligible delay the outdoor and indoor delay spectra resemble, thus

are omitted here. The indoor elevation spectrum is assumed to resemble (3.7 ) with LOS

condition, Figure 13 above. Of big interest is the indoor azimuth spectrum for internally

periodic walls, Figure 13 below. The formula provided above gives the appropriate power

spectra of the space-harmonics induced. It can be seen that already the 4th and 5th space

harmonics are expected to carry negligible power.




       Figure 13: Outdoor (left) and Indoor (right) Normalized Power Spectra for f=5.2GHz
                        Upper: Power Elevation Spectrum (axis in degree)
                        Lower: Power Azimuth Spectrum (axis in degree)




                                                 35
                                                                                                     Chapter 3



3.4. Outdoor Transmitter and Indoor Receiver
Let us assume an outdoor transmitter and an indoor receiver. Furthermore the transmitter

ought to be remote enough to consider the impinging wave as a plane wave. Due to

multipath propagation there exists a certain amount of impinging waves with different

angles of incidence and time delays.

                                            z
                                                          Antenna array consisting of
                                                          M antenna elements




                                                                      y
                                      x


         Figure 14: Antenna array consisting of M antenna elements


If the receiver consists of an antenna array as depicted above in Figure 14, the received

field-strength can be expressed as follows:

E received (t ) = òòò h(t ,τ , φ ,θ ) • c(φ ,θ ) ⋅ E transmitted (t − τ ) ⋅ dτ ⋅ dφ ⋅ dθ + N(t ) ,
  array                                                                                               (3.13)

where

               æ E 1 (t ) L E xM (t ) ö
               ç 1 x
                                      ÷
  received                     M
E array (t ) = ç E y (t ) L E y (t ) ÷                                                                (3.14)
               ç E 1 (t ) L E M (t ) ÷
               è z            z       ø

is the matrix of the received spatial signal components of the appropriate antenna element

(M number of antenna elements forming the antenna-array),

           æ c1     L cx ö
                         M
           ç 1 x
                           ÷
c(φ ,θ ) = ç c y         M
                    L cy ÷                                                                            (3.15)
           ç c1     L c zM ÷
           è z             ø

is the array steering matrix for the spatial components,



                                                            36
                                                                                                               Chapter 3



                 æ hxx        0       0ö
                 ç                       ÷
h(t ,τ ,φ ,θ ) = ç 0         hyy         ÷ with                                                                 (3.16)
                 ç 0          0      hzz ÷
                 è                       ø

                       L
hςς (t ,τ , φ ,θ ) = å aςς (t ,τ , φ ,θ ) ⋅ δ (τ − τ l ) ⋅ δ (φ − φ l ) ⋅ δ (θ − θ l ) and ςς ∈ {xx, yy, zz}
                        l

                      l =1


is the time-dependent radio channel delay-azimuthal-elevation spread function for a

                                                             l
linear medium, where L is the number of appearing paths and aςς the channel response of

the lth path. N(t) is the noise vector implying the independent complex white Gaussian

noise components of the antenna elements. In general, the channel-spread function can be

resolved in its participating components, e.g.

h(t ,τ ,φ ,θ ) = h outdoor (t ,τ ,φ ,θ ) ∗ h interface (t ,τ ,φ ,θ ) ∗ h indoor (t ,τ ,φ ,θ ) .                 (3.17)
Simply multiplying the components in the frequency domain can perform the

convolution. Assuming the receiver is not deep in the indoor environment, the last

formula (3.17 ) can be drastically eased to

h(t ,τ ,φ ,θ ) = h outdoor (t ,τ ,φ ,θ ) ⋅ h interface (φ ,θ ) .                                                (3.18)
If the antenna is a vertically aligned uniform linear antenna array with λ/2 element

spacing the steering matrix takes the following form:

           æ 0       L     0 ö
           ç                    ÷
c(φ ,θ ) = ç 0       L     0 ÷ with c zm (φ ) = f m (φ ) ⋅ e − j ( m−1)π sin φ ,                                (3.19)
           ç c1 (φ ) L c M (φ ) ÷
           è z           z      ø

where f m (φ ) is the complex field pattern of the mth array element.




                                                                   37
                                                                                    Chapter 4




4. Application

4.1. Introduction
The theory and formulas provided in the previous Chapters 2 and 3 give sufficient insight

into the physical nature of the outdoor-indoor propagation process, yet are quite useless

regarding an engineering benefit. A method has to be elaborated to use the acquired

physical knowledge and to apply given formulas. This Chapter provides therefore two

approaches, which can be used for simulation platforms or rough power estimations.

4.2. The Generic Cell
To cover the large number of possible window-wall-configurations a rough grid is laid

over a building dividing the surface into cells. These cells should at least be small enough

to cover typical configurations and at most big enough to allow the field strength over the

cell to be assumed constant. Some typical configurations would be: (1) wall, (2) wall with

interiorly periodic structure, (3) wall with lattice, (4) single window, (5) double window,

(1)-(3) with single window, (1)-(3) with double window. The formulas can now

theoretically be applied to a three-dimensional measure cell as depicted in Figure 15. The

depth of this measure cell should be big enough to cover a room or parts of it. The height

should embed the height of the basic window/wall-cell and the width should seize

diffracted rays. The measure-cell should not be confused with the basic window/wall-

cell, since the latter captures the structure of the building whereas the first allows one to

calculate the power-distribution in a room by overlapping the shares of the appropriate

measure-cells. The data-base for a chosen environment is now scanned and all occurring




                                             38
                                                                                            Chapter 4


cell configurations computed. The time-consuming calculation is done once and the

obtained power distribution being tabled.




                                                                            z       y
Measure cell
                                                                                x
depth: 5m
                                                              Cell width: 2m

                                                                   Measure cell width: 6m

     Typical Cell Configurations:
             Plain Wall (concrete/brick, different thickness)
             Internally periodic Wall (usually brick)
             Single/Double glazed Window
             Wall with Window (comprising the above mentioned configurations)

        Figure 15: Proposed measurement cell configuration


In practice, however, the three-dimensional calculation is reduced to two dimensions,

thus the power is available in the (x,z)-plane. Once the power is computed in each

measure cell for each basic window/wall-configuration, these cells are overlapped. The

idea of overlapped measure-cells is reflected in Figure 16, whereas Figure 17 shows the

top-view of merely one floor. Once the power shares are added up the overall power-

distribution can be predicted quite precisely in a room or in a whole floor. It should be

borne in mind that this overlapping does NOT include multiple reflected rays within the

room. Therefore, in Figure 17 it is presumed that the room is open-end. Figure 18

displays the three-dimensional power-distribution for a single measure-cell consisting of

a wall cell and Figure 19 for a window/wall cell. Both distributions are needed for the

room proposed in Figure 17. Finally in Figure 20 the overall power-distribution in the

room resulting from all the single measure-cells can be seen.




                                                 39
                                                                                          Chapter 4




2nd floor




1st floor




Basement


 Figure 16: Overlapped measure-cells (grid) for adjacent basic window/wall cells (gray)




Basic Wall Cell


Basic Wall/Window Cell




                                                                     Basic Wall Cell
                             30°
                                                   Basic Wall/Window Cell

 Figure 17: Top view of a room consisting of different basic cells




                                             40
                                                                                        Chapter 4




Figure 18: Specular Propagation in a Cell consisting of a plain Wall (f=5.2GHz, constant Loss of -
           13dB, Impinging Angle 30 degree)




Figure 19: Specular Propagation and Diffraction in a Cell consisting of a plain Wall with window
           (f=5.2GHz, Impinging Angle 30 degree, averaged)




Figure 20: Specular Propagation and Diffraction in a room proposed in Figure 17 consisting of the
           measure cell power distribution of Figure 18 and Figure 19.



                                           41
                                                                                  Chapter 4


       From Figure 20 the window/wall-diffracted components from the single windows

can conspicuously be seen, there the room is assumed open-end. To overcome this ‘open-

end’ room problem, first the specular components are multiple reflected in the room

using ray tracing methods and later the diffracted part is added. The advantage of this

method is that, if the impinging angle is fixed, the relative power distribution remains

constant. Hence, if the distribution in dB was computed for an impinging wave with unity

field strength, the impinging field strength in dB has merely to be added. A further

advantage is that the entire site has been reduced to a small number of tractable cells.

Furthermore, it allows one to calculate the average power in the cell. This can be used to

get an approximated power margin in the cell, room or even floor. The disadvantage is

that as soon as the impinging angle changes all the calculations have to be redone.

Furthermore, it is cumbersome to calculate the power distribution in the secondary

penetrated rooms in the same floor or adjacent floors, where this approach simply fails.


4.3. The modified COST 231 – Motley Model
The aforementioned problems are solved with loss in accuracy assuming that diffraction

plays a dominant role exclusively in the primary penetrated rooms, i.e. the Base Station

facing rooms. This allows using the Cost 231 – Motley penetration loss model for both

primary and secondary penetrated rooms, where the primary room is corrected with an

diffraction term. To save calculation time this term can be approximated considering

merely the adjacent two or three optical boundaries. The adopted micro cell Cost 231 –

Motley model is itself based on measurements for frequencies around 1.8GHz with an

averaged output. Therefore, the assumed attenuation loss table for this model has to be re-

completed with measurements for all frequencies concerned unless it is interpolated with



                                            42
                                                                                         Chapter 4


theoretical curves. Using the latter, i.e. the theoretical interpolation, a very gross

estimation yields the corrections summarized in Table 2. The interpolation was roughly

performed by estimating the input parameter in equations (3.1) through (3.5) with given

losses for f=1.8GHz. Afterwards the dependencies of these parameters from the

frequency were applied, e.g. the alteration of the permittivity. The parameters obtained

were finally inserted back into equations (3.1) through (3.5) to give the appropriate

losses for 2.4GHz, 5.2GHz and quite inaccurately for 60GHz. In fact these figures are

easily obtained through elementary measurements and there is no need to use possibly

incorrect figures. Table 2 is merely given for comparison with measurements performed

later and to demonstrate its applicability.


                                                 Absolute losses in dB for frequencies
              Object                   f=1.8GHz
                                                        f=2.4GHz     f=5.2GHz       f=60GHz
                                         (given)

 Thick concrete, no windows                 13             17            36              400

 Glass wall                                 2              13            15              15

 Wall with window
                                          2…13           13…17        15…36         15…400
 For a given wall-window-ratio the
 appropriate figure can be estimated
                                            Additional losses in dB relative to the tabled
                                                            f=1.8GHz case

 Thick concrete, no windows                 0              4             23              390

 Glass wall                                 0              11            13              13

 Wall with window
                                            0            4…11         13…23         13…390
 For a given wall-window-ratio the
 appropriate figure can be estimated

       Table 2: dB-correction for higher frequencies




                                                   43
                                                                                                                      Chapter 4


Once the losses of all the materials are obtained, either through measurements or

theoretically, the figures are put into the model formula:

LCost 231− Motley = L(d outdoor ) + Lwall , external / cos(φ ) + α ⋅ d indoor + n wall ⋅ Lwall , internal − n f ⋅ G    (4.1)

LCost 231− Motley , modified = LCost 231− Motley + s ⋅ D( x, y, z ) .                                                  (4.2)
The parameters are defined as:

                                               L                        path loss in dB

                                               L(d outdoor )            path loss up to the building

                                               Lwall , external         penetration loss of the external wall (Tabled)

                                              φ                         external angle of incidence

                                              α                         specific internal attenuation

                                               d indoor                 distance travelled inside the building

                                               nwall                    number of penetrated internal walls

                                               Lwall , internal         penetration loss of the internal wall (Tabled)

                                               nf                       number of penetrated floors

                                               G                        gain per floor (0dB micro cells, 2dB else)

                                               s                        switch (1 for primary penetrated rooms, 0

                                                                        elsewhere)

                                               D ( x, y , z )           diffraction loss in primary penetrated

                                                                        rooms.

The original Cost 231 – Motley model (4.1) requires the approximate position of the

receiver within the building since only the number of penetrated walls are of importance.

The modified model needs a precise position in the primary penetrated rooms to give


                                                                  44
                                                                                                 Chapter 4


exact predictions about possible power-drops caused by diffraction. However, as

formulated in Shannon’s information theory, the less information given, the less that can

be obtained from it, and vice versa. Hence, if the precise position of the receiver is not

known, a specific or general diffraction margin has to be used depending on the general

appearance of the external wall. This margin hardly depends on the window/wall

materials itself, but on the number of possible diffraction sources, e.g. window-wall

transitions. Some often-necessary margins were calculated and are given in Table 3. It

has been distinguished between two frequencies (2.4GHz & 5.2GHz) and additionally

between the number of illuminated wall-surfaces. The actual altering parameter is the

number of diffractive sources, i.e. the number of irradiated windows.

