A NOVEL WAVELET-GALERKIN METHOD FOR MOD- ELING RADIO WAVE PROPAGATION IN TROPO- SPHERIC DUCTS

Transcription

1 Progress In Eectromagnetics Research B, Vo 36, 35 52, 2012 A NOVEL WAVELET-GALERKIN METHOD FOR MOD- ELING RADIO WAVE PROPAGATION IN TROPO- SPHERIC DUCTS A Iqba * and V Jeoti Department of Eectrica and Eectronics Engineering, Teknoogi PETRONAS, Tronoh, Perak 31750, Maaysia Universiti Abstract In this paper, a nove Waveet-Gaerkin Method (WGM) is presented to mode the radio-wave propagation in tropospheric ducts Gaerkin method, with Daubechies scaing functions, is used to discretize the height operator Later, a marching agorithm is deveoped using Crank-Nicoson (CN) method A new fictitious domain method is aso deveoped for paraboic wave equation to incorporate the impedance boundary conditions in WGM In the end, resuts are compared with those from Advance Refractive Effects Prediction System (AREPS) Resuts show that the waveet based methods are indeed feasibe to mode the radio wave propagation in troposphere as accuratey as AREPS and proposed method can be a good aternative to other conventiona methods 1 INTRODUCTION Paraboic Wave Equation (PWE) method is a widey used technique for modeing tropospheric anomaous radio wave propagation To date, severa numerica techniques have been deveoped to sove PWE However, the most popuar among them is the Spit-Step Fourier transform Method (SSFM) [1] Though it is quite popuar among researchers, this agorithm is not effective for modeing arbitrary boundary conditions An important advancement in this technique is the introduction of the Discrete Mixed Fourier Transform (DMFT) [2] DMFT based SSFM aows one to hande finitey conductive surfaces Finite Difference Method (FDM) is considered effective for a types of boundary conditions [1] But for non-standard atmosphere or Received 12 September 2011, Accepted 27 October 2011, Schedued 3 November 2011 * Corresponding author: Asif Iqba (asifiqba@msncom)

2 36 Iqba and Jeoti compex refractive structure, the matrices become more and more dense So, it takes ong to sove these matrices and it aso uses up too much memory Finite Eement Method (FEM) has aso been empoyed for the soution of paraboic equation [3 7] Effective boundary handing makes FEM attractive to numerica soution of partia differentia equations (PDE) Recenty, Apaydin and Sevgi [8] extended the soution for the appication to surface wave propagation But in their work, modified refractivity is assumed inear which is not a suitabe assumption for higher frequencies From iterature review, it is aso found that the finite methods deveoped so far are accurate ony up to the maximum of second order To hande singuarities and abnorma environment, higher order methods are not ony computationay more efficient but aso more accurate Advance Refractive Effects Prediction System (AREPS) software is used as a benchmark in this work, because it is optimized for tropospheric radio-wave propagation modeing and considered to be most suitabe software package The Advance Propagation Method (APM) is the core of AREPS APM is a hybrid mode which contains four modes These are fat earth (FE), ray optics (RO), extended optics (XO), and the Spit-Step Paraboic Equation (PE) Agorithm PE Method is the primary mode around which the other three sub-modes are buit [9] In this work, a nove more accurate WGM is presented to mode radiowave propagation in troposphere Waveet based numerica methods are most often used to sove inear and non-inear differentia equations [10 14] Orthogonaity, compact support and exact representation of poynomias of a fixed degree make them usefu for representing the soutions of PDEs According to the best of authors knowedge, appication of waveets to mode radio wave propagation is not addressed in iterature so far In the formuation of WGM, Daubechies scaing functions are used as basis functions in Gaerkin method to sove 2D PWE The discretization process is appied to height operator Resutant system of agebraic equations is soved by CN Method In this paper, a new fictitious domain method based on Lu et a [15] is aso deveoped for paraboic equations to hande arbitrary boundary conditions without osing the simpicity of WGM Parametric refractivity M-profie mode is used to generate verticay modified refractivity profie to mimic rea environment [16] The proposed method not ony provides higher order accurate soution, it can aso hande any type of ducting profie The rest of the paper is organized as foows: Brief background of radio refractive index, an overview of PWE and Basic Waveet theory is given in Section 2 Detaied WGM formuation is described in Section 3 In Section 4, numerica impementation of WGM aong

