Radiowave Propagation Modelling using the Uniform Theory of Diffraction

Transcription

1 Deptment of lecticl nd lectonic ngineeing Pt IV Poject Repot Ye 2003 inl Repot Rdiowve Popgtion Modelling using the Unifom Theoy of Diffction chool of ngineeing The Univesity of Aucklnd Cho-Wei Chng Poject ptne: tnley Mk upeviso: D. M. Neve D. G. Rowe

2 Decltion of Oiginlity This epot is my own unided wok nd ws not copied fom o witten in collbotion with ny othe peson igned: Dte: i

3 Abstct The gowth in dio communictions nd wieless technology hs pomoted the development of electomgnetic techniques fo modelling diowve popgtion. By modelling the tnsmitted fields with y optics theoy, we will be ble to undestnd how diowves popgte in indoo envionments nd pedict the powe distibution, s might be impotnt fctos in the development of indoo wieless LANs. This epot povides n oveview of ou poject, which is to implement nd investigte computtionl electomgnetic technique the Unifom Theoy of Diffction (UTD) with y optics theoy to model the diowve popgtion in n indoo envionment. The pefomnce of UTD is nlysed by comping modelling esults with expeimentl mesuements. This epot stts by intoducing the bckgound of electomgnetic wves theoy nd descibing esons fo why we use UTD s the technique to model diowve popgtion fo indoo envionments. Resech undetken on electomgnetic wve theoy is essentil fo modelling electomgnetic wve popgtion. A hlf-plne exmple illusttes the use of diffction coefficients of UTD nd the behviou of field components between the incident nd eflection shdow boundies. A cubic spce poblem demonsttes the modelling of diowve popgtion in n simple indoo envionment. The ctul indoo envionment we ttempted to model fo electomgnetic field popgtion ws simplified fo efficiency nd divided into diffeent zones fo diffeent types of fields. The modelling esults e in compison with expeimentl mesuements, diffeences between esults will be nlysed nd used to judge the pefomnce of UTD. ii

4 Acknowledgements ist nd foemost, I would like to cknowledge nd thnk D. Michel J. Neve fo his guidnce duing the pst six months. I lso wish to thnk my poject ptne, tnley Mk, fo his coopetion. iii

5 Contents Abstct Acknowledges Contents ii iii iv 1. Intoduction 1 2. The Bckgound of Optics nd lectomgnetic Wve Theoy 3 3. Ry Theoy nd Geometicl Optics ield oundtions fo Geometicl Optics Diect Rys Reflected Rys Loction of the Reflection Point Clculting the Reflected ield Diffcted Rys Loction of the Diffction Point Clculting the Diffcted ield Hlf-plne Diffction Poblem ield Popgtion in Cubic pce loo Modelling of Rdio Wve Popgtion Intoduction to the Indoo nvionment Diect ield Popgtion Modelling Reflected ield Popgtion Modelling ingle Reflected ield Double Reflected ield Diffcted ield Popgtion Modelling Diffcted Plus Reflected ields Popgtion Modelling Totl ield Popgtion Modelling xpeimentl Mesuements nd Anlysis Conclusions Refeences 34 iv

6 Appendices A Loction of the Reflection Point 35 B Reflected ield Clcultion 36 C Locting the Diffction Point 37 D Diffcted ield Clcultion 39 Deivtion of Pth Gin 41 The Rough loo Plne of the 8 th of chool of ngineeing 42 v

7 1. Intoduction The development of wieless technology epesents new e of telecommunictions, especilly the sevices which e povided by dio communictions nd d technology hve been used fo ove centuy. cientists nd enginees hve been tying to impove the electomgnetic techniques fo modelling diowve popgtion. By modelling the tnsmitted electomgnetic fields in the indoo envionments, we will be ble to undestnd how diowves popgte, s might be impotnt fctos in the development of indoo wieless LANs. ctos such s the physicl envionment nd the fequency of electomgnetic wve e usully tken into considetion to incese the ccucy of modelling, but implementtion nd efficiency e lso impotnt if the technique is going to be pplied in the pcticl sitution. The im of this poject is to esech, implement nd investigte the pefomnce of computtionl electomgnetic technique the Unifom Theoy of Diffction (UTD) with y optics theoy to model the diowve popgtion in n indoo envionment. Then the modelling esults fom this poject wee comped with the expeimentl mesuements nd the pefomnce of UTD ws nlysed. The pupose of the epot is to pesent ou chievement of using UTD nd MATLAB to model the diowve popgtion nd the powe distibution in n indoo envionment. The chosen envionment is the 8 th floo of chool of ngineeing, which is cubic spce with concete coe in the middle. The modelling we hve chieved ws using UTD nd MATLAB to simulte the popgtion of electomgnetic fields ound the coido nd clculting the powe distibution, especilly t cones when fields e eflected nd diffcted. ection 2 stts by intoducing the bckgound of optics nd wve theoy. The esons fo why diffeent methods hve been developed to explin high-fequency electomgnetic field phenomen nd impotnt fctos of ech theoy will lso be intoduced in this section. Redes should be ble to undestnd why UTD is chosen s the technique we used to model the diowve popgtion. ection 3 pesents the methods we used to clculte diffeent types of electomgnetic fields diect, eflected nd diffcted fields. ctos involved in clculting field popgtion nd the min technique we used to model this poject, UTD, e intoduced in this section. In section 4, hlf-plne diffction exmple illusttes the use of UTD, the diffction coefficient nd the clcultion fo field components. This exmple will pesent the mplitude of ech field component vies with diffeent ngles nd it lso poves tht UTD cn be used s tool to model popgtion of diffcted field on shdow boundies.. 1

8 ection 5 pesents using MATLAB to model the popgtion of diect nd eflected fields in cubic spce. The gph shows the distibution of pth gin long the smpling e. Redes will see the mplitude of pth gin chnges due to intefeence of diect nd eflected fields. ection 6 contins how we modelled the popgtion of diffeent fields in the indoo envionment, which is the 8 th floo of chool of ngineeing. Distibutions of pth gin of diffeent fields wee plotted long the smpling e. ection 7 descibes the expeiment we did on the 8 th floo of chool of ngineeing. We mesued sevel men pth gins long the coido nd comped ou modelling esults with mesuements nd nlysed the pefomnce of UTD in modelling diowve popgtion in indoo envionments ection 8 contins conclusions of this poject nd epot. Min findings fom sections of this epot nd ou modelling esults will be concluded, the futhe developments fo this poject will lso be ecommended. 2

