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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 Bibliography [1] Monica Dell’Anna, A. Nix, T. Harrold, Roger Cheung, Michael Döhler. Propagation Models, Deliverable D4, Radio Environment Work Area, 1999. [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. 97 Bibliography [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 Electromagnetic Wave Series I, Exeter, 1986. [17] R.J.Luebbers. Finite Conductivity Uniform GTD versus Knife Edge Diffraction in Prediction of Propagation Path Loss, IEEE Transactions on Antennas and Propagation, vol. AP-32, no. 1, pp. 70-76, January 1984. [18] Klaus I. Pedersen, P.E. Mogensen, B.H. Fleury. A Stochastic Model of the Temporal and Azimuthal Dispersion seen at the Base Station in Outdoor Propagation Environments. 1999, not published yet. [19] A.H. Матвеев. Электродинамика и Теория Относительности, Moscow- Press, 1964. [20] S.T.Peng, T.Tamir, H.L.Bertoni. Theory of Periodic Dielectric Waveguides, IEEE Transactions on Microwave Theory and Technologies, vol. MTT-23, no. 1, pp. 123-133, January 1975. [21] S. .W.Lee, G.Zarrillo, C.L.Law. Simple Formulas for Transmission Through Periodic Metal Grids or Plates, IEEE Transactions on Antennas and Propagation, 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 MMI

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