 Frequency         Number of illuminated         Number of windows per             Average diffraction
  in GHz              wall-surfaces                  wall-surface                    margin in dB
                                                           1                              1.3
                                                           2                              1.9
                             1
                                                           4                              2.8
                                                           6                              3.5
    2.4
                                                           1                              2.4
                                                           2                              3.4
                             2
                                                           4                              4.7
                                                           6                              5.7
                                                            1                              8.8
                                                            2                             10.9
                             1
                                                            4                             13.4
                                                            6                             15.7
    5.2
                                                            1                              8.8
                                                            2                             11.4
                             2
                                                            4                             14.4
                                                            6                             15.7
          Table 3: Average diffraction margins for several window-constellations


Table 3 reveals that the most influential parameter appears to be the frequency with up to

10dB difference. This is in accordance with the expectation of a rising number of fades

with increasing frequency. The number of diffractive sources, expressed through the



                                                    45
                                                                                                Chapter 4


number of illuminated walls and windows per wall, appears to be less influential. Here

the average margin increases slowly, almost monotonically with 0.5dB and 1dB per

window for 2.4GHz and 5.2GHz, respectively. For link-budget calculations network

operators should make extensive use of Table 3, which offers them the possibility of

accounting for occurring signal fades. Furthermore, the weak dependence of the margin

from the number of diffractive sources can be used to give rough margins mainly

depending on the frequency. Table 3 suggests to use a margin of D=3.2dB for 2.4GHz

and a margin of D=12.4dB for 5.2GHz.




       Figure 21: Standard deviation of the diffracted field vs. impinging angle for an assumed case with
                  f=5.2GHz, two illuminated right-angled wall-surfaces with 6 windows each.




       Figure 21 depicts the averaged deviation of the diffracted field from a purely

optical field vs. impinging angle for 5.2GHz, two illuminated right-angled wall-surfaces


                                                  46
                                                                                  Chapter 4


with 6 windows each. Disregarding normal and grazing incidence, it can be seen that

diffraction causes almost uniform deviation that fluctuates with less than 1dB around a

mean value of 15.7dB. The physical reason behind this is that an incident ray is diffracted

in all directions independently from the impinging angle. The power-drops do not depend

on the magnitude of the diffracted rays, rather on their mutual interference, which occurs

for all impinging angles.

       The original Cost 231 – Motley outdoor-indoor model includes diffraction via

increased outer-wall penetration losses. The values were obtained through numerous

measurement campaigns. These were averaged over a large number of sites, buildings,

floors and rooms. The tolerated fault is obvious: In reality primary penetrated rooms

suffer a much higher diffraction fade than secondary penetrated rooms, where the fade is

actually much less than predicted. The model introduced in          (4.2)   overcomes this

inaccuracy. It distinguishes between primary and secondary penetrated rooms through an

additional diffraction margin for the former ones. This can be backed up with the fact that

the diffractive impact weakens with increasing penetration depth. It must be noted that

now the outer-wall penetration losses differ from the original Cost 231 – Motley model.

       Disadvantageous is that both Cost 231 – Motley models (4.1) and (4.2) fail as

soon as the indoor environment appears to be highly reflective or highly obstructive. The

models give overestimation loss for the former and underestimation for the latter. To

overcome this problem an additional gain has to be added for a highly reflecting

environment depending on the passed rooms and floors. For the case of highly

obstructing, the corners act as signal sources. Thus, all corners have to be included into

the overall model.



                                            47
                                                                                  Chapter 5




5. Conclusions

5.1. Conclusion
The outdoor-indoor model developed embraces the most important propagation effects

through a windowed wall, which are specular transmission, non-specular transmission,

attenuation due to internal lattices and diffraction. The general path-loss-coefficient was

obtained following an approximated deterministic approach. This loss is dependent on the

impinging and emitting angles that requires knowledge about the outdoor conditions and

the indoor position of the receiver. If these figures are available, either an indoor ray-

launching model for precise predictions or the suggested cell philosophy can be used. The

former method is extremely time-intensive in terms of computation-time extensive since

any change in the parameters requires a complete re-calculation. The latter requires re-

calculations once the impinging angle changes. A trade-off between those methods is the

modified Cost 231 – Motley model, which is used in its original formulation for the

secondary penetrated rooms added with a diffraction correction coefficient for primary

penetrated rooms. This coefficient depends most on the frequency and less on the number

of diffraction sources, i.e. the number of windows in a wall, and is calculated for some

illuminating constellations.

       Furthermore, the model allows transformation of the known outdoor pdf’s to

calculate the appropriate indoor pdf’s. For a given outdoor delay-azimuth-elevation

power spectrum the transformation rule is given, where a spread of the spectrum is

caused due to diffraction.




                                            48
                                                                                   Chapter 5



5.2. Further Outlook
As mentioned in the introduction the engineer’s effort to gain dBm, whether in coding or

prediction accuracy, reaches saturation. The meticulous methods used today to predict

field-distribution probably won’t be necessary in a couple of decads. But until this

turning point some enhancements could be achieved. In principle, research can be

classified into two categories. If you appear to be in the first one then you do research to

please yourself, yet nobody can apply it. The second way to do it is to get a good

applicable idea and then call the fuss around it research.

        Following the first approach, the spread of the indoor azimuthal and elevation

spectrum introduced by diffraction should be obtained with the help of generic

calculations. Furthermore, a closed diffraction formula for oblique incidence in case of

non-perfectly conducting edges would save many measurements. And finally, the TM-

case for periodic structures should be studied.

        The second approach should concentrate more on the random character of the

interface channel, caused by site-specific irregularities. A more recent challenge would

be to verify the suggested models through measurements that have already been carried

out under the Radio Environment work area of the Mobile VCE. Unfortunately, they

haven’t been processed yet, which leaves the engineering approach developed in this

thesis still a theoretical piece of art.




                                             49
                                                                                               Chapter 6




6. Appendix I (Graphics)

6.1. Introduction
The large amount of graphics surely would have disturbed the readability of the actual

workout, the reason why they were taken out and placed into a separate Appendix. The

numeration follows the one of Chapter 2 to maintain a clear overview. To ease the access

to the individual graphics a separate table-of-figures follows:

       Figure 22: Transmitted power vs. impinging angle in dependency of pane-thickness for
                  single-glazing and f=2.4GHz (not shifted), f=5.2GHz (shifted by 10dB), f=60GHz
                  (shifted by 20dB) and given pane-thickness of 0.5, 1.0, 1.5 and 2.0 mm for the TE
                  (upper) and TM (lower) field components.
       Figure 23: Transmission through a lossy wall in dependency of the wall-thickness and angle
                   of incidence for the TE and TM components for f=2.4GHz and f=5.2GHz giving
                   Transmission Coefficients (upper) and the Transmitted Power (lower).
       Figure 24: Coupled space harmonics in dependency of the inbound angle providing the
                  harmonic-number and the proper outbound angle for f=2.4GHz (upper) and f=5.2GHz
                  (lower).
       Figure 25: Reflection and Transmission of the coupled space harmonics for f=2.4GHz.
                  Coefficients (upper): Reflection (left) & Transmission (right), Power (lower): absolute
                  power (coloured lines) & relative power w.r.t. specular component (black).
       Figure 26: Reflection and Transmission of the coupled space harmonics for f=5.2GHz.
                  Coefficients (upper): Reflection (left) & Transmission (right), Power (lower): absolute
                  power (coloured lines) & relative power w.r.t. specular component (black).
       Figure 27: Diffraction at a semi-infinite wall for impinging angle φ. The following graphs were
                   taken on a measurement track in a distance of 0.5m from the edge.
       Figure 28: Diffraction in dB at a perfectly conducting wedge with normal incidence of field-
                  strength 1 for f=2.4GHz utilizing the approximated modified Fresnel Integral.
       Figure 29: Diffraction in dB at a window-wall transition with normal incidence of field-strength
                   1 for f=2.4GHz utilizing the approximated modified Fresnel Integral. The window
                   attenuates the incident field by 2dB, the wall by 10.5dB. The method suggested in
                   Subsection 7.6.-D were applied.
       Figure 30: The same as Figure 29 with the only difference in the impinging angle: φ=30˚.
       Figure 31: Non-Specular Propagation in a Cell consisting of an internally periodic Wall
                  (f=5.2GHz, Impinging Angle 30 degree)
       Figure 32: Non-Specular Propagation in a Cell consisting of an internally periodic Wall
                  (f=5.2GHz, Impinging Angle 80 degree)
       Figure 33: Non-Specular Propagation in a Cell consisting of an internally periodic Wall with
                  Window (f=5.2GHz, Impinging Angle 80 degree)
       Figure 34: Non-Specular Propagation and Diffraction in a Measuring Cell consisting of Wall
                   Cells and Window/Wall Cells (f=5.2GHz, Impinging Angle 30 degree)



                                                  50
                                                                                         Chapter 6



6.2. Transmission Coefficients
     Figure 22: Transmitted power vs. impinging angle in dependency of pane-thickness for single-
                glazing and f=2.4GHz (not shifted), f=5.2GHz (shifted by 10dB), f=60GHz (shifted by
                20dB) and given pane-thickness of 0.5, 1.0, 1.5 and 2.0 mm for the TE (upper) and
                TM (lower) field components.




                  TM




     Figure 23: Transmission through a lossy wall in dependency of the wall-thickness and angle of
                incidence for the TE and TM components for f=2.4GHz and f=5.2GHz giving
                Transmission Coefficients (upper) and the Transmitted Power (lower).




                                              51
     Chapter 6




52
                                                                                         Chapter 6



6.3. Non-specular Transmission due Interior Periodic Structures
     Figure 24: Coupled space harmonics in dependency of the inbound angle providing the harmonic-
                number and the proper outbound angle for f=2.4GHz (upper) and f=5.2GHz (lower).




     Figure 25: Reflection and Transmission of the coupled space harmonics for f=2.4GHz.
                Coefficients (upper): Reflection (left) & Transmission (right), Power (lower):
                absolute power (coloured lines) & relative power w.r.t. specular component (black).


                                              53
                                                                                    Chapter 6




Figure 26: Reflection and Transmission of the coupled space harmonics for f=5.2GHz.
           Coefficients (upper): Reflection (left) & Transmission (right), Power (lower):
           absolute power (coloured lines) & relative power w.r.t. specular component (black).



                                         54
                                                                                                Chapter 6



6.4. Scattering due to Surface Roughness
Since scattering due to rough surfaces is neglected no graphs are produced.

6.5. Scattering due to Wall Interior Metallic Lattices and Meshes
Except a certain constant loss the meshes have a neglecting influence on the propagation

effects. There was no need to underpin it with the aid of graphs.

6.6. Diffraction

                                                           Semi-infinite wall
                                                   φ                        x

                     Measurement Track                        0.5 m


                                              z
       Figure 27: Diffraction at a semi-infinite wall for impinging angle φ. The following graphs were
                  taken on a measurement track in a distance of 0.5m from the edge.




       Figure 28: Diffraction in dB at a perfectly conducting wedge with normal incidence of field-
                  strength 1 for f=2.4GHz utilizing the approximated modified Fresnel Integral.



                                                  55
                                                                                           Chapter 6




Figure 29: Diffraction in dB at a window-wall transition with normal incidence of field-strength 1
            for f=2.4GHz utilizing the approximated modified Fresnel Integral. The window
            attenuates the incident field by 2dB, the wall by 10.5dB. The method suggested in
            Subsection 7.6.-D were applied.




Figure 30: The same as Figure 29 with the only difference in the impinging angle: φ=30˚.



                                           56
                                                                                           Chapter 6



6.7. The Generic Cell




     Figure 31: Non-Specular Propagation in a Cell consisting of an internally periodic Wall
                (f=5.2GHz, Impinging Angle 30 degree)




     Figure 32: Non-Specular Propagation in a Cell consisting of an internally periodic Wall
                (f=5.2GHz, Impinging Angle 80 degree)




                                               57
                                                                                     Chapter 6




Figure 33: Non-Specular Propagation in a Cell consisting of an internally periodic Wall with
           Window (f=5.2GHz, Impinging Angle 80 degree)




Figure 34: Non-Specular Propagation and Diffraction in a Measuring Cell consisting of Wall
           Cells and Window/Wall Cells (f=5.2GHz, Impinging Angle 30 degree)




                                          58
                                                                                 Chapter 7




7. Appendix II (Formulas)

7.1. Introduction
The difference between a Master thesis in Mathematics and a thesis in Engineering

merely lies in the order of the chapters’ appearances. The former dedicates the entire

work to pedantic formulas and just at the end awkwardly tries to find remote applications.

Whereas the latter lives from approximations and relegates the exact foundations with

heavily loaded mathematics to the appendix. The reason might be that any operation that

is not applicable to money, e.g. square-root, integrals, etc., is highly suspicious to the

industry. Therefore most Engineering thesises end up with the four basic arithmetic

operations: addition & multiplication, subtraction & division. A compromise has been

found here, where the mathematics is banished to the appendix, yet remained heavily

loaded.

7.2. Transmission Coefficients
The transmission coefficients for a single and double glazed window and a lossy wall are

obtained using the transmission line model. The theory of transmission lines can be

gleaned from any physics book, e.g. [19]. Merely the formulas are given here.