9 2. The Bckgound of Optics nd lectomgnetic Wve Theoy emt s pinciple postulted in 1654 sying tht no mtte wht kind of eflection o efction to which y is subjected, the y tkes minimum time to tvel fom one point to nothe, then clssicl geometicl optics ws estblished when emt s pinciple ws completed in mthemticl theoy. This is the fundmentl theoy descibing the phenomenon of y [1, p3]. Howeve, clssicl geometicl optics is limited on solving high-fequency electomgnetics poblems becuse the concepts of phse, wvelength, poliztion nd diffction e not included. Physicists intoduced the electomgnetic wve theoy with mthemticl equtions cn explin wht clssicl geometicl cn not, but it hs pcticl disdvntges nd the concepts of geometicl optics wee lost [1, p4]. ome time lte, Lunebeg nd Kline extended the clssicl geometicl optics to moden geometicl optics (GO) by estblishing connection between Mxwell s electomgnetic field equtions nd geometicl concepts [1, p4]. Howeve, GO is incpble of pedicting the field in the shdow egions [1, pp19]. o exmple, t shdow boundies, the field intensity chnges pidly nd GO cnnot descibe this behviou coectly [2, p121]. lectomgnetic fields hve to be continuous eveywhee, the discontinuities coss the shdow boundies does not exist in ntue, nd tht is nothe eson why GO fils to descibe the totl electomgnetic field [1, p160]. Diffction is the pocess wheeby light popgtion diffes fom the pedictions of geometicl optics. [3] Joseph B. Kelle explined the phenomenon of diffction with this desciption. He intoduced diffcted ys tht behve like the odiny ys of GO once they leve the edge nd this lid the bsis fo wht hs become geometicl theoy of diffction (GTD) [4, p130] [1, p4]. igue 1 nd 2 e used to illustte the effect of diffcted field. The egion in spce illuminted by given field is efeed to s the lit egion nd the egion not illuminted is shdow egion. These two egions e septed by incident shdow boundy i (IB). The incident wve in the egion x > 0 cn be expessed s [1, p163] i jkx ( x, y) = 0 e, in the egion y > 0 i ( x, y) = 0, in the egion y < 0 ig 1. Diffction fom pefect conducting hlf-plne illuminted by plne wve t noml incidence. [1, p163] 3

10 igue 2 shows plot of totl field t ( x, y) with x = 100 m nd fequency of 10 GHz. It is cle to see tht field does exist in the shdow egion (x > 0, y < 0). Howeve, thee is no incident field in the shdow egion, so tht must be the diffcted field d. We cn conclude tht the totl field in the egion x > 0 cn be expessed s [1, p164] + in the egion y > 0 t d ( x, y) = in the egion y > 0 t i d ( x, y) =, ig 2. Diffction fom hlf-plne illuminted by plne wve t noml incidence; the field is evluted t x = 100 m t fequency of 10 GHz [1, p165] In the lit egion (y > 0) thee is ipple in the totl field tht is cused by the intefeence of the diffcted field with the incident field. The ipple deceses when the field is moving wy fom IB becuse the stength of diffcted field is decesing nd the mplitude of the totl field ppoches the incident plne wve. Kelle extended the GO to GTD by dding diffcted fields nd he succeeded in coecting the deficiency in the GO tht pedicts zeo fields in the shdow egions. Howeve, the oiginl fom of the GTD suffes mny poblems. The most seious one is t shdow boundies, whee the GO fields fll buptly to zeo nd the pedicted diffcted fields become infinity [1, p4]. The esons fo this is oughly descibed in efeence 1, pge 5. Kelle s oiginl GTD is not unifom solution. It cn pedict the diffcted fields in egions wy fom the shdow boundies, but become singul in the tnsition egions suounding such boundies. The unifom theoy of diffction (UTD) developed by Kouyoumjin ws divegent fom Kelle s GTD theoy but ovecomes GTD s defects tht GTD is inpplicble in the vicinity of shdow boundies [1, p175]. UTD pefoms n symptotic nlysis nd by incopoting tnsition function into the diffction coefficient, the diffcted fields emin bounded coss the shdow boundies [5, 1 p175]. In this poject we hve pplied UTD with y optics theoy to investigte the diowve popgtion (diect, eflected nd diffcted fields) nd powe distibution in indoo envionments. Methods will be descibed in the following pgphs. 4

11 3. Ry Theoy nd Geometicl Optics ield This section contins techniques we used to clculte electomgnetic fields popgtion using y methods. In this poject we only consideed spheicl wve, which is electomgnetic enegy being dited eqully in ll diection. Tems usully ppe fo descibing electomgnetic fields will be explined hee fo bette undestnding. 3.1 oundtions fo Geometicl Optics lectomgnetic fields t high fequencies cn be expessed s Lunebeg-Kline expnsions [1, p17] ( ) H ( ) ~ ω ~ ω H 0 ( ) 0 ( ) e e jk ψ ( ) jk ψ ( ) whee is the sptil coodinte, ω is the din fequency, k = ω µ 0ε, ϕ() is phse function [5, p3]. qutions (1) nd (2) cn be substituted into Mxwell s equtions nd the vecto Helmholtz eqution to find solutions [5, p4]. o the detils ede cn efe to [1, pp15-18]. The genel fom of GO field long y tjectoy cn be witten s [1, p26] jks ( s) = (0) ρ 1ρ2 /( ρ1 + s)( ρ2 + s) e (3) (1) (2) (0 ) 0 (0 ) jk ψ ( 0 ) whee () = e, the field t the efeence (4) point s = 0. (b) s is the distnce long the y pth fom the efeence point s = 0. jks (c) e is the phse shift eltive to the efeence point s = 0 (d) A s) = ρ ρ /( ρ + s)( ρ + ) is the divegence o speding (5) ( s fcto which govens the mplitude vition of the field long the y pth. (e) ρ 1 nd ρ 2 e the pinciple dii of cuvtue of the wvefont t the sufce point s = 0. The sign convention is tht positive (negtive) dius of the cuvtue implies diveging (conveging) ys in the coesponding pinciple plne. [1, p34] 5

