An Algorithm for Rough Surface Generation with Inhomogeneous Parameters

Transcription

1 Journal of Algorithms & Computational Technolog Vol 5 No An Algorithm for Rough Surface Generation with Inhomogeneous Parameters Kazunori Uchida, Junichi Honda 2, and Kwang-Yeol Yoon 3 Fukuoka Institute of Technolog, Information and Communication Engineering Fukuoka, Japan k-uchida@fitacjp 2 Fukuoka Institute of Technolog Intelligent Information Sstem Engineering Fukuoka, Japan k-uchida@fitacjp 3 Keimung Universit, Department of Electronic Engineering Daegu, , Korea oonk@kmuackr Received: 22/02/200; Accepted: 07/04/200 ABSTRACT This paper is concerned with an algorithm for generating the inhomogeneous random rough surface (RRS) that can simulate the phsical environments such as desert, vegetable field, sea surface and others, better than the homogeneous RRS First, we discuss the statistics for twodimensional (2D) RRSs with arbitrar tpes of spectra Second, we review the essence of the convolution method which is fleible and effective for generation of various kinds of RRSs Finall, we propose an algorithm for inhomogeneous RRS generation b modifing the convolution method Numerical eamples are also given for some inhomogeneous RRSs INTRODUCTION Recentl, a rapid progress has been made in the area of wireless sensor networks to gather phsical data and to control natural environments Sensors are usuall distributed randoml on terrestrial surfaces such as deserts, vegetable fields, sea surfaces and others, which are considered to be RRSs As a result, studies on propagation characteristics along RRSs are strongl required In the past, however, most of the investigations were focused on electromagnetic wave

2 260 An Algorithm for Rough Surface Generation with Inhomogeneous Parameters scattering b RRSs from the viewpoint of radar and remote sensing technologies [,2,3,4,5,6] An empirical formula was proposed b Hata [7] to estimate propagation loss in urban areas for cellular communications, but it seems difficult to appl it straightforwardl to wireless sensor networks In the last five ears, we have investigated propagation characteristics of electromagnetic waves traveling along RRSs with uniform statistical parameters such as spectrum, standard deviation of height and correlation length [8,9,0,,2] In real situations, however, these parameters are not uniform and the var from place to place [3] Thus it is necessar to estimate the propagation characteristics along inhomogeneous RRSs from the viewpoint of wireless channel modeling corresponding to the phsical laer of the OSI model We can simulate electromagnetic wave propagation along an inhomogeneous RRS b assuming that it is composed of several homogeneous RRSs In this contet, the first step to wireless channel modeling is to numericall generate inhomogeneous RRSs So far the direct DFT method has been utilized as a numerical generation tool for homogeneous RRSs with arbitrar spectrum and statistical parameters Disadvantage of the method, however, is its infleibilit and it cannot be applied to more realistic inhomogeneous RRSs Recentl, we have proposed the convolution method for generating homogeneous RRSs b modifing the direct DFT method [0] One of the advantages of the convolution method is that we can simulate arbitraril long or wide RRSs b successive computations The other is its fleibilit and feasibilit resulting in generation of inhomogeneous RRSs in a ver simple fashion We propose here an algorithm for generating 2D inhomogeneous RRSs based on the convolution method In this paper, we first describe statistics of RRSs and show three tpes of spectral densit functions, that is, Gaussian, N-th order of Power-Law and Eponential spectra Second, we review the algorithm of the direct DFT method, and we discuss the convolution method b modifing the direct DFT method Finall, based on the convolution method, we propose an algorithm to generate inhomogeneous RRSs Numerical calculations are also carried out for some inhomogeneous RRSs 2 HOMOGENEOUS RRS GENERATION 2 Spectral densit function The spectral densit function W(K) of 2D RRS is defined b using height function f(r) = f(, ) as follows:

