ix. modeling propagation in residential...

Polytechnic University,Brooklyn, NY

©2002 by H.L. Bertoni 1

IX. Modeling Propagation inResidential Areas

•Characteristics of City Construction

•Propagation Over Rows of Buildings Outside the Core

•Macrocell Model for High Base Station Antennas

•Microcell Model for Low Base Station Antennas



Characteristics of City Construction

• High rise core surrounded by large areas of lowbuildings

• Street grid organizes the buildings into rows



High Core & Low Buildings in New York



High Core & Low Buildings in Chicago, IL



Rows of Houses in Levittown, LI - 1951



Rows of Houses in Boca Raton, FL - 1980’s



Rows in Highlands Ranch, CO - 1999



The EM City - Ashington, England



Rows of Houses in Queens, NY



Rectangular Street Geometry in Los Angeles, CA



Uniform Height Roofs in Copenhagen



Predicting Signal Characteristic for DifferentBuilding Environments

• Small area average signal strength– Low building environment: Replace rows of buildings by

long, uniform radio absorbers

– High rise environment: Site specific predictions accountingfor the shape and location of individual buildings

• Time delay and angle of arrival statistics– Site specific predictions using statistical distribution of

building shapes and locations



Summary of Characteristics of the UrbanEnvironment

• High rise core surrounded by large area having low buildings

• Outside of core, buildings are of more nearly equal height withoccasional high rise building– Near core; 4 - 6 story buildings, farther out; 1 - 4 story buildings

• Street grid organizes building into rows– Side-to-side spacing is small

– Front-to-front and back-to-back spacing are nearly equal (~50 m)

• Taylor prediction methods to environment, channel characteristic– Small area average power among low buildings found from simplified

geometry

– High rise environments and higher order channel statistics needs raytracing



Propagation Past Rows of LowBuildings of Uniform Height

• Propagation takes place over rooftops

• Separation of path loss into three factors

• Free space loss to rooftops near mobile

• Reduction of the rooftop fields due to diffraction past previous rows

• Diffraction of rooftop fields down to street level

• Find the reduction in the rooftop fields using:– Incident Plane wave for high base station antennas

– Incident cylindrical wave for low base station antennas



Three Factors Give Path Gain for PropagationOver Buildings

Path Gain

Free space path gain

Reduction in the field at the roof top just before the mobile due to

propagation past previous rows of buildings given by a factor

Diffraction of the roof top field down to the mobile (add ray power

to get the small area average)

PG PG PG PG

PGR

Q

PG Q

PGk k

= ( )( )( )

=

=

= −−

+ −

−

0 1 2

0

2

12

21 1 1

2 2

2 2 2

2

4

12

1 12 2

1 12

λπ

π ρ θ π θ π ρ θ π θ

Γ

αθ

dR

y

HBhS



Roof Top Fields Diffract Down to Mobile(First proposed by Ikegami)

hB

Because θ2 ∼ θ1/3 and |Γ|2 ~ 0.1,rays and have nearly equalamplitudes. Adding power isapproximately the same asdoubling the power of .

PGk k

H h H hH h y

PGH h

B m B mB m

B m

2

2

2 2 2

1 2 2

2 2 2

1 1 12

12

2

≈ −−

≈ =

=−

≈

−= −( ) +

=−

−

π ρ θ π θ π ρθλ

π ρθ

θρ ρ

ρ

λρπ

where

and sin

( )

θ



Comparison of Theory for Mobile AntennaHeight Gain with Measurements

Median value of measurements made at many locations for 200MHz signalsin Reading, England, whose nearly uniform height <HB>=12.5 m.



Summary of Propagation Over Low Buildings

• A heuristic argument has been made for separatingthe path gain into three factors– Free space path gain to the building before the mobile

– Reduction Q of the roof top fields due to diffractionpast previous rows of buildings

– Diffraction of the rooftop fields down to the mobile

• Diffraction of the rooftop gives the observedheight gain for the mobile antenna.



Computing Q for High Base Station Antennas

• Approximating the rows of buildings by a series ofdiffracting screens

• Finding the reduction factor using an incidentplane wave

• Settling behavior of the plane wave solution andits interpretation in terms of Fresnel zones

• Comparison with measurements



Approximations for Computing QEffect of previous rows on the field at top of last row of building before mobile

• External and internal walls of buildings reflect and scatter incident waves -waves propagate over the tops of buildings not through the buildings.