12 ig 3. Genel diveging stigmtic y tube, fo which both p1 nd p2 e positive [1, p28] 3.2 Diect Rys o spheicl wve tube ρ 1 = ρ 2 = ρ. The divegence fcto becomes ρ A( s) = (6) ρ + s The expession fo the spheicl wve is [1, p36] s) = A 0 e s jks ( (7) 3.3 Reflected Rys Loction of the Reflection Point [5, p7] Accoding to emt s pincipl, egdless of wht kind of eflection, the y tkes the minimum distnce fom one point to nothe [1, p3] A C B B B ig. 4 emt s pinciple of eflection. Pth ABC is the minimum pth of eflection. [1, p2] 6

13 C n A - ψ n R B Oigin ig. 5 The field eflected by pln sufce [5, p7] ig. 5 is used to illustte the method we used to find the eflection point R on pln sufce P when y tvels fom souce point to field point vi R [5, p7]. o the full method we used to clculte the loction of the eflection point plese efe to Appendix A. The min concept to clculte the eflection point R on plne P is fist by woking out the imge of the souce point in the plne, nmely, then the intesection of the plne P nd the vecto fom to is the loction of the eflection point R. ist of ll, the plne is defined s P = A + sb + tc, nd we clculted the pependicul pojection point of on the plne, nmely, nd the eltionship between nd is = + u n (8) whee u is numbe of quntity nd n is unit vecto noml to the plne expessed s B C n = (9) B C nd the full clcultion fo u is in Appendix A. When u is solved, substitute into (10) to clculte the position vecto. The imge of the souce point in the plne cn be expessed s = + 2( ) (10) The position of R cn be expessed s R = + u ( ) (11) nd R = A + s B + t C (12) 7

14 The clcultions fo s, t nd u e shown in Appendix A. Then substitute u into (13) to find the eflection point R Clculting the Reflected ield [5, p8] The eflection pocesses cn be descibed s ˆ ˆ ( ) = R ˆ ( ) ˆ i i ( R) AR ( ) e ( R) jk (13) The pllel nd pependicul components of eflected field is equl to the pllel nd pependicul components of incident field times the divegence fcto, which is 1, times the phse shift in tems of pth length, nd multiplied by eflection mtix which is used to clculte the pllel nd pependicul components of the incident field septely. which the field quntities i ( R) nd ˆ ( R) e the pllel nd pependicul components of the incident field t the point of eflection. The full clcultion of i ( R) nd ˆ i ( R) is shown in Appendix B. ˆ ˆ The totl eflected field t, ˆ ( ), is ˆ = ( ) + ( ) (14) The divegence fcto A R () fo the spheicl wve incidence is sme s (8). The GO eflection mtix R is given by [5, p8] nd [1, p77] i R R = = (15) 0 R 0 1 The eflection mtix indictes the pllel nd pependicul components of the incident field e eflected independently of ech othe [5, p8] 3.4 Diffcted Rys Diffction is phenomenon tht electomgnetic enegy bends ound edges nd sufces [5, p9]. In this poject we only consideed diffction on stight edges. The fist step to ppoch diffcted fields is to detemine the y tjectoy nd the point of diffction on the edge. ig. 6 illusttes the point of diffction nd the diffcted y pth. The full method fo clculting the loction of the diffction point nd diffcted field is shown in Appendix C nd Appendix D espectively. 8

15 3.4.1 Loction of the Diffction Point [5, p9] υ β ns φ β bs φ β β ' φ c s D Q n φ ig. 6 The field diffcted by stight edge [5, p10] The position of ny point P on the edge cn be defined s P = D + u (16) Assume diffcted y tvels fom souce point to field point vi diffction point P = Q on the edge. The wedge is configued by vectos D, ; efeence plne vectos n nd the intenl wedge ngle v. To clculte Q it is fist necessy to clculte the pependicul pojections of the souce nd field points upon the edge, nd espectively [5, p9] Appendix C When nd e found, two simil tingles cn be used to detemine the loction of Q. Q' nd Q' shown in ig. 7 β β Q ig. 7 Diffction Point Clcultion [5, p11] 9

16 If q epesents the function of the totl distnce fom to of Q, tht ' ' ' ) (1 ' ' ' q q = (17) ' ' ' q + = (18) Then the diffction point Q is found s Q = + q( ) (19) Clculting the Diffcted ield [5, p11] The diffction pocess cn be descibed s [5, p11] jk D i i d d e A D = ), ( ˆ ˆ ) ( ˆ ) ( ˆ φ β φ β (20) ) ( ˆ Q i β nd ) ( ˆ Q i φ e the soft nd hd polised components of the field t the point of diffction. The full clcultion of components in ig.6 is shown in Appendix D. o spheicl wve incidence, the divegence fcto ), ( A D [5, p12] is ) ( ), ( A D + = (21) The diffction coefficient mtix D fo the y-fixed coodintion system is given by [5, p12] = h s D D D 0 0 (22) This mtix is digonl so the pllel nd pependicul components of the incident field e diffcted independently of ech othe. The totl diffcted field t, is φ φ β β d d d ) ( ˆ ) ( ˆ ) ( ˆ + = (23) A numbe of eches hve ttempted the deivtion of diffction coefficients fo the pefectly conducting wedge. In the next section hlf-plne diffction poblem is used to illustte tht by combining tnsition function with the Unifom Theoy of Diffction (UTD) into the diffction coefficient, the diffcted fields emin bounded coss the shdow boundies. 10

