Methods for Path loss Prediction

Transcription

1 School of Mathematics and Systems Engineering Reports from MSI - Rapporter från MSI Methods for Path loss Prediction Cem Akkaşlı October 9 MSI Report 967 Växjö University ISSN SE VÄXJÖ ISRN VXU/MSI/ED/E/--967/--SE

2 Abstract Large scale path loss modeling plays a fundamental role in designing both fixed and mobile radio systems. Predicting the radio coverage area of a system is not done in a standard manner. Wireless systems are expensive systems. Therefore, before setting up a system one has to choose a proper method depending on the channel environment, frequency band and the desired radio coverage range. Path loss prediction plays a crucial role in link budget analysis and in the cell coverage prediction of mobile radio systems. Especially in urban areas, increasing numbers of subscribers brings forth the need for more base stations and channels. To obtain high efficiency from the frequency reuse concept in modern cellular systems one has to eliminate the interference at the cell boundaries. Determining the cell size properly is done by using an accurate path loss prediction method. Starting from the radio propagation phenomena and basic path loss models this thesis aims at describing various accurate path loss prediction methods used both in rural and urban environments. The Walfisch-Bertoni and Hata models, which are both used for UHF propagation in urban areas, were chosen for a detailed comparison. The comparison shows that the Walfisch-Bertoni model, which involves more parameters, agrees with the Hata model for the overall path loss. Keywords: path loss, prediction, wave propagation, rural, urban, Hata, Walfisch-Bertoni. jamakkashly@hotmail.com

3 Acknowledgments I would like to express my sincere thanks to my supervisor Prof. Sven-Erik Sandström, Växjö University for his support and helpful suggestions for this thesis work. I also would like to express my special thanks to my family and friends.

4 Table of Contents. INTRODUCTION.... THEORETICAL BACKGROUND.... RADIATED AND RECEIVED POWER..... Radiated Power..... Radiation Resistance and Received Power Friis Transmission Equation PROPAGATION MODELING..... Overview of Channel Modeling..... Path loss Models due to Propagation Mechanisms a Path loss due to reflection and the Two Ray model b Path loss due to diffraction PROPAGATION MODELS PROPAGATION MODELS FOR RURAL AREAS Deterministic Multiple Edge Diffraction Models Approximate Multiple Edge Diffraction Models a The Bullington method b The Epstein Petersen method c The Japanese method d The Deygout method e The Giovanelli method The Slope UTD method The Integral Equation approach The Parabolic Equation method PROPAGATION MODELS FOR URBAN AREAS The Okumura model The Hata model The Walfisch - Bertoni model SIMULATION The Hata model The Walfisch - Bertoni model Comparison of the Hata model and the Walfisch - Bertoni model APPENDIX A APPENDIX B APPENDIX C... 8 APPENDIX D REFERENCES... 89

5 . INTRODUCTION In radio propagation channels, spatial and temporal variations of signal levels are usually observed on three main scales. Signal variation over small areas (fast fading), variations of the small area average (shadowing), and variations over very large distances (path loss). Path loss prediction plays a crucial role in determining transmitter-receiver distances in mobile systems []. This thesis aims at describing various accurate path loss models that are used in rural and urban areas. In Chapter, the theoretical origins of the propagation phenomena and the received power concept are introduced. The difference between the range dependent path loss and fast/slow fading, as well as the basic path loss models are introduced. Basic models are also simulated by using MATLAB. Chapter 3 focuses on describing various accurate models for rural and urban areas, respectively. For the rural case, deterministic and approximate multiple edge diffraction models, as well as models with other approaches, are introduced. For the urban case the Walfisch-Bertoni and the Okumura-Hata models are chosen for a detailed comparison. The Walfisch-Bertoni model is described in detail via MATLAB simulations. A detailed description of the repeated Kirchoff-Integral which is used in this model is given in Appendix A. The standard Hata model is chosen as a reference for the results of the Walfisch-Bertoni model. In Chapter 4, the proposed models are compared for various parameters and simulated in MATLAB.

6 . THEORETICAL BACKGROUND. RADIATED AND RECEIVED POWER.. Radiated Power From the Maxwell equations for a homogeneous, isotropic, linear and lossless dielectric medium, the magnetic vector potential A can be obtained as [, 3]: where, jkr A e dv ', 4 J (..) v R R r r', (..) is the distance from the source point to the field point. The free space wave number is, k. (..3) Assuming a Hertzian dipole source one has the current as, i( t) Re{ Ie jt }. (..4) Figure.. The Hertzian dipole. Considering Figure.. and replacing Jdv ' in equation (..) with z ˆ Idz ' it can be written, I e jkr A dz ' ˆ. 4 z (..5) c R Assuming that the current is constant over the infinitesimal length l of the dipole ( R r), and that the point of observation is far away, one has,

7 Il e jkr A z ˆ. (..6) 4 r This expression suggests that the wave is propagating radially, in the direction of ˆr, with the phase constant k. The amplitude of the wave is inversely proportional to the distance. Since, B H A, (..7) and in spherical coordinates, one can write, zˆ (cos rˆ sin θ ˆ), (..8) Il e jkr A (cos rˆ sin θ ˆ), (..9) 4 r and obtain the following expression for the magnetic field, jkil jkr H [ A] sin e ˆ 4r jkr. (..) For the electric field far away from the dipole, Maxwell s equation ( J ) yields, which can be written, E [ H ], (..) j E Il jkr jkil jkr cos e ˆ sin e ˆ, r jkr 4r jkr k r r θ (..) where the intrinsic impedance of free space is introduced as,. (..3) From the above equations it is understood that the electric and magnetic fields vary with the distance r. If the kr product in the equations is much smaller than, i.e. kr<<, the terms with r and r 3 are predominant and the term jkr e goes to unity. After approximations the following expression can be written for the magnetic and electric field: 3

8 Il sin H ˆ. (..4) 4 r For the electric field some extra approximations can also be made for kr<<. The term ( ) jkr can be approximated by. Furthermore, substituting jkr k in equation (..) with, the electric field can be written as, Il cos sin ˆ ˆ E r θ, 3 3 (..5) 4 j r r where the term with r 3 is the electric field intensity produced by a static electric dipole, and hence called the electrostatic field component. Similarly the term with r for H, represents the magnetic field intensity produced by the very short filament of current, and is called as the induction field component. Since r << /k this zone is called the near field. The time dependent instantaneous vectors are, e( t) Re{ E e jt } and h( t) Re{ H e jt }. (..6) For the near field the instantaneous Poynting vector, which corresponds to the vector power density (W/m ), is, ( ) ( ) ( ) Re j t j t t t t e w e h {E } Re{ H e }. (..7) One may write a vector depending on (x,y,z,t) in phasor form as a vector depending on (x,y,z) independent of time. By writing the real magnetic and electric field as, jt * jt h( t) [ [ He H e ], (..8) jt * jt e( t) [ [ Ee E e ], (..9) the instantaneous Poynting vector can then be rewritten as follows: j t * w( t) e( t) h( t) Re{[ E H] e [ E H ]}. (..) The time averaged vector power density then becomes, * w( t) Re{ E H }, (..) 4