The characteristic impedance is defined as η = E x H y = µ ε . For oblique incidence

the TE and TM polarized waves have an intrinsic impedance Z TE = η ⋅ secα

and Z TM = η ⋅ cosα , respectively. The input impedance Z i into an intervening dielectric

with Z intervening loaded with Z load at distance l , is given through:

                 Z load ⋅ cos(kl ) + jZ intervening sin (kl )
          Zi =                                                  .                 (7.1)
                 Z intervening ⋅ cos(kl ) + jZ load sin (kl )



                                                         59
                                                                                     Chapter 7


In case of more than one intervening medium as depicted in Figure 35 the process is

repeated as the input impedance for one region becomes the load value for the next, until

one arrives at the region in which reflection is to be computed. The reflection coefficient

for the electric field components referred to medium 0 is given as:

             Z i − Z Medium 0
        ρ=                    ,                                                       (7.2)
             Z i + Z Medium 0

where Z i is calculated with (7.1) . The overall transmitted power is know readily

available:

                 (        )(
       T ⋅T * = 1− ρ 2 ⋅ e å i i ,
                          − (δ ⋅t )
                                    2
                                          )                                           (7.3)

where δ i and t i are the loss coefficient and thickness of the i th medium. δ i can be

approximated with:

                     (
       δ i = imag ε − sin (α ) .      )                                               (7.4)


                                  α                   Air, semi-infinite thickness
                                                      Medium 0

                                                      Dielectric, thickness t




               Air, thickness d



               Dielectric, thickness t


              Air, semi-infinite thickness


       Figure 35: Generic 5-layer structure




                                              60
                                                                                                    Chapter 7



7.3. Non-specular Transmission due Interior Periodic Structures
The theory of the specular and non-specular transmission and reflection can be gleaned

from references [14] and [20], which unfortunately are faulty. Therefore the whole theory

is expatiated.

Lets assume the periodic structure as it has been done in [14] and which is redrawn in

more detail below (Figure 36).

                                                                  Mode n=-1

                     Specular impinging wave
                                                                              Mode n=0, specular reflection
                                                            φ
      y          x
                               εL                                                 Mode n=1

                                                                                           h1   …Outer Wall
      z

                          ε2                                                               2h2 …Periodic Structure

                                                                                           h1   ...Inner Wall
ε1
                                                                                Mode n=1
                     d1                      d2

                                    d                                       Mode n=0, specular transmission

                                                                Mode n=-1

          Figure 36: Detailed horizontal cross-section of a typical periodic wall structures


Lets assume an impinging wave in the x-z plane at some angle φ being polarized along y

(TE-case). Hence we find the impinging wave and the reradiated reflected waves in the

upper half-space (z<0). The transmitted part through the outer wall continues propagating

in the x-z plane and is being partly reflected at the outer-wall/periodic-structure transition

region what leads to forward and backward travelling electromagnetic waves along z

within the outer wall. Whereas, due to Floquet’s Theorem, the inner periodic structure



                                                       61
                                                                                               Chapter 7


supports an infinite set of modes with different wavenumbers κm ( m = 0,±1,±2,... )

travelling along z. These modes can be decomposed into a series of space harmonics

( n = 0,±1,±2,... ) each having a wavenumber k nx = k 0 ⋅ sin φ + 2π ⋅ n             along x. Of course
                                                                                 d

just space harmonics with k nx ≤ k 0 can be coupled to air and propagate into the far-zone

of the external and internal walls since otherwise the waves are evanescent. The inner

wall obeys the same propagation effects as the outer wall.

Hence, for the TE-case, we conceive the following propagation formulas, there the time

dependence exp(-j⋅ω⋅t) is suppressed and an impinging wave amplitude E0 assumed:

Air region in front of the wall ( z < 0 ):

Einc ( x, z ) E0 = e j⋅( k0 ⋅x +k0 ⋅z ) + å rn ⋅e j⋅( kn ⋅x−kn ⋅z )
                              x     z                     x    z
  y
                                                                                                (7.5)
                                                n



− H inc ( x, z ) E0 = Y0 ⋅ e j⋅( k0 ⋅x +k0 ⋅z ) − å Yn rn ⋅e j⋅( kn ⋅x−kn ⋅z )
                                        x       z                     x   z
     y
                                                                                                (7.6)
                                                      n



          k nx = k 0 ⋅ sin φ + 2π ⋅ n
                                            d

          k nz = k 02 − (k nx ) 2

                  k nz
          Yn =
                 ω ⋅ µ0

          −∞ < n < ∞

          k0 = ω
                    c

          c= 1
                        µ0 ⋅ε0

          rn ... reflection coefficient of the space harmonics




                                                              62
                                                                                                                                       Chapter 7


Outer wall region:

Eouter ( x, z ) E0 = å (Voutern e j⋅β n ⋅z + Voutern e − j⋅β n ⋅z ) ⋅ e j⋅kn ⋅x
                                                                                               x
  y                       +                    −
                                                                                                                                        (7.7)
                             n

                                  βn +
− H outer ( x, z ) E0 = å                                                                                                               (7.8)
                                                                                                    x
    x
                                     (Vouter e j⋅β      n ⋅z
                                                               − Voutern e − j⋅β n ⋅ z ) ⋅ e j⋅kn ⋅ x
                                                                   −

                              n   ωµ           n




           β n = k 02 ⋅ ε L − (k nx ) 2

            +        −
          Voutern ,Voutern ... forward and backward travelling waves in the outer wall

Periodic structure:

In the periodic structure the field is expressed in the general Floquet form, where the

expansion coefficients anm are obtained below.

E periodic ( x, z ) E0 = å å anm ( f m e j⋅κ m ⋅( z −h1 −h2 ) + bm e − j⋅κ m ⋅( z −h1 − h2 ) ) ⋅ e j⋅kn ⋅x
                                                                                                                        x
  y
                                                                                                                                        (7.9)
                                  n    m



− H periodic ( x, z ) E0 = å å anmη m ( f m e j⋅κ m ⋅( z −h1 − h2 ) − bm e − j⋅κ m ⋅( z − h1 −h2 ) ) ⋅ e j⋅kn ⋅x
                                                                                                                                   x
    x
                                                                                                                                        (7.10)
                                      n    m


           anm ... nth Fourier component of the mth Floquet mode (expansion coefficient)


          η m = κ m ωµ … admittance of the mth Floquet mode
                             0



          κ m … modal wavenumber of the mth Floquet mode

Inner wall region:

Einner ( x, z ) E0 = å (Vinnern e j⋅β n ⋅( z − h1 −2 h2 ) + Vinnern e − j⋅β n ⋅( z − h1 −2 h2 ) ) ⋅ e j⋅kn ⋅x
                                                                                                                    x
  y                       +                                   −
                                                                                                                                        (7.11)
                             n


                                      β n + j⋅β ⋅( z −h −2 h )
− H inner ( x, z ) E0 = å
                                                                                                                            x
     x
                                         (Vinner e             − Vinner e − j⋅β ⋅( z −h −2 h ) ) ⋅ e j⋅k
                                                                 n −     1        2                     n   1   2           n ⋅x
                                                                                                                                        (7.12)
                                  n   ωµ            n                                           n




           β n = k 02 ⋅ ε L − (k nx ) 2

            +        −
          Vinnern ,Vinnern ... forward and backward travelling waves in the inner wall


                                                                             63
                                                                                                     Chapter 7


Air region behind the wall ( z > 2 ⋅ h1 + 2 ⋅ h2 ):

Etran ( x, z ) E0 = å t n ⋅e j⋅( kn ⋅x + kn ⋅( z −2 h2 −2 h1 ))
                                           x       z
  y
                                                                                                      (7.13)
                            n



− H tran ( x, z ) E0 = å Yn t n ⋅e j⋅( kn ⋅x +kn ⋅( z −2 h2 −2 h1 ))
                                                        x   z
     y
                                                                                                      (7.14)
                                    n


           t n ... transmission coefficient of the space harmonics

Floquet expansion coefficients for the periodic structure:

The Helmholtz equation in the source-free periodic structure for the TE-case is given as:

           ∇ 2 E + k x2 ( x) ⋅ E = 0 .                                                                (7.15)
Furthermore, for the TE-case we obtain:

          E = y⋅E                       H= 1
                                               j ⋅ ωµ 0
                                                            ∇× E                                      (7.16)

           k x2 ( x) = k 02 ⋅ ε ( x)

Presuming the periodic structure depicted at the right being continued till infinity, what

can be justified with λ << extensions of the periodic structure, ε(x) appears to be periodic

as is k(x), hence can be decomposed into its Fourier components such that:

                                           j⋅2π ⋅n⋅ x
           k x2 ( x) = k 0 ⋅ å ε n ⋅ e
                         2                              d
                                                                    (7.17)
                                n


                      d                                                      ε1,d1   ε2,d2   ε1,d1   ε2,d2
                  1              − j ⋅2π ⋅n x d
          εn =
                  d0ò ε ( x) ⋅ e                dx .


With the given periodic structure we get for the Fourier coefficients:

                             d1                               d2
                   d1 sin( nπ d ) − jnπ d1 d      d 2 sin( nπ d ) jnπ d 2 d
          ε n = ε1               e           + ε2                e                                    (7.18)
                   d nπ d 1                       d         d
                                                         nπ 2
                              d                                d




                                                                   64
                                                                                                         Chapter 7


Knowing about the periodicity of the structure we represent the field vector E in its

spatial Fourier components:

        E = å q n ( z ) ⋅ e j ⋅k n ⋅ x                                                                    (7.19)
                                   x



                n



        k nx = k 0 ⋅ sin φ + 2π ⋅ n
                                         d

Introducing the representations of E(x,z) and k2(x) given in                              (7.19)   and    (7.17) ,
respectively, into the source-free Helmholtz-equation (7.15) yields:

        æ ∂2 ∂2  ∂2 ö                                         j ⋅2π ⋅ x d
        ç 2 + 2 + 2 ÷å qn ( z ) ⋅ e j⋅k n ⋅x + k 02 å ε l ⋅ e             ⋅å qn ( z ) ⋅ e j⋅kn ⋅x = 0     (7.20)
                                        x                                                    x

        ç ∂x ∂y     ÷
                 ∂z ø n
        è                                           l                      n



Executing the derivations and changing the order of summing one may obtain:

          ∂2                é
                                         (
        å ∂z 2 qn ( z ) + å êå k 02 ⋅ ε l −n − {k lx }2 ⋅ δ ln   )ù ⋅ q ( z ) = 0
                                                                  ú    n                                  (7.21)
        n                 n ë l                                   û

This can be written in matrix form, like:

        ∂2
             I⋅q + I⋅P⋅q = 0                                                                              (7.22)
        ∂z 2

                         æ q−n ö              æ k 02ε −l + n − k −l
                                                                 x
                                                                      ... k 02ε −l    ...    k 02ε −l −n ö
                         ç ÷                  ç                                                            ÷
                         ç ... ÷              ç          ...          ...     ...     ...         ...      ÷
I1n = (1 1 … 1 1), qn1 = ç q0 ÷ , Pln =       ç       k 02ε n         ... k 0 ε n=0
                                                                            2
                                                                                      ...     k 0 ε −n ÷ (7.23)
                                                                                                  2
                         ç ÷                  ç                                                            ÷
                         ç ... ÷              ç          ...          ...     ...     ...         ...      ÷
                         çq ÷                 ç                                                           x÷
                                              è k0 ε l +n             ... k 02ε l     ... k 0 ε l −n − k l ø
                                                       2                                    2
                         è nø

Since both decompositions have to be assumed infinite there exists merely the trivial

solution, what is that the particularly adequate components of the sum have to resemble.

Hence the I-matrix can be omitted and the components regarded separately:

        ∂2
             q + P⋅q = 0.                                                                                 (7.24)
        ∂z 2




                                                        65
                                                                                  Chapter 7


This coupled system of differential equations with constant coefficients is easily solved

assuming the solution of the form:

       q = a⋅ e j⋅κ ⋅ z                                                            (7.25)
Finally we get:

       P⋅a = κ2⋅a,                                                                 (7.26)

what is nothing else than an eigenvalue problem of the matrix P with the eigenvalues κ m
                                                                                       2




and the column-eigenvectors am (with elements anm) as solutions for the mth Floquet

mode. Each of these modes is thus decomposed in an infinite set of space harmonics with

an amplitude anm.