• Gaps between buildings are usually not aligned with path from base stationto mobile - replace individual buildings by connected row of of buildings.

• Variations in building height effect the shadow loss, but not the rangedependence - take all buildings to be the same height.

• Forward diffraction through small angles is approximately independent ofobject cross section - replace rows of buildings by absorbing screens.

• For high base station antenna and distances greater than 1 km, the effect ofthe buildings on spherical wave field is the same as for a plane wave - Q(α)found for incident plane wave.

• For short ranges and low antennas, the effect of buildings on spherical wavefield is the same as for a cylindrical wave - find QM for line source.



Method of Solution

• Physical Optical Approximations– Walfisch and Bertoni - IEEE/AP, 1988

Repeated numerical integration, Incident plane wave for α > 0.– Xia & Bertoni - IEEE/AP, 1992

Series expansion in Borsma functions, screens of uniform height, spacing.– Vogler - Radio Science, 1982

Long computation time limits method to 8 screens

– Saunders & Bonar - Elect. Letters, 1991

Modified Vogler Method

• Parabolic Method– Levy, Elect. Letters, 1992

• Ray Optics Approximations– Anderson - IEE- µwave, Ant., Prop., 1994; Slope Diffraction– Neve & Rowe - IEE µwave, Ant., Prop., 1994; UTD



Plane Wave Solution for High BaseStation Antennas

–Reduction of rooftop fields for a spherical wave incidenton the rows of buildings is the is the same as the reduction for an incident plane wave after many rows.

–Reduction is found from multiple forward diffractionpast an array of absorbing screens for a plane wave withunit amplitude that is incident at glancing the angle α.



Physical Optics Approximations for Reductionof the Rooftop Fields

I. Replace rows of buildings by parallel absorbing screens

II. For parallel screens, the reduction factor is found by repeated application of the Kirchhoff integral. Going from screen n to screen n+1, the integration is

H x y H x yjke

rdy dzn n n n n n

jkr

n

h

n

n

( , ) cos cos ( , )+ +

−∞

−∞

∞

= +( )∫∫1 1 4α δ

π

α δnρn

yn

x

n=1 n=2 n=3 n n+1

EH

Incidentwave

yn+1



Paraxial Approximation for RepeatedKirchhoff Integration

For small angles and , cos 2. Let

Then for integration over , +z

2 so that

(cos

n n n

nn2

n

n-

α δ α δ ρ

ρ ρρ

α δπ πρ

ρ

+ ≈ = −( ) + −( )

= + ≈

+ ≈

+ +

−

∞

∞ −

∫

cos

cos ) ( , ) ( , ) exp(

n n n n n n

n n n

n n n

jkr

n

jk

nn n

x x y y

z r z

H x yjke

rdz

jkeH x y

n

1

2

1

2

2 2

4 2−−

≈

=

−∞

∞

−−

+ +

∞ −

∫

∫

jkz dz

jkeH x y e

k

H x ye

H x ye

dy

n n n

jk

nn n

j n

n n

j

n nh

jk

n

n

n

n

n

2

4

1 1

4

2

22

ρ

πρπρ

λ ρ

ρπ

π ρ

)

( , )

( , ) ( , )

/

/

Thus



Paraxial Approximation forRepeated Kirchhoff Integration - cont.

For uniform building height and uniform row spacing

so that

and

h x x d

d y y dy y

d

H x ye e

dH x y jk y y d dy

H x y

n n n

n n nn n

n n

j jkd

n nh

n n n

N

n

= − =

= + −( ) ≈ +−

= − −[ ]

+

++

+ +

− ∞

+

+

∫

0

2

2

1

21

2 12

1 1

4

12

1

,

( )

( , ) ( , )exp ( ) /

( ,

/

ρ

λ

π

NN

j N jkNd

N N n nn

N

n n nj

n

N N

e e

ddy dy H d y j

k

dy y

v yjk

ddy e

ddv

H x y

+

− ∞ ∞

+=

−

+ +

= ••• − −( )

= =

∫ ∫ ∑1

4

2 10 0

1 1

2

1

4

1 1

2

2

)( )

( , )exp

;

( , )

/

/

π

π

λ

λπ

Let

then

== ••• − −( )

− ∞ ∞

+=

∫ ∫ ∑edv dv H d y v v

jkNd

N N n nn

N

π / ( , )exp2 10 0

1 1

2

1



Rooftop Field for Incident Plane Wave

H d y e e e e

H d y e e eq

jky

gd

yjk

d

H d y e e eq

g j

jkd jky jkd jky

jkd jky jkd q

q

p

jkd jky jkdp

( , )