9 which for the near field eventually yields, I l w r (..) ˆ r( t) Re{ j sin }. 5 6 r In equation (..), the -j indicates that the near zone has capacitive behavior, which means that the dominant field is purely reactive and hence has zero average power. On the other hand, assuming that the observation point is far away from the dipole (kr>>), the terms r and r 3 get extremely small. The far field components then become, jkil sin jkr H e ˆ, (..3) 4 r jkil sin jkr E e θ ˆ. (..4) 4 r From this, it is seen that the far field is a spherical wave with H and E fields that are perpendicular, transverse and propagating in the r direction. In this case the medium s intrinsic impedance can be written as, E, (..5) H and from equation (..3) it is found that, 377. (..6) For the far field time averaged vector power density one has, ˆ * w( t) Re{ E H } E r, (..7) I l w( t) k sin r ˆ (W/m ). (..8) 3 r Equation (..8) states that the power density in the far field is purely real and directed radially outwards. Thus, this is also called the radiated power per unit area. The total radiated power then becomes, P rad w ( t) ds. (..9) s Substituting equation (..8) into (..9) and integrating over a large sphere one has, 5

10 3 d k I l 3 k I l P sin rad In free space ( ) one has,. (..3) l Prad 4 I. (..3).. Radiation Resistance and Received Power One may seek a relation between the current and the total radiated power from the dipole. Since the far field is purely real, the far field can be linked to a resistance R called radiation resistance, Thus, P rad I Rrad. (..3) rad R rad l 3, (..33) and in free space, R rad l 8. (..34) Receiver Z A Rrad R Loss X A V Z L Transmitter r E (9 ) V s I Figure.. Voltage induced at the receiver antenna. 6

11 The scenario in Figure.. assumes that the antennas are lossless Hertzian dipoles ( RLoss ) polarized in z direction. The electric field at distance r is, E jkil sin jkr e. (..35) 4 r The electric field impinges on the receiving antenna at an angle of (9 ) and induces a voltage proportional to vertical component of the electric field and the length of the receiver antenna l. So the induced voltage is, V E l sin, (..36) where E from equation (..4) is, jk Il E sin. (..37) 4 r To find out the average power delivered to the load in Figure.. one can assume that the load is matched to the antenna. That is to say that the antenna impedance Z A Z * L. The total impedance is R rad which yields maximum power to the load according to Jakobi s law. The power delivered to the load then becomes, P [ V / R ] R E l sin. (..38) r rad rad 8Rrad..3 Friis Transmission Equation In antenna theory, the receiving antenna s power capturing capability A er Pr, (..39) w () t is defined as the ratio of average power received by the antenna s load, to the time average power density at the antenna, and is called effective area. Earlier the average power density for the far-field was introduced as, E w ( t). (..4) From the equations (..38), (..39) and (..4) the effective area of the receiving antenna then can be written as, A er 4R l sin. (..4) rad 7

12 By using equation (..33) one obtains, A er.5sin Gr 4 4, (..4) where, Gr.5sin, (..43) is the directive gain of the Hertzian dipole. Equation (..4) shows that the receiving antenna s effective area is independent of its length and inversely proportional to the square of the carrier frequency. At this point one can realize that the term frequency dependent propagation loss is not the effect of wave propagation but the receiving antenna itself. The average power density in terms of radiated power, transmitter gain G t and the distance r can be written as, Considering, Prad Gt w () t. (..44) 4 r P r ( t) Aer w, (..45) the equations (..4), (..44) and (..45) yield the following formula: P P ( ) 4 r r rad GG t r, (..46) which is called the Friis transmission formula and gives a relation between the power radiated by the transmitting antenna and the power received by the receiving antenna. The Path loss for the free space in db then can be written as follows: P GG PL( db) log log[ ]. (..47) rad t r P r (4 ) r The far-field (Fraunhofer) distance transmitter antenna [], R f depends on the maximum linear dimension D of the R f D. (..48) For the distance R f to be in the far-field zone it should also satisfy R D and R. Another practical choice is to determine a received power reference point R, which is chosen smaller than any distance practically used in the mobile propagation, and also satisfies f f 8

13 R R. Applying Friis formula provides a relation between power P r( R ) at any arbitrary f distance and power P r( R ) at the reference point, R r R r R P ( ) P ( ) R, (..49) where R R emphasizes the inverse square law. The power level in dbm is defined as the received power with respect to milli-watt [4]: P ( R ). R R r P r( R) dbm log log( ). (..5) R In practice, could be m for indoor environment and m- km for outdoor environment, for frequencies in the range - GHz. 9

14 . PROPAGATION MODELING.. Overview of Channel Modeling Traditionally, mobile radio propagation models are categorized in two groups based on the fading phenomena. The models that predict the overall average of the received signal strength at a distance from the transmitter are called large-scale propagation models [4]. In general the amount of damping is then called the path loss. Path loss is crucial in determining link budgets, cell sizes and reuse distances (frequency planning). In this kind of modeling, the mean signal strengths for arbitrary transmitter-receiver separation and large distances are predicted. Therefore, large-scale propagation models are useful for estimation of radio coverage areas, since they provide a general overview over long T-R distances. Basically, as the mobile moves away from the transmitter, the local mean of the received signal decreases gradually with slight variations. Here the local average of the received signal power is the issue. Large scale models account for reflectors, but not dense reflecting and scattering environments. The local average received power is computed by averaging signal measurements over a measurement track of 5 to 4 moving radially away from the base station [4]. Measurement tracks for personal communications is generally from m up to meters long. Large-scale propagation curves are plotted based on these measurements. One important thing to point out is that the dominant mechanism in large scale models is reflection. Generally path loss models assume that the path loss is the same at a given T-R distance without including the middle scale shadowing effects [5]. Shadowing (slow-fading, log-normal fading) is caused by obstacles between transmitter and receiver. These obstacles attenuate the propagating signal power by absorbing, reflecting, diffracting or scattering. Shadowing occurs over tens or some hundred meters and is also seen as a part of large scale fading, which in fact can be also categorized as middle-scale fading. Variation of the signal due to path loss occurs over very large distances, m up to km. However in the case of shadowing the variation occurs over distances proportional to the length of the obstructing object, typically in the order of building dimensions, m up to meters. Especially in cellular system design, when modeling the coverage area of a cell, path loss and a shadowing model is combined. Figure..3 depicts contours of constant received power due to path loss plus random or average shadowing.

15 Path loss + random shadowing Path loss + average shadowing R Figure..3 Combined path loss and shadowing. The second group is called small-scale models that focus on the instantaneous behavior of the received signal. These models are called small scale because here the issue is the variations of instantaneous signal strength over very short time intervals or very short distances with respect to the wave length. Moving the mobile the distance of one wavelength may cause a variation in the received signal strength of up to 4 db. Unlike the predictions in large scale models, the variation of the signal is not a gradual decrease. The speed of motion is also a parameter that effects the variation in the signal level. Especially in urban and heavily populated areas the received signal is the vector sum of the scattered, reflected and diffracted signals which arrive at the receiving point from various directions with various propagation mechanisms. These propagation mechanisms cause rapid and dramatic variations for the resultant received signal as the mobile moves a few wavelengths. It is also important to point out that the dominant propagation mechanism in small scale fading is not reflection but scattering. Due to the fact that the phases of the received signals are random, the sum of all these different components behaves like noise. In this case the Rayleigh fading model can be used. To overcome the fading, MIMO technology may be used to achieve diversity by combining signals. The signal attenuation can be either Rayleigh distributed or Rician distributed, depending on whether there is line of sight (LOS) or not. If there are a lot of scattered components but no LOS then the attenuation coefficient can be effectively modeled as Rayleigh distributed. If there is LOS then one can apply the Rician distribution. The small scale fading model is a short term model and the severe signal fluctuations usually happen around a slowly varying mean. Figure..4 illustrates the received power variations with distance for both types of fading.