3 Progress In Eectromagnetics Research B, Vo 36, with fictitious domain method is presented Resuts comparison and discussion is provided in Section 5 2 BACKGROUND 21 Radio Refractive Index The radio refractive index (n) is caused due to the moecuar constituents of the air [17] Normay, the numerica difference in refractivity is a very sma fraction of unity So, a convenient way of expressing the refractive index is in terms of refractivity (N) is given by, N = (n 1) 10 6 (1) There are four refractive conditions which depend upon refractive gradient The reations of refractivity gradient and reated refractive condition are summarized in Tabe 1 [18] Trapping condition, often caed ducting phenomenon, causes anomaous radiowave propagation We known tropospheric ducts are surface duct (ground-based duct), surface-based duct and eevated duct as shown in Figure 1 22 Paraboic Wave Equation (PWE) A two dimensiona scaar wave equation in the cartesian coordinates system can be written as: 2 ψ x ψ z 2 + k2 n 2 ψ = 0 (2) In the equation mentioned above, k = 2π/λ is the wave number in vacuum, and n is the refractive index To reduce (2) into PWE, a function associated with the paraxia direction x is taken as [1], ψ(x, z) = u(x, z)e jkx (3) Tabe 1 Refractive condition Condition N-Gradient M-Gradient (N Unit/km) (M Unit/km) Trapping dn/dh 157 dm/dh 0 Supper Refraction 157 < dn/dh 79 0 < dm/dh 78 Standard 79 < dn/dh 0 78 < dm/dh 157 Sub Refraction dn/dh > 0 dm/dh > 78

4 38 Iqba and Jeoti Substituting (3) into (2) we obtain 2 u u 2jk x2 x + 2 u z 2 + ( k2 n 2 1 ) u = 0 (4) Using paraxia approximation, ie, 2 u/ x 2 2jk u/ x, (4) wi be reduced to the famiiar PWE in two-dimensiona space: 2 u u 2jk z2 x + ( k2 n 2 1 ) u = 0 (5) This is caed standard paraboic equation (SPE) For Earth fattening, the factor (n 2 1) in (5) shoud be repaced by (n z/R) [8], where, R is the Earth s Radius and z the height above the ground eve 23 Boundary Conditions 231 Impedance Boundary Conditions With the assumption that the skin depth of eectromagnetic radiation within the earth is sma as compared to the earth s radius of curvature, the boundary conditions (BC) at smooth earth surface can be approximated by [2]: α 1 (x) z u(x, z) z=0 + α 2 (x)u(x, z) z=0 = 0 (6) where, α 1 (x) and α 2 (x) are constants For a perfecty conducting surface, α 1 (x) = 0 (Dirichet BC) and α 2 (x) = 0 (Neumann BC) for horizonta and vertica poarization respectivey For finitey conducting earth surface, α 1 (x) = 1, whie α 2 (x) = (j/µ 0 ω)η for (a) (b) (c) Figure 1 Type of Ducts (a) Surface duct (b) Surface-based duct (c) Eevated duct

5 Progress In Eectromagnetics Research B, Vo 36, horizonta poarization and α 2 (x) = jωε s η for vertica poarization, where ε s is the permittivity of the surface medium, µ 0 the free-space permeabiity, and ω the radia frequency 232 Absorbing Boundary Conditions In order to imit the height of computationa domain there is a need to truncate the domain at finite height Artificia truncation can cause strong refections To avoid these non-physica refections, a windowing function, perfecty matched ayer (PML) termination, or ocating an absorbing ayer is used [1, 8, 19] In this paper, a windowing function is used as an absorbing ayer 24 Initia Fied PWE is an initia vaue probem It is necessary to specify the fied at range x = 0 There are severa ways that the starting fied can be generated, eg, by some anaytica function or by using antenna theory The fied at range x = 0 is essentiay the antenna aperture distribution, and the far-fied antenna pattern f(p) and its aperture distribution A(z) are a Fourier transform pair [9], A(z) F f(p) (7) where F is Fourier transform Appying the boundary condition that the fied vanishes at the surface, we use image theory to obtain: ψ(0, z) = A(z z 0 ) + Γ A (z + z 0 ) (8) where Γ is the refection coefficient Equation (8) is written as the sum of the source and image fieds, and z 0 represents the antenna height Since the antenna pattern is what is normay deat with, one can simpy transform it to obtain ψ(0, p) = f(p)e jpz 0 + Γ f (p)e jpz 0 (9) Pattern factor f(p) for omnidirectiona antenna is equa to 1, and for a normaized Gaussian antenna pattern: f(p) = e p2 w 2 /4, where w = 2 n 2/k 0 sin(θ bw /2) (10) In (10), θ bw /2 is beam-width ange Eevation ange can easiy be incorporated by repacing f(p) by f(p p 0 ) Where p 0 = k 0 sin θ 0