17 4. Hlf-plne Diffction Poblem The following exmple illusttes the use of the diffction coefficients of UTD nd the clcultions fo field components [5, pp12-14]. Mtlb ws used fo pogmming nd clculting the nswe. cenio: A point souce diting n 850 MHz veticlly-polised -field component is locted t n incident ngle φ ' = 75 nd distnce '=1000λ fom hoizontl edge. The totl field (compising diect, eflected nd diffcted ys) is mesued ove the nge of diffction ngle φ = t distnce = 3λ fom the edge of pefect conducting plne. The pictue is shown in ig. 8 ield Point z y = 1000λ = 3λ φ x ig. 8 Hlf-plne diffction Poblem [5, p14] Ê φ = 75 Tnsmitting Antenn ist, we define the shdow boundies fo the hlf-plne diffction poblem. hdow boundies divide the spce into diffeent zones with diffeent fields, nd they shown in ig. 9. Reflected hdow Boundy z Zone II Diect Diffcted Incident hdow Boundy y Zone I Diect Reflected Diffcted x Zone III D iffcted ig. 9: hdow boundies fo the hlf-plne diffction poblem [5, pp 15] 11

18 Incident shdow boundy is t 180 degees plus the incident ngle ( π + φ ) fom the efeence plne nd the eflection shdow boundy is t 180 degees minus the incident ngle ( π φ ) fom the efeence plne. + β = φ φ' nd β = φ + φ' e defined fo the unifom diffction coefficient epesenting the ngle of diffcted field plus nd minus the incident ngle of souce field septely. The unifom diffction coefficients e defined s [5, p13] D s, h ( d + ( L, φ, φ ' β ') + ( β ) ( κ + 1 = [ d sin β ' + ( β )) + d + ( β ) ( κ + ( β ) ( κ ( β )) + d xplntions fo ech component e descibed below. We stted by woking on fou diffeent combintions of functions (.) with β. The function (.) is defined s [5, p13] + ( β + )))] ( β ) ( κ ( β )) (24) ± ± ( β ) = 1+ cos( β 2nN π ) (25) ± N is the intege which most closely stisfied the equtions N = ( β π ) / 2nπ (26) + N = ( β + π ) / 2nπ (27) nd υ n = 2 [5, p12], whee υ is the inteio wedge ngle. π The pmete κ = kl, whee k = 2π / λ [5, p13]. ' sin 2 β ' L is known s the displcement pmete given by L = fo spheicl ' + wve. nd e the distnce fom souce field to the diffction point on edge nd the distnce fom the diffction point to the field point. The poduct of κ nd function (.) e then substituted into (.) function The eson why diffcted fields with coefficients of UTD emin bounded coss the shdow boundies is the following -function will tends to zeo to cncel the singulity diveged fom d-function in the tnsition egions suounding such boundies jζ 1 1 ( ζ ) = j 2πζ e [( C( ζ )) j( ( ζ ))] (28)

19 whee C(.) nd (.) e the esnel cosine nd sine integls espectively [6, pp ]. The esnel integl f(x) defined by Boesm [7] is f ( x) = x 0 jt e dt = C( x) 2πt j( x) (29) The function d(.) cn be witten s [5, p13] d ± e ( β ) = 2n π j 4 β π ± cot 2πk 2n (30) It chnges with vibles k, n nd cotngent of vibles β nd n. Vibles β e + β = φ + φ' nd β = φ φ'. (31) Afte woking out the UTD coefficient, it is substituted in to the diffction pocess descibed in ection nd the totl diffcted field fellows (23) The totl field is yielded by dding the diffeent field components in the coect sense, which cn be detemined by exmining the field component diections t the incident nd eflected shdow boundies (ig. 10) [5, pp14]. The totl field cn be expessed s = + (32) tot di ef di ig. 10 ield component diections fo the hlf-plne diffction poblem [5,p15] 13

20 ig. 11 Plot of the mgnitude of the field components vs. ngle fom the efeence plne ig. 11 contins fou plots showing the mgnitude of ech field component nomlised with espect to the vlues of totl field. We cn see the diect field is constnt nd does not exist in the egion fte incident shdow boundy ( = 255 ). The eflected field is constnt nd does not exist in the egion fte eflection shdow boundy ( = 105 ). The diffcted field is bounded coss shdow boundies. The totl field plot shows fo φ > 255 the totl field is just the diffction field nd fo φ < 105 the totl field oscilltes minly due to the stnding wve poduced by the incident nd eflected wve. o 105 < φ < 255, the totl field oscilltes due to the intefeence between the incident field nd the diffction field 14

21 5. ield Popgtion in Cubic pce This section pesents the modelling of wve popgtion in cubic spce, simil to n empty oom, nd we wnt to obseve the wve popgtion nd powe distibution on smpling e t cetin height fom gound. Thee e only diect nd eflected fields popgting in this spce nd we ssume fields e totlly eflected when they hit the wlls ound this cubic spce. The physicl dimensions of this cubic spce e shown in ig 12 below. The cubic spce hs 10 m 10 m 10 m volume, the left bottom cone is set to be oigin nd the tnsmitting ntenn is 2 m high plced t 2 m wy fom x nd y- xis efeed to oigin. The tnsmitting ntenn dites spheicl wves nd fields on the sufce 2 m high fom the gound is smpled. The smpling e is t sme height with the tnsmitting ntenn s illustted in ig 12. ig. 12 The physicl dimension of the cubic spce qution (7) ws used fo clculting diect field. o clculting the eflected field, we used the method mentioned in section 3.3. This time we clculted the eflection points on 6 plnes nd eflected fields vi 6 plnes. We e inteested in obseving how much powe we eceive t the field point compe with the powe t souce point theefoe we need to convet ou modelling esults fom field to powe. In ode to know the tio between the eceiving end nd the tnsmitting end of ntenns, we need to clculte the pth gin. The pth gin in db between 2 ntenns is defined s P Pth Gin db = 10log10 ( ) (33) Pt whee P is the eceived powe (W) P t is the tnsmitted powe (W) 15

22 The field t the eceiving ntenn is [5, p20] ηp 2π ˆ t = tot z (34) whee Ê is the field t the field point. is the mgnitude of totl field t the field point. tot The eceived open-cicuit voltge is defined s [5, p20] Vˆ = hˆ ˆ (35) OC whee ĥ is the complex effective length of the eceiving ntenn defined s whee R hˆ π = 2 z [5, p20] (36) k η R is the ntenn dition esistnce nd we ssumed is 1 The ms powe t the eceiving ntenn is 2 VOC P = 8R [5, p20] (37) whee we ssumed R is 1. inlly we got the esult pth gin in db expessed s P λ 2 10 log10 = 10 log10 tot ( db) P 4π t 2 (38) The full deivtion of bove steps is shown in Appendix The modelling esults of the pth gin distibution on the smpling e e shown in ig. 13 nd 14. We cn clely see tht cicul e ound the souce point hs no ipple nd the est e is full of ipples. The e with no ipple indictes the diect field is the dominnt pt of the totl field nd the pth gin deceses pidly fom the cente of the cicul e (souce point) outwds indictes the mgnitude tnsmitted powe deceses with pth length. The e with ipples indictes the intefeences between diect nd eflected fields. The ipple goes up when fields e in phse nd powe of ech field dds up, when ipple goes down when fields e out phse so powe of fields cncel out. 16

