IX. Modeling Propagation in Residential Areas

Transcription

1 IX. Modelig Propagatio i Residetial Areas Characteristics of City Costructio Propagatio Over Rows of Buildigs Outside the Core Macrocell Model for High Base Statio Ateas Microcell Model for Low Base Statio Ateas 00 by H.L. Bertoi

2 Characteristics of City Costructio High rise core surrouded by large areas of low buildigs Street grid orgaizes the buildigs ito rows 00 by H.L. Bertoi

3 High Core & Low Buildigs i New York 00 by H.L. Bertoi 3

4 High Core & Low Buildigs i Chicago, IL 00 by H.L. Bertoi 4

5 Rows of Houses i Levittow, LI by H.L. Bertoi 5

6 Rows of Houses i Boca Rato, FL s 00 by H.L. Bertoi 6

7 Rows i Highlads Rach, CO by H.L. Bertoi 7

8 The EM City - Ashigto, Eglad 00 by H.L. Bertoi 8

9 Rows of Houses i Quees, NY 00 by H.L. Bertoi 9

10 Rectagular Street Geometry i Los Ageles, CA 00 by H.L. Bertoi 0

11 Uiform Height Roofs i Copehage 00 by H.L. Bertoi

12 Predictig Sigal Characteristic for Differet Buildig Eviromets Small area average sigal stregth Low buildig eviromet: Replace rows of buildigs by log, uiform radio absorbers High rise eviromet: Site specific predictios accoutig for the shape ad locatio of idividual buildigs Time delay ad agle of arrival statistics Site specific predictios usig statistical distributio of buildig shapes ad locatios 00 by H.L. Bertoi

13 Summary of Characteristics of the Urba Eviromet High rise core surrouded by large area havig low buildigs Outside of core, buildigs are of more early equal height with occasioal high rise buildig Near core; 4-6 story buildigs, farther out; - 4 story buildigs Street grid orgaizes buildig ito rows Side-to-side spacig is small Frot-to-frot ad back-to-back spacig are early equal (~50 m) Taylor predictio methods to eviromet, chael characteristic Small area average power amog low buildigs foud from simplified geometry High rise eviromets ad higher order chael statistics eeds ray tracig 00 by H.L. Bertoi 3

14 Propagatio Past Rows of Low Buildigs of Uiform Height Propagatio takes place over rooftops Separatio of path loss ito three factors Free space loss to rooftops ear mobile Reductio of the rooftop fields due to diffractio past previous rows Diffractio of rooftop fields dow to street level Fid the reductio i the rooftop fields usig: Icidet Plae wave for high base statio ateas Icidet cylidrical wave for low base statio ateas 00 by H.L. Bertoi 4

15 Three Factors Give Path Gai for Propagatio Over Buildigs Path Gai PG PG PG PG Free space path gai PG 0 = ( )( )( ) = λ 4πR 0 h S α d R Reductio i the field at the roof top just before the mobile due to propagatio past previous rows of buildigs give by a factor Q y θ H B PG = Q Diffractio of the roof top field dow to the mobile (add ray power to get the small area average) PG = k + k Γ πρ θ π θ πρ θ π θ 00 by H.L. Bertoi 5

16 Roof Top Fields Diffract Dow to Mobile (First proposed by Ikegami) θ Because θ θ /3 ad Γ ~ 0., rays ad have early equal h B amplitudes. Addig power is approximately the same as doublig the power of. PG PG where k k = λ πρ θ π θ πρθ πρθ H h H h θ = si ρ ρ λρ = π ( H h ) B m ad ( ) + ρ = H h y B m B m B m 00 by H.L. Bertoi 6

17 Compariso of Theory for Mobile Atea Height Gai with Measuremets Media value of measuremets made at may locatios for 00MHz sigals i Readig, Eglad, whose early uiform height <H B >=.5 m. 00 by H.L. Bertoi 7

18 Summary of Propagatio Over Low Buildigs A heuristic argumet has bee made for separatig the path gai ito three factors Free space path gai to the buildig before the mobile Reductio Q of the roof top fields due to diffractio past previous rows of buildigs Diffractio of the rooftop fields dow to the mobile Diffractio of the rooftop gives the observed height gai for the mobile atea. 00 by H.L. Bertoi 8