6 40 Iqba and Jeoti 25 Basic Waveet Theory Ingrid Daubechies constructed the cass of compacty supported scaing and waveet function in 1988 [20] Briefy, scaing function (ϕ) satisfies the foowing expression, caed Diation Equation: ϕ(z) = M 1 k=0 h k ϕ(2z k) where M is the order of Daubechies waveet, and ϕ(z) is caed scaing function The associated waveet function ψ is required to satisfy the foowing equation caed waveet equation: where h k condition ψ(z) = M 2 k= 1 ( 1) k h k+1 ϕ(2z + k) is the set of nonzero constant scaing coefficients with M 1 k=0 h k = 2 The orthogona waveet and scaing functions generated with these scaing coefficients wi have a supp(ϕ) = [0, M 1] For j, k Z, the diation and transations of scaing function (ϕ(z)) and waveet function (ψ(z)) can be written as: ϕ j,k (z) = 2 j/2 ϕ ( 2 j z k ) ψ j,k (z) = 2 j/2 ψ ( 2 j z k ) Let V be the set of a scaing functions and W the set of a waveet functions A brief summary of waveet properties is given as foows For more detais, see [20 22] (1) {ϕ j,k } j 0,k Z is an orthonorma basis for L 2 (R) (2) V j+1 = V j W j (3) L 2 (R) = cos L 2(V 0 j=0 W j) (4) {ϕ 0,k, ψ j,k } j 0,k Z is an orthonorma basis for L 2 (R) (5) ϕ(z)dz = 1 (6) k Z ϕ 0,k = 1 (7) ψ(z)zk dz = 0: k = 0,, L 1 (8) {z k } L 1 k=0 V N, where L is an positive integer

7 Progress In Eectromagnetics Research B, Vo 36, Figure 2 Daubechies scaing and waveet function for M = 6 with support [0, 5] Based on property (2), the mutiresoution anaysis is a nested sequence, V 0 V 1 L 2 (R) satisfying the foowing properties [22] (1) j Z V j = 0 (2) cos L 2( j Z V j) = L 2 (R) (3) f(z) V j f(2z) V j+1 (4) There is a function ϕ V 0 such that {ϕ 0,k (z) = ϕ(z k)} form a Riesz basis for V 0 For arbitrariy arge even M, the Daubechies famiy of waveet have fundamenta support in the interva [0, M 1] In Figure 2 an exampe of a compacty supported Daubechies scaing and waveet functions for M = 6 are shown 3 WAVELET GALERKIN METHOD (WGM) Soution domain is divided into severa subdomains in WGM The approximate soution is written in the form of, N z f(z) = f ϕ (z) (11) =1

8 42 Iqba and Jeoti where N z is number of grid points, f the unknown coefficients, and ϕ the waveet scaing basis functions Inner product of basis functions can be defined as, ϕ k, ϕ = ϕ k ϕ dz (12) Inner product of waveet scaing basis functions with or without derivatives is caed connection coefficients Connection coefficients for unbounded intervas were described by Latto et a [23] In Genera form, n-term connection coefficients can be defined as n Ω d 1,,d n k 1,,k n = ϕ dn k n dz = ϕ d i k i dz (13) ϕ d 1 k 1 i=1 where superscript d i refers to the derivative of the scaing function ϕ(z) with respect to z If the discretizing operator is inear, then integra wi have the form as hereunder, Ω d 1,d 2 k, = ϕ d 1 k ϕd 2 dz (14) here, Ω d 1,d 2 k, is known as 2-term connection coefficients In the formuation of PWE, maximum of 2-term connection coefficient wi be used By change of variabe, 2-term connection coefficient Ω d 1,d 2 k, can be turned into Ω d 1,d 2 0, k := Ωd 1,d 2 0,, where = k For simpicity, Ω d 1,d 2 wi be used throughout this document, Ω d 1,d 2 = More detais can be found in [23, 24] 31 WGM Formuation for PWE ϕ d 1 ϕ d 2 dz (15) In the formuation of WGM, weak form of (5) can be obtained by mutipying (5) with basis function ϕ k and integrating over the domain: ( 2 u u ϕ k 2jk z2 x + ( k2 n 2 1 ) ) u dz = 0 (16) In discrete space, u(x, z) = a (x)ϕ (z) (17) where, a = u(x, z), ϕ