3 Journal of Algorithms & Computational Technolog Vol 5 No W( K) dk = h 2 () and L / 2 L / 2 W( K) = lim f() r e L, L ( ) 2 2π LL L / 2 L / 2 jkr dr 2 (2) where <> indicates the ensemble average and h is a standard deviation of height The position vector r and the spatial angular frequenc vector K are defined b r = (, ), r= +, K = ( K, K), K= K + K (3) The auto-correlation function is given b the Fourier transform of the spectral densit function as follows: jkr ρ( r) = W( K) e dk (4) We summarize three tpes of spectral densit functions for numerical simulations Gaussian tpe of spectrum: The spectrum densit function is defined b clclh Kcl Kcl W( K ) = 2 ep 2 2 4π 2 2 (5) where cl and cl are correlation length in - and -directions, respectivel The auto-correlation function is given b ρ( r ) = ep h cl cl (6) 2 N-th Order Power-Law Spectrum: The spectrum densit function is given b

4 262 An Algorithm for Rough Surface Generation with Inhomogeneous Parameters cl cl h W( K ) = 4π Γ ( N ) Kcl Kcl + + ( N) 2 Γ (7) where Γ(N) is the Gamma function and N > is assumed The auto-correlation function is given b N 2 ρ( r ) = h N cl cl (8) 3 Eponential Spectrum: The spectrum densit function is given b { } clclh W( K ) = 2 Kcl Kcl ( ) + ( ) π (9) The auto-correlation function is given b 2 ρ( r ) = h ep cl cl (0) 22 Discrete spectral densit function In this paper we use the 2D DFT in the form F = DFT(f) and the arra components are calculated as follows: N N nv nv j2π j2π N N vv nn n = 0 n = 0 F = f e ( v = 0,,, N ) ( p=, ) p p () where N and N are truncation numbers in - and -directions, respectivel The inverse DFT is epressed as f = DFT (F) and the arra components are calculated as follows: f N N nv nv j2π + j2π N N = Fv ve ( np = 0,,, Np ) ( p=, ) N N nn v = 0 v = 0 (2)

5 Journal of Algorithms & Computational Technolog Vol 5 No Assuming the length of the 2D RRS to be L and L in - and -directions, respectivel, the discretized spatial angular frequencies are given as follows: K m 2πm 2πm =, Km = ( mp = 02,,,, Mp) ( p=, ) L L (3) where M = N /2 and M = N /2 Consequentl we can constitute an arra w as follows: w = { w } ( m = 02,,,, 2M ) ( p =, ) mm p p (4) The arra components W(K) as follows: w mm are given b using the spectral densit function w 4 2 mm = π LL WK ( m K m ) (5) where mp ( 0 mp < Mp) m p = 2Mp mp ( Mp mp < 2Mp) ( p=, ) (6) It should be noted that the DFT of this weighting arra corresponds to the autocorrelation function as DFT( w) ρ( r) and this relation is useful for checking the accurac of the numerical results based on the DFT calculations Finall we make another weighting arra v b etracting square root of eqn (5) as follows: υ = { w } ( m = 02,,, L, 2M ) ( p=, ) mm p p (7) Performing DFT of the arra leads to DFT(υ) = V, and this relation will be used as a weighting arra of the convolution method described later 23 Gaussian random numbers We consider a random number generator necessar for computer simulations C programming language provides a function rand(a) which produces a sequence of random numbers ranging in [0, a] [4] This function enables us to generate a Gaussian random number X in the following wa u = rand( 2π ), u = rand( ), X = 2log( u )cos( u ) 2 2 (8)

6 264 An Algorithm for Rough Surface Generation with Inhomogeneous Parameters As a result, we can construct two Gaussian random number sets with zero mean and unit standard deviation as follows: { X }, { Y } N( 0, ) ( m = 02,,, L, M ) ( p=, ) mm mm p p (9) where N(0,) indicates the normal distribution, and M = 2M and M = M for X-arra and M = M and M = 2M for Y-arra Now we constitute a comple arra u = Re (u) + jim(u) in the following form mm mm u= { u + u } ( m = 02,,, L, 2M ) ( p=, ) p p (20) The real part of the first arra is given b X0m / m 2 for = 0 Re( umm ) = X / m = M 2 for Xmm / 2 otherwise N m (2) where m for 0 m < M 2m for 0< m < m m = = = M m M < m < M m 0 for m M 4 2 for 2 2M m for M < m < 2M (22) And the imaginar part of the first arra is given b 0 for m = 0 Im( umm ) = 0 for m = M ± Xmm / 2 otherwise (23) where refer to m < M and m > M respectivel, and 2m for 0< m < M m = 4M 2m for M < m < 2M m for 0 m < M m = 0 for m = M 2M 2m for M < m < 2M (24)