Final solution for the reflection and transmission coefficients:

The Maxwell equations lead to boundary conditions that can be expressed as Et ,1 = Et , 2

and H t ,1 − H t , 2 = 0 assuming current-free surfaces. Those conditions should be hold on

each boundary for each space harmonics, that is:

        Etn,1 = Etn, 2                                                             (7.27)

        H tn,1 = H tn, 2                                                           (7.28)
                           z=0

                           z=h1

                           z=h1+2⋅h2

                           z=2⋅h1+2⋅h2

Applying those conditions to the formulas (7.5) through (7.14) we obtain:

                      +           −
       δ n 0 + rn = Vouter ,n + Vouter ,n                                          (7.29)

                            βn +
       Yn (δ n 0 − rn ) =
                            ωµ
                               (Vouter ,n − Vouter ,n )
                                              −




                                                          66
                                                                                                                                                Chapter 7


                                   −
                                                                                (
       Vouter ,n ⋅ e j⋅β n ⋅h1 + Vouter ,n ⋅ e − j⋅β n ⋅h1 = å a nm ⋅ f m ⋅ e − j⋅κ m ⋅h2 + bm ⋅ e j⋅κ m ⋅h2
         +
                                                                                                                            )
                                                               m


        βn
           ⋅ (Vouter ,n ⋅ e j⋅β ⋅h − Vouter ,n ⋅ e − j⋅β ⋅h ) = å anm ⋅ η m ⋅ ( f m ⋅ e − j⋅κ
                +                 n   1−                       n   1                                               m ⋅h2
                                                                                                                           − bm ⋅ e j⋅κ m ⋅h2   )
        ωµ                                                      m


                     −
                                                  (
       Vinner ,n + Vinner ,n = å a nm ⋅ f m ⋅ e j⋅κ m ⋅h2 + bm ⋅ e − j⋅κ m ⋅h2
         +
                                                                                                )
                                      m


        βn
           ⋅ (Vinner ,n − Vinner ,n ) = å a nm ⋅η m ⋅ ( f m ⋅ e j⋅κ
                +           −                                                       m ⋅h2
                                                                                            − bm ⋅ e − j⋅κ m ⋅h2   )
        ωµ                              m



       Vinner ,n ⋅ e j⋅β n ⋅h1 + Vinner ,n ⋅ e − j⋅β n ⋅h1 = t n
         +                         −




        βn
           ⋅ (Vinner ,n ⋅ e j⋅β
                +                 n ⋅h1
                                                                       )
                                          − Vinner ,n ⋅ e − j⋅β n ⋅h1 = Yn ⋅ t n
                                              −

        ωµ



                                                                    +
                                                                       , − , inner −
To solve this system of equations for the unknowns rn, tn, fm, bm,VouterVouterV+ ,Vinner the

infinite sums are truncated and the above set of equations put into matrix form to apply

standard matrix calculations.

                    +       −
       δ n0 + R = Vouter + Vouter                                                                                                                (7.30)
                          +       −
        Y(δ n0 − R) = B(Vouter − Vouter )

             +         −    −           −
        E β Vouter + E β 1 Vouter = A(E κ1f + E κ b)

               +         −     −               −
        B(E β Vouter − E β 1 Vouter ) = A η (E κ1f − E κ b)

          +        −                  −
        Vinner + Vinner = A(E κ f + E κ1b)

            +        −                       −
        B(Vinner − Vinner ) = A η (E κ f − E κ1b)

              +        −    −
        E β Vinner + E β1 Vinner = T

                +        −    −
        B(E β Vinner + E β1 Vinner ) = YT




                                                                           67
                                                                                               Chapter 7


                        R, T     … column vectors with elements rn , t n

                                 … (0 ← n  → 0)
                                                                    T
                        δ n0           1 n

                        B, Y     … diagonal admittance matrices with δ nm β n ωµ , δ nm k nx ωµ 0

                        E β , E κ … diagonal matrices with δ nm e jβ h , δ nm e jκ
                                                                        n 1          m h2




                         +        −         +        −
                        Vouter , Vouter , Vinner , Vinner …amplitudes of the respective forward and

                                                             backward propagating Fourier components

                        f, b     … column matrices with the amplitudes of the forward and

                                     backward propagating Floquet modes

                        A        … Floquet expansion matrix with the Floquet modes in columns

                                    and the respective Fourier components in rows

                        Aη       … columns of A multiplied by the correspondent admittance η m

This set of matrix equations can be put in a hyper-matrix, which can be facilely solved by

standard matrix manipulations.

æ1 0     0             0           −1        −1              0       0 ö æ R ö æ − δ n0 ö
ç                                                                         ÷ ç        ÷ ç   ÷
çY 0     0             0           B         −B              0       0 ÷ ç T ÷ ç Yδ n0 ÷
ç 0 0 AE −1           AE κ        − Eβ          −
                                            − E β1           0       0 ÷ ç f ÷ ç 0 ÷
ç          κ
            −                                   −
                                                                          ÷ ç        ÷ ç   ÷
ç 0 0 A η E κ1      − A ηEκ      − BE β     BE β 1           0       0 ÷ ç b ÷ ç 0 ÷
ç 0 0 AE                                                                  ÷⋅ +        =         (7.31)
ç                    AE κ1−
                                    0         0             −1      − 1 ÷ ç Vouter ÷ ç 0 ÷
           κ                                                                ç        ÷ ç   ÷
ç 0 0 A ηEκ                 −
                    − A η E κ1      0         0            −B        B ÷ ç Vouter ÷ ç 0 ÷
                                                                                −

ç                                                                      − ÷ ç         ÷ ç   ÷
ç0 1     0             0            0         0            − Eβ    − E β1 ÷ ç Vinner ÷ ç 0 ÷
                                                                                +

ç0 Y
è        0             0            0         0           − BE β   BE β1 ÷ ç Vinner ÷ ç 0 ÷
                                                                       −
                                                                          ø è
                                                                                −
                                                                                     ø è   ø

                        1        … diagonal identity matrix

                        0        … zero matrix

Therefore we obtain the hyper matrix equation: (H ) ⋅ (S ) = (C) , which has the solution:

        (S ) = (H )−1 ⋅ (C)                                                                     (7.32)


                                                     68
                                                                                      Chapter 7


       The first two sub-columns in the hyper-column (S) in               (7.32)   include the

reflection and transmission coefficients rn , t n , which can be inserted into equations (7.13)

and (7.14) to obtain the (far-) field components of the transmitted wave. The individual

components propagating into different directions depending on their wave-vector can

now be decomposed and handled separately. Furthermore, inserted into (7.9) and (7.10)

the 3rd and 4th sub-columns give the Floquet Modes. A glance at them reveals that their

periodicity does match to the periodic structure of the wall, yet is far from being

harmonic. The last 4 sub-columns give the field distribution inside the outer and inner

walls, which is not of big interest since cannot be accessed anyway.

       The inversion performed in (7.32) comes along with certain problems. First, it is

arduous to program matrix operations in C++, not to talk about hyper-matrixes.

Fortunately, Matlab spared no effort to take it from the mortal programmer. A second

problem arises from the singularity, which the matrix is close to due to the enormous

number of zeros. This leads to possible incisions in accuracy since Eigenvalue problems

in general are unstable. Luckily even this is solved in Matlab within the bounds of

possibility. Setting a certain parameter Matlab does account for the large number and the

uneven distribution of zeros. This was the main reason why Matlab was chosen to

develop and test the theory given above. The m-files are given in Appendix III (Matlab).




                                              69
                                                                                   Chapter 7



7.4. Scattering due to Surface Roughness
As mentioned before scattering is neglected for f=2.4GHz and f=5.2GHz. The f=60GHz

case demands special care since for this frequency scattering can lead to a severe spread

of the impinging power. The overall propagation formula has to be enriched with the

scattering matrix S, which is given for the reflected part in [2].

7.5. Scattering due to Wall Interior Metallic Lattices and Meshes
The derivation of the formulas has been omitted since this can be gleaned from reference

[21]. In general the theory of section 7.3. can be used, where only the permittivities need

to be altered. Furthermore, since the presence of an internal metallic lattice can merely be

guessed, a mean propagation loss factor has to be added for the specular 2.4GHz and

5.2GHz and non-specular 60GHz propagation.

7.6. Diffraction
Throughout the history of electromagnetic waves many endeavours have been made to

solve the problem of diffraction at an impedance wedge in a closed form. Many

approximations had to be assumed to give a manageable formulation of the problem.

Firstly the wedge was assumed to be perfectly conducting, what led to closed formulas

for arbitrary incidence. Just very recently closed formulas were found for the impedance

case, but merely for normally incident plane waves.

       In the case of a windowed wall the problem should be eased as far as possible.

Therefore it might be advantageous firstly to consider the window edges as perfectly

conducting half-planes. If this approximation is too gross the problem could be extended

to a perfectly conducting edge. If even this is too coarse the formulas for the impedance

half-plane should be used, what could be extended to the impedance edge.



                                              70
                                                                                            Chapter 7



                                                  y

                           Diffracted wave
                           with polarization Ez                      Impinging wave
                                                                     with polarization Ez

                                  ρ
                                                  φ    φ0
                                                                          x
                                                  z
                                                            Half-Plane


       Figure 37: Canonical Problem of the half-plane diffraction



A. Perfectly conducting half-plane:

The perfectly conducting half-plane as depicted in Figure 37 belongs to the canonical

problems of geometrical optics. Its facile manageability allows one to gain a profound

insight into the entire class of diffraction problems. Merely few extensions lead to other,

related problems as the window-wall diffraction. Serious discussion follow here that are

completed with further suggestions in Subsections B & C, whereas Subsection D deals

with the practical applications.

       Since the region is assumed to be charge-free and infinite in z the field solution

has to obey the two-dimensional homogenous scalar Helmholtz equation:

        (∇ 2 + k 2 )u = 0 ,                                                                  (7.33)
       with       ∇ … Nabla operator

                  k … wave-number

                  u … scalar function describing the field.

It can be easily shown that

        u=e
              − j (kx x+k y y)
                                                                                             (7.34)


                                                      71
                                                                                                                 Chapter 7


satisfies the above mentioned Helmholtz equation                                       (7.33)     and thus all its linear

combinations:

        u = å a n ⋅ e − jk ( x cos φn + y sin φn )                                                                (7.35)

and     u=ò
                 Contour
                           f (φ ) ⋅ e − jk ( x cos φn + y sin φn ) dφ .                                           (7.36)

It turned out to be advantageous to choose the integral form (7.36) for the half-plane

problem. Hence the integration contour and the weighting function f(φ) have to be found.

Let’s solve this configuration for the electric polarization as depicted in Figure 37. The

incident wave of unit amplitude can be represented in the following form:

         E   i
             z   = e       jk ( x cos φ 0 + y sin φ 0 )
                                                            .                                                     (7.37)
The Maxwell equations yield for the magnetic term:

         jωµ ⋅ H = e z × ∇E z , thus                                                                              (7.38)

                     ε
        Hi =           (−e x sin φ 0 + e y cos φ 0 )e jk ( x cos φ + y sin φ ) ,
                                                                               0   0
                                                                                                                  (7.39)
                     µ

where e x , e y , e z are the unit vectors spanning the space.

This incident field induces electric currents that reradiate a scattered field Es and Hs. To

calculate the scattered field the boundary conditions have to be applied. This means for

the y=0 plane that for:

        0< x<∞                     Þ             E zi + E zs = 0           due to the perfectly                   (7.40)
                                                                           conducting surface,

        −∞ < x < 0                 Þ             H xs = 0                  due to absent surface currents,        (7.41)

                                                 H zs = 0                  due to TE polarization.                (7.42)



                                                                          72
                                                                                     Chapter 7


We seek a solution for the scattered field on the y=0 in terms of the integral

representation    (7.36) .          The y=0 plane was chosen since some conditions on the

electromagnetic field are available there. Therefore, using the boundary conditions (7.40)

and (7.42) stated above we readily obtain:

        E zs ( x,0) = ò
                       Contour
                                    f (ξ ) ⋅ e − jkx cos ξ dξ = −E zi ( x,0)   x>0    (7.43)

        H xs ( x,0) = ò
                          Contour
                                    sin ξ ⋅ f (ξ ) ⋅ e − jkx cos ξ dξ = 0      x<0    (7.44)

Effectively we got now two dual integral equations with three unknowns, i.e. f(ξ) and the

complex contour C. Hence it would be advantageous to transform those equations into a

problem with two unknowns, which could be f(ξ) and a general rule for the behavior of

the contour C. One approach is to construct a Fourier transform of the scattered field on

the half-plane using an auxiliary function g(ν),

                       +∞
          s
        E ( x ,0 ) =
          z            ò g (ν ) ⋅ e
                                       − jxν
                                               dν         for x > 0 .                 (7.45)
                       −∞


To equate it with the above written integral representation we must allow

       ν = k cos ξ , g (ν ) = f (ξ ) / k sin ξ                                        (7.46)
and the contour C in the complex ξ-plane to include all real values of k·cos ξ from -∞ to

∞, as depicted in Figure 38.

Extending the Fourier transform representation (7.45) to the whole space and taking care

about some branch cuts due to the multi-valued square-root function one might get

                       +∞
          s
        E ( x, y ) =
          z            ò g (ν ) ⋅ e
                                    − j ( xν + y    k 2 −ν 2 )
                                                                 dν .                 (7.47)
                       −∞




                                                                        73
                                                                                                      Chapter 7




               Due to the fact
               that cosξ is an
               even function

                                            0                                                     0
                 -π                                                            -π
                          Contour C                                                 Contour C
                          For Im(k)=0                                               For Im(k)≠0




         Figure 38: Contours in the complex ξ-plane for the integral representation




The back transform with

         ν = k cos ξ , g (ν ) = f (ξ ) / k sin ξ and x = ρ ⋅ cos φ , y = ρ ⋅ sin φ                     (7.48)
yields

         E zs ( ρ , φ ) = ò
                            Contour
                                      f (ξ ) ⋅ e − jkρ cos(φ ±ξ ) dξ ,                                 (7.49)

where the combined plus and minus sign arises from the ambiguous square root.