16 Received Power (dbm) T - R distance (meters) Figure..4 Large scale versus small scale fading over distance. Figure..5 illustrates the received signal strength variations in large and small scale models over time. Red and green curves represent the means in large and small scale fading, respectively. One can imagine that a person is holding a power-meter and taking signal strength measurements while moving along. On top in Figure..5 large scale fading is plotted while small scale fading is plotted at the bottom. Both fadings have slowly varying means. But one can also see that even the mean variations in the small scale fading are faster than in the large scale fading. Received Power strength (dbm) Figure..5 Large scale versus small scale fading over time. Time (µs) Figure..6 illustrates the same scenario and depicts the relation between the received signal strength and the distance from the transmitter. If one takes a small segment from the path loss curve and magnifies it, as in the middle figure, one finds a slow-fading (shadowing) curve. However if one again magnifies a small section from the slow-fading curve and analyzes it in detail one finds a fast-fading (short-term fading) curve. So fading can be subdivided into two additional types of fading which of course have different mathematical descriptions.

17 Received Signal Strength (db) Fast-fading Slow-fading Path loss Figure..6 Path loss vs. Slow-fading vs. Fast-fading. As the mobile user moves away from the transmitter, measurements with a power-meter probably yield a graphic similar to Figure..4 with a decreasing trend with fluctuations around the mean. If the same user repeats this experiment he will get a similar but not the same curve. Figure..7 also shows the received power variations of the overall signal and its components as the receiver moves away from the transmitter. The horizontal axis corresponds to the T-R distance and the perpendicular axis corresponds to the power in db. Figure..7 Components of Overall Signal as a function of distance. 3

18 .. Path loss Models due to Propagation Mechanisms...a Path loss due to reflection and the Two Ray model Reflection of an electromagnetic wave occurs when it impinges upon an object with different electrical properties and very large dimensions compared to the wavelength [4]. The wave impinging upon a new medium is partially transmitted into the second medium and partially reflected back to the first medium. If the second medium is perfectly dielectric there is no energy loss during reflection. If one assumes a second medium which is a perfect conductor, all the incident energy is reflected back into the first medium without any loss. When the first medium is free space and the permeabilities of two media are, the electric field intensity of the reflected and the transmitted waves are related by the simplified Reflection Coefficient [4, 5]: and, Er j e, (..) E i r sini r cos i v r sini r cos i sini r cos i h sini r cos i for vertical polarization for horizontal polarization (..) where, i is the grazing angle as shown in Figure.. and r is the complex dielectric constant of the ground. Reflection causes multipath effects. It may also be used (passive metallic reflectors) for covering areas which are normally not covered. To know the electrical properties of the reflector surface is important when it comes to modeling. Materials in rooms that cause reflections are made out of dielectrics (walls, tables etc.). There are also nearly perfect conductors such as metal frames in the windows that form perfect reflectors. At this point it is important to note that the statistical models begin to fail above GHz. Above GHz one has to search for a deterministic model in terms of rays. 4

19 h t T T '' T ' E d d i r i Figure.. Geometry of the two ray ground reflection model. E O E R h r R ' The two ray ground reflection model is a simple but useful model for predicting large scale signal strength over long intervals. Ground reflection occurs when one normally has a LOS. This model is accurate for predicting the large scale path loss for mobile radio systems with large T-R distances (without any obstruction) and tall transmitter masts (exceeding 5 m) [4]. The surface wave is usually negligible in two ray models, since the surface wave extends about λ above ground and gives no contribution to the received field strength. Figure.. shows the geometry of the two ray ground reflection model in the case of horizontal polarization [4, 5]. To find out the total field strength at the reception point R one must first seek a formula for the phase difference between the direct field E and the ground reflected field E r. Since obviously depends on the difference in path lengths the geometry in Figure.., and imaging, yields: ht hr TR d ( ht hr) d ( ), (..3) d d ht hr TOR TO OR TR ' d ( ht hr) d ( ), (..4) d where d = T'' R '. Additionally one can obtain the Taylor series of x for small x as, x x 8 x... (..5) For x<< one has, 5

20 ( ) x. (..6) / x In this case, x h h d t r ( ) or x h h d t r ( ) and t r d h h. (..7) The difference in length then can be written as, h h h h h h d d d d d t r t r t r ( ) ( ( ) ). (..8) The phase difference then becomes [6], 4hh t r d d. (..9) Here is the phase delay due to the path length difference. But there is also another phase delay which occurs at the point of reflection O. The reflection coefficient, j e, (..) changes the amplitude and the phase of the impinging wave on reflection. In the case of vertical polarization, for very large distances, the vertical vector components E and E becomes almost the same. It can be seen from the geometry that, d i sin d TR ' and sin d. (..) TR Assuming i for very large distances one can see that TR TOR and sin sin which results in equal vertical components of the electric fields in the z direction. In the case of horizontal polarization, the incident and reflected electric field are also assumed to remain the same, independent of the grazing angle. The equality of the angles i and can be derived by using boundary conditions from Maxwell s equations in a dielectric. In the case of vertical and horizontal polarization, when i ( =8 ), the reflected wave Er magnitude with i i is assumed to be 8 out of phase but equal in j j E. The reflection then gives a phase shift e e when, for both vertical and horizontal polarization, in the case of grazing incidence. 6

21 Total electric field strength at the point R is, j j E E E E ( e ) E ( e ), (..) tot d r d d E E ( cos jsin ). (..3) tot d For the amplitude one can write, which simplifies to, Etot Ed {( cos ) (sin ) }, (..4). (..5) Etot 4 Ed sin Thus, the received signal can be written as, hh t r Etot Ed sin( ). (..6) d The average power density for the direct path was introduced as, Erms PG t t w () t, (..7) 4d so for the direct wave, E d PtGt, (..8) 4 d and in free space, E d 3PG t t. (..9) d The resultant field can then be written as, 3PG t t hh t r Etot sin( ). (..) d d 7

22 Figure.. Received field strengths in the Two Ray model. Figure.. was created in MATLAB and shows the change in total and direct field strengths at the receiver due to distance. The chosen parameters are: transmitter height receiver height h 6m, h m, transmitted power P 5W, and the transmitter gain G,5 and an r t ideal reflection coefficient. The first graph depicts the scenario for 9 MHz, while in the second graph the frequency is 9 MHz. Many observations can be made in equation (..) and the corresponding Figure... It is obvious that the heights of the antennas, the T-R distance and the wavelength determine the maxima and minima of the field strength. One can also say that as the frequency is increased, more rapid variations occur in the resultant field due to distance. The maximum received field strength caused by the interference is twice the field strength of the direct wave. The dips in the graphs actually represent zeros ( db ), and cannot be shown clearly in Figure... For d h, h one has, and the field becomes, ht hr ht hr sin( ), (..) d d E hh t r tot 4 3 PG t t d. (..) t r t t 8

23 At this point an expression for the total average received power can be derived as follows: P r hh t r (4 ) 3PG 4 E t t d Gr Aer, (..3) 4 which simplifies to the following equation: hh. (..4) d t r Pr PG t tgr 4 Equation (..4) shows that for very large distances the resultant field strength falls faster than the direct field. Independent of frequency, the received power decays as 4 [4]. This is d good for avoiding interference in cellular systems. This simplistic but realistic model can hold good for mobile application at GSM frequencies. The model is applicable in LOS scenarios but LOS is seldom found as one moves very large distances away from the transmitter. Since the antenna gains are fixed, the way to adjust the signal is usually to change the antenna heights....b Path loss due to diffraction Geometrical optics doesn t include the field in the shadow of an obstruction, since it is based on rays. However, to some agree the E-M waves can propagate into the geometrical shadow depending on the wavelength, shape, height and location of the obstruction. Especially in the case of deterministic modeling diffraction models should be used. A knife-edge geometry, depicted in Figure..3, is a special simplistic case of diffraction where there is a single obstacle which is assumed to be extremely sharp so that no significant reflections can occur. It has infinite width along z-axis (perpendicular to the page). Assuming diffraction without any transverse effects one may reduce the scenario to -D. It is also assumed that the obstacle in Figure..3 absorbs all the upcoming waves below its tip [4]. T d ' h ' h d R ' d d Figure..3 Knife-edge geometry for a single obstacle. 9