( , )!( sin )

sin

( , )!

cos sin sin

sin

sin

1

1 10

1 1

1

1 1

1

1

1

2

12

= ≈

= =

= =

= = ( )

− −

− −

=

∞

− −

∑

α α α

α

α

α

αλ

ν

π

Use Taylor series expansion

Define and since , then

qqq

q

N N

N

jk N d

N N p

qq

qn n n

n

N

n

y

H xe

d dq

g j

ν

ν

πν ν π ν ν ν ν ν

10

1 1

1

1

2 10 0

10

12

12

2

0 0

01

2 2 2

=

∞

+ +

+

− + ∞ ∞

=

∞

+=

∑

∫ ∫ ∑ ∑

= =( )

= ••• ( )

− + −

Then the field at is

( , )!

exp( )

/==

−

+− +

=

∞

∑

∑

= ( )

1

1

11

0

01

2 1

1

N

Njk N d

p

q

N qq

N q

H x eq

g j I

I

( , )!

( ),

( )

( ),

,

π

where is a Borsma function defined in the next slide.



I d d d

q

IN q

NI

N q N Nq

n nn

N

nn

N

N q N q

, /

, ,

exp

( )( )

11

2 2

2

12 1

2 1 200 0

1 12

11

12

2

1 2

( ) = ••• − + −

≥

( ) =−

+

∞∞ ∞

+=

−

=

− −

∫∫ ∫ ∑ ∑π

ν ν ν ν ν ν ν ν

β β

Recursion relation for

ββπ

β

π

ββ

( ) ++

( )−

==

>

= =−

−

−

= −

−

=

−

∑

∑

1

2 1

11 0

0 0

11 2

11

2

1 2

1 2

1

1

1

1

0

0 10

1

( )

( )

( )( / )

!; ( )

( / )

!

( / )

,

,

, ,

N

I

N n

I q

q

IN

In N n

n q

n

N

q

NN

Nn

n

N

n

where

for

for

The term represents Pockhammer's Symbol for a ==

= = = + + −

1/2, where

( ) ; ( ) ; ( ) ( ) ( )a a a a a a a nn0 11 1 1L

Borsma Functions for β = 1



Field Incident on the N + 1 Edge for α = 0

1 3 N+12

E in

H in

x

y

Since

Amplitude decrease monotonically with

gd

H x e I

eN

eN

N

p

Njk N d

N

jk N d N

jk N d

= =

=

=

≈+

+− +

− +

− +

sin

( , ) ( )

( / )!

( ),

( )

( )

αλ

π

0

0 1

1 2

1

1

11

0

1

1



Field Incident on the N + 1 Edge for α ≠ 0

After initial variation, field settles to a constant value Q(gp) for N > N0

20 1;sin pp gN

dg ==

λα

N0

SettledFieldQ(gp)

Angles indicatedare ford =200λ

α = 2.0ο

α

1 2 …. n n+1

…



Explanation of the Settling Behavior in Terms of theFresnel Zone About the Ray Reaching the N+1 Edge

Only those edges that penetrate the Fresnel zone affect the field at the N +1 edge

Nd

gp0 2 2

1= =

λα

/

sin

α

d

n=1 n=3 n=5 n=N n=N +1

E1

n=2 n=4

H1

N0

n=N -1

αλ secNd

αtanNdN d N d0 0tan secα λ α=

Fresnel zone half width

W sF = λ



Settled Field Q(gp) and Analytic Approximations

0.01 0.02 0.05 0.1 0.2 0.5 1.00.03

0.05

0.1

0.2

0.5

1.0

1.5

Q

gp

Straight line approximation

for

where

0 015 0 4

010 03

0 9

. g .

Q(g ) .g

.

gd

p

pp

.

p

< <

≈

= sinαλ

Polynomial fit for g

Q(g ) g g g

p

p p p p

≤

= − +

1 0

3 502 3 327 0 9622 3

.

. . .



Path Gain/Loss for High Base Station Antenna

Comparison with measurements made in Philadelphia by AT&T

Q gg

g dh H

R

d

PGR

QH h R

h H

R

d

pp

pBS B

B m

BS B

( ) ≈

= ≈

−

=

( )

−

=

−

0 10 03

4 2 40 01

0 03

0 9

22

2 2

2 1 8

..