19 Computig Q for High Base Statio Ateas Approximatig the rows of buildigs by a series of diffractig screes Fidig the reductio factor usig a icidet plae wave Settlig behavior of the plae wave solutio ad its iterpretatio i terms of Fresel zoes Compariso with measuremets 00 by H.L. Bertoi 9

20 Approximatios for Computig Q Effect of previous rows o the field at top of last row of buildig before mobile Exteral ad iteral walls of buildigs reflect ad scatter icidet waves - waves propagate over the tops of buildigs ot through the buildigs. Gaps betwee buildigs are usually ot aliged with path from base statio to mobile - replace idividual buildigs by coected row of of buildigs. Variatios i buildig height effect the shadow loss, but ot the rage depedece - take all buildigs to be the same height. Forward diffractio through small agles is approximately idepedet of object cross sectio - replace rows of buildigs by absorbig screes. For high base statio atea ad distaces greater tha km, the effect of the buildigs o spherical wave field is the same as for a plae wave - Q(α) foud for icidet plae wave. For short rages ad low ateas, the effect of buildigs o spherical wave field is the same as for a cylidrical wave - fid Q M for lie source. 00 by H.L. Bertoi 0

21 Method of Solutio Physical Optical Approximatios Walfisch ad Bertoi - IEEE/AP, 988 Repeated umerical itegratio, Icidet plae wave for α > 0. Xia & Bertoi - IEEE/AP, 99 Series expasio i Borsma fuctios, screes of uiform height, spacig. Vogler - Radio Sciece, 98 Log computatio time limits method to 8 screes Sauders & Boar - Elect. Letters, 99 Modified Vogler Method Parabolic Method Levy, Elect. Letters, 99 Ray Optics Approximatios Aderso - IEE- µwave, At., Prop., 994; Slope Diffractio Neve & Rowe - IEE µwave, At., Prop., 994; UTD 00 by H.L. Bertoi

22 Plae Wave Solutio for High Base Statio Ateas Reductio of rooftop fields for a spherical wave icidet o the rows of buildigs is the is the same as the reductio for a icidet plae wave after may rows. Reductio is foud from multiple forward diffractio past a array of absorbig screes for a plae wave with uit amplitude that is icidet at glacig the agle α. 00 by H.L. Bertoi

23 Physical Optics Approximatios for Reductio of the Rooftop Fields I. Replace rows of buildigs by parallel absorbig screes H α E y δ y + Icidet wave ρ x = = =3 + II. For parallel screes, the reductio factor is foud by repeated applicatio of the Kirchhoff itegral. Goig from scree to scree +, the itegratio is ( ) Hx (, y ) = cosα + cos δ Hx (, y) + + h jke jkr 4πr dy dz 00 by H.L. Bertoi 3

24 Paraxial Approximatio for Repeated Kirchhoff Itegratio For small agles α ad δ, cosα + cosδ. Let ρ = x x y y The for itegratio over, + z z r= ρ + z ρ ρ so that ( ) + ( ) (cosα jkr jkρ jke + cos δ) H( x, y) (, ) exp( πr dz jke H x y jkz 4 πρ Thus jπ / 4 e e H( x+, y+ ) = H( x, y) λ h jkρ ρ jke πρ dy jkρ H( x, y ) e jπ / 4 πρ k ρ ) dz 00 by H.L. Bertoi 4

25 Paraxial Approximatio for Repeated Kirchhoff Itegratio - cot. For uiform buildig height h = 0, ad uiform row spacig x x = d ( y+ y) ρ = d + ( y+ y) d+ so that d jπ / 4 jkd e e H( x+, y+ ) = H( x, y)exp [ jk( y + y ) / d] dy λd ad H( x, y Let v the N + N + h + jnπ 4 jknd N e e ) = dy dy H( d, y )exp j k y y ( λd) d jk = y ; dy = e d N / N = jπ / 4 λd dv π jknd N e H( xn+, yn+ ) = dv dv H( d, y )exp v + v π / 0 0 = ( ) N N ( ) 00 by H.L. Bertoi 5