7 Journal of Algorithms & Computational Technolog Vol 5 No The real part of the second arra is given b Ym / m 0 2 for = 0 Re( umm ) = Y / m = M 2 for Ymm / 2 otherwise m N (25) where m for 0 m < M 2m for 0< m < M m = = = M m M < m < M m 0 for m M 4 2 for 2 2M m for M < m < 2M (26) And the imaginar part of the second arra is given b 0 for m = 0 Im( umm ) = 0 for m = M ± Ymm / 2 otherwise (27) where refer to m < M and m > M respectivel, and 2m for 0< m < M m = 4M 2m for M < m < 2M m for 0 m < M m = 0 for m = M 2M 2m for M < m < 2M (28) 24 Convolution Method We consider the product of the weighting and Gaussian random number arras in the following form z = vu= { υ u } ( m = 02,,, K, 2M ) ( p=, ) mm mm p p (29) where the arra z is also comple The DFT of the product arra is epressed as follows: Z = DFT() z = DFT( υu) (30)

8 266 An Algorithm for Rough Surface Generation with Inhomogeneous Parameters The arra Z is real and it corresponds to the height function as shown b Z f(, ), and thus this is the direct DFT method for 2D RRS Now we discuss the convolution method in relation to the direct DFT method Appling the DFT convolution theorem to eqn(30) leads to Z = N N DFT( υ)* DFT( u) (3) and b using the notations V = DFT(v) and U = DFT(u), we have the following relation Z = N N V N N U * (32) It can be proved that the second term of the right hand side constitutes an arra with components in the normal distribution as follows: N N U N (, 0 ) (33) Net we constitute a new weighting arra b permuting the order of the first term of the right hand side of eqn(32) in the following form w = { w kk} = { } (,,, ) (, ) N N V kk k p = M p p = 02L 2 (34) where kp + Mp for 0 kp < Mp kp = kp Mp for Mp kp < 2Mp (35) As a result, we can generate an tpes of 2D RRSs based on the convolutional computations epressed b 2M 2M fnn = wkk Xnn np np kp np N = + = 02,,, L, p k = 0 k = 0 ( ) (36) We call this algorithm a convolution method One of the advantages of the present method is that once the weighting arra is computed, we can w kk

9 Journal of Algorithms & Computational Technolog Vol 5 No generate an size of continuous RRSs because we can choose N and N arbitraril The other is that we can reduce the size of the weighting arra to save computation time when the correlation length of a RRS is small 3 INHOMOGENEOUS RRS GENERATION Based on the convolution method, we can generate inhomogeneous RRSs of which parameters are continuousl varied from place to place We assume that the statistical propert of the RRS is such that there eists one tpe of RRS in one region, another tpe of RRS in other region and mied tpe of RRS in their transition region 3 Plate-Oriented Method for Generating Inhomogeneous 2D RRS Now we assume two rectangular RRS regions denoted b µ and v which have different statistical properties Then we can define a 2D weighting arra as follows: w kk w kk( µ ) Region µ = w kk( µ ) gnn ( µ v) + w kk() v gnn ( vµ ) Region µ v w kk() v Region v (37) The transition functions g nn ( µν) and gnn ( vµ ) are linearl interpolated between µ and v as follows: Region µ N n N n gnn ( µ v ) 2 2 = or Region µ v N2 n N2 N 0 Region v Region µ n N n N gnn ( v µ ) = or Region µ v N2 N N2 N 0 Region v (38) (39) where N,2 and N,2 are the numbers for the boundaries of the two µ-th and v-th regions The above discussion is focused on the rectangular case, but it is worth noting that the present algorithm can easil be applied to other cases such as a circular region