, sin /

( ).

.

.

.

α λλ

λπ

λρπ

λπ λ

−

=−( )

−

0 9

2 2

4

1 8 0 9

2

2 1

3 8

2

5 5132

.

. . .

.

( )

.( )

λρπ

πρ λ

H h

h H d

H h R

B m

BS B

B m



Comparison Between Hata Measurement Modeland the Walfisch-Ikegami Theoretical Model

For in MHz and in km

Assume m m

m

f R

L . dH h

h H f R

L f h h R

h H

h d

M k

B mBS B M k

M BS BS k

BS B

M

Theory :

Hata :

= − −−

− − + +

= + − + −

= =

= =

89 5 9 10 18 21 38

69 55 26 16 13 82 44 9 6 55

30 12

1 5 50

2log log( )

log( ) log log

. . log . log ( . . log )log

.

ρ

mm

If Theory dB

Hata dB

Theory :

Hata :

L f R

L f R

f L

R L

M k

M k

M

k

= + +

= + +

= =

= =

57 7 21 38

49 2 26 2 35 2

1 000 147 3

5 152 4

. log log

. . log . log

, ; .

; .



Comparison of Theory for Excess Path Losswith Measurements of Okumura, et al.

Path Loss

Excess Path Loss depeneds on and

only through the angle

= = − ( ) − −

= − = − − −

L R Q PG

L L Q PG R h HBS B

10 4 10 10

10 10

2 22

02

2

log log log

log log

λ π

α

f = 922 MHz



Walk About From Rooftop to Street Level

f h H

h d d H h

PGh H d

H h R R

BS B

m B m

BS B

B m

= = = =

= = = + − =

=−

−

= ≥ ×

450 2 3 20 7

1 5 50 2 25 6

5 5132

0 2181 6

2 2

4

1 8 0 9

2

2 1

3 8 3 8

MHz m m m

m m m

λ

ρ

πρ λ

/

. ( / ) ( ) .

. ( )( )

..

. . .

. . 1010

1 36 10 4 45 10

14

133 8 3

−

+ +≤ × = ×or mR . ..

R



Summary of Q for High Base Station Antennas

• Rows of buildings act as a series of diffractingscreens

• Forward diffraction reduces the rooftop field by afactor that approaches a constant past many rows

• The settling behavior can be understood in termsof Fresnel zones, and leads to the reduction factorQ, which depends on a single parameter gp

• Good comparison with measurements is obtainedusing a simple power expansion for Q



Cylindrical Wave Solution for LowBase Station Antennas

• Finding the reduction factor Q using an incidentcylindrical wave

• Q is shown to depend on parameter gc and thenumber of rows of buildings

• Comparison with measurements

• Mobile-to-mobile communications



Cylindrical Wave Solutions for Microcells UsingLow Base Station Antennas

Microcell coverage out to about 1 km involves propagationover a limited number of rows.

Must account for the number of rows covered, and hence forthe field variation in the plane perpendicular to the rows ofbuildings.

Therefore use a cylindrical incident wave with axis parallel tothe array of absorbing screens to find the field reduction due topropagation past rows of buildings.



Physical Optics Approximations for Reductionof the Rooftop Fields

I. Replace rows of buildings by parallel absorbing screens

II. For parallel screens, the reduction factor will apply for a spherical wave and for acylindrical wave. For 2D fields, Kirchhoff integration gives

H x y H x yjke

rdy dz

eH x y

edy

n n n n n n

jkr

n

h

n

j

n n

jk

nn

h

n n

n

n

n

( , ) cos cos ( , )

( , ) cos cos/

+ +

−∞

−∞

∞

−∞

= +( )

≈ + ≈

∫∫

∫

1 1

4

4

2

α δπ

λ ρα δ

π ρ

, since

α δnρn

yn

x

n=1 n=2 n=3 n n+1

EH

Incidentwave

yn+1



Paraxial Approximation for Repeated KirchhoffIntegration and Screens of Uniform Height

For uniform building height and uniform row spacing

h x x d

d y y dy y

d

H x ye

de dy dy H d y j

k

dy y

n n n

n n nn n

N N

j N

NjkNd

N n nn

= − =

= + −( ) ≈ +−

= ••• − −( )

+

++

+ +−

∞ ∞

+=

∫ ∫

0

2

2

1

21

2 12

1 1

4

2 10 0

1 1

2

,

( )

( , )( )