26 Rooftop Field for Icidet Plae Wave Hdy (, ) = e e e e jkd cosα jky si α jkd jky si α Use Taylor series expasio si α Hdy (, ) = e e = e q! ( jky si α ) jkd jky jkd q q= 0 d jk Defie gp = siα ad sice ν = y, the λ d jkd jky si jkd q α Hdy (, ) = e e = e ( q gp jπ ) ν q! N + N+ N+ q= 0 The the field at y = 0 ν = 0 is H( x, 0) H( x, 0) N + ( ) jk( N + ) d N N e q q = dν dν N N gp jπ ν ν ν ν ν ( ) exp + + / π 0 0 q= 0 q! = = q jk( N + ) d = e ( p π ) q= 0 q! g j I (), where I () is a Borsma fuctio defied i the ext slide. Nq, Nq, 00 by H.L. Bertoi 6

27 N N q INq, ()= d d d N / N exp ν ν ν ν ν + ν + ν ν π = = Recursio relatio for q I Nq ( ) ( β)= β ( N + ) Nq, Nq, where I I 0, q Borsma Fuctios for β = for q = 0 () = 0 for q > 0 I ( β)+ (/) N () = ; IN ( ) = N! π N, 0, π ( N + ) N = 0 (/)! N β N = β q, The term (/) represets Pockhammer's Symbol for a = /, where I ( β) N ( a) = ;( a) = a;( a) = a( a+ ) L( a+ ) 0 00 by H.L. Bertoi 7

28 Field Icidet o the N + Edge for α = 0 y Sice g p d = siα = 0 λ E i x jk( N + ) d H( x, 0) = e I ( ) N + = e e N, 0 (/) N! jk( N + ) d N jk( N + ) d πn + H i 3 N+ Amplitude decrease mootoically with N 00 by H.L. Bertoi 8

29 Field Icidet o the N + Edge for α 0 After iitial variatio, field settles to a costat value Q(g p ) for N > N 0 α =.0 ο Agles idicated are for d =00λ Settled Field Q(g p ) N 0 α d g p = siα ; N = λ 0 g p by H.L. Bertoi 9

30 Explaatio of the Settlig Behavior i Terms of the Fresel Zoe About the Ray Reachig the N+ Edge Ndλ secα E Fresel zoe half width Nd taα H Ndtaα = Ndλsecα 0 0 α W F = λ s N λ / d = = si α 0 g p d = = =3 =4 =5 N 0 =N - =N =N + Oly those edges that peetrate the Fresel zoe affect the field at the N + edge 00 by H.L. Bertoi 30

31 Settled Field Q(g p ) ad Aalytic Approximatios Q Straight lie approximatio for < g < gp Q(g p) where g p d = siα λ p Polyomial fit for g g p Q(g ) = 3. 50g 3. 37g g p 0. 3 p p p p 00 by H.L. Bertoi 3

32 Path Gai/Loss for High Base Statio Atea ( p) Qg 09. g p h 0., gp = si α d/ λ 003. PG = λ R Q ( ) = H h λρ λ 4π π ( ) 4πR = ( ). 55. hbs HB ρd λ 4. 3π ( HB hm) R B m 38 BS H R B d λ hbs H R 8 B. d λ 09. λρ π ( HB hm) Compariso with measuremets made i Philadelphia by AT&T 00 by H.L. Bertoi 3

33 Compariso Betwee Hata Measuremet Model ad the Walfisch-Ikegami Theoretical Model For fm i MHz ad Rk i km Theory : ρ L = logd 0log 8log( h H ) + log f + 38logR ( H h ) Hata : L= log f 3. 8log h + ( log h )logr Assume hbs = 30m HB = m hm = 5. m d = 50m Theory : L= log fm + 38logRk Hata : L= log f logr B m BS B M k M BS BS k M k If f =, 000; Theory L= db R M k = 5; Hata L= 5. 4 db 00 by H.L. Bertoi 33

34 Compariso of Theory for Excess Path Loss with Measuremets of Okumura, et al. Path Loss = = ( λ π ) L 0log 4 R 0logQ 0logPG Excess Path Loss = L L0 = 0logQ 0logPG depeeds o Rad hbs H oly through the agle α B f = 9 MHz 00 by H.L. Bertoi 34

35 Walk About From Rooftop to Street Level f = 450 MHz λ = / 3 m h = 0 m H = 7 m h = 5. m d = 50m ρ = ( d/ ) + ( H h ) = 56. m m B m PG = hbs HB d. 55. ( ) ρ λ 0. 8 = π ( H h ) R R B or R = m m BS B 4 R 00 by H.L. Bertoi 35