Examining the behavior of the exponential term in infinity one easily obtains that the plus

sign holds for 0<φ<π, whereas the minus sign for π<φ<2 π.

Now lets return to the dual integral equations (7.43) , (7.44) and (7.37 ) , which, being

transformed by ν=k·cosξ, yield:

         ∞
              f (arccos(ν / k ))
         ò                            e − jxν dν = −e jxν 0              x>0                           (7.50)
         −∞        k −ν
                      2       2



         ∞

         ò f (arccos(ν / k )) ⋅ e
                                         − jxν
                                                 dν = 0                  x<0                           (7.51)
         −∞


with ν 0 = k cos φ 0 . These dual integrals are easier tractable, the more so as the Wiener-

Hopf technique allows them to be solved for the unknown f(arccos(ν/k)). The procedure


                                                                   74
                                                                                                        Chapter 7


is omitted here, but can be gleaned from [16]. As a result this contour integration method

yields:

                        1 ì 1                   1          ü
          f (ξ ) = −       ísec (ξ − φ 0 ) − sec (ξ + φ 0 )ý .                                           (7.52)
                       4πj î 2                  2          þ

Substituting this into (7.49) we get:

                                1           ì 1                    1          ü − jkρ cos(φ ±ξ )
          E zs ( ρ , φ ) = −       òContour ísec 2 (ξ − φ0 ) − sec 2 (ξ + φ0 )ý ⋅ e              dξ .    (7.53)
                               4πj          î                                 þ

Let’s firstly calculate the following fraction of the above stated integral, i.e.

                   1               1
          I =−        òContour sec 2 (ξ − φ0 ) ⋅ e
                                                   − jkρ cos(φ +ξ )
                                                                    dξ for 0 < φ < π .                   (7.54)
                  4πj

The substitution ξ ' = ξ + φ and some rearrangements lead to:

                   1                 1
          I =−        òContour ' sec 2 (ξ '−φ − φ0 ) ⋅ e
                                                         − jkρ cos ξ '
                                                                       dξ '                              (7.55)
                  4πj

                  1             ì 1                        1              ü − jkρ cos ξ '
          I =−       òContour ' ísec 2 (ξ '−φ − φ0 ) + sec 2 (ξ '+φ + φ0 )ý ⋅ e           dξ '           (7.56)
                 8πj            î                                         þ

                                1         1
                             cos ξ '⋅ cos (φ + φ 0 )
                1               2         2                                                              (7.57)
               2πj òContour ' cos ξ '+ cos(φ + φ 0 )
          I =−                                       ⋅ e − jkρ cos ξ ' dξ ' .


                                                            ξimag


                                                                              SDP
                                              Contour C


                                            -π                           φ      π            ξreal
                                                                    0


                                    Contour C’



          Figure 39: Contours in the complex ξ-plane




                                                              75
                                                                                        Chapter 7


The contour C’ is the contour C shifted by φ and including k⋅cosφ0 as can be seen in

Figure 39. The last integral is of the form:

        I =ò
               Contour
                         f ( z ) ⋅ e jkg ( z ) dz , with z = x + jy .                    (7.58)

g(z) may be expressed as g ( z ) = u ( x, y ) + jv( x, y ) , where u and v are real functions and

both satisfy the Cauchy-Riemann equations. Substituting g ( z ) = u ( x, y ) + jv( x, y ) into

the integral (7.58) gives:

        I =ò
               Contour
                         f ( z ) ⋅ e jku ⋅ e − kv dz .                                   (7.59)

From (7.59) it can be seen that the magnitude of the integral will change most rapidly

along the path where ∂v / ∂C is a maximum, whereas the phase along the path where

∂u / ∂C is a maximum. Employing the Cauchy-Riemann equations we get:

        ∂u
           =0                   for a maximum change in ν                                (7.60)
        ∂C

        ∂v
           =0                   for a maximum change in u.                               (7.61)
        ∂C

The Method of Steepest Descent tries to ease the original contour C in equation (7.58)

into a path which passes through the saddle point z0 : g’(z0)=0. This path includes the

region where the magnitude of exp(jkg(z)), which is exp(-kv), changes most rapidly and

thus contributes most to the integral (7.58) for large k. Equation (7.60) gives the

requirement for u(x,y):

        u ( x, y ) = const. ,                                                            (7.62)
whereas the saddle point claims:

        v ( x, y ) ≥ v ( x 0 , y 0 ) .                                                   (7.63)



                                                             76
                                                                                                    Chapter 7


Before applying the above scribed theory to equation (7.57 ) , lets separate the argument

of the exponential term in (7.57 ) into real and imaginary components ξ | = ξ real + ξ imag :
                                                                              |         |




                                                                                                     (7.64)
                                          |           |                      |           |
                              − jkρ cos ξ real cosh ξ imag        − kρ sin ξ real sinh ξ imag
        e − jkρ cos ξ ' = e                                  ⋅e

From (7.62) and (7.63) we now can easily deduce:

        cos ξ real cosh ξ imag = C 0
              |            |
                                                                                                     (7.65)

        sin ξ real sinh ξ imag ≥ 0 ,
              |            |
                                                                                                     (7.66)
Where C is an arbitrary constant which is determined to be one since the path has to

intersect the origin due to (7.66) . Defining a new variable

                                1
        ν = 2 ⋅ e − jπ / 4 ⋅ sin ξ |                                                                 (7.67)
                                2

we can see that along the PSD this variable embraces all real values between ±∞. Now

we are ready to transform (7.57 ) to the PSD with the change of variable, what yields:

                                π     ∞                 2
            1 − j ( kρ − 4 ) a ⋅ e − kρν
        I=    e              ò 2 2 dν + {pole residue at ξ ' = φ + φ 0 − π }                         (7.68)
           2π               −∞ν + ja


               1
where a = 2 cos (φ + φ 0 ) . Equation (7.68) is of the form of the modified Fresnel
               2

integral; hence the notation can be simplified to:

        I = sgn( a ) ⋅ K − ( a                       k ρ ) ⋅ e − jk ρ + {residue                }    (7.69)

where the residue is identified with incident field E zi and calculated as:

        {residue} = −U (π − φ − φ 0 ) ⋅ e jkρ cos(φ +φ )                         0
                                                                                                     (7.70)
        U (•) = unit step function.




                                                                                 77
                                                                                                               Chapter 7


The symmetry of the problem allows us now to put equation (7.56) in a mathematically

tractable form:

        E z ( ρ , φ ) = U (ε i ) ⋅ u 0 ( ρ , φ ) − U (ε r ) ⋅ u 0 ( ρ , φ ) + u d ( ρ , φ ) − u d ( ρ , φ )
                                     i                          r               i               r
                                                                                                                 (7.71)

                      with       u 0,r ( ρ , φ ) = e jkρ cos(φ mφ0 )
                                   i



                                 ε i ,r = sgn(a i ,r )

                                 u d,r ( ρ ,φ ) = −ε i ,r ⋅ K − ( a kρ ) ⋅ e − jkρ
                                   i



                                               1
                                 a i ,r = 2 cos (φ m φ0 )
                                               2
                                                               ∞
                                                   j jx 2         2
                                 K − ( x) =         ⋅ e ⋅ ò e − jt dt … modified Fresnel integral
                                                  π       x


The magnetic components are found using equation (7.38) . To calculate the TM-case the

above undergone procedures have to be repeated.

For the engineering approach, of course, equation (7.71) is still far too intricate. We may

find an asymptotic solution for the modified Fresnel integral. The asymptotic expansion

for sufficiently large arguments x leads to:

                             1
        K − ( x) ∝                          for x > 3                                                            (7.72)
                        2 x jπ

The argument of the modified Fresnel integral in (7.71) is                                2kρ ⋅ cos1 2 (φ m φ 0 ) , which

is large for remote ρ and for φ removed from the optical boundaries where

cos(φ ± φ 0 ) = −1 . Holding these conditions the diffraction terms in equation (7.71) may

be expressed as:

                                 1    1                      φ m φ0    e − jkρ
        u   i ,r
            d      ∝−
                         1
                                   ⋅       ⋅e − jkρ
                                                    = − sec(        )⋅         .                                 (7.73)
                      cos (φ m φ0 ) 8 jπkρ                     2       8 jπkρ
                         2




                                                                   78
                                                                                   Chapter 7


Therefore the total electromagnetic diffracted field removed from the optical boundaries

for the TE and TM case can be summarized as followed:

       TE polarization:

                          e − jkρ
       E ≈ D (φ , φ 0 ) ⋅
         d
         z
                  TE
                                                                                       (7.74)
                          8 jπkρ

                       ε
       H φd ≈ −          ⋅ E zd
                       µ
        d
       Hρ ≈ 0

       TM polarization:

                                  e − jkρ
       H zd ≈ D TM (φ , φ0 ) ⋅                                                         (7.75)
                                  8 jπkρ

                   µ
       Eφd ≈ −       ⋅ H zd
                   ε
        d
       Eρ ≈ 0

       with the edge diffraction coefficients for a perfectly conducting half-plane:

                   ì     φ − φ0          φ + φ0 ü
       D TE ,TM = −ísec(        ) m sec(       )ý                                      (7.76)
                   î       2               2 þ



It should be noted that the above given approximations are not valid for indoor

considerations. Since the receiver is expected to appear right on the optical boundaries,

another expansion should be used given in the last Subsection D.




                                               79
                                                                                                         Chapter 7


B. Perfectly conducting wedge:

Following the same approach as for the perfectly conducting half-plane similar formulas

for the diffraction coefficients for a perfectly conducting and impedance wedge can be

obtained, [16]. They are merely summarized here.

                                {               } {
              D TE ,TM = h(Φ i ) + h(−Φ i ) m h(Φ r ) + h(−Φ r )          }                               (7.77)
              Φ   i ,r
                         = φ − φ0

                                                                              π + Φ i ,r
                                                              − cscθ 0 ⋅ cot(            )
              h(Φ i ,r ) = −ε i ,r ⋅ ρ ⋅ K − (ν i ,r )Λi ,r ≈                   2N         ,ν i ,r > 3
                                                                     N 8 jπk
                          2π − β
              N=
                            π
              ε i ,r     = sgn(a i ,r )                                                                   (7.78)
                                 1
              a i ,r = 2 ⋅ cos( ⋅ (Φ i ,r + 2nπN ))
                                 2
                       a i ,r
                                   π + Φ i ,r
              Λi ,r =         cot(            )
                        2N           2N
             ν i ,r = kρ a i ,r ⋅ sin(θ 0 )


C. Impedance half-plane and impedance edge:

                                          {            [                      ]                     [
D TE ,TM = 1 / Ψ TE ,TM (φ 0 ) ⋅ Ψ TE ,TM (φ + π ) h(Φ i ) − h(−Φ r ) + Ψ TE ,TM (φ − π ) h(−Φ i ) − h(−Φ r )      ]}
Φ   i ,r
           = φ − φ0

Ψ TE ,TM ( z ) = ψ N ( z + υ B ,TM ) ⋅ψ N ( z − υ B ,TM ) ⋅ψ N ( z − Nπ + υ TE ,TM ) ⋅ψ N ( z − Nπ − TE ,TM )
                             TE                   TE
                                                                            A                        A

ψ 2 ≈ 1 − 0.0139 z 2 , y ≤ 8
                                                                          … impedance half-plane
ψ 2 ≈ 1.05302 cos1 / 4( z − j 0.69315), y > 8

ψ 3 / 2 = [4 / 3 cos(( z − π ) / 6) cos(( z + π ) / 6)] / cos( z / 6) … impedance edge

ν TEB = sec(Z 0 / Z A, B )
  A,                                                                      … complex Brewster angles

ν TM = sec(Z A, B / Z 0 )
  A, B                                                                        for given impedances




                                                              80
                                                                                                             Chapter 7



h(Φ i ,r ) = −ε i ,r ⋅ ρ ⋅ K − (ν i ,r )Λi ,r ≈ − cscθ 0 ⋅ cot((π + Φ i ,r ) / 2 N ) / N 8 jπk ,ν i ,r > 3

N = 2...3 / 2, ε i ,r = sgn(a i ,r )
a i ,r = 2 ⋅ cos(1 / 2 ⋅ (Φ i ,r + 2nπN ))
Λi ,r = a i ,r / 2 N cot((π + Φ i ,r ) / 2 N ),ν i ,r = kρ a i ,r ⋅ sin(θ 0 )


D. Practical Approach:

The complicated a diffraction problem might occur, it always leads after some more or

less rough approximations to the Modified Fresnel Integral:

                                        ∞
                          j jx 2
                                                                                                              (7.79)
                                         2
          K − ( x) =       ⋅ e ⋅ ò e − jt dt .
                         π       x



Equation (7.79) finds an almost perfect approximation for x > 0 through:

                                (      (            )
                             − j arctan x 2 +1.5 x +1 −π       )
                    1 e               4
          K − (x ) ≈ ⋅                                                                                        (7.80)
                    2   π ⋅ x2 + x +1

The term given in (7.80) looks a little ponderous but can be easily tabled to allow a rapid

access during simulations. Within the scope of this work it has been suggested to use this

approximation given in (7.80) to compensate for the discontinuity between the window

and wall transmitted waves. To do so the role of K − ( x ) is thoroughly examined herein.