( , )exp/

ρ

λ

π

11

4

1 1 2 10 0

1 1

2

1

2

N

n n nj

n

N N

jkNd

N N n nn

N

v yjk

ddy e

ddv

H x ye

dv dv H d y v v

∑

∫ ∫ ∑

= =

= ••• − −( )

−

+ +

− ∞ ∞

+=

Let ;

( , ) ( , )exp

/

/

π λπ

π



Approximation for Cylindrical Wave of aLine Source

H d ye

d y y

dy y

d

H d ye e

de e

gy

dv y

jk

d

H d ye e

jk

jkd jky djky y d jky d

c

jkd jky

( , )

( , )

( , )

// /

11

12

1 0

2

1 0

2

1

22

01 1

1

1

02

0 1 12

02

2

2

=

= + −( )

≈ +−( )

≈

= =

≈

−

− −−

− −

ρ

ρ

ρ

ρ

λ

where

In exponent

Define and

Then

1

// 22 1 1

2d

g j

de ec πν ν−

1

y

x

y0

d

2 3 4 N N+1

d



Integral Representation for Field at the N+1 Edge

At the roof top of the + row of buildings N y

H xe e

ddv dv dv e v v v v

N N

N

jk N d jky d

N Ng j

n nn

N

nn

Nc

1 0 0

0 2 2 2

1 1

1

1 2

2 1 200 0

212

11

12

2

02

1

+ +

+

− +( ) − ∞∞ ∞

+=

−

=

= =( )

= ••• − + −

∫∫ ∫ ∑ ∑

ν

ππν( , ) exp

( )

/

= ( )

= ( )

=

∞

+

− +( ) −

=

∞

∑

∑

Use Taylor series expansion

Then

where are Borsma functions

eq

g j v

H xe e

d qg j I

I

g j vc

qq

q

N

jk N d jky d

c

q

N qq

N q

c21

0

1

1 2

0

1

02

12

01

2 2

2

π π

π

!

( , )!

( )

( )

( )

,

,



I d d d

q

IN q

NI

N q N Nq

n nn

N

nn

N

N q N q

, /

, ,

exp

( )( )

21

2 2 2

2

12 1

2 1 200 0

1 12

11

12

2

1

( ) = ••• − + −

≥

( ) =−

+

∞∞ ∞

+=

−

=

− −

∫∫ ∫ ∑ ∑π

ν ν ν ν ν ν ν ν

β β

Recursion relation for

22 1

1

1

1

0 32

1 23

321

1

2 1

21

12

1

4

1

1

βπ

β

π

ββ

( ) ++

( )−

=+( )

=+ −( )

−

−

= −

−

=

∑

∑

( )

( ) ; ( )

,

, ,

N

I

N n

IN

In N n

n q

n

N

N Nn

N

where

Borsma Functions for Line Source Field



Rooftop Field Reduction Factor for LowBase Station Antenna

Reduction factor found from cylindrical wave field

where

In terms of Boersma functions

For and

Q gH x

eN d y N d

Q g Nq

g j I

y gy

d

Q g N

N cN

jk

N c cq

q

N q

c

N c

++

−

+=

∞

+

= = +[ ] + ≈ +

= + ( )

= = =

= +

∑

11 2

02

10

00

1

01 1

11

2 2

0 0

( )( , )

/( ) ( )

( )!

( )

,

( )

,

ρ ρρ

π

λ

1111

113 2( ) /N N+

=+



Field Reduction Past Rows of Buildings

Field after multiple diffraction over absorbing screens. Values of y0 are for a frequency of 900MHz and d=50 m.

λ=−= dygHhy cBBS 00 ,

Number of Screens M = N+11 10 100

10

1

0.1

0.01

0.001

0.0001

QM

y0= +11.25m

y0= –11.25m

y0= 0m

1/M



sQ Q

M M

Q Q

M MM M M M= −

−+ −

−( )

+( )[ ]+ +log log

log( ) log

log

log1 1

1 1

Slope of field H(M) vs. Number of Screensfor different Tx heights at 1800MHz

0 10 20 30 40 50 60 70 80 90 1000.5

0.6

0.7

0.8

0.9

1

1.1

1.2

Number of ScreensM

Slop

e of

Fie

ld, s

y0 < 0

y0 > 0



Modifications for Propagation Oblique to theStreet Grid

Base Station

φ

x

R

x=0

mobile

Radio propagation with oblique incidencex = Md + d/2

PGkr

g ydc2

2

0101 1 1

2= −

+

=⊥ ⊥ ⊥

logcos

cosπ φ θ π θ

φλ



Comparison of Base Station Height Gain withHar/Xia Measurement Model

f

d

y h H

R

NR

d

NR

d

BS B

=

=

= −

=

=

+ = =

= °

+ = =

1 8

50

1

0

1 20

60

1 10

0

.

