•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
Polytechnic University,
Brooklyn, NY
©2002 by H.L. Bertoni 1

Characteristics of City Construction
• High rise core surrounded by large areas of low buildings
• Street grid organizes the buildings into rows
Polytechnic University,
Brooklyn, NY
©2002 by H.L. Bertoni 2

High Core & Low Buildings in New York
Polytechnic University,
Brooklyn, NY
©2002 by H.L. Bertoni 3

High Core & Low Buildings in Chicago, IL
Polytechnic University,
Brooklyn, NY
©2002 by H.L. Bertoni 4

Rows of Houses in Levittown, LI - 1951
Polytechnic University,
Brooklyn, NY
©2002 by H.L. Bertoni 5

Rows of Houses in Boca Raton, FL - 1980’s
Polytechnic University,
Brooklyn, NY
©2002 by H.L. Bertoni 6

Rows in Highlands Ranch, CO - 1999
Polytechnic University,
Brooklyn, NY
©2002 by H.L. Bertoni 7

The EM City - Ashington, England
Polytechnic University,
Brooklyn, NY
©2002 by H.L. Bertoni 8

Rows of Houses in Queens, NY
Polytechnic University,
Brooklyn, NY
©2002 by H.L. Bertoni 9

Rectangular Street Geometry in Los Angeles, CA
Polytechnic University,
Brooklyn, NY
©2002 by H.L. Bertoni 10

Uniform Height Roofs in Copenhagen
Polytechnic University,
Brooklyn, NY
©2002 by H.L. Bertoni 11

Predicting Signal Characteristic for Different
Building Environments
• Small area average signal strength
– Low building environment: Replace rows of buildings by long, uniform radio absorbers
– High rise environment: Site specific predictions accounting for 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
Polytechnic University,
Brooklyn, NY
©2002 by H.L. Bertoni 12

Summary of Characteristics of the Urban
Environment
• High rise core surrounded by large area having low buildings
• Outside of core, buildings are of more nearly equal height with occasional 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 ray tracing
Polytechnic University,
Brooklyn, NY
©2002 by H.L. Bertoni 13

• 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
Polytechnic University,
Brooklyn, NY
©2002 by H.L. Bertoni 14


Three Factors Give Path Gain for Propagation
Over Buildings a
Path Gain
PG
 
PG
0

PG
1

PG
2
 h
S
Free space path gain
PG
0



4


R


2
R d
Reduction in the field at the roof top just before the mobile due to
propagation past previous rows of buildings given by a factor Q
PG
1

Q
2
Diffraction of the roof top field down to the mobile (add ray power
to get the small area average)
PG
2


1

 2
 k

1
1
 
1

1
2
  
1
 2


 2
2
 k

2
1
 
2

1
2
  
2
 2




 y

Polytechnic University,
Brooklyn, NY
©2002 by H.L. Bertoni

15
H
B


Roof Top Fields Diffract Down to Mobile
(First proposed by Ikegami)


 h
B
Because 

  and || 2 ~ 0.1, rays  and  have nearly equal amplitudes. Adding power is approximately the same as doubling the power of  .
PG
2


1


 k



1


1
2
  
 2




1
 k
 2


2
 2  2 where
  sin

1


H
B

 h m



H
B

 h m
and
  
H
B
 h m
 2  y
2
PG
2


 2

2 ( H

B
 h m
) 2


Polytechnic University,
Brooklyn, NY
©2002 by H.L. Bertoni 16

Comparison of Theory for Mobile Antenna
Height Gain with Measurements
Median value of measurements made at many locations for 200MHz signals in Reading, England, whose nearly uniform height  H
B
 =12.5 m.
Polytechnic University, ©2002 by H.L. Bertoni
Brooklyn, NY
17