24 According to the knife-edge diffraction geometry in Figure..3 the path difference can be found as follows where h' d ', d ' [4, 5]: d ' h' d ' h' ( d ' d '), (..5) which can also be written as, h' h' d ' ( ) d ' ( ) ( d ' d '), (..6) and the approximate path length is, d' d' h' h' d '( ( ) ) d '( ( ) ) ( d ' d '), (..7) d' d' h' ( d' d'). (..8) d ' d ' Since in practice h d, d, the equations can be approximated in terms of distances parallel to and normal to the ground, one has, h h', d d ' and d d '. Hence the phase difference becomes, h ( d d). (..9) dd Assuming that, and, the pitch angle, are very small, so that approximately, in radians, d ' d ' d ' d ' tan tan h'( ), (..3) d d dd h( ). (..3) It is important to mention the Fresnel ellipsoids since obstructions that penetrate the first Fresnel zone play a significant role in diffraction loss. If the propagating waves are considered to bend at every point between T and R then the same path length forms an ellipsoid. A Fresnel ellipsoid is a surface where the excessive path length is constant and has values, n, (..3)

25 where n is an integer. Figure..4 depicts Fresnel ellipsoids where TOR TO ' R TO '' R form the n th ellipsoid and T and R are the focal points of the ellipsoids. The n th Fresnel zone is usually defined with its maximum radius, r n nd d d d, (..33) which actually is the length of its minor half axis as represented by O'' M in Figure..4. O '' O T O ' M R Figure..4 Fresnel Ellipsoids. Since the phase difference is, the paths with yield a phase difference 4 9 and the resultant phasor is affected destructively and becomes smaller than the LOS phasor. The paths with yield a phase difference 9 and the resultant phasor is 4 affected constructively and becomes greater than the LOS phasor. Figure..5 shows this, secondary wave resultant wave secondary wave resultant wave LOS wave LOS wave Figure..5 Destructive and constructive interferences due to the difference in path lengths.

26 Often the Fresnel-Kirchhoff diffraction parameter is used as a normalized phase difference, ( d d) dd h d d ( d d ), (..34) or simply, v. (..35) By applying Huygens principle to radio waves, one can predict the received signal in the shadow region as the phasor sum of the electric field components of all secondary wavelets from the obstacles [7]. Since Huygens principle assumes all these incoming and outgoing wavelets to be spherical, one should use Fresnel diffraction. It is important to emphasize that in Fraunhofer diffraction the incoming and outgoing wavefronts are assumed to be planar and the propagation paths after obstruction are assumed to be parallel. So the phase differences between each consecutive element of the wave front increases linearly which results in constant phase differences between consecutive elements of the wave front. Also the paths taken by the waves are considered to be the same for very large distances, again because of the parallelism, which results in equal amplitudes at a point of observation. Thus the phasors of these elements form an arc of a circle at a point of observation. In Fresnel diffraction the outgoing wave-fronts are considered to be composed of tiny spherical wavelets and the propagating wavelets rays from the obstruction axis to the point of observation are not parallel. The phase differences do not increase linearly between two consecutive elements. y ' y h 3 d 3 h h P x h d P Figure..6 Huygen s principle and Fresnel diffraction geometry.

27 In Figure..6, at point P, the phase differences of the wavelets from the elements at h, h, h3,..., h n, where h h h3 h... hn hn differences,, is directly related to the path, (..36) dn ( hn h ) d d which can be written in the form, h h d n dn d( ( ) ). (..37) Binomial approximation for hn d path yields, hn h ( h h ) dn d d d n, (..38) and the phase difference for the n th element is approximately, ( h h ) d n n. (..39) This shows that the linear increase in height causes a quadratic increase in phase difference. If one marks the consecutive points with a constant phase increment ' C, i.e. the points ' ' ' ' '... ' ', one can write the following h', h'... h ' n where 3 n n expression: ' ' ' [( h' h ) ( h' h ) ] C n n d n n. (..4) To satisfy the equation (..4), one can see that h' h' h' 3 h'... h' n h' n. This means that for a linear step in phase, the step in height must decrease with n. Here the height difference h' n h' n is proportional to the radius and the effect of each wavelet at the observation point. Hence, the contribution of each wavelet reduces for a linear increase in phase difference [7]. At a point of observation, phasors of these elements form a curve called the Cornu spiral. In Figure..6 the incident plane wave can be defined as, jkx E ( x, y) E e. (..4) inc Applying the Kirchhoff integral and Huygens s principle to the geometry in Figure..6 for the region x, the total electric can be written as [8, 6], 3

28 where, j( kx /4) Ee j( /) u Etot ( x, y) e du, y (..4). (..43) x and y is the axis of the point P in Figure..6. The integral can be written in terms of Fresnel Integrals with the cosine integral, and the sine integral, v ( ) cos( ) C v u du, (..44) v ( ) sin( ) S v u du, (..45) and the Fresnel-Kirchhoff parameter, Thus, v y'. (..46) v j( /) u. (..47) C( v) js( v) e du The asymptotic expansions for v yield, The limit, yields, C v, (..48) v ( ) cos( v ) S v. (..49) v ( ) sin( v ) lim[ C( v) js( v)] j, (..5) v j( /) u e du ( j ), (..5) 4

29 and the expression for the total field can be rewritten as [8], j( kx /4) Ee Etot( x, y) j [ C( y) js( y)]. (..5) To illustrate the amplitude changes due to height for a fixed distance x from the edge, the absolute value of the conjugate is of interest L j [ C ( y ) js ( y )]. (..53) Figure..7 is a plot of the Cornu Spiral F( v) C( y) js( y) in MATLAB. Figure..7 The Cornu Spiral and the corresponding field strengths. In Figure..7, for any value of v y, the distance L from the point Fv ( ) to (.5 j.5) is proportional to the resultant signal. This is because the Cornu spiral in this case represents the vector sum of the sources from the tip y ' to +. The length of the blue line from (.5 j.5) to (.5 j.5) represents the condition of LOS where v i.e. y in Figure..6. The blue line from the origin to (.5 j.5) represents the condition where v which in Figure..6 corresponds to the point at the shadow boundary y. The green Etot line represents the first point where the field strength is equal to the LOS condition E over the shadow boundary where y and v<. The red line represents the point P in Figure..6, somewhere below the shadow boundary where y, v. inc 5

30 The total field relative to the incident field is, E E tot inc j 4 e j [ C ( y ) js ( y )], (..54) and using the identity, j 4 e j, (..55) it becomes, Etot [ C( v) S( v)] j[ S( v) C( v)]. (..56) E inc Figure..8 represents the plots of the absolute values of the total field strengths relative to Etot the direct field both in linear and decibel scales due to the parameter v in MATLAB. E inc Large positive values of v correspond to deep shadow. Figure..8 Relative field strength E E tot inc due to the parameter v in linear and db scales. The terms with Fresnel integrals in equation (..56) makes the computation difficult. Therefore, when predicting the knife-edge diffraction loss, equation (..56) can be approximated by the expression given by Recommendation ITU [8] for v.7, E tot log J ( v) 6.9 log [ ( v.) v.] [db] Einc. (..57) 6