10 268 An Algorithm for Rough Surface Generation with Inhomogeneous Parameters 32 Point-Oriented Method for Generating Inhomogeneous 2D RRS We assume that a required RRS consists of M kinds of homogeneous RRSs having M tpes of weighting arras w kk (m) with M representative points at n m = (n m,n m ) where m =, 2,, M Then we can define distances r nn (m) from the m-th representative point to an observation point n = (n,n ) as follows: rnn ( m) = n n m ( m= 2L,, M) (40) Among the M distances, we can alwas find the minimum distance (m ) at m = m Then we select the points out of the M representative points which satisf the following relation: r nn τ( n,n m,nm ) T (4) where τ is the distance from an observation point to the bisector of the line from n m to n m calculated b 2 2 nm nm 2( nm nm ) n τ( n,n m,nm ) = 2 n n m m (42) The constant T corresponds to half of the width of the transition region, and its value should be appropriatel chosen We assume that the number of the points satisfing the above relations is M (< M) Now we define the new discrete weighting spectral functions follows: g nn (m) as g nn 0 for τ( n,n m,nm ) > T ( m) = g nn ( m) for τ( n,n,n ) T m m (43) where g ( m) nn = τ( n,n ) M m,nm T (44) When m = m, the weighting function is given b g nn 0 for M = ( m ) = Σ mg n n ( m) for M > (45)

11 Journal of Algorithms & Computational Technolog Vol 5 No where Σ shows that m = m is ecluded in the summation As a result, the final epression for the weighting arra of an inhomogeneous RRS at the n-th point is summarized as follows: M w w ( m) g ( m) kk = kk nn m= (46) It is needless to sa that the present point-oriented method is more simplified and fleible than the former plate-oriented method 4 NUMERICAL EXAMPLES In this section we show some numerical eamples for inhomogeneous 2D RRSs These numerical eamples are not necessaril corresponding to an real situations, but the parameters used are chosen as the values applicable to vegetable fields including a pond The computation time of the present algorithm depends strongl on the correlation length, because it is proportional to the size of the weighting arra as is evident from eqn(5) When we simulate a RRS with a large correlation length, we need much computation time However, profile of the RRS with a large correlation length becomes ver smooth, and the RRS can be approimated b a completel flat ground Figure shows a 2D RRS with the same Gaussian spectrum but different parameters, h = 0 and cl = cl = 40 in the first quadrant, h = 5 and cl = cl = 60 in the second, h = 20 and cl = cl = 80 in the third, and h = 5 and cl = cl = 60 in the fourth, respectivel Figure 2 shows a 2D RRS with different spectra, Gaussian spectrum with h = 0 and cl = cl = 40 in the first quadrant, the second order Power-Law with h = 5 and cl = cl = 60 in the second, Eponential spectrum with h = 20 and cl = cl = 80 in the third, and Figure Inhomogeneous 2D RRS with same spectrum and three different parameters Figure 2 Inhomogeneous 2D RRS with four different spectra and parameters