Summary of Propagation Over Low Buildings
• A heuristic argument has been made for separating the path gain into three factors
– Free space path gain to the building before the mobile
– Reduction Q of the roof top fields due to diffraction past previous rows of buildings
– Diffraction of the rooftop fields down to the mobile
• Diffraction of the rooftop gives the observed height gain for the mobile antenna.
Polytechnic University,
Brooklyn, NY
©2002 by H.L. Bertoni 18

Computing Q for High Base Station Antennas
• Approximating the rows of buildings by a series of diffracting screens
• Finding the reduction factor using an incident plane wave
• Settling behavior of the plane wave solution and its interpretation in terms of Fresnel zones
• Comparison with measurements
Polytechnic University,
Brooklyn, NY
©2002 by H.L. Bertoni 19

Approximations for Computing Q
Effect 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 station to mobile - replace individual buildings by connected row of of buildings.
• Variations in building height effect the shadow loss, but not the range dependence - take all buildings to be the same height.
• Forward diffraction through small angles is approximately independent of object cross section - replace rows of buildings by absorbing screens.
• For high base station antenna and distances greater than 1 km, the effect of the buildings on spherical wave field is the same as for a plane wave Q ( a
) found for incident plane wave.
• For short ranges and low antennas, the effect of buildings on spherical wave field is the same as for a cylindrical wave - find
Q
M for line source.
Polytechnic University,
Brooklyn, NY
©2002 by H.L. Bertoni 20

Method of Solution
• Physical Optical Approximations
– Walfisch and Bertoni - IEEE/AP, 1988
Repeated numerical integration, Incident plane wave for a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 - IEEm wave, Ant., Prop., 1994; Slope Diffraction
– Neve & Rowe - IEE m wave, Ant., Prop., 1994; UTD
Polytechnic University,
Brooklyn, NY
©2002 by H.L. Bertoni 21

Plane Wave Solution for High Base
Station Antennas
–Reduction of rooftop fields for a spherical wave incident on 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 diffraction past an array of absorbing screens for a plane wave with unit amplitude that is incident at glancing the angle a.
Polytechnic University,
Brooklyn, NY
©2002 by H.L. Bertoni 22



Physical Optics Approximations for Reduction of the Rooftop Fields
I. Replace rows of buildings by parallel absorbing screens
E
H a
Incident wave y n
 n
 n y n+ 1 x n= 1 n= 2 n= 3 n n +1
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 n

1
, y n

1


 h n


)
  cos a n
 cos
 n

H ( x n
, y n
) jke
 jkr
4
 r dy n dz n
Polytechnic University,
Brooklyn, NY
©2002 by H.L. Bertoni 23


Paraxial Approximation for Repeated
Kirchhoff Integration
For small angles a n
and
 n
, cos a n
 cos
 n

2. Let
 n
  x n

1
 x n
 2   y n

1
 y n
 2
Then for integration over z n
, r
  n
2
 z n
2
  n
+
2 z
 n
2 n
so that


-

(cos a n
 cos
 n
) H ( x n
, y n
) jke
 jkr
4
 r dz n

 jke
 jk
 n
2
 n jke
 jk
 n
2
 n
H ( x n
, y n

) exp(
 jkz n
2


H ( x n
, y n
) e
 j

/ 4
2
 n k
Thus
H ( x n

1
, y n

1
)
 e j

/ 4


 h n
H ( x n
, y n
) e
 jk
 n
 n dy n
2
 n
) dz n
Polytechnic University,
Brooklyn, NY
©2002 by H.L. Bertoni 24


Paraxial Approximation for
Repeated Kirchhoff Integration - cont.
For uniform building height
 n
 d 2
  y n