23 ig. 13 Distibution of pth gin esulted by diect nd eflected fields. ig. 14 Mgnitude of pth gin esulted by diect nd eflected fields. 17

24 6. loo Modelling of Rdio Wve Popgtion This is the min investigtion in the poject nd ou inteest is modelling the wve popgtion in n indoo envionment with the computtionl electomgnetic techniques we hve lened nd pedicting the powe distibution in the envionment. The othe topic we investigted is fo the fields ound the edge, does the diffcted field fom the edge diectly contibute moe powe o the field diffcted fom the edge fist then eflected fom the wll hs moe contibution. We lso modelled the double-eflected fields ound the cone to see how much they contibute to the totl powe. An expeiment ws done by mesuing the men pth gin within some es then we comped the modelling esults with ou expeimentl mesuements. 6.1 Intoduction to the Indoo nvionment ist of ll, we need to know wht kind of indoo envionments we e going to model. The chosen envionment is the 8 th floo of chool of ngineeing nd its ough floo pln is shown in ig. 15, Appendix The ough floo pln ws simplified nd lbelled fo moe convenient nd efficient modelling s shown in ig. 16. The floo hs volume ound 10 m long 10 m wide 2.5 m high nd thee is concete coe ound 7 m long 7 m wide 2.5 m high ight in the middle on the floo. We ssume ll the plnes on this floo e mde of concete. ig. 16 The simplified nd lbelled floo plne fo modelling 18

25 6.2 Diect ield Popgtion Modelling o modelling the diect field popgtion in the floo, fist we need to know the es tht diect field cn ech nd theefoe we need to find the incident shdow boundies fo diect fields tnsmitted fom the souce ntenn on the floo. IB Concete Coe IB Tnsmitting Antenn ig. 17 Incident hdow boundies fo the diect field fom souce ntenn. In ig. 17 the incident shdow boundies (IBs) indicte the e tht only diect fields cn popgte. We ssumed tht fields tnsmitted fom the souce ntenn cn not penette concete theefoe the IBs e lines fom tnsmitting ntenn to points P2 nd P3 to plne 4 nd plne 2 espectively. (ig 16) ig. 18 nd 19 disply the distibution nd mgnitude of pth gin in the e of diect field popgtion. We cn see fom the gphs tht pth gin dops significntly when the field point moves wy fom the souce point. Accoding to the eqution (7) we used to model the diect field, the mplitude of the spheicl wve vies invesely with the distnce, tht implies the powe mesued by the eceiving ntenn vies invesely with the distnce squed. 19

26 ig. 18 Distibution of pth gin esulted by diect field popgtion. ig. 19 Mgnitude of pth gin esulted by diect field popgtion. 20

27 6.3 Reflected ield Popgtion Modelling It is moe complicted to del with eflected field thn diect field in n indoo envionment due to the complexity of considetions on modelling. In n indoo envionment, fields cn be eflected sevel times between plnes nd theefoe the difficulty to clculte the eflection pths inceses. Reflection shdow boundies will become moe difficult to be detemined; the oienttion of ech eflected field chnges; the mteil (eflection coefficient) of ech plne my be diffeent; the mgnitude of eflected field chnges when it eflects ech time. Compe this simplified 8 th floo of chool of ngineeing with the pevious cubic spce poblem, thee is n dditionl sque concete coe in the middle fom ceiling to gound nd we ssume ll the plnes, including ceiling nd gound, e mde of concete nd fields cn not penette though ingle Reflected ield In this section we will only conside single eflected fields nd obseve the pth gin ound coido. The fist thing is to detemine eflection shdow boundies constined by the concete coe fo single eflected fields. ig. 20 shows eflection shdow boundies on ech plne nd this cn help us divide the coido into sections nd in ech section we cn know the limit tht single eflected fields cn ech fom ech plne. o exmple, the top-left digm in ig. 20 shows the egion tht single eflected fields cn ech vi plne 1. The eltion between the tnsmitted powe nd the eceived powe cn be expessed s Two-Ry Model [8] s illustted in ig. 21. The eceived signl P fo isotopic ntenns, obtined by summing the contibution fom ech y, cn be expessed s: P 2 λ 1 1 = Pt exp( jk1 ) + Γ( α) exp( jk 4π ) 2 (39) whee P t is the tnsmitted powe (W). P is the eceived powe (W). 1 is the diect distnce fom tnsmitte to eceive (m). 2 is the distnce though eflection on plne (m) Γ (α) is the eflection coefficient depending on the ngle of incidence α nd poliztion. 21

28 ig. 20 Reflection shdow boundies fo single eflected fields α Tnsmitting Antenn θ 2 Receiving Antenn ig. 21 Two-Ry Model of eflected field 22

29 The eflection coefficient is given by cosθ Γ( θ ) = cosθ + ε sin ε sin whee θ = 90 α = 1 o 1 fo veticl o hoizontl poliztion, espectively ε ε = eltive dielectic constnt of the pln, which we use ε = 6 fo concete. Compe the Two-Ry Model eqution (39) with the pth gin eqution (38) we hve deived in the cubic spce exmple, we cn see the eltion between eceived powe nd the tnsmitted powe is equl to constnt of field t eceiving point. 2 2 θ θ 2 (40) λ multiplied by the mgnitude 4π Double Reflected ield In this section we will obseve how much powe the doubled eflected field contibute to the pth gin in the model. If the effect on pth gin is significnt, we will continue to model the tiple eflected field. The fist thing fo modelling double eflected fields is to know thei eflection pths nd shdow boundies, theefoe, we need to define the eflection points on plnes. The concepts e simil to wht we hve done fo modelling single eflections nd this time we will pply these concepts on two plnes. Accoding to emt s pinciple, we need to find the shotest double eflection pth between two points, nd this is illustted in ig. 22 The imge points of point A nd point B in plne 1 nd plne 2 e lbelled with A nd B. A line is dwn fom A nd B nd the intesections with plne 1 nd plne 2 e the eflection points fo double eflection, nmed R1 nd R2, nd the pth tht links A, R1, R2 nd B is the pth of double eflection. ig. 23 shows two kinds of eflection shdow boundies of double eflection. In this model we consideed the es (shded with lines) tht only double eflected fields cn ech becuse diect nd single eflected fields e dominnt fields in the egions ovelpped with the egions within double eflected fields. 23