36 Summary of Q for High Base Statio Ateas Rows of buildigs act as a series of diffractig screes Forward diffractio reduces the rooftop field by a factor that approaches a costat past may rows The settlig behavior ca be uderstood i terms of Fresel zoes, ad leads to the reductio factor Q, which depeds o a sigle parameter g p Good compariso with measuremets is obtaied usig a simple power expasio for Q 00 by H.L. Bertoi 36

37 Cylidrical Wave Solutio for Low Base Statio Ateas Fidig the reductio factor Q usig a icidet cylidrical wave Q is show to deped o parameter g c ad the umber of rows of buildigs Compariso with measuremets Mobile-to-mobile commuicatios 00 by H.L. Bertoi 37

38 Cylidrical Wave Solutios for Microcells Usig Low Base Statio Ateas Microcell coverage out to about km ivolves propagatio over a limited umber of rows. Must accout for the umber of rows covered, ad hece for the field variatio i the plae perpedicular to the rows of buildigs. Therefore use a cylidrical icidet wave with axis parallel to the array of absorbig screes to fid the field reductio due to propagatio past rows of buildigs. 00 by H.L. Bertoi 38

39 Physical Optics Approximatios for Reductio of the Rooftop Fields I. Replace rows of buildigs by parallel absorbig screes H α E y δ y + Icidet wave ρ x II. For parallel screes, the reductio factor will apply for a spherical wave ad for a cylidrical wave. For D fields, Kirchhoff itegratio gives jke Hx ( +, y+ ) = ( cosα + cos δ) Hx (, y) 4πr = = =3 + h h jkr dy dz jπ / 4 jkρ e e Hx (, y) dy, sice cosα + cosδ λ ρ 00 by H.L. Bertoi 39

40 Paraxial Approximatio for Repeated Kirchhoff Itegratio ad Screes of Uiform Height For uiform buildig height h = 0, ad uiform row spacig x x = d + ( y+ y) ρ = d + ( y+ y) d+ d jnπ 4 e H x y e dy dy H d y j k N jknd ( N+, N+ ) = y y N (, )exp / + ( λd) 0 0 d = ( ) N jk jπ / 4 λd Let v = y ; dy = e dv d π jknd N e H( x, y ) = dv dv H( d, y )exp v v ( ) N+ N+ N / N + π 0 0 = 00 by H.L. Bertoi 40

41 Approximatio for Cylidrical Wave of a Lie Source e Hdy (, ) = jkρ ρ ( ) where ρ = d + y y I expoet ρ d + 0 ( y y0) d y 0 y x e Hdy (, ) y0 Defie gc = ad v = y λd The e jkd jky / d d 0 0 e e jky y / d jky / d jk d d d 3 4 N N+ Hdy (, ) e jkd e jky d 0 /d e e g j c πν ν 00 by H.L. Bertoi 4

42 Itegral Represetatio for Field at the N+ Edge ( ) At the roof top of the N + row of buildigs y = 0 ν = 0 N+ N+ jk( N + ) d jky0 e e ( d ) N N gc jπν H( xn +, 0) = dv dv dv e v v v v N N exp + + / π d = = Use Taylor series expasio e g c The jπ v H( x, 0) N + = ( c π ) q= 0 q! g j v ( ) jk N + d jky0 e e ( d ) = ( c π ) d q q q= 0 q g j INq, () q! where I () are Borsma fuctios Nq, 00 by H.L. Bertoi 4

43 Borsma Fuctios for Lie Source Field N N q INq, ()= d d d N / N exp ν ν ν ν ν + ν + ν ν π = = Recursio relatio for q I Nq ( ) ( β)= β ( N + ) I ( β)+ Nq, Nq, β π ( N + ) N = β I ( β) N q, where I () = ; I () = N + ( ) N, 0 3 N, 3 4 π 3 = N ( N + ) 00 by H.L. Bertoi 43

44 Rooftop Field Reductio Factor for Low Base Statio Atea Reductio factor foud from cylidrical wave field Q N+ H( xn +, 0) ( gc) = where ρ = [( N + ) d] + y N + d jk 0 ( ) ρ e / ρ I terms of Boersma fuctios q N+ c = + ( c ) q= 0 Q ( g ) N y0 For y0 = 0, gc = = 0 ad λd QN+ ( gc) = N + ( N + ) / g jπ INq, () q! 3 = N + 00 by H.L. Bertoi 44