1
 y n
 2  h n d

0, and uniform row spacing x n

1
 x n

( y n

1
 y n
)
2
2 d
so that
H ( x n

1
, y n

1
)
 e j

/ 4 e
 jkd
 d h n


H ( x n
, y n
) exp

 jk ( y n

1
 y n
)
2
/2 d
 dy n
 d and
H ( x
N

1
, y
N

1
)
 e j N

4 e
(
 jkNd
 d )
N / 2
Let v n
 y n
0

 dy
1
 

0
 dy
N


H ( d , y
1
)exp

 j k
2 d
N  n

1
 y n

1
 y n
 2
  jk
2 d
; dy n
 e
 j

/ 4
 d
 dv n then
H ( x
N

1
, y
N

1
)
 e
 jkNd

N / 2
0

 dv
1
 

0
 dv
N


H ( d , y
1
)exp


N  n

1
 v n

1
 v n
 2
 

Polytechnic University,
Brooklyn, NY
©2002 by H.L. Bertoni 25


Rooftop Field for Incident Plane Wave
H ( d , y
1
)
 e
 jkd cos a e jky
1 sin a  e
 jkd e jky
1 sin a
Use Taylor series expansion
H ( d , y
1
)
 e
 jkd e jky
1 sin a  e
 jkd q



0
1 q !
( jky
1 sin a
) q
Define g p
 sin a d
 and since

1
 y
1
H ( d , y
1
)
 e
 jkd e jky
1
Then the field at y
N

1 sin a  e
 jkd

0
 
N

1
  q

0
1 q !

0

is

2 g p
H ( x
N

1
, 0)
 e
 jk ( N

1) d
 N / 2
 
0 d

1
 
 
0 jk
2 d
, then j

 q 
1 q d

N





  q

0
1 q !

2 g p j

 q 
1 q

 exp

 
1
2 
2
N

1 
 n

1 n

1
 n

2
N 
 n

2 n
2
 


H ( x
N

1
, 0)
 e
 jk ( N

1) d
  q

0
1 q !

2 g p j

 q
I
N , q
(1) , where I
N , q
(1) is a Borsma function defined in the next slide.
Polytechnic University,
Brooklyn, NY
©2002 by H.L. Bertoni 26


Borsma Functions for  = 1
I
N , q
  
1

N / 2


0 d

1


0 d

2
Recursion relation for q

2

  d

0

N



1 q exp




2
1

2
N

1  n

1
 n

1
 n

2
N  n

2
 n
2
 
 
I
N , q
  
N ( q

1)
2( N

1)
 
1
I
N , q

2
  
2
1

( N

1)
 
1 n

N

1 
 
1
I n , q

1
N
 n where
I
0, q
(1)



1 for q

0
0 for q

0
I
N ,0
(1)

(1/2)
N
N !
The term (1/2) n
; I
N ,1
(1)

2
1


1 N  n

0
(1/2) n n !
N
 n
represents Pockhammer' s Symbol for a

1/2, where
( a )
0

1; ( a )
1
 a ; ( a ) n
 a ( a

1) ( a
 n

1)
Polytechnic University,
Brooklyn, NY
©2002 by H.L. Bertoni 27




Field Incident on the N + 1 Edge for a
= 0 y
Since g p
 sin a d


0
H ( x
N

1
,0)
 e
 jk ( N

1) d
I
N ,0
(1)
 e
 jk ( N

1) d
(1/2)
N
N !
 e
 jk ( N

1) d
1

N

1
Amplitude decrease monotonically with N
E in
H in
1 
3 N+ 1 x
Polytechnic University,
Brooklyn, NY
©2002 by H.L. Bertoni 28

Field Incident on the N + 1 Edge for a ≠ 0
After initial variation, field settles to a constant value Q ( g p
) for N > N
0 a.0

Angles indicated are for d =
200

N
0 g p
 sin a d

;
Polytechnic University,
Brooklyn, NY
N
0

1 g
2 p
©2002 by H.L. Bertoni
Settled
Field
Q ( g p
) a
…
1 2 ….
n n+1
29




Explanation of the Settling Behavior in Terms of the
Fresnel Zone About the Ray Reaching the
N+ 1
Edge
Nd
 sec a
Nd tan a
E
1
H
1
N
0 d tan a 
N
0 d
 sec a a
Fresnel zone half width
W
F
  s
N
0