30 P l n e 2 B B ' A R 2 P l n e 1 R 1 A ' ig. 22 Illusttion of eflection points nd the pth of double eflection. Concete Coe Tnsmitting Antenn ig. 23 Two kinds of double eflection shdow boundies ig. 24 nd ig. 25 show the distibutions nd mgnitudes of pth gins esulted fom single nd double eflected fields. In ig. 25 we cn see the pth gins esulted fom single eflected fields e in the nge between 40 db to 60 db gdully decesing with distnce. The es only contin double eflected fields e eltively smll with pth gins ound 67 db. We concluded tht double eflected fields do not contibute significnt powe in the es outside incident shdow boundy. 24

31 ig. 24 Distibution of pth gin esulted by single nd double eflected field popgtion. ig. 25 Mgnitude of pth gin esulted by single nd double eflected field popgtion. 25

32 6.4 Diffcted ield Popgtion Modelling Points on the smpling e ll hve fields tht diffcted fom the edges of the concete coe. ig. 26 below shows field points hve diffcted fields coming fom diffeent edges of the concete coe. Pth 2 Concete Coe Pth 3 Pth 1 Tnsmitting Antenn ig. 26 Diffcted fields fom the edge of the concete coe. Distibution of pth gin esulted by diffcted fields e shown on ig. 27 nd the mgnitude cn be seen moe esily on ig. 28. We cn see fom the gphs tht es ound the cones of the concete coe hve highe pth gin esulted by diffcted fields thn pth gin long coido, tht is becuse the diffcted fields e diffcted on the edges of the cones. Pth gin of diffcted field ound the cone P1 cn go to ound 50 db; ound the cones P2 nd P3 dops to 60 db; nd ound the fthest cone fom the souce point is ound 110 db. 26

33 ig. 27 Distibution of pth gin esulted by diffcted field popgtion. ig. 28 Mgnitude of pth gin esulted by diffcted field popgtion. 27

34 6.5 Diffcted Plus Reflected ields Popgtion Modelling As illustted in ig. 29, nothe condition would be field point hs fields diffcted fom the edge nd lso fields eflected vi plne fom the edge. Comped these two components, we found tht the diffcted fields fom the edge diectly to the field point e dominnt. Diffction only Concete Coe Diffction + Reflection P2 Incident ield ig. 29 A field point hs fields diffcted fom n edge nd lso eflected vi plne. 6.6 Totl ield Popgtion Modelling o modelling the distibution of pth gin of totl field long the coido, we dded up powe distibuted by ech component field (diect, single nd double eflected, diffcted, diffcted nd eflected) then conveted to pth gin. The distibution of totl pth gin is shown on ig. 30 nd mgnitudes cn be seen moe esily on ig. 31. Genelly we cn see tht the e close to the souce point hs lge pth gin, then it deceses with the distnce wy fom the souce point. Pth gin dops significntly when diect nd single eflected fields cn not ech, nd diffcted field become dominnt. 28

35 ig. 30 Distibution of pth gin esulted by totl field popgtion. ig. 31 Mgnitude of pth gin esulted by totl field popgtion. 29

36 7. xpeimentl Mesuements nd Anlysis Ou modelling esults wee comped with expeimentl esults to nlyse is UTD suitble technique fo modelling diowve popgtion in indoo envionments. The expeiment ws set up with signl geneto connected to souce ntenn, eceiving ntenn nd compute fo collecting dt. In the modelling thee is no obstcle on coido so the time we did the expeiment ws when not mny people wee wlking ound coido in ode to hve moe simil conditions. As illustted in ig. 32, the souce ntenn ws plced t the middle of one cone of coido. The eceiving ntenn ws fist plced t the middle of coido 0.7 m wy fom the souce ntenn nd otting 360 degees in ode to collect 360 pth gin smples then the compute clculted the men pth gin of tht 360 smples, then the eceiving ntenn ws plced nothe 0.7 m futhe wy nd nothe men pth gin ws clculted fom nothe 360 smples. me pocesses wee epeted long the coido until 41 men pth gins wee collected. Men pth gins t the sme positions in the modelling wee picked nd comped with the dt fom expeiment. ctos such s ttenution nd cble loss wee eliminted nd the compisons e shown on ig.33 nd ig m 1.5m 7m 1.5m Concete Coe 7m 1.5m Tnsmitting Antenn Receiving Antenn ig. 32 Men pth gins wee mesued ound the coido. Compison shows in the egion tht diect field cn ech, pth gins obtined fom ou modelling nd expeiment hve no much diffeences, but in the egion ound cone, pth gin in ou model is ound 10 db highe thn expeiment, fte some distnces, pth gin in ou expeiment becomes ound 20 db less thn expeiment, thn is bout 100 times diffeence in tems of powe. 30

37 0 Compison between the model nd the expeimentl esults Pth Gin - db Model xpeiment Distnce between the souce nd the eceiving ntenn in Log scle ig.33 Pth gin of model nd expeimentl mesuements long coido fom souce point to left-top cone Compison between the model nd the expeimentl esults Pth Gin - db Model xpeiment Distnce between the souce nd the eceiving ntenn in Log scle ig.34 Pth gin of model nd expeimentl mesuements long coido fom lefttop cone to the ight-top cone. 31