12 270 An Algorithm for Rough Surface Generation with Inhomogeneous Parameters Figure 3 Inhomogeneous 2D RRS with a circular region Figure 4 Inhomogeneous 2D RRS with a circular region and three sectors the third order Power-Law with h = 5 and cl = cl = 60 in the fourth, respectivel These two figures were obtained b the plate-oriented method Figure 3 shows a 2D RRS with Eponential spectrum of h = 02 and cl = cl = 50 inside the circle of radius 500 and Gaussian spectrum of h = 0 and cl = cl = 50 outside it The transition width was selected as T = 00 Figure 4 shows a 2D RRS generated b the point-oriented method with nine points at n (i) = cos(2πi/9) and n (i) = sin(2πi/9) for i =,2,, 9 together with the tenth point at the origin n (0) = 00 and n (0) = 00 The spectra and parameters are selected as Gaussian spectrum with h = 0 and cl = cl = 50 for i =, 2, 3, Gaussian spectrum with h = 5 and cl = cl = 75 for i = 4, 5, 6, Gaussian spectrum with h = 20 and cl = cl = 00 for i = 7, 8, 9, and Eponential spectrum with h = 05 and cl = cl = 00 for i = 0 5 CONCLUSION Based on the convolution method, we have introduced an algorithm to generate inhomogeneous 2D RRSs Some numerical eamples have been shown for 2D RRSs in order to demonstrate the availabilit of the proposed method The introduced algorithm can provide us a useful tool to simulate electromagnetic wave propagation along the inhomogeneous RRSs simulating the phsical environments composed of several homogeneous RRSs such as desert, vegetable field, sea surface and so on Consequentl, we can deal with a more realistic channel modeling than ever b using the propagation characteristics obtained b the simulations dealing with more complicated natural environments than known before Such numerical simulation and channel modeling deserve as a future investigation in relation to wireless sensor networks Acknowledgment The work was supported in part b a Grand-in Aid for Scientific Research (C) (256042) from Japan Societ for the Promotion of Science

13 Journal of Algorithms & Computational Technolog Vol 5 No REFERENCES [] Thoros, EI, The validit of the Kirchhoff approimation for rough surface scattering using a Gaussian roughness spectrum, J Acoust Soc Am, 988, 83(), [2] Thoros, EI, Acoustic scattering from a Pierson-Moskowitz sea surface, J Acoust Soc Am, 990, 88(), [3] Phu, P, Ishimaru, A and Kuga, Y, Co-polarized and cross-polarized enhanced backscattering from two-dimensional ver rough surfaces at millimeter wave frequencies, Radio Sci, 994, 29(5), [4] Tsang, L, Chan, CH and Pak, K, Backscattering enhancement of a twodimensional random rough surface (three-dimensional scattering) based on Monte Carlo simulation, J Opt Soc Am A, 994, (2), 7 75 [5] Yoon, KY, Tateiba, M and Uchida, K, FVTD simulation for random rough dielectric surface scattering at low grazing angle, IEICE Trans Electron, 2000, E83-C(2), [6] Yoon, KY, Tateiba, M and Uchida, K, A numerical simulation of low-grazing-angle scattering from ocean-like dielectric surfaces, IEICE Trans Commun, 2002, E85- B(0), [7] Hata, M, Empirical formula for propagation loss in land mobile radio services, IEEE Trans Veh Technol, 980, VT-29(3), [8] Uchida, K, Fujii, H, Nakagawa, M, Honda, J and Morikawa, T, On random surface generation and FVTD analsis of propagation characteristics, IEEJ Technical Reports, 2005, EMT-05-47, [9] Honda, J, Morikawa, T, Nakagawa, M, Fujii, H and Uchida, K, Effect of rough surface spectrum on propagation characteristics, IEEJ Technical Reports, 2006, EMT-06-28, [0] Uchida, K, Fujii, H, Nakagawa, M, Li, XF and Maeda, H, FVTD analsis of electromagnetic wave propagation along random rough surface, IEICE Trans Commun, 2007, J90-B(), [] Uchida, K, Fujii, H, Nakagawa, M, Honda, J and Yoon, KY, Analsis of electromagnetic wave propagation along rough surface b using discrete ra tracing method, Proceedings of ISAP 2008, 2008, [2] Honda, J, Uchida, K and Yoon, KY, Estimation of radio communication distance along random rough surface, IEICE Trans Electron, 200, E93-C(), [3] Uchida, K, Honda, K and Yoon, KY, Distance characteristics of propagation in relation to inhomogeneit of random rough surface, Proceedings of ISMOT 2009, 2009, [[4] Johnsonbaugh, R and Kalin, M, C for Scientists and Engineers, Prentice-Hall, Inc, New Jerse, 997, 9 95