/ d sin
2 a

1 g p
2 d n= 1 n= 2 n= 3 n= 4 n= 5
N
0 n=N 1 n=N n=N +1
Only those edges that penetrate the Fresnel zone affect the field at the N +1 edge
Polytechnic University,
Brooklyn, NY
©2002 by H.L. Bertoni 30

Settled Field Q ( g p
) and Analytic Approximations
1.5
1.0
Straight line approximation for 0 .
015
 g p

0 .
4
0.5
0.2
0.1
0.05
0.03
0.01
0.02
Polynomial fit for
0.05
g p

1.0
0.1
g p
0.2
Q(g p
)

3.502
g p

3.327
g p
2

0.962
g p
3
Polytechnic University,
Brooklyn, NY
0.5
©2002 by H.L. Bertoni
1.0
Q(g p
)

0 .
1

 g p
0 .
03


0 .
9 where g p
 sin a d

31


Path Gain/Loss for High Base Station Antenna
 

0.1
g p

0.03
 0.9

, g p
 sin a d /
  h
B S

H
B
R d

PG



4


R


2
 

 2

2
( H

B
 h m
)
2




4


R


2

0.01

 h
B S

0.03
H
R
B


1.8
 d
 


0.9
 

2

2
( H

B
 h m
)
2



5.51
32
 4
  h
B S

H
B
( H
B

1.8
 d 0.9
 h m
)
2
 
2.1

R
3.8
Comparison with measurements made in Philadelphia by AT&T
Polytechnic University,
Brooklyn, NY
©2002 by H.L. Bertoni 32


Comparison Between Hata Measurement Model and the Walfisch-Ikegami Theoretical Model
For f
M
in MHz and R k
in km
Theory :
L

89 .
5

9log d

10log

( H
B

 h m
)
2

18log( h
B S

H
B
)

21log f
M

38log R k
Hata :
L

69.55

26.16log
f
M

13.82log
h
B S

(44.9

6.55log
h
B S
)log R k
Assume h
B S
Theory :
Hata : h
M
L



30m
1.5 m
57.7

21log f
M
H
B

12m d

50m
L

49.2

26.2log
f
M

38log R k

35.2log
R k
If f
M
R k

1,000; Theory L

147.3 dB

5; Hata L

152.4 dB
Polytechnic University,
Brooklyn, NY
©2002 by H.L. Bertoni 33


Comparison of Theory for Excess Path Loss with Measurements of Okumura, et al.
Path Loss

L
 
10log
 
4

R
 2 
10log Q 2 
10log PG
2
Excess Path Loss

L

L
0
 
10log Q
2 
10log PG
2
depeneds on R and h
B S
only through the angle a

H
B f = 922 MHz
Polytechnic University,
Brooklyn, NY
©2002 by H.L. Bertoni 34


Walk About From Rooftop to Street Level f

450 MHz
 
2 /3 m h
B S

20 m H
B

7 m h m

1.5 m d

50 m
 
( d /2)
2 
( H
B
 h m
)
2 
25.6 m
PG

5.51
32

4

( h
B S
(

H
H
B
B
)
 h
1.8
m
)
 d
0.9
2
  2.1

R
3.8

0.218
R
3.8

1.6

10

14 or R

3.8
1.36

10

13

4.45

10

3 m
Polytechnic University,
Brooklyn, NY
R
©2002 by H.L. Bertoni 35

Summary of Q for High Base Station Antennas
• Rows of buildings act as a series of diffracting screens
• Forward diffraction reduces the rooftop field by a factor that approaches a constant past many rows
• The settling behavior can be understood in terms of Fresnel zones, and leads to the reduction factor
Q, which depends on a single parameter g p
• Good comparison with measurements is obtained using a simple power expansion for Q
Polytechnic University,
Brooklyn, NY
©2002 by H.L. Bertoni 36

• Finding the reduction factor Q using an incident cylindrical wave
• Q is shown