38 Diffeences between the esults of ou modelling nd expeiment my due to vious fctos in the ctul physicl envionments nd modelling techniques we used. In the hlf-plne diffction poblem, UTD ws pplied on clculting the diffcted fields on the edge of pefect conducting mteil, but in the physicl envionment, fields wee diffcted on dielectic mteil (concete in ou expeiment). In the modelling, we only consideed diffcted fields on the edges t positions with sme height s the souce point nd eceiving ntenn, but in elity, fields cn be diffcted t evey point long the edges. In the egions tht diect fields cn ech, ou modelling esults e quite close to the expeimentl mesuement. Howeve, in the egions tht diect fields cn not ech, we only consideed single, double eflected fields nd diffcted fields. In the ctul envionment, thee must moe efected nd diffcted field. 8. Conclusions emt s pinciple estblished the foundtion of clssicl geometicl optics nd descibed the behviou of y, but it is limited on descibing the phenomenon of high-fequency electomgnetic field. Moden geometicl optics (GO) fils to descibe the behviou of totl electomgnetic field in spce due to the concept of field in the shdow egion is not included. Kelle extended the GO to geometicl theoy of diffction (GTD) by dding diffcted fields, but the pedicted fields t shdow boundies become infinite which is impossible in ntue, theefoe GTD is not unifom solution fo solving ll high-fequency electomgnetic field poblems. Kouyoumjin extended GTD to unifom theoy of diffction (UTD) by dding tnsition function into the diffction coefficient nd the diffcted fields emin bounded coss the shdow boundies, theefoe we chose UTD s the technique fo modelling diowve popgtion in indoo envionments. Thee types of fields wee investigted in this poject: diect, eflected nd diffcted fields nd we ssumed ll these fields e spheicl wves dited fom souce ntenn. The clcultions of these thee types of fields wee bsed on using ymethods. o diect field, the mplitude vies invesely popotionl with distnce. o eflected field, the fist thing is to find the loction of the eflection point on plne, then the eflection pocess cn be descibe s the pllel nd pependicul components of the incident field t the point of eflection, multiplied by the divegence fcto of spheicl wve, nd the phse shift in tems with eflection pth length, nd the eflection mtix which is used to indicte tht the pllel nd pependicul components of the incident field e eflected independently of ech othe. o diffcted field, the fist thing is to find the loction of the diffction point on the edge, then the diffction pocess cn be descibe s the components of the incident field t the point of diffction, multiplied by the divegence fcto of spheicl wve, nd the phse shift in tems with diffction pth length, nd the diffction mtix which is used to indicte tht the pllel nd pependicul components of the incident field e diffcted independently of ech othe. 32

39 The hlf-plne exmple ws used to illustte tht with the implementtion of UTD, diffcted fields emin bounded coss shdow boundies. In the egion behind the incident shdow boundy, thee e only diffcted fields. In the egion befoe the eflection shdow boundy, diect, eflected nd diffcted fields exist, diect nd eflected fields e dominnt nd cusing intefeences. In the egion between two boundies, thee e diect nd eflected fields nd diect fields e dominnt. A simple model ws built to demonstte diowve popgtion in cubic spce. We ssume thee is no obstcle in the cubic spce nd fields e totlly eflected when they hit the wlls, so thee e only diect nd eflected fields. The modelling esult shows the smpling sufce is smooth due to the dominnt diect field in the e close to the souce point, then the smpling sufce contins ipples due to intefeences between diect nd eflected fields. Then the cubic spce ws extended to moe complicted indoo envionment. In ou modelling, the 8 th floo of chool of ngineeing is simplified to cubic spce with concete coe in the middle of the floo. All wlls wee ssumed to be concete nd plt lnes nd we modelled the field popgtion ound the coido. The im of this modelling is to undestnd powe distibution in n indoo envionment by modelling the wve popgtion. Modelling ws pocessed by defending the envionment into diffeent zones with diffeent field popgtion. The modelling esults wee comped with expeimentl mesuements we took on 8 th floo of chool of ngineeing. We found tht in the egions tht diect fields cn ech, ou modelling esults e quite close to the mesuements, but in the egions tht diect fields cn not ech, expeimentl mesuements of powe distibution cn go up to 20 db highe thn wht we pedicted in model. The diffeences my be due to fctos such s in the modelling of UTD, fields e diffcted on the edge of pefect conducting mteil, but in ou mesuement envionment fields e diffcted t the edge mde by concete. Diffcted fields in ou modelling we only consideed those t positions on the edges sme height s the souce nd eceiving ntenn, double nd diffcted fields in the egions tht does not include diect field. In elity, fields e diffcted t eveywhee long the edges nd thee must be moe fields thn single nd double eflected fields in the egions tht diect fields cn not ech. om the modelling nd expeiment, we concluded tht UTD is not suitble technique fo modelling diowve popgtion in indoo envionments. Recommendtions fo futhe development of this poject cn be including moe types of fields the thn just double eflected fields in ode to incese ccucy, o seching othe techniques fo modelling diowve popgtion in indoo envionments. Modelling nd expeiment cn be pocessed with obstcles between souce nd eceiving ntenns in ode to be moe simil to the sitution when wieless LANs e usully used duing dy time in elity. 33

40 9. Refeences [1] D. A McNm, C. W. I. Pistoius, nd J. A. G. Mlhebe, Intoduction to The Unifom Geometicl Theoy of Diffction. Boston: Atech House, 1990 [2] M. Bon,. Wolf. Pinciple of Optics lectomgnetic Theoy of Popgtion Intefeence nd Diffction of Light. London: Pegmon, 1959 [3] J. B. Kelle, One Hunded Yes of Diffction Theoy I Tnsctions on Antenns nd Popgtion., vol. AP-33, No. 2, ebuy 1985 [4] G. L. Jmes, Geometicl Theoy of Diffction fo lectomgnetic Wves. nglnd: Pete Peeginus Ltd [5] M. J. Neve, High equency Asymptotic Techniques, Applied lecto mgnetics Couse Notes. Deptment of lecticl nd lectonic ngineeing, The Univesity of Aucklnd, New Zelnd, [6] M.Abmowitz, I. tegun, Hndbook of Mthemticl unctions. New Yok: Dove, fifth ed., My 1986 [7] J. Boesm, Computtionl of esnel Integls, J. Mth. Comp., vol. 14, p.380, [8] A. Neskovic, N. Neskovic nd G. Punovic Moden Appoches in Modeling of Mobil Rdio ystems Popgtion nvionment, in I Communiction uveys & Tutoils: The lectonic Mgzine of Oiginl Pe-Reviewed uvey Aticles. Univesity of Belgde, Thid Qute