/cos

GHz

m

km

For perpendicular propagation

For oblique propagation

φ

φ

φ-8 -6 -4 -2 0 2 4 6 8

-50

-45

-40

-35

-30

-25

-20

-15

-10

-5

0

Q in

dB

y0

Q10

Q20

Qexp



Experimentally Based Expression for Qexp

We can compare the theoretical with the Har/Xia measurements using

The Har/Xia formulas for path loss on staircase and transverse paths give

, so that

Substituing their expression for gives

Q

L PG PG Q PG

L

Q L PG PG

L

Q f

f y y

y y R

exp

exp G

G

= − = − − −

= − − −

= − +[ ]{− −[ ] +( )+ − +( )[ ]

0 2

0 2

0 0

0 0

20

20

20 138 3 38 9

13 7 4 6 1

40 1 4 4 1

log

log

log . . log

. . log sgn( )log

. . sgn( )log log kk

k B mR H h

}

−×

−

−

10

4 1010

23

2

2 2log log( )

λπ

λρπ



Comparison of Range Index n with Har/XiaMeasurement Model

km

m

GHz

2/1

50

8.1

=

=

=

R

d

f

n=2+2s

-8 -6 -4 -2 0 2 4 6 83.4

3.5

3.6

3.7

3.8

3.9

4

4.1

4.2

4.3

4.4

n

y0

theoryexp



Q for Mobile to Mobile Communications

θ0 θ1

ρ1

ρ0

h0 h1Rn=1 2 M

HB

Peak of first building acts as line source of strength

Propagation past remaining peaks gives factor 1/(M-1)Effective reduction factor

D

QD

Me

θ ρ

θ

ρ

0 0

0

0 1

( )

=( )

−( )



Comparison of Q Factors for Plane Waves,Cylindrical Waves and Mobile-to-Mobile

λ

λ

αλ λ

λ

λ

λ

= = =

=

= =−

= =

>

< −

1 3 50 20

1 1

0

0

0

0

/

sin

m, m,

Use plane wave factor for

use mobile - to - mobile factor for

d M

g y d

gd h H

Md

d

M

y

d Mg

y d

y d

c

pBS B

c

-15 -10 -5 0 5 10 15-60

-50

-40

-30

-20

-10

0

Q (

dB)

y0

Q20(gc)

Q (gp)

Qe

dλ− dλ



Path Loss for Mobile-to-Mobile Communication

LR

QD

R Md

L dM MD D

D

k k d

d

H h

e

B m

= −

− −

=

= −

+ −[ ]− −

≈ ≈−

204

20 10

204

20 1 10 10

1

2

1 1

2 2

2

1

2

1

0

2

0

1

2

1

2

2

log log log

log log ( ) log log

( / )

/

λπ ρ

λπ ρ ρ

ρ π ρ θ π

Since

If both mobile are at same height and in the middle of the street, using22

2 2

3

8

20 16 20 1 40

=−

= + −[ ] +−

d

H h

L M MH h

B m

B m

λπ

πλ

( )

log( ) log ( ) log

gives



θ0 θ1

ρ1

ρ0

h0 h1Rn=1 2 M

HB

PG M M H h

H h

PG M M

M M M M M

d

dB B m

B m

dB

≈ − ( ) − −[ ]− −[ ]= = =

≈ − − −[ ]− > −

−[ ] < − < <

=

20 16 20 1 40

2 3 10 2

53 9 20 1 43 2 138

20 1 40 9 1 111 11

50

3log log ( ) log ( )

/

. log ( ) .

log ( ) . ( )

π λ

λFor m, m, m

Thus or or

For m, RR Md= = 550 m = 0.55 km

Walk About Range for Low Buildings



Summary of Solution for Low BaseStation Antennas

• Reduction factor found using an incidentcylindrical wave

• QM depends on parameter gc and the number ofrows of buildings M over which the signal passes

• Theory gives the correct trends for base stationheight gain and slope index, but is pessimistic forantennas below the rooftops

• Theory give simple expressions for path gain inthe case of Mobile-to-mobile communications

ix. modeling propagation in residential...

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