First, lets recall the main diffraction formulas given in Subsection 3.2.- D:

          D TE ,TM = d in ( ρ in , φin ) m d out ( ρ in , φin )                                               (7.81)
                                            (              )           (         )
         d in ,out ( ρ in , φin ) = − sgn a in ,out ⋅ K − a in ,out ⋅ k ⋅ ρ in ⋅ e − j⋅k ⋅ρ in                (7.82)

                            2
                                (
         a in ,out = 2 cos 1 ⋅ (φ in m φ out )                     )                                          (7.83)

          K − (x) ≈
                      1
                        ⋅
                            1                                                                                 (7.84)
                      2 x ⋅ j ⋅π

                2π
         k=                                                                                                   (7.85)
                 λ


                                                                           81
                                                                                                               Chapter 7


Equations             (7.80)       to   (7.85)        hold for a perfectly conducting edge. Through their

appearance they form one of the most aesthetic set of formulas to describe the obscure

phenomena of diffraction. The beauty is their most facile intelligibility.

        As mentioned in Subsection 7.6.-A, the cylindrical symmetry of this particular

diffraction constellation suggests to use cylindrical coordinates given through the angle φ

and the distance ρ. Equation (7.81) expresses the diffraction coefficient that has a similar

role as transmission or reflection coefficients. It is a linear combination of two parts, one

embracing the diffraction of the specular transmitted, and the other of the specular

reflected waves. Both are dependent on the sources’ and receivers’ distance and angle

from the diffraction edge, what is explicitly given in (7.82) . It consists of three parts, an

                               (           )                                  ( (              ))
exponential term e − j⋅k ⋅ρin , a weighting term K − a in ,out ⋅ k ⋅ ρ in and a sign-giving term

(− sgn (a   in ,out
                      )). The first accounts for the phase of the propagating wave. The second
weights the emanated diffracted rays with magnitude and phase. Remote from optical

boundaries, its asymptotic behaviour is characterized by (7.84) , which describes an

outgoing cylindrical wave. Physically the edge thus acts as a radiating filament. The

weighting term             (K ( a  −
                                        in ,out
                                                  ⋅ k ⋅ ρ in   ))   is parameterized with the wave-vector k , the

distance from the edge ρ in and the angular distance between the optical boundaries and

the receiver position, expressed through (7.83) . K − ( x ) is a decaying function as depicted

in Figure 40. Thus, with increasing distance from the boundaries the diffraction effect

vanishes. The essential sign-giving term − sgn a in ,out             (         (    )) can be explained utilizing Figure
41, where the angular parameter-dependency of K − ( x ) is resolved.




                                                                         82
                                                                                         Chapter 7




Figure 40: The approximated modified Fresnel integral given in equation (7.80) .




Figure 41: Diffraction Coefficients vs. Indoor Angle for an impinging grazing angle φ0 = 30° in a
           distance of ρ in = 10 ⋅ λ from the edge:   Transmitted Component (blue, solid)
                                                      Reflected Component (red, dotted)




                                           83
                                                                                    Chapter 7


Depicted are the absolute values of the diffraction coefficients measured in a distance of

ρ = 10 ⋅ λ from the edge in case of a grazing incidence under φ 0 = 30° . The diffraction

coefficients have their peaks for angles φ = 210° and φ = 150° for the transmitted and

reflected rays, respectively. The diffraction coefficient’s task is to compensate either, the

perfectly transmitted optical field of strength 1 and the perfectly obstructed optical field

of strength 0. Therefore its maximum value can be found along the optical boundaries

and, as can be conspicuously seen in Figure 41, the absolute value does not exceed 0.5.

                                      (      (
Furthermore, the sign-giving term − sgn a in ,out   ))   does change its sign from 1 to -1

moving across the optical boundary out of the shadowed region into the lucid one. This

causes the diffracted part to be added in the shadow region and to be subtracted in the

illuminated region. The maximum value of 0.5 guarantees the required steadiness.

       An illuminated windowed wall does differ from the above handled case, since

both wall and window allow rays to penetrate. Again, the sole Fresnel Theory yields

discontinuities that now can be eliminated by compensating both fields with the

approximated modified Fresnel Integral. To do so, the weaker region is strengthened by

the half of the difference between both fields, and the stronger region is weakened by the

same amount. As previously mentioned, this is backuped by the fact that most diffraction

problems lead to the modified Fresnel Integral. Another mentioned approximation to

disregard remote optical boundaries can now be assessed with the aid of Figure 41.

Already angular alterations of less than 20˚ let drop the influence of diffraction below the

10 percent threshold. It does then play a minor role because interior multiple-reflections

maintain an indoor-level afar from zero.




                                             84
                                                                                  Chapter 7



7.7. Proof of absence of side lobes for the cell-philosophy
The proposed cell-philosophy demands to overlap and to add up the single cell power-

distributions to get an overall power-distribution for a room, floor, etc. There arises the

question if the coupling effects between two or more window/wall-cells can be neglected.

It is exceedingly awkward to give a mathematically correct answer, hence a suggestively

proof is given here. Lets assume the window/wall-cells being extended to infinity as can

be seen from Figure 42.


                                                   Window/Wall Cell



                     x
                                                 d1 , ε 1 d 2 , ε 2
              z

       Figure 42: Infinite window/wall-cell structure



Assuming that a wave impinges under a certain angle φ for z<0 the field for z>0 can be

expressed as follows:

        Etran ( x, z ) E0 = å t n ⋅ e jk n ⋅z ⋅ e j⋅kn ⋅x
                                             z            x
          y
                                                                                   (7.86)
                                  n



        k nx = k 0 ⋅ sin φ + 2π ⋅ n
                                      d
                                                                                   (7.87)

        k nz = k 02 − (k nx ) 2                                                    (7.88)
The transmission coefficients tn are obtained using the Floquet theorem. As for the before

mentioned periodic case the Helmholtz equation in the source-free periodic structure for

the TE-case has to satisfy:

        ∇ 2 E + k x2 ( x) ⋅ E = 0 , with k x2 ( x) = k 02 ⋅ ε ( x)                 (7.89)
The decomposition k(x) yields:


                                                              85
                                                                                                       Chapter 7


                                          j ⋅2π ⋅n⋅ x
         k x2 ( x) = k 0 ⋅ å ε n ⋅ e
                       2                                d
                                                            .                                           (7.90)
                             n


Again, knowing about the periodicity of the structure we represent the field vector E in its

spatial Fourier components:

         E = å q n ( z ) ⋅ e j ⋅k n ⋅ x                                                                 (7.91)
                                     x



                 n



         k nx = k 0 ⋅ sin φ + 2π ⋅ n
                                             d
                                                                                                        (7.92)

The decomposition of both values for an infinite periodic structure is depicted in Figure

43, dotted arrows.

If the this infinite periodicity is now violated by cutting it to some window/wall-cells, the

Delta-function broadens the appropriate values of the coefficients decrease, except the

mean, and, since the periodicity d increases all the coefficients approach the mean value.

The resultant coefficient distribution can be seen from Figure 43. As a consequence one

obtains a result which resembles the purely diffraction calculations. Thus the cell

philosophy is consistent since diffraction is being taken into account.




 ε       Specular                                                        Specular
         Component                                                       Component
                     1st side-lobe                                            1st side-lobe
                     Component                                                Component



         æ 2π ö æ 2π ö                     Spatial Components            æ 2π ö æ 2π ö        Spatial Components
     0   ç    ÷ 2⋅ç  ÷                                               0   ç    ÷ 2⋅ç  ÷
         è d ø è d ø                                                     è d ø è d ø

                     Figure 43: Spread of the spatial decomposition due to non-periodicity




                                                                86
                                                                                    Chapter 8




8. Appendix III (Matlab)

8.1. Introduction
Matlab itself is an exceptionally powerful tool to evaluate and depict formulas fast and

precisely. The disadvantage is that it has to be interpreted, thus is much slower than a

well-compiled C++ program. Therefore, the indoor-outdoor model itself is embedded

into a whole C++ Simulation Platform entirely written by Roger Cheung. Since it is

impossible to include the long C++ source code, the more clear Matlab source code is

given below. What follows are the names of and the links to the according m-files:



       8.2 Transmission Coefficients                                      page 88

       8.3 Non-specular Transmission due Interior Periodic Structures     page 89

       8.4 Scattering due to Surface Roughness                            page 91

       8.5 Scattering due to Wall Interior Metallic Lattices and Meshes   page 91

       8.6 Diffraction                                                    page91




                                                 87
                                                                 Chapter 8



8.2. Transmission Coefficients
%   5 Layer Transmission Line Concept!
%   t=pane thickness
%   d=pane separation
%   epsilon = permittivity

% single glazing: t=0.5*pane-thickness, d=0, epsilon=19-0.1*j
% double glazing: t=pane-thickness, d=pane-separation, epsilon=19-0.1*j
% lossy wall: t=0.5*wall-thickness, d=0, epsilon=3.5-0.9*j

clear;

t=3/1000;
d=2/100;
epsilon=19;

f=5.2e9;
c=3e8;
l=c./f;
k=2*pi./l;

y1=1;
y2=sqrt(1./epsilon);

ZL4      =   y1;
ZL3      =   imp(k./y2,t,y2,ZL4);
ZL2      =   imp(k,d,y1,ZL3);
ZL1      =   imp(k./y2,t,y2,ZL2);

R=(ZL1-y1)./(ZL1+y1);
T=sqrt(1-R.^2);

TdB=10*log10(1-abs(R.^2));




                                     88
                                                                    Chapter 8



8.3. Non-specular Transmission due Interior Periodic Structures

      Routine of the induced Space Harmonics

inbound = [-90:1:90];
moden = [-5:1:5];

[I,M] = meshgrid(inbound*pi/180,moden);

outbound = asin(sin(I)+2*pi*M/16.3362818);

for n=-5:1:5
    plot(inbound,180/pi*real(outbound(n+6,:)));
    hold on;
end

axis([-90 90 -90 90]);
title('Coupled space harmonics depending on the inbound angle for
f=5.2GHz');
xlabel('Inbound angle');
ylabel('Outbound angle');
grid;

      Routine of the carried Power of the Space Harmonics

clear;
f = 5.2e+9;
dim = 11;

% For f=2.4GHz dim has to be put to 5 !!!

for theta = -90:1:90,
    [R,T,k0, kxn, Cn] = kof3(theta,f,dim);
    zxz = theta+91;
    TT(:,zxz)=T.*exp(j*kxn.'+j*Cn.').*conj(T.*exp(j*kxn.'+j*Cn.'));
    RR(:,zxz)=R.*exp(j*kxn.'+j*Cn.').*conj(R.*exp(j*kxn.'+j*Cn.'));
end

for bild=1:1:dim,
    subplot(dim,1,bild);
    plot([-90:1:90],20*log10(TT(bild,:)),'k');
    hold on;
    plot([-90:1:90],20*log10(TT(bild,:)./TT((dim+1)/2,:)),'k:');
    plot([-90:1:90],zeros(1,181),'k');
    axis([-90 90 -30 30]);
end

gtext('Relative Power in dB vs. deg for f=5.2GHz');
gtext({'solid line ... transmitted power',
   'dashed line ... relative transmitted power with
   respect to the transmitted specular component'});




                                       89
                                                                     Chapter 8


       Subroutine kof3.m

function [R,T, k0,kxn,Cn] = kof(theta, f, dim);

% theta in degree
j=sqrt(-1);
theta = theta*pi/180;
e1 = 3.5;             % permeability
h1 = 0.0135;          % breadth of the exterior wall in m
h2 = 0.065/2;         % half breadth of the interior wall in m
k0 = 2*pi/3e+8*f;     % wave number
y0 = pi*4e-7;         %
d1 = 0.04;
d2 = 0.11;
d = d1+d2;
e2 = 1;
n=[-dim:1:dim];       % dimension of the mode-Fourier-transformation
nn=[-dim/2+0.5:1:dim/2-0.5];
p=d2/d*e2*sinc(n*d2/d).*exp(-
j*pi*n*d2/d)+d1/d*e1*sinc(n*d1/d).*exp(j*pi*n*d1/d); % Fourier
coefficients

pnl = zeros(dim,dim);

for x = 1:dim,
    for y = 1:dim,
        pnl(x,y)=p(x-y+dim+1)*k0^2;
    end
end

kxn = k0*sin(theta)+2*pi*nn/(d1+d2);       % Floquet wave-number
zw = diag(kxn.^2,0);

pnl = pnl - zw;
[A,km2] = eig(pnl.','nobalance');          % eigenvalue and eigenvector of
the coupling diff. equation

bn = sqrt(k0^2*abs(e1)-kxn.^2);
Cn = sqrt(k0^2-kxn.^2);

Eb = diag(exp(j*bn*h1),0);
Ek = exp(j*sqrt((km2))*h2).*diag(ones(1,dim));

B = diag(bn/(2*pi*f*y0));
Y = diag(Cn/(2*pi*f*y0));

vor = (1/(2*pi*f*y0)*ones(dim,1)*(diag(sqrt(((km2)))).'));
An = vor.*A;

dn1=zeros(dim,1);
dn1((dim+1)/2,1)=1;

I        =   diag(ones(1,dim));
N        =   zeros(dim,dim);
Nn       =   zeros(dim,1);




                                      90
                                                                                  Chapter 8


MATRIX       = [I   N   N        N           -I    -I        N                N
                Y   N   N        N           B     -B        N                N
                N   N   A*inv(Ek) A*Ek       -Eb   -inv(Eb) N                 N
                N   N   An*inv(Ek) -An*Ek    -B*Eb B*inv(Eb) N                N
                N   N   A*Ek A*inv(Ek)       N N      -I       -I
                N   N   An*Ek -An*inv(Ek)    N N      -B       B
                N   I   N     N N            N -Eb    -inv(Eb)
                N   Y   N     N N            N -B*Eb B*inv(Eb)];

CONST        = [-dn1
                Y*dn1
                Nn
                Nn
                Nn
                Nn
                Nn
                Nn];

SOLUTION =       inv(MATRIX)*CONST;

R            =   SOLUTION([1:1:dim],1);
T            =   SOLUTION([dim+1:1:2*dim],1);



8.4. Scattering due to Surface Roughness
Since scattering is neglected no program has been produced.

8.5. Scattering due to Wall Interior Metallic Lattices and Meshes
A constant loss has been added in the programs, yet is not repeated herein.