41 Appendix A: Loction of the Reflection Point [5, p7] A point on the plne is defined s P = A + sb + tc nd the pependicul pojection of souce point on the plne is. The unit vecto n noml to the plne is C B C B n = (A1) Othe position vectos cn be expessed s = + u n (A2) = A + s B + t C (A3) ubstitute (A3) into (A2), we get = A + s B + t C + u n (A4) A = s B + t C + u n (A5) Renge qution (A5) in mtix fom: = z z y y x x n z z n y y nx x x A A A C B C B C B u t s z y ' ' ' (A6) When (A6) is solved fo u, substitute into (A2) to clculte the position vecto. The imge of the souce point in the plne cn be expessed s = + 2( ) (A7) The position of R cn be expessed s R = + u ( ) (A8) nd R = A + s B + t C (A9) Renge (A8) nd (A9) we get s B + t C + u ( ) = - A (A10) qution (A10) expessed in mtix fom is = z z y y x x z z z z y y y y x x x x A A A C B C B C B u t s " " " " " " " " " (A11) Then substitute u into (A8) to find the eflection point R 35

42 Appendix B: Reflected ield Clcultion [5, p8] The eflection pocesses cn be descibed s ˆ ˆ ( ) ˆ = R ( ) ˆ i i ( R) AR ( ) e ( R) jk (B1) whee the field quntities i ( R) nd ˆ ( R) e the pllel nd pependicul ˆ components of the incident field t the point of eflection. They cn be expessed s ˆ i ( R) ˆ i ˆ ( i R i ( R) ) = (B2) = ' ˆ i ( R) (B3) ' nd ' e the unit vectos cn ffect the polistion of y, they cn be clculted fom the coss poducts tht exist between ' nd n shown in ig. 5 ' n ' = (B4) The totl eflected field t, ˆ ( ), is whee ˆ ' ' ' n = (B5) ' = ( ) + ( ) (B6) n = (B7) n = (B8) The divegence fcto A R () fo the spheicl wve incidence is sme s (6) The GO eflection mtix R is given by [5, p8] nd [1, p77] R R = = (B9) 0 R 0 1 The eflection mtix indictes the pllel nd pependicul components of the incident field e eflected independently of ech othe [5, p8] 36

43 Appendix C: Locting the Diffction Point [5, p9] The position of ny point P in ig. 6 on the edge cn be defined s P = D + u (C1) Assume diffcted y tvels fom souce point to field point vi diffction point P = Q on the edge. The wedge is configued by vectos D, ; efeence plne vectos n nd the intenl wedge ngle v. To clculte Q it is fist necessy to clculte the pependicul pojections of the souce nd field points upon the edge, nd espectively [5, p9] Unit vectos ns, bs nd cs cn be clculted s s = = (C2) n c b s s = c n n s s s ( D ) ( D ) n s (C3) = (C4) The pojection of D on whee is ( D ) bs b s (C5) ' = ( ( D )) b s (C6) = + ( ( D )) b b s b s s The simil wy to find [5, p9] nd we cn get ' = + ( ( D )) (C7) Two simil tingles the loction of Q. b f b f Q' nd Q' shown in ig. 7 cn be used to detemine β β Q ig. 7 Diffction Point Clcultion [5, p11] 37

44 If q epesents the function of the totl distnce fom to of Q, tht ' ' ' ) (1 ' ' ' q q = (C8) ' ' ' q + = (C9) Then the diffction point Q is found s Q = + q( ) (C10) 38

45 Appendix D: Diffcted ield Clcultion [5, p11] The unit vectos φ, β, φ nd β in ig.6 cn be detemined when we know (= s n = f n ) is the unit vecto in the diection of the edge, then c s = = φ (D1) φ β = (D2) c f = = φ (D3) β φ = (D4) The diffction ngles β, β, φ nd φ cn be clculted s = = Q Q ) ( ccos β β (D5) = n b n b s s ) ( tn 2 φ (D6) = n b n b f s ) ( tn 2 φ (D7) whee tn2 is the fou-qudnt ctngent [5, p11] The diffction pocess cn be descibed s [5, p11] jk D i i d d e A D = ), ( ˆ ˆ ) ( ˆ ) ( ˆ φ β φ β (D8) ) ( ˆ Q i β nd ) ( ˆ Q i φ e the soft nd hd polised components of the field t the point of diffction, which ) ( ˆ ) ( ˆ Q Q i i = β β (D9) ) ( ˆ ) ( ˆ Q Q i i = φ φ (D10) 39

46 o spheicl wve incidence, the divegence fcto A D (, ) [5, p12] is A D (, ) = (D11) ( + ) The diffction coefficient mtix D fo the y-fixed coodintion system is given by [5, p12] D = D 0 s 0 D h (D12) This mtix is digonl so the pllel nd pependicul components of the incident field e diffcted independently of ech othe. The totl diffcted field t, is ˆ d ( ) = ˆ ( ) + ˆ ( ) (D13) d β β d φ φ 40

47 Appendix : Deivtion of Pth Gin The field t the eceiving ntenn is The eceived open-cicuit voltge is ηp 2π ˆ t = tot z (1) ˆ V OC = hˆ ˆ (2) whee ĥ is the complex effective length of the eceiving ntenn R hˆ π = 2 z (3) k η ˆ 2 k πr η ηp 2π P R 2 t t VOC = z tot z = tot = tot 2 2 k 1 k P R t (4) 2π k = λ so P = V 2 λ 2 2 tot OC 2π 8R = 8R 2P R t nd ssume R nd R = 1 (5) P P t = λ 4π 2 tot 2 (6) so Pth Gin = 2 P λ 10 log10 ( ) = 10 log10 tot Pt 4π 2 db (7) 41

48 Appendix : The Rough loo Plne of the 8 th of chool of ngineeing 1.44m 6.77m 1.44m 1.45m Concete Block 6.77 m 1.5m 1.6m 6.5 m 1.6m ig. 15 The floo plne of the 8 th floo of chool of ngineeing. 42