45 Field Reductio Past Rows of Buildigs 0 y 0 = 0m y 0 = +.5m 0. Q M 0.0 /M 0.00 y 0 =.5m Number of Screes M = N+ y = h H g = y dλ 0 BS B, c 0 Field after multiple diffractio over absorbig screes. Values of y 0 are for a frequecy of 900MHz ad d=50 m. 00 by H.L. Bertoi 45

46 Slope of field H(M) vs. Number of Screes for differet Tx heights at 800MHz. Slope of Field, s y 0 < 0 y 0 > Number of Screes M logq Q Q Q M + log log M ( M + M) s = log( M + ) log M log M + M [( ) ] 00 by H.L. Bertoi 46

47 Modificatios for Propagatio Oblique to the Street Grid Base Statio φ R mobile x=0 x = Md + d/ Radio propagatio with oblique icidece x PG = 0log πkr cosφ θ π + θ g c = y 0 cosφ λd 00 by H.L. Bertoi 47

48 Compariso of Base Statio Height Gai with Har/Xia Measuremet Model f = 8. GHz d = 50 m y = h H 0 BS R = km B -5 Q i db Q 0 Q exp For perpedicular propagatio φ = 0 R N + = = 0 d Q y 0 For oblique propagatio φ = 60 R N + = = 0 d /cosφ 00 by H.L. Bertoi 48

49 Experimetally Based Expressio for Q exp We ca compare the theoretical Q with the Har/Xia measuremets usig L= PG= PG 0logQ PG The Har/Xia formulas for path loss o staircase ad trasverse paths give L, so that 0logQ = L PG PG exp Substituig their expressio for L gives 0log Q = log f exp { [ G] [ ] log f sg( y )log y G 0 ( 0) ( ) [ ] sg( y0)log + y0 logr k λ λρ 0log 0log 4πR 0 π ( H h ) 3 k B m } 00 by H.L. Bertoi 49

50 Compariso of Rage Idex with Har/Xia Measuremet Model =+s theory exp f =.8 GHz d = 50 m R = / km y 0 00 by H.L. Bertoi 50

51 Q for Mobile to Mobile Commuicatios ρ 0 θ 0 θ ρ H B = M h 0 R h Peak of first buildig acts as lie source of stregth ( ) D θ ρ 0 0 Propagatio past remaiig peaks gives factor /(M-) Effective reductio factor Q e = ( ) D θ 0 ρ ( M ) 0 00 by H.L. Bertoi 5

52 Compariso of Q Factors for Plae Waves, Cylidrical Waves ad Mobile-to-Mobile 0 λ = 3 / m, d = 50m, M = 0-0 g = y λd c 0 Q (db) Q 0 (g c ) Q (g p ) g p d hbs HB d = siα = λ Md λ y0 = = M λd M g c λd Q e y 0 λd Use plae wave factor for y 0 0 > < λd use mobile - to - mobile factor for y λd 00 by H.L. Bertoi 5

53 Path Loss for Mobile-to-Mobile Commuicatio λ L = 0log 0logQe 0log 4πR Sice R = Md L λ D0 D = 0log + 0log [ dm( M ) ] 0log 0log 4π ρ ρ If both mobile are at same height ad i the middle of the street, usig D d / ρ πkρ θ πk( d/ ) H h gives B m D ρ 3 = [ ]+ 40 H L log( π ) log M( M ) log 0 dλ = 8π ( H h ) B m B h λ m 00 by H.L. Bertoi 53

54 Walk About Rage for Low Buildigs ρ 0 θ 0 θ ρ H B = M h 0 R h [ ] 3 PGdB 0log( 6π ) 0log[ M( M ) ] 40log ( HB hm) λ For λ = / 3 m, HB = 0 m, hm = m PGdB log [ M( M ) ] 43. > 38 Thus 0log [ MM ( ) ]< or MM ( ) < or M< For d = 50 m, R= Md =550 m = 0.55 km 00 by H.L. Bertoi 54

55 Summary of Solutio for Low Base Statio Ateas Reductio factor foud usig a icidet cylidrical wave Q M depeds o parameter g c ad the umber of rows of buildigs M over which the sigal passes Theory gives the correct treds for base statio height gai ad slope idex, but is pessimistic for ateas below the rooftops Theory give simple expressios for path gai i the case of Mobile-to-mobile commuicatios 00 by H.L. Bertoi 55