8.6. Diffraction
         Routine of the 3D-Diffraction-Graph

clear;
f=5.2e9;                          %frequency
phi0 = [0:1:360];              %impinging angle
phi = [0:1:360];                  %diffracted angles

epsilon = 19-j*0.1;

[PHI0,PHI]=meshgrid(phi0,phi);

e1       =   3.5-0.9*j;
k0       =   2*pi*f/3e+8;
Z0r      =   sqrt(1/1);
ZAr      =   sqrt(1/e1);
ZBr      =   sqrt(1/e1);

vA       =   1/cos(Z0r/ZAr);
vB       =   1/cos(Z0r/ZBr);



                                            91
                                                                        Chapter 8


DparallelImpedanceHalf =         De(PHI*pi/180,PHI0*pi/180,vA,vB,k0);

rho=ones(size(PHI0));
D = DparallelImpedanceHalf.*1./sqrt(rho).*exp(-j*k0*rho);

TT=D;

%imperfect approximation of the transition region

for nn=1:1:length(phi)
    for nnn=1:1:length(phi0)
        if abs(TT(nnn,nn)) > 1
            TT(nnn,nn) = NaN;
        end
    end
end

TTdB=10*log10(TT.*conj(TT));
mesh(phi,phi0,abs(TT));
axis([0 360 0 360 0 1]);
colormap('gray')
title('Diffraction Coefficient for the TE wave');
xlabel('Outbound Angle (0-360 degree)');
ylabel('Inbound Angle (0-360 degree)');
zlabel('Diffraction Coefficient');

         Routine of the 2D-Diffraction-Graphs

clear;
f=2.4e9;
phi = [0:1:360];

epsilon = 19-j*0.1;

e1       =   3.5-0.9*j;
k0       =   2*pi*f/3e+8;
Z0r      =   sqrt(1/1);
ZAr      =   sqrt(1/e1);
ZBr      =   sqrt(1/e1);

vA       =   1/cos(Z0r/ZAr);
vB       =   1/cos(Z0r/ZBr);

for nn=1:1:length(phi)

      DparallelImpedanceHalf(nn) =      De(phi(nn)*pi/180,45*pi/180,vA,vB,k0);

end

rho=1;
D = DparallelImpedanceHalf.*1./sqrt(rho).*exp(-j*k0*rho);

TT=D;




                                          92
                                                                 Chapter 8


for nn=1:1:length(phi)
 if abs(TT(nn)) > 1
        TT(nn) = NaN;
    end
end

TTdB=10*log10(TT.*conj(TT));
plot(phi,TTdB,'k');
grid;
axis([0 360 -45 0]);

title('Diffraction Coefficient for the TE wave and -45 degree inbound
angle');
xlabel('Outbound Angle (0-360 degree)');
ylabel('Diffraction Coefficient');

f=5.2e9;
phi = [0:1:360];

epsilon = 19-j*0.1;

e1       =   3.5-0.9*j;
k0       =   2*pi*f/3e+8;
Z0r      =   sqrt(1/1);
ZAr      =   sqrt(1/e1);
ZBr      =   sqrt(1/e1);

vA       =   1/cos(Z0r/ZAr);
vB       =   1/cos(Z0r/ZBr);

for nn=1:1:length(phi)

      DparallelImpedanceHalf(nn) =
      De(phi(nn)*pi/180,135*pi/180,vA,vB,k0);

end

rho=1;
D = DparallelImpedanceHalf.*1./sqrt(rho).*exp(-j*k0*rho);

TT=D;

for nn=1:1:length(phi)
 if abs(TT(nn)) > 1
        TT(nn) = NaN;
    end
end

TTdB=10*log10(TT.*conj(TT));
hold on;
plot(phi,TTdB,'k--');

gtext({'solid line ... Diffraction Coefficient for f=2.4GHz','dashed
line ... Diffraction Coefficient for f=5.2GHz'});




                                      93
                                                                      Chapter 8


         Subroutine De.m

function DeBer =         De(phi,phi0,vA,vB,k0)

DeBer = 1./Ksi(phi0,vA,vB).*(Ksi(phi+pi,vA,vB).*(h(phi-phi0,k0)-
h(phi+phi0,k0))+Ksi(phi-pi,vA,vB).*(h(phi0-phi,k0)-h(-phi-phi0,k0)));
         Subroutine Ksi.m

function KsiBer          =   Ksi(z,vA,vB)

KsiBer       =   ksiN(z+vB).*ksiN(z-vB).*ksiN(z-1.5.*pi+vA).*ksiN(z-1.5*pi-
vA);

         Subroutine ksiN.m

function ksiNBer         =   ksiN(z)

ksiNBer          =   1.05302*sqrt(cos(0.25*(z-j*0.69315)));

         Subroutine h.m

function hBer        =   h(Phi,k0)

hBer     =   cot((pi+Phi)/3)/(1.5*sqrt(8*j*pi*k0));




                                            94
                                                                                                                                        Table of Figures




Table of Figures
FIGURE 1: DECOMPOSITION OF THE PROPAGATION ALLOTMENTS .................................................................. 15
FIGURE 2:         HORIZONTAL CROSS-SECTION OF A TYPICAL WALL-WINDOW CONFIGURATION SHOWING BOTH
                  SOLID AND PERIODIC WALL STRUCTURES ON THE LEFT AND RIGHT PART OF THE PICTURE,
                  RESPECTIVELY ........................................................................................................................... 15

FIGURE 3: ELEVATION AND AZIMUTHAL DECOMPOSITION FOR WINDOW AND WALL...................................... 16
FIGURE 4: GENERIC 5-LAYER STRUCTURE USED FOR DERIVATION OF THE PROPAGATION FORMULAS............ 16
FIGURE 5: TE TRANSMITTED POWER VS. IMPINGING ANGLE FOR SINGLE GLAZING; F=2.4GHZ (NOT SHIFTED),
           F=5.2GHZ (SHIFTED BY 10DB), F=60GHZ (SHIFTED BY 20DB) PANE-THICKNESS 2T: 0.5MM,
           1.0MM, 1.5MM, 2.0MM ............................................................................................................... 18
FIGURE 6: TE TRANSMITTED POWER VS. PANE SEPARATION FOR DOUBLE GLAZING; F=2.4GHZ (NOT
           SHIFTED), F=5.2GHZ (SHIFTED BY 10DB), F=60GHZ (SHIFTED BY 20DB) PANE-THICKNESS 2T:
           1.5MM ........................................................................................................................................ 18
FIGURE 7: TOP VIEW OF A HORIZONTAL CROSS-SECTION OF AN INTERNALLY PERIODIC WALL....................... 19
FIGURE 10: COMMON METALLIC LATTICE (LEFT) AND COMMON REINFORCING WIRE MESH (RIGHT)............. 24
FIGURE 11: TRANSMITTED POWER OF THE INDUCED SPACE-HARMONICS FOR A METALLIC MESH AT F=2.4GHZ
           (CIRCLE) AND F=5.2GHZ (STAR)................................................................................................ 25
FIGURE 12: TRANSMITTED POWER OF THE INDUCED SPACE-HARMONICS VS. LATTICE-PERIODICITY AT
           F=2.4GHZ (DASHED) AND F=5.2GHZ (SOLID)............................................................................ 25

FIGURE 13: OUTDOOR (LEFT) AND INDOOR (RIGHT) NORMALIZED POWER SPECTRA FOR F=5.2GHZ .. ABOVE:
           POWER ELEVATION SPECTRUM (AXIS IN DEGREE) BELOW: POWER AZIMUTH SPECTRUM (AXIS
           IN DEGREE)                                                      35
FIGURE 14: ANTENNA ARRAY CONSISTING OF M ANTENNA ELEMENTS ......................................................... 36
FIGURE 15: PROPOSED MEASUREMENT CELL CONFIGURATION ...................................................................... 39
FIGURE 16: OVERLAPPED MEASURE-CELLS (GRID) FOR ADJACENT BASIC WINDOW/WALL CELLS (GRAY)...... 40
FIGURE 17: TOP VIEW AT A ROOM CONSISTING OF DIFFERENT BASIC CELLS .................................................. 40
FIGURE 18: SPECULAR PROPAGATION IN A CELL CONSISTING OF A PLAIN WALL (F=5.2GHZ, CONSTANT LOSS
           OF -13DB, IMPINGING ANGLE 30 DEGREE)................................................................................. 41

FIGURE 19: SPECULAR PROPAGATION AND DIFFRACTION IN A CELL CONSISTING OF A PLAIN WALL WITH
           WINDOW (F=5.2GHZ, IMPINGING ANGLE 30 DEGREE, AVERAGED) ........................................... 41

FIGURE 20: SPECULAR PROPAGATION AND DIFFRACTION IN A ROOM PROPOSED IN FIGURE 17 CONSISTING OF
           THE MEASURE CELL POWER DISTRIBUTION OF FIGURE 18 AND FIGURE 19. ................................ 41

FIGURE 21: STANDARD DEVIATION OF THE DIFFRACTED FIELD VS. IMPINGING ANGLE FOR AN ASSUMED CASE
           WITH F=5.2GHZ, TWO ILLUMINATED RIGHT-ANGLED WALL-SURFACES WITH 6 WINDOWS EACH.
           ................................................................................................................................................... 46
FIGURE 22: TRANSMITTED POWER VS. IMPINGING ANGLE IN DEPENDENCY OF PANE-THICKNESS FOR SINGLE-
           GLAZING AND F=2.4GHZ (NOT SHIFTED), F=5.2GHZ (SHIFTED BY 10DB), F=60GHZ (SHIFTED BY
           20DB) AND GIVEN PANE-THICKNESS OF 0.5, 1.0, 1.5 AND 2.0 MM FOR THE TE (ABOVE) AND TM
           (BELOW) FIELD COMPONENTS. ................................................................................................... 51
FIGURE 23: TRANSMISSION THROUGH A LOSSY WALL IN DEPENDENCY OF THE WALL-THICKNESS AND ANGLE
           OF INCIDENCE FOR THE TE AND TM COMPONENTS FOR F=2.4GHZ AND F=5.2GHZ GIVING
           TRANSMISSION COEFFICIENTS (ABOVE) AND THE TRANSMITTED POWER (BELOW). .................. 51


                                                                              95
                                                                                                                                   Table of Figures


FIGURE 24: COUPLED SPACE HARMONICS IN DEPENDENCY OF THE INBOUND ANGLE PROVIDING THE
           HARMONIC-NUMBER AND THE PROPER OUTBOUND ANGLE FOR F=2.4GHZ (ABOVE) AND
           F=5.2GHZ (BELOW). .................................................................................................................. 53

FIGURE 25: REFLECTION AND TRANSMISSION OF THE COUPLED SPACE HARMONICS FOR F=2.4GHZ.
           COEFFICIENTS (ABOVE): REFLECTION (LEFT) & TRANSMISSION (RIGHT), POWER (BELOW):
           ABSOLUTE POWER (COLOURED LINES) & RELATIVE POWER W.R.T. SPECULAR COMPONENT
           (BLACK). .................................................................................................................................... 53
FIGURE 26: REFLECTION AND TRANSMISSION OF THE COUPLED SPACE HARMONICS FOR F=5.2GHZ.
           COEFFICIENTS (ABOVE): REFLECTION (LEFT) & TRANSMISSION (RIGHT), POWER (BELOW):
           ABSOLUTE POWER (COLOURED LINES) & RELATIVE POWER W.R.T. SPECULAR COMPONENT
           (BLACK). .................................................................................................................................... 54
FIGURE 27: DIFFRACTION AT A SEMI-INFINITE WALL FOR IMPINGING ANGLE Φ. THE FOLLOWING GRAPHS
           WERE TAKEN ON A MEASUREMENT TRACK IN A DISTANCE OF 0.5M FROM THE EDGE.................. 55