9 Progress In Eectromagnetics Research B, Vo 36, After substituting the waveet expansion of u(x, z) into (16) and rearranging, we have a (x) ϕ k ϕ dz + ( j a (x) ϕ k ϕ 2k dz jk ( k 2 (n 2 1) ) ) ϕ k ϕ dz = 0 (18) 2 In matrix notations, (18) can be written as where, [δ k, ]{a (x)} + [L k, + S k, ]{a (x)} = 0 (19) δ k, = I k, = L k, = j 2k ϕ k ϕ dz, ϕ k ϕ dz = j ( 2k Ω 0,2 ), S k, = jk 2 ( k 2 (n 2 1) ) ϕ k ϕ dz, δ k, is known as Kronecker deta function, and Ω 0,2 are the connection coefficient briefy described in previous section If whoe domain is divided into N z number of grid points, then = k = 0,, N z 1 In order to satisfy the BC, Equation (6) can be incorporated to PWE of (5) with fieds approximated as in (17) Assume BC k, to be resutant BC matrix, that is given by, [ α 1 (x)ω 0,1 z=0 α 2 (x)δ k, z=0 ] {a (x)} = 0, [BC k, ]{a (x)} = 0 (20) Combining (20) with (19) gives us the required system of inear equations that satisfy both the PWE and its boundary conditions [δ k, ]{a (x)} + [L k, + S k, + BC k, ] {a (x)} = 0 (21) Finay, the resutant inear system given in (21) is soved using CN method The genera soution of Equation (21) can be written as, {a (x + x)} = ( ) 2[Ik, ] x [L k, + S k, + BC k, ] {a (x)} (22) 2[I k, ] + x [L k, + S k, + BC k, ]

10 44 Iqba and Jeoti It is an iteractive agorithm Once the system is initiated with starting fied, fied at the next range step can be cacuated from the previous one It is an unconditionay stabe, a second order accurate system in range operator The accuracy of height operator is dependent on the choice of waveet Genus Higher order waveet provides higher accuracy Overa compexity of agorithm is same as conventiona finite methods Athough, CN Method provides fast soutions but the step-size shoud be chosen as sma as necessary to overcome numerica osciation probems [8] 4 NUMERICAL IMPLEMENTATION Connection Coefficients derived by the Latto et a [23] are for unbounded intervas In order to dea with bounded intervas, fictitious domain method and capacitance matrix method are more common [10, 15] However, Green function soution is required for the impementation of capacitance matrix method But Green function is not readiy known for PWE [25] Due to this, fictitious domain approach is used to hande boundary condition in the soution of PWE In this approach, interva is modified by adding a fictitious interva outside of interested domain Same connection coefficient can be used to hande rea boundary conditions as cacuated by Latto et a 41 Fictitious Domain Method For the case of PWE, overa scenario is iustrated in Figure 3 Bottom surface (Z min ) is finitey conductive surface whie an absorbing ayer is paced at the top (Z req ) of domain of interest Tota atitude is divided into N z number of grid points, therefore, ϕ has anaytica Figure 3 Computation domain for WGM