31 Approximations for E E tot inc in db are also given by Lee [4], J ( v) v - J ( v) log (.5.6 v) - v J v.95v ( ) log (.5 e ) v J v v v ( ) log (.4.84 (.38. ).4.5 J ( v) log ( ) v>.4 v (..58) Figure..8 shows the diffraction loss due to height for a fixed frequency. However if the frequency is increased for the same scenario the ratio E E tot inc decreases more quickly. Figure..9 shows the diffraction losses for GHz and 4 GHz at a point of observation x= m and -5<y<5 meters. According to the figure, a higher frequency produces a more distinct shadow. Figure..9 - Relative field strength E E tot inc for GHz and 4 GHz. The first Fresnel zone is crucial for diffraction loss. If the frequency is increased for an obstructed path, due to equation (..33), the first zone gets narrower and the diffraction loss may be reduced. However, as can be understood from Figure..9 and considering the geometry in Figure..6, at higher frequency, for a fixed observation point x, moving the point in -y direction or increasing the height of obstruction causes diffraction loss to happen more quickly. The related parameters are, ( d d ) v y y h x dd. (..59) 7

32 3. PROPAGATION MODELS 3. PROPAGATION MODELS FOR RURAL AREAS 3.. Deterministic Multiple Edge Diffraction Models When predicting the path loss between two fixed stations for large distances, the path profile between the stations is often reduced to single knife edges since the wavelength is short compared to the size of obstacles such as hills. The total path loss is then the free space loss plus the predicted obstruction loss of the two-dimensional multiple edge diffraction model. For multiple knife-edge geometry the waves incident on the edges will not be plane after the first edge and the Fresnel integral should be evaluated consecutively for each edge to obtain a deterministic loss prediction. Hence, for n edges an n-dimensional Fresnel integral must be evaluated. This is very difficult to implement. A method with extreme accuracy for the attenuation of electromagnetic waves by multiple edges is given by L. E. Vogler. Starting from Furutsu s generalized residue series formulation for electromagnetic waves over a sequence of smooth cylindrical obstacles, Vogler derived an attenuation function for multiple knife edges using Fresnel-Kirchhoff theory []. In his derivation he replaces the residue series in Furutsu s method with a contour integral since the series converges slowly in the case of a grazing incidence. Vogler obtains an N-dimensional integral I N by considering the cylindrical surfaces with zero that correspond to N multiple knife-edge geometry and describes C as the spreading loss factor over the propagation path in his formula [9]. The N attenuation A N, which is the ratio of total loss relative to the free space loss for N edges (see Figure 3..), is given as follows [9, ]: where, A C e I, (3..) N / N N (db) N N C N dd3... dnd T N ( dm dm ) m, (3..) for N ; and N f xm m I... e dx... dx N, (3..3) N xnn x C for N where T n N d d... d and, N f ( x, x,..., x ) ( x )( x ), (3..4) N m m m m m m 8

33 for N ; and f for N. N N, (3..5) m dd m m m, (3..6) ( dmdm )( dm dm) jkd d r, (3..7) m m m m m ( dm dm ) where r is the radius of the first Fresnel zone. Figure 3.. depicts the geometry in Vogler s method where the given, 3, n are taken positive since h h, h3 > h4 and hn> hn and is taken negative since h h3. d n is the distance between ( ) th th n and n edges. 3 h h h h 3 n hn hn d d 3 d dn Figure 3.. Multiple edge geometry for Vogler s integral method. Vogler transforms the multiple integral m m I N in equation (3..3) into an infinite summation I which can be computed numerically. The simplified solution for the attenuation N becomes, A where, N / N N N m m A C e I, (3..8) m m I I( m m, ) C(, m, m ), m m mm (3..9) 9

34 j ( k i)! ji C( N L, j, k) N L I( k i, N L) C( N L, i, j), (3..) ( j i)! i with the repeated integrals of the error function, n x (3..) I( n, ) ( x ) e dx, n! and, m C( N, m, m ) ( m )! N I( m, ) I( m, ), (3..) N N3 N3 N N3 N N N where the notations are given [9], i mn L, j mn L, k mn L L N, N 4 N, m m, mk when k N- (3..3) 3.. Approximate Multiple Edge Diffraction Models Because of its computational rigor, Vogler s method is usually used when comparing the accuracy of approximate methods. Commonly used approximate methods are the Bullington, Epstein-Petersen, Japanese, Deygout, and Giovanelli methods []. 3...a The Bullington method The terrain profile must be reduced to knife edges before the Bullington method is implemented. To compute the diffraction loss, an equivalent knife edge is determined by reducing all knife edges to a single knife edge. To determine the equivalent knife edge, two lines are drawn joining transmitter and receiver to with respective dominant edges which has the greatest angle of elevation. The intersection point of these two lines is the top of the equivalent knife edge and the diffraction loss is calculated as if it were the only obstacle [7]. In Figure 3.. the dashed red lines depicts the Bullington geometry. 3

35 3 T R d d d 3 d3 Figure 3.. Geometry for the Bullington, Epstein Petersen and Deygout methods. 3...b The Epstein-Petersen method Here, attenuation due to each obstacle is calculated and summed to find the total diffraction loss. For instance, in Figure 3.. the geometries of each triangle T, 3 and 3R are taken as a simple knife-edge geometry and diffraction losses are computed individually and summed. For each case the obstruction lengths above the lines T, 3 and R can be found by using simple geometrical laws. This method results in large errors when the obstacles are close []. In this case the Millington correction is added to the original loss, L' log (cosec ), (3..4) where is the spacing parameter for the found from the formula [], th n and ( n ) th edges. The spacing parameter is ( d d )( d d ) n n n n cosec. dn ( dn dn dn) / (3..5) The Epstein-Petersen method calculates the diffraction loss for N successive knife edges and can be written as [], where, N L ( N) L( v ), (3..6) epstein i i 3

36 j j t L( vi ) e dt, (3..7) vi is the diffraction loss for the i th knife edge. Since Vogler s simplified attenuation function is, N Lvogler ( N) ( ) CN Im, (3..8) m then for m= the Epstein-Petersen loss can be described in terms of Vogler s loss as, L epstein ( N) L ( N) vogler m, (3..9) C N which shows that Epstein-Petersen method turns out to be a first order approximation of Vogler s method and underestimates the C spreading loss factor. N 3...c The Japanese method From the top of each obstruction a horizon ray is drawn which also passes from the top of the previous obstruction. The intersection point of this horizon ray with the plane of the terminal is then seen as the effective source. For instance, in Figure 3..3 the diffraction losses from each obstruction are calculated due to the geometries of T, T'3 and T''3R. The sum of these gives the total diffraction loss []. Actually this method is equivalent to the Epstein- Petersen method plus the Millington correction given in equation (3..4). T '' 3 T ' T R Figure 3..3 Geometry for the Japanese method. 3

37 3...d The Deygout method First the v parameter is calculated for each obstacle as if there were no other diffracting obstacles. Secondly, the dominant edge which has the maximum v parameter is determined and the diffraction loss is calculated as if it were the only edge. The dominant edge now becomes the terminal point of two sections divided by it. Then the process is repeated recursively by finding out the maximum v parameter and determining loss until all edges are considered. The total diffraction loss is then the sum of all these losses. For the geometry in Figure 3.. the total excess loss L in single edge diffraction terms can be computed by the following expressions [3]: where L L L L (3..) (db) total 3 L i log A i (3..) i x Ai e e dx (3..) i where is calculated as in Vogler's method. i If the edges are too many or too close, the Deygout method overestimates the path loss. In these cases Causebrook introduces a correction which reduces the overestimated path loss as follows: where, L L L (6 L L )cos L (6 L L )cos, (3..3) total d ( d d ) 3 4 cos, ( d d)( d d3 d4) d ( d d ) 4 cos 3. ( d d d3)( d3 d4) (3..4) 3...e The Giovanelli method Figure 3..4 relates to this method. As in the Deygout method firstly the dominant edge is determined due to the v parameter. Edge M represents the dominant edge. Then a reference point R is found by projecting a line on to the RR plane which starts from M and passes through the adjacent edge N. The loss due to the v parameter for the TMR geometry is then 33

38 calculated by obtaining the excessive effective height [, 3]. The effective height for edge M which is the excess height above TR is, h dh '' h, d d d3 (3..5) where, H h md 3 and m h h. (3..6) d So the loss from edge M can be written as a function of the TMR geometry parameters as, L f ( d, d d, h h ''). (3..7) M 3 After computing the loss from the dominant edge the dominant edge now becomes the terminal point for two sections divided by it, as in the Deygout method. The same process is applied to the remaining secondary edge N. The effective height for edge N which is the excess height above MR is, h dh 3 ' h. d d3 (3..8) So the loss from edge N can be written as a function of the MNR parameters as, L f ( d, d, h h '). (3..9) N 3 The total loss can then be written as, Ltotal LM LN. (3..3) For n edges this process is repeated until there are no edges left to be considered. M h '' h h ' N R '' R ' T h R d d d3 Figure 3..4 Geometry for the Giovanelli method. 34

39 3..3 The Slope UTD method Unlike the methods introduced previously this method uses the Uniform Theory of Diffraction (UTD). The Slope UTD method estimates the propagation loss with much better accuracy than the approximate multiple knife edge methods and with much shorter computation time than Vogler s method. The integral equation and parabolic equation methods for terrain propagation uses the actual terrain profile but they do not provide an optimum solution for a point to point prediction. The Helmholtz-Kirchoff theory of waves has many computational difficulties and especially in the case with a large number of obstacles it gets too complex to apply [4, 5, 6]. Geometrical optics (GO) is a useful method since it views propagating radio waves as light rays for very small wavelengths, but neglects diffraction and predicts zero fields in the shadow regions. A high frequency technique, the Geometrical theory of diffraction (GTD) developed by J.B. Keller, is an extension of geometrical optics, basically by inclusion of additional diffracted rays. The GTD method can accommodate 3D environments and can predict diffracted rays very rapidly for complicated geometries. But because the GTD method is a ray-optical theory it is valid only when outside the transition region which is the region within the GO incidence and reflection shadow boundaries. That is because around shadow boundaries the field has rapid spatial variations. For a perfectly conducting edge Keller's GTD results in discontinuity problems in the vicinity of incidence and reflection shadow boundaries. The UTD method developed by Kouyoumjian and Pathak is a development of GTD for perfectly conducting wedges which overcomes GTD s transition region limitations. By identifying a region around the shadow boundary, performing an asymptotic analysis, and multiplying the diffraction coefficient by a transition function F( X ), which involves a Fresnel integral, they created a ray based theory valid at all spatial locations. It is proven that UTD provides a more accurate prediction than GTD. Luebbers also developed this theory for wedges with different dielectric materials by modifying the UTD coefficients, reducing the errors in the case of grazing angles. The UTD method can also be extended to multi-shaped canonical objects such as wedges, edges and cylinders. The UTD generally fails when the transition regions or GO caustics overlap. For such regions the Physical Theory of Diffraction (PTD) may be used for more accurate calculations. However, UTD and GTD both fail when the incident field is not a ray optical field. For example in the case of double wedge diffraction the wave impinging upon the second wedge is already a diffracted field from the first wedge. One then cannot just apply UTD theory for this wedge since it is illuminated by the transition region field, not the incident field from the transmitter. Therefore, the Slope UTD approach was introduced by Andersen as an extension of UTD for the cases where the incident field has a rapid spatial variation i.e. has non-zero gradient and even zero amplitude [, 7]. The slope UTD uses a high order term asymptotic expansion 35

40 and includes higher-order diffraction unlike classical UTD. Hence, just after diffraction, the total created field at a point is composed of a first order diffracted field which is proportional to the amplitude of the incident field and a slope diffracted field which is proportional to the derivative of the incident field. Figure 3..5 depicts a slope UTD diffraction geometry with a wedge, where incident shadow boundary and TT ' is the reflection boundary [7]. R R TT ' is the n T R ' T ' T s ' T R ' s R Figure 3..5 UTD diffraction geometry. According to the slope UTD theory the total diffracted field strength behind a single absorbing half-plane is [4]: Ei s Ed EiD( ) d ( ) A( s) e n jks, (3..3) where E i is the incident field and, ', (3..3) is the angle between the incident and the diffracted ray, i.e. the angle above the shadow boundary. j /4 e D( ) F[kLcos ( / )], (3..33) k cos( / ) is the amplitude diffraction coefficient. As () s s( s s ), (3..34) is the spreading factor and, s D( ) d ( ), jk (3..35) 36

41 is the slope diffraction coefficient. The UTD transition function is given by, ju jx F( x) j xe e du. (3..36) x By using the derivative of F( x, ) F( x) F '( x) j[ F( x) ], (3..37) x the slope diffraction coefficient s d then can be rewritten as, j /4 s e d ( ) Ls cos( / )( F( x)) (3..38) k where the L and L s parameters are the distance factors calculated according to some continuity equations that vary for each ray due to geometry. T s s ' E E s n ' W W P, s s s E P, P, R Figure 3..5 Slope UTD geometry for wedges. Figure 3..5 depicts a multiple-diffraction scenario that is suitable for the slope UTD method. One can describe the field impinging on the left face of the first wedge W in terms of the radiated field strength for s m which is defined as E s. That is to say the incident field at the vertex point of W then can be described as, E s E( W), s (3..39) since the distance from the transmitter T to the vertex point of the first edge is s. The fields from the first wedge due to classical UTD method can then be described [4]. Multiplying the 37

42 diffraction coefficients as follows: D i and spreading factors E ( W ) E ( W ) D ( W ) A ( s ) Ai for each wedge the fields can be written (3..4) is the field computed at W which is diffracted by the vertex of W. E ( R) E ( W ) D ( R) A ( s ) (3..4) is the field computed at the receiver R which is diffracted by the vertex of W. Once E is diffracted, E impinges on W as an already diffracted field since W is in the shadow. In the classical UTD method the slope diffraction term is neglected. However, in this double diffraction scenario, better accuracy is obtained, if slope diffraction is taken into account. Therefore the equations can be rewritten as follows for the slope UTD case as follows: E ( W ) E ( W ) D ( W ) A ( s ), (3..4) is the field computed at W which is diffracted at the vertex of W. There is no slope term in equation (3..4) as in equation (3..4), since the wave diffracted at the vertex of W is not an already diffracted wave before impinging on W. E( W) s E( R) [ E( W ) D( R) d ( R)] A ( s), n (3..43) is the field computed at the receiver R which is diffracted at the vertex of W. E( W) n the directional derivative of E in the normal direction of the incident shadow boundary of s E and d ( ) R is the slope diffraction coefficient which is derived from the amplitude diffraction coefficient as, is s d ( ) D( ) jk where '. (3..44) Since, j /4 e D( ) F[kLcos ( / )], (3..45) k cos( / ) and the derivative of the transition function is, 38

43 F( x) F '( x) j F( x), (3..46) x the slope diffraction coefficient is written as, j /4 s e d Ls sin( / )[ F( x)]. (3..47) k The directional (normal) derivative of the incident field cylindrical coordinates for the D case as follows: E i n can also be defined in which reduces to, Ei Ei Ei ( ˆ ˆ Ei ˆ) ˆ s z s, (3..48) n z s Ei Ei Ei n s s s. (3..49) Therefore, the second diffracted field in slope UTD can be rewritten as: E ( W ) E ( R) [ E ( W ) D ( R) d ( R)] A ( s ) s s, (3..5) which then also can be written as, E ( W ) D ( W ) E ( R) [ E ( W ) D ( R) d ( R) A ( s )] A ( s ). s s s (3..5) This expression indicates that the received field at R, which is generated at the second wedge W, can be written as a compound of two different waves, an amplitude wave and a slope wave. For computing the distance factors L and L s, the points on the shadow boundaries for each successive edge are determined so that their distances are the same as the distances between the wedges, as shown in Figure Therefore, according to the method, one can write the following expression for the field at point P, which is diffracted at W, E ( P ) E ( W ) D ( P ) A ( s ). (3..5),, At this point a known fact from knife-edge diffraction theory, stating that the diffracted field on the shadow boundary is half of the incident field, can be employed. That is to say, if there 39

44 was no diffraction, the incident field on P, the continuity equation is,,, would be twice that of the current situation. Thus, E ( P ) E ( W ) D ( P ) A ( s ), (3..53) and for Andersen introduces, D( ) L/, (3..54) so the equation becomes, E ( P ) E ( W ) L A ( s ). (3..55),, The amplitudes of E( W ) and E ( P ), are inversely proportional to the distances from the transmitters which are consecutively s and ( s s). Therefore, s e e L e s s s s s s jk ( s s ) jks jks, ( ), (3..56) with the distance factor, L, ss s s, (3..57) and similarly for the next point, L, s( s s). s s s (3..58) But for the point P, there should be two distance factors, since the field before diffraction s from W is an already diffracted wave. Therefore L, for the amplitude wave and L, slope of the diffracted wave are to be calculated. L, continuity equation, which becomes,,,, for the is calculated in the same way from the E ( P ) E ( P ) E ( W ) D ( P ) A ( s ), (3..59) E ( P ) E ( W ) L A ( s ). (3..6),, Since E( P, ) and E( W ) are already known, and by using the previously determined L, s and L,, L, can then be calculated. For L, of the field incident upon W, one needs the continuity equation for the slope 4

45 E( P, ) E( W) d ( P, ) A( s), n n n (3..6) indicating that the diffracted field s slope for point P, without the existence of W is twice the slope-diffracted field s slope for point P, when diffracted from W. In terms of the distance factor, E( P, ) E( W) s 3/ L, A ( s ). n n (3..6) The same scenario can be extended for three wedges as in Figure The previous scenario is modified by replacing the receiver R with another wedge W3 with the same height. And the new receiver R ' is placed again in the shadow of the closest edge. T s E ' W s E ' W n s 3 E s 3 WE 3, 3 R ' Figure 3..6 Slope UTD geometry for 3 wedges. In this case the received field at the receiver sum of two waves created at W R ' can be written as E ( R '). Since E ( ) 3 R is the and computed at R in the previous scenario, it is an already diffracted field when it impinges on W 3. It will create a field strength E ( ') 3 R at the point R ', E( R) s which is now composed of D3( R') E( R) and d 3 ( R '), which requires the data of n normal and slope diffraction coefficients of the third wedge computed at R '. Hence, the new received field at receiver will be, E( R) s E3( R') D3( R') E( R) d3 ( R'). n (3..63) From the first term of the expression one has one amplitude wave and one slope wave and from the second part of the equation one has two slope waves. The second part of the s d ( ) expression requires the second derivative of the diffraction coefficient which can be n found in related literature. This scenario can be extended to many wedges since the computing process is a recursive algorithm. If there are reflections to the receiver they should be considered as extra, additional rays. 4

46 A comparison of these models is implemented by Tzaras, detailed in [7]. Figure 3..7 is a comparison of the methods and includes the deterministic Vogler s method. Implementing the Vogler method is a huge undertaking, therefore in this thesis the data of Tzaras are used. Tzaras tested the methods in various parts of the UK for frequencies between 4 MHz and 9 MHz. The results in Figure 3..7 show a cumulative distribution function of the diffraction loss over the paths studied on one of the sites, containing 39 records. The y-axis gives the probability that the diffraction loss is less than the corresponding value on the x- axis. For example, approximately 3% of the measured paths have a diffraction loss of less than db. Figure 3..7 Comparison of described models The Integral Equation approach The previous methods reduce the actual terrain profile to canonical objects. For accuracy Vogler s method can be used but it gets too complex as the number of canonical objects increases. Hata presents a simple parametric model, but his model lacks dynamic variations of the average signal due to shadowing and reflection. Without reducing the actual surface geometry, Hufford derived an integral equation by employing the Green s theorem. But his integral equation method starts to experience numerical instabilities over MHz and in the UHF band and above these instabilities result in significant loss of accuracy. By evaluating Hufford s approach, Hviid presents another method which can overcome the instabilities at high frequencies. Hviid derives his method by using some important assumptions. The surface is assumed to be a perfect magnetic conductor with a reflection coefficient of -. To avoid problems in integration the surface has to be 4

47 smooth which means that the method excludes steep slopes and cliffs. Since the terrain profile is two-dimensional side scattering effects are also neglected. Back scattering which happens beyond and before the surface is also neglected because the method is developed to predict the mean average field. Back scattering is negligible for slow fading and is only taken into account for fast fading which is not an issue of interest for this method [8, 7, 8]. Since the surface is assumed to be a perfect magnetic conductor (PMC) the boundary conditions are, ne ˆ and nˆ H. (3..64) The free space spherical Green s function is defined as jkr e G( rr, '), R (3..65) which satisfies the homogenous scalar Helmholtz equation (see Appendix A), ( k ) G with a source point at, (3..66) r' r. R r r' is the distance from the source point to the observation point as shown in Figure By using the second Green s theorem an integral equation is obtained, T nˆ E( r) Tnˆ E ( ) ˆ ( ˆ i r n ' ) ' Gds ' 4 n E, (3..67) S where T when r is on the surface S and T otherwise. The electric field at the point r can be written as a sum of the incident field Ei () r total scattered field Escat () r, and the nˆ E( r) Tnˆ E ( r) Tnˆ E ( r ). (3..68) In terms of magnetic currents, i scat M nˆ E, (3..69) the integral equation becomes, M( r) TM ( r) TM ( r ), (3..7) i scat T M( r) TM ( ) ˆ i r n ' Gds ', 4 M (3..7) S T ˆ ˆ jkr jkr M( r) TM ( ) (( ˆ ) ( ˆ i r ) ) e ds', 4 n R M n M R (3..7) S R 43

48 where nm ˆ =. Hviid simplifies this expression by assuming the surface to be constant transverse to the direction of propagation. The integration is reduced to x-z pane for a vertically polarized source at r. The function z = c(x, y) = c(x) can be used to describe the -D surface. Figure 3..8 shows the geometry for -D scattering occurring at a point r ' where the observation point is at r and the transmitter is at r. The scattering points are treated as source points for the observation point. z r R n ˆ ' r ' R ˆn E r x Figure Source point r, scattering point r ', and observation point r in x-z plane. For each scattering source point r', there are other contributions ( y ' Figure ). This is shown in z r R ( x',) E R ( x ', y ') r ' R ( x', y') ˆn R (x',) r x x ' y y ' Figure 3..9 Scattering geometry for a scattering point on y ' -axis. 44

49 Taking the x-z plane as reference, one may find relations between the path lengths R( x', y ') and corresponding phases ( x', y') along y '. The path difference for a scattering point on written as, y ' -axis with respect to Rx ( ',) then can be R( x', y') R( x', y') R( x',). (3..73) The phase difference becomes, ( x', y') k R( x', y ') (3..74) From Figure 3..9 the geometry yields, y ' R( x ', y ', z ') R ( x',) y ' R( x',) (3..75) Rx ( ',) Assuming y ' R( x',) and r' r one obtains, y ' R( x ', y '). (3..76) Rx ( ',) Similarly for the path length difference between the source and the scattering point, For a point ( x', y ') becomes, y ' R ( x ', y ') R ( x',) y ' R ( x',). (3..77), the total difference in path length with respect to the point ( x ',) then R( x ',) R ( x',) Rtot R R y '. (3..78) R( x ',) R ( x',) For a point on the path, one can compute the scattering integral along the y ' -axis, considering only the phase variation. This is because the amplitude variation is negligible compared to the phase variation. The reduced integral is given by [8], ˆ jkr( x', y ') jk ( R( y ') R ) ( ) ' ( ') ˆ ( ', ') tot scat '. 4 nr R ( x ', y ') M x dl M x x y e dy (3..79) In equation (3..79), M scat gives the magnetic current induced by the scattering surface at the observation point. The integral with respect to y ' corresponds to the total contribution from a point x '. One can note that nr ˆ ˆ ( x', y') when the observation and scattering points are on a planar surface and no scattering occurs (specular reflection). The equation 45

50 nr ˆ ˆ ( x', y') also holds for r' r on smooth surface. Therefore, Hviid approximates the jkr( x ', y ') term R ( x ', y ') amplitude in the by jk R and hence neglects the near field term assuming constant y ' direction. The integral then is reduced to, The integral with respect to k RR j y' jkr R R ( ) ' ( ') ˆ ˆ jk M scat x dl M x ( x ', y ') e e dy '. 4 nr R (3..8) y ' is a Fresnel integral and can be reduced to, ˆ jk R R M x M x n R x y e e dl (3..8) j jkr ˆ 4 scat( ) ( ') ( ', ') '. 4 R R R Equation (3..8) is the integral along the path in x-z plane. For the total field at the observation point one obtains the integral equation, where, T jk R R M x TM x M x n R e e dl (3..8) j ˆ jkr ˆ 4 ( ) i( ) ( ') ', 4 R R R M ( x) nˆ E ( x), (3..83) i i is the magnetic current induced by the incident field at the observation point. Numerically, this integral is replaced by a finite summation over surface elements along the x-axis. The increments at a point xm in the summation are smaller than x n x from the source point, m n i, n m m 4 m. The result is a linear system for M n n T M TM M f ( n, m) x, (3..84) where m, N and n, N and, where l m ( ) ˆ jk RR j kr l ˆ 4 m f ( n, m) n R e R R R x, (3..85) is a line element along the surface and the distances are, m R x z h (3..86) m ( m T ), R x x h z (3..87) ( N m) ( R m), 46

51 where h T and respectively. h T, h R represent the heights of the source point and the observation point, h R and numerical parameters for finding x N (the end point) are the fixed parameters, and M n at an observation point m x m and z are m x N x. For each increment, the distances R and R are updated and employed in equation (3..85) to obtain f ( n, m ). The parameters are defined using Figure 3... h T h R zn z m x m xn x N N x m Figure 3.. Geometry for the -D integral equation method. Smoothness of the surface is important for the approximations. Since the integral equation in Hviid s approach has no singularity for smooth surfaces, suitable assumptions yield a simple numerical solution without divergences which makes the method a good alternative to UTD. On the other hand, since there is a linear system, the conditioning of the system might deteriorate if the system is large. More recent work by Carin and Li, which aims at validating ray tracing models for complex environments, uses the Multilevel Fast Multipole algorithm based on integral equations [9]. Especially in large and complex environments the algorithm seems to be more accurate and applicable than Hviid s method. 47

52 3..5 The Parabolic Equation method The Parabolic Equation Method is used for long range forward propagation problems and often for irregular terrains with buildings sprawling over them. The method takes reflection, diffraction and forward scattering into consideration. The PE method was developed for problems where the propagation has a paraxial direction as shown in Figure 3... By calculating the dominant initial field distribution u( x, z ), the next distribution u( xn x, z) is found by propagating it in x. The distribution is computed by marching forward step by step ( x) in Cartesian coordinates. Each step in the calculation depends on the previous step [8, j t ]. Assuming the time dependence e, Helmholtz equation for a field component is given by, where,, (3..88) k n is the refractive index and, k n, (3..89) k k, (3..9) is the wave-number. k is the free space wave-number. Since the earth is spherical, a spherical coordinate system would seem appropriate for propagation in the troposphere, but one can simplify the problem by considering the earth as flat and use cylindrical coordinates ( r,, z). Another assumption for this case is that the field is constant in the azimuth. One can then write the Helmholtz wave equation in cylindrical coordinates in terms of range r and height z as, k n r r r z (3..9) Figure 3.. illustrates paraxial propagation with a propagation angle in cylindrical coordinates. 48

53 z r x Figure 3..- Paraxial propagation with a narrow beam-width in cylindrical coordinates. To find a solution in the paraxial direction, one may write the field component in terms of an envelope function u( r, z ) which is assumed to be slowly varying in ( rz,, ) and a Hankel function H ( k r ) which represents the rapid oscillations in range, () ( r, z) u( r, z) H ( k r). (3..9) () In the far field kr and the asymptotic expression of the Hankel function can be used, H k r e. (3..93) j( kr ) () 4 ( ) kr Substituting the modified uh into the Helmholtz equation, neglecting small terms, ( uh ) u H k n uh, (3..94) r z and taking the second derivatives yields, ( urh uh r) uzzh uhk n, (3..95) r ( u H u H u H uh ) u H uhk n, (3..96) rr r r r r rr zz ( u u ) H u H u( H Hk n ). (3..97) rr zz r r rr Neglecting the small terms yields, u u u jk k ( n ) u. (3..98) r z r For the -D problem, one can replace r with x in equation (3..98), 49

54 u u u jk k ( n ) u. x x z (3..99) This equation describes the propagation in both the +x and x directions. By factorizing the equation (3..99), (3..) jk jk ( n ) jk jk ( n ) u, x k z x k z and discarding the backward propagation factor, one has the expression for the forward propagation, jk jk ( n ) u. x k z (3..) Defining a differential operator, P x z k (, ) ( n ) z, (3..) one can write, u jk P ( x, z ) u, (3..3) x and by using Taylor expansion, P( x, z) P( x, z) P. (3..4) The narrow angle ( 5 5 ) parabolic equation is then obtained as, u u k ( n ) u. x jk z (3..5) This partial differential equation can be solved by the split-step Fourier Transform method. By defining, u x ( a( x, z) bdz ) u, (3..6) where, 5

55 k a( x, z) ( n ( x, z) ) and j b, jk (3..7) one transforms u( x, z ) to U ( x, k ) z by using the Fourier transform and its derivatives, ( D u) jk U. (3..8) Here the domain of the transform, z z k z, is the transverse wave-number which can be written, k ksin, (3..9) z where is the propagation angle relative to the horizontal. The equation then becomes, U x z a k b U, (3..) where a is constant if a small variation is assumed for the refractive index. The solution to this ordinary differential equation yields, ( a k b) z x U( x x, k ) U( x, k ) e, (3..) z z and by taking the inverse Fourier transform, the solution for u( x x, z) in terms of Fourier and inverse Fourier transforms is, ax bk z x u( x x, z) e e u( x, z) (3..) The solution can be marched forward numerically along the x-axis, for a given value of z. Figure Solution domain of the Parabolic Equation method. 5