FIGURE 28: DIFFRACTION IN DB AT A PERFECTLY CONDUCTING WEDGE WITH NORMAL INCIDENCE OF FIELD-
           STRENGTH 1 FOR F=2.4GHZ UTILIZING THE APPROXIMATED MODIFIED FRESNEL INTEGRAL. .... 55

FIGURE 29: DIFFRACTION IN DB AT A WINDOW-WALL TRANSITION WITH NORMAL INCIDENCE OF FIELD-
           STRENGTH 1 FOR F=2.4GHZ UTILIZING THE APPROXIMATED MODIFIED FRESNEL INTEGRAL. THE
           WINDOW ATTENUATES THE INCIDENT FIELD BY 2DB, THE WALL BY 10.5DB. THE METHOD
           SUGGESTED IN SUBSECTION 7.6.-D WERE APPLIED. ................................................................... 56

FIGURE 30: THE SAME AS FIGURE 29 WITH THE ONLY DIFFERENCE IN THE IMPINGING ANGLE: Φ=30˚........... 56
FIGURE 31: NON-SPECULAR PROPAGATION IN A CELL CONSISTING OF AN INTERNALLY PERIODIC WALL
           (F=5.2GHZ, IMPINGING ANGLE 30 DEGREE).............................................................................. 57
FIGURE 32: NON-SPECULAR PROPAGATION IN A CELL CONSISTING OF AN INTERNALLY PERIODIC WALL
           (F=5.2GHZ, IMPINGING ANGLE 80 DEGREE).............................................................................. 57
FIGURE 33: NON-SPECULAR PROPAGATION IN A CELL CONSISTING OF AN INTERNALLY PERIODIC WALL WITH
           WINDOW (F=5.2GHZ, IMPINGING ANGLE 80 DEGREE) .............................................................. 58
FIGURE 34: NON-SPECULAR PROPAGATION AND DIFFRACTION IN A MEASURING CELL CONSISTING OF WALL
           CELLS AND WINDOW/WALL CELLS (F=5.2GHZ, IMPINGING ANGLE 30 DEGREE) ..................... 58
FIGURE 35: GENERIC 5-LAYER STRUCTURE ................................................................................................... 60
FIGURE 36: DETAILED HORIZONTAL CROSS-SECTION OF A TYPICAL PERIODIC WALL STRUCTURES ................ 61
FIGURE 37: CANONICAL PROBLEM OF THE HALF-PLANE DIFFRACTION .......................................................... 71
FIGURE 38: CONTOURS IN THE COMPLEX Ξ-PLANE FOR THE INTEGRAL REPRESENTATION ............................. 74
FIGURE 39: CONTOURS IN THE COMPLEX Ξ-PLANE ........................................................................................ 75
FIGURE 40: THE APPROXIMATED MODIFIED FRESNEL INTEGRAL GIVEN IN EQUATION (7.80) . ....................... 83

FIGURE 41: DIFFRACTION COEFFICIENTS VS. INDOOR ANGLE FOR AN IMPINGING GRAZING ANGLE φ0 = 30° IN
           A DISTANCE OF ρ in = 10 ⋅ λ FROM THE EDGE: ............... TRANSMITTED COMPONENT (BLUE, SOLID)
                 REFLECTED COMPONENT (RED, DOTTED) .................................................................................. 83
FIGURE 42: INFINITE WINDOW/WALL-CELL STRUCTURE ................................................................................ 85
FIGURE 43: SPREAD OF THE SPATIAL DECOMPOSITION DUE TO NON-PERIODICITY ......................................... 86




                                                                           96
                                                                          Bibliography




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[1]    Monica Dell’Anna, A. Nix, T. Harrold, Roger Cheung, Michael Döhler.
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[2]    B. Fleury. Mobile Radio Channels, MPCS99, Ulm, Germany, 1999.

[3]    D. Molkdar. Review on radio propagation into and within buildings, IEE-
       Proceedings-H, Vol. 138, No. 1, February 1991, pp. 61-73.

[4]    J. Horikoshi et al., 1.2GHz Band Wave Propagation Measurements in Concrete
       Building for Indoor Communications, IEEE Transactions on Vehicular
       Technology, VOL. VT-35, No. 4, November 1986, pp. 146-152.

[5]    J.-E. Berg. Angle Dependent Building Penetration Loss along LOS Street
       Microcells, COST 231 document TD96(006) Belfort France, January 1996.

[6]    R. Gahleitner, E. Bonek. Radio Wave Penetration into Urban Buildings in Small
       Cells and Microcells, Proceedings IEEE VTC94, March 1994, pp.887-891.

[7]    S.J. Hong, K.J. Kim, J.R.Lee. Moisture effects on the Penetration Loss through
       Exterior Building Walls, Proceedings IEEE VTC98, April 1998, pp.860-864.

[8]    A.M.D. Turkmani, A.F. de Toledo. Radio transmission at 1800MHz into, and
       within, multistory buildings, IEE Proceedings-I, Vol. 138, No. 6, December
       1991, pp. 577-584.

[9]    A.M.D. Turkmani, A.F. de Toledo. Propagation into and within Buildings at
       900, 1800 and 2300MHz, Proceedings IEEE VTC92, February 1992, pp. 633-
       636.

[10]   W.J. Tanis II, G.J. Pilato. Building Penetration Characteristics of 880MHz and
       1922MHz Radio Waves, Proceedings IEEE VTC93, 1993, pp. 206-209.

[11]   G. Durgin, T.S. Rappaport, H. Xu. Measurements and Models for Radio Path
       Loss and Penetration Loss in and around Homes and Trees at 5.85GHz, IEEE
       Transactions on Communications, VOL. 46, No. 11, November 1998, pp. 1484-
       1495.


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[12]   B. De Backer et al. The study of wave-propagation through a windowed wall at
       1.8GHz, Proceedings IEEE VTC96, June 1996, pp. 156-169.

[13]   COST 231 Subgroup on Propagation Models. Indoor Propagation Models for
       1.7-1.9GHz, COST 231 TD(90)114, Firenze, January 1991.

[14]   H. Bertoni, W. Honcharenko et al. UHF Propagation Prediction for Wireless
       Personal Communications, Proceedings of the IEEE, vol.82, no.9, pp. 1333-
       1359, September 1994.

[15]   W. Honcharenko, H.L. Bertoni, Transmission and Reflection Characteristics at
       Concrete Block Walls in the UHF Bands Proposed for Future PCS, IEEE
       Transactions on Antennas and Propagation, vol. 42, no. 2, pp. 232-239, February
       1994.

[16]   G.L. James. Geometrical Theory of Diffraction for Electromagnetic Waves, IEE
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[17]   R.J.Luebbers. Finite Conductivity Uniform GTD versus Knife Edge Diffraction
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[18]   Klaus I. Pedersen, P.E. Mogensen, B.H. Fleury. A Stochastic Model of the
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[19]   A.H. Матвеев. Электродинамика и Теория Относительности, Moscow-
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[20]   S.T.Peng, T.Tamir, H.L.Bertoni. Theory of Periodic Dielectric Waveguides,
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[21]   S. .W.Lee, G.Zarrillo, C.L.Law. Simple Formulas for Transmission Through
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       vol. AP-30, no. 5, pp. 904-909, September 1982.




                                          98
                                                                                                                                                              Index




Index
                                                                                            Method of Moments .............................................10
                                                                                            Ray tracing ...........................................................10
A                                                                                        Floquet
                                                                                            Floquet Mode ...........................................64, 67, 69
Abstract ..................................................................... vi        Frauenhofer-Criterion ...............................................23
Acknowledgments.....................................................iii                  Frequency dependent Loss........................................11
Angle                                                                                    Further Outlook ........................................................49
  Grazing Angle.................................. 8, 9, 10, 11, 84
  Inbound Angle ............................. 21, 50, 53, 93, 97
  Normal Angle .......................... 8, 17, 24, 50, 56, 57                          G
  Outbound Angle................................. 20, 27, 50, 53
Appendix                                                                                 Gaussian Scattering Matrix.......................................14
  Appendix I (Graphics) ..................................... 7, 50
  Appendix II (Formulas) ................................... 7, 60
  Appendix III (Matlab)................................ 7, 70, 88                        H
Application............................................................... 38
  Azimuth ............................... 16, 32, 33, 34, 49, 102                        Half Plane .............................27, 71, 72, 73, 74, 80, 81
                                                                                         Hand-Over ................................................................13


B                                                                                        L
Bibliography........................................................... 101
                                                                                         Lattice .....................................................14, 24, 30, 48
Brewster Angles ....................................................... 81


C                                                                                        M
                                                                                         Margin
C++ Program............................................................ 88
Cauchy-Riemann Equations ..................................... 77                          Fading Margin........................................................8
Cell                                                                                     Matlab.............................................................7, 70, 88
  Generic Cell ................................................... 38, 58                Measurements.......6, 11, 13, 17, 31, 32, 34, 42, 44, 49
                                                                                         Mesh ...................................................................30, 56
  Measure Cell............................................ 38, 39, 41                    Method of Steepest Descent .....................................77
  Micro Cell...................................................... 42, 44                Models
Conclusion ............................................................... 48              Angle dependent Path-Loss Model ........................9
                                                                                           COST 231 Keenan and Motley Model ...................9
                                                                                           Linear Path-Loss Model .........................................8
D                                                                                          modified COST 231 – Motley Model...................42
                                                                                           Outdoor-Indoor Model ..................vi, 4, 6, 7, 47, 48
Database ............................................................... 4, 13
                                                                                           Propagation Model ...............................28, 101, 102
                                                                                           Transmission Line Model...............................16, 60
E                                                                                        Moisture Effects .......................................................12

Edge
   Impedance Edge............................................. 71, 81                    N
   Perfectly Conducting Edge ...................... 27, 71, 83
Elevation .......................................... 16, 32, 34, 35, 49                  Network operators ..............................................12, 46



F                                                                                        O

Field-strength predicting Methods                                                        Optical Boundaries .....................26, 42, 79, 80, 83, 85



                                                                                    MM
                                                                                                                                                            Index


Outdoor Transmitter and Indoor Receiver................ 36                              Penetration Loss Statistics....................................12
                                                                                        Rayleigh Distribution .............................................8
                                                                                        Rice ........................................................................8
P                                                                                       Small-Scale Fading ................................................8
                                                                                        Time of Arrival (TOA)...........................................8
Parameter
  Dependencies and Tendencies ............................. 11
  general ................................................. 5, 43, 45, 70           T
  Penetration Loss Model Parameter ...................... 11
Path of steepest Descent (PSD) ................................ 78                  Tables
Polarization                                                                          Table of Contents ..................................................iv
  TE Polarization .............................................. 73, 80
                                                                                      Table of Figures ...................................................99
  TM Polarization ................................................... 80            Transformation
Power Budget ........................................................... 23           Deterministic Transformation ..............................29
Power Control ...................................................... 3, 13
Propagation Allotments............................................ 13                 Transformation of the Probability Functions........31
                                                                                    Transmission
                                                                                      general ...vi, 2, 4, 13, 14, 15, 16, 17, 19, 24, 27, 29,
                                                                                         34, 48, 60, 62, 65, 67, 70, 83, 101
R
                                                                                      Non-specular Transmission ..........19, 53, 62, 88, 93
Receiver Height........................................................ 12            Transmission Coefficients......16, 50, 51, 52, 60, 91
Reflection ................. 12, 14, 26, 61, 62, 63, 67, 70, 83

                                                                                    U
S
                                                                                    UMTS.....................................................................1, 2
Scattering                                                                          Universal Theory of Diffraction (UTD)....................14
   Coherent Scattering.............................................. 23
   Diffuse Scattering ................................................ 23
   due to Surface Roughness ............ 23, 56, 71, 88, 95
                                                                                    V
   due to Wall Interior Metallic Lattices and Meshes
      ................................................. 24, 56, 71, 88, 95          Virtual Centre of Excellence (VCE) ..............vi, 13, 49
Simulation Platform ..................................... 13, 38, 88
Space Harmonics19, 20, 21, 35, 50, 53, 54, 55, 63, 65,
   67, 93                                                                           W
Spectrum
   Azimuthal Spectrum ...................................... 32, 33                 Wall
   Delay Spectrum.............................................. 32, 33                Bricked Wall ........................................................22
   Elevation Spectrum.................................. 32, 34, 35                    Lossy Wall .............................16, 17, 26, 50, 52, 60
Spread                                                                                Periodic Structure...........................................19, 20
   Azimuthal Spread ................................................ 32             Wavenumber.......................................................63, 64
   Delay Spread........................................................ 32          Wiener-Hopf Technique ...........................................75
                                                                                    Window
   Elevation Spread .................................................. 32
                                                                                      Closed Window....................................................17
Statistics
   Angle of Arrival (AOA)......................................... 8                  Double Glazed Window ...........................16, 60, 90
   Large-Scale Fading ................................................ 8              Single Glazed Window...................................16, 60
   Log-normal Distribution .................................. 8, 12




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