11 Progress In Eectromagnetics Research B, Vo 36, boundaries at 0 and N z 1 To satisfy the support of expansion in (17), ie, [ M + 1, M 2 + N z ], there is a need to add M 1 functions both at the top and bottom of domain of interest, where, M is the genus of waveet scaing function With this approach, the rea domain wi be stretched and anaytica boundaries wi be shifted to M + 1 and M 2 + N z The dimensions of resutant inear system wi be N z + 2 (M 1) by N z + 2 (M 1) After discretizing with Daubechies-6 (D6), the matrices in (22) are obtained as: S k, = diag S 5, 5 S 0,0 S Nz 1,N z 1 S Nz 1+5,N z 1+5, BC k, = diag L k, is a 9-diagona matrix with = 4,, 0,, 4, and L k, = Ω 0,2 0 Ω 0,2 1 Ω 0,2 2 Ω 0, Ω 0,2 1 Ω0,2 0 Ω 0,2 1 Ω 0, Ω 0,2 4 Ω0,2 3 0 Ω0, Ω 0,2 0 Ω 0,2 3 Ω 0, Ω 0, Ω 0,2 4 BC 5, 5 BC 0,0 BC Nz 1,N z 1 BC Nz 1+5,N z 1+5 Ω0,2 1 Ω0,2 0 Ω 0,2 1 Ω0,2 2 Ω0,2 1 Ω0,2 0 I k, = [Identity Matrix] Nz +2(5) N z +2(5), a (z) = [a 5 a 4 a 0 a Nz 1 a Nz 1+4 a Nz 1+5] T A symmetric extension of refractive profie is taken in extended region at bottom whie same profie is extended at top, as shown in Figure 3 We can write, S m, m = S m,m for m = 1, 2,, 5 It can be seen that rea boundaries exist at BC 0,0 and BC Nz 1,N z 1 It shoud aso be noted that, BC k, = 0 for k, 0 & N z 1,,

12 46 Iqba and Jeoti For Dirichet BC, coefficients (a (z)) shoud equa to zero at Z min Hence, [BC k, ] and first M rows and coumns of inear system, given in (22), wi be zero For Neumann BC, coefficients (a (z)) are unknown at Z min but the impementation of Neumann BC wi aso yied [BC k, ] = 0 For finitey conductive surfaces, an assumptions is made that the extended domain at bottom consists of the same materia as at BC 0,0 We can write: BC m,m = j 2k 0 α 2 (x) for m = 5, 4,, 0 After reaching the desired range step using marching soution given in (22), the required coefficients, [a 0,, a Nz 1] T, wi be extracted from the extended domain soution 42 Propagation Loss In rectanguar coordinate system, propagation factor can be cacuated by F = x u(x, z) where, x and z is in meters x is the distance between the transmitting antenna and receiving antenna, and z is the atitude measured with respect to the mean sea surface Once propagation factor F is obtained, the propagation oss in db is determined by the basic radio wave theory in the form of [26], L = L f 20 og F where, L f is free space propagation oss and computed from the formua given by, L f = 20 og f(mhz) + 20 og x(m) NUMERICAL RESULTS To vaidate the efficiency of proposed method, the propagation path oss is cacuated for standard and ducting environment conditions Normaized Gaussian antenna pattern is used for a simuations The bottom boundary is assumed to be fat sea surface The resuts from AREPS package are obtained with ony PE mode and with same parameters 51 Standard Environment In the first case, path oss resuts for 50 km transmission range are obtained using WGM as shown in Figure 4(a) Horizontay poarized transmitter antenna height is chosen at 50 m above the ground eve

13 Progress In Eectromagnetics Research B, Vo 36, (a) (b) (d) (c) Figure 4 (a) Range (km) vs height (m) pathoss (db) diagram of WGM for standard environment (b) Pathoss vs height for range of 20 km (c) Path-oss vs range for height of 30 m (d) Standard environment refractive profie Beam-width is set to be 3 degrees Detaied comparison of resuts with AREPS package is iustrated in Figure 4(b) (c) Figure 4(b) shows the comparison of pathoss versus height at range of 20 km whie Figure 4(c) shows pathoss versus range comparison at the height of 15 m It is observed that for norma gradient of refractivity, energy moves away from the earth s surface as shown in Figure 4(a) The resuts aso show that for ower receiving antenna height pathoss is very high after a few kiometers Communication range can ony be improved by increasing the atitude of receiver and/or transmitter antenna 52 Ducting Environment Two different ducting environments are studied to vaidate the performance of WGM in ducting environment 521 Surface Duct To demonstrate the behavior of radiowave in surface duct, propagation oss is cacuated with 58 GHz frequency Simuations were performed with horizontay poarized antenna Transmitter antenna is taken

14 48 Iqba and Jeoti (a) (b) (d) (c) Figure 5 (a) Range (km) vs height (m) pathoss (db) diagram of WGM for surface duct (b) Pathoss vs height for range of 30 km (c) Path-oss vs range for height of 15 m (d) Surface duct refractive profie at 25 m above the ground eve Beam-width is set to be 3 degrees Anomaous radio wave propagation can be seen in Figure 5(a) Comparison of WGM resuts with AREPS is given in Figures 5(b) (c) Ducting profie is iustrated in Figure 5(d) Resut shows that propagation has been extended to severa kiometers due to ducting phenomenon As the tota occurrence time of surface-based duct is very sma, we cannot take benefit from this phenomenon for a reiabe ong range communication But this phenomenon becomes a cause of strong interference to other communication systems even at far distances 522 Evaporation Duct Evaporation duct is one of the most important ducting phenomenon 105 GHz is chosen for further path oss measurement because the effect of evaporation duct is more dominant in this range of frequencies Simuations are carried out with horizontay poarized transmitter antenna with 15 m height above the ground eve Beam-width is set to be 2 degrees Figure 6(a) shows the trapping of signas in evaporation duct A detaied comparison of resuts with AREPS package is provided in Figures 6(b) (c) for specific range and height From resuts, it can be seen that in trapping condition, the signa attenuation is much ess even at 100 km as shown in Figure 6(c)

15 Progress In Eectromagnetics Research B, Vo 36, (a) (b) (d) (c) Figure 6 (a) Range (km) vs height (m) pathoss (db) diagram of WGM for evaporation duct (b) Pathoss vs height for range of 45 km (c) Path-oss vs range for height of 10 m (d) Evaporation duct refractive profie Tabe 2 Mean reative squared error (MRSE) wrt AREPS Environment Condition Case RMSE Standard Figure 4(b) Figure 4(c) Surface Duct Figure 5(b) Figure 5(c) Evaporation Duct Figure 6(b) Figure 6(c) Discussion In brief, the performance of WGM is measured under different compex environments and the accuracy of proposed method with respect to AREPS is summarized in Tabe 2 In Tabe 2, mean reative squared error (MRSE) is computed for the cases discussed in previous section Tabe 2 shows that the maximum of mean squared reative difference of WGM and AREPS is of the order of 10 4 Hence, resuts give a strong agreement with AREPS package The formua used to cacuate MRSE

16 50 Iqba and Jeoti is given by, k ( ) AREPSdatai WGMdata 2 i MRSE = 1 n 6 CONCLUSION i=0 AREPSdata i In this paper, a nove waveet-gaerkin method is presented for the numerica soution of two dimensiona paraboic equation A new fictitious domain method is aso introduced for paraboic wave equation to incorporate the impedance boundary conditions A brief discussion on the behavior of radiowave propagation in troposphere is aso provided At the end, resuts are compared with those from AREPS for both environment conditions standard and ducting The resuts show that the proposed agorithm is neary as good as AREPS and it can be a better aternative to other we-known methods From the simuation resuts, it is aso found that the grid size for height and range operator is chosen carefuy to make WGM computationay efficient Smaer grid size is required for higher frequencies due to which WGM cannot save significant computation cost, reative to the AREPS In spite of this, waveet methods sti have specia properties ike muti-resoution anaysis and exact soution of connection coefficient which make them superior to other conventiona methods and aow to provide higher accurate soutions REFERENCES 1 Levy, M, Paraboic Equation Methods for Eectromagnetic Wave Propagation, Vo 45, Inst of Engineering & Technoogy, Kutter, J and G Dockery, An improved-boundary agorithm for fourier spit-step soutions of the paraboic wave equation, IEEE Transactions on Antennas and Propagation, Vo 44, No 12, , Isaakidis, S A and T D Xenos, Paraboic equation soution of tropospheric wave propagation using FEM, Progress In Eectromagnetics Research, Vo 49, , Deshpande, V and M Deshpande, Study of eectromagnetic wave propagation through dieectric sab doped randomy with thin metaic wires using finite eement method, IEEE Microwave and Wireess Components Letters, Vo 15, No 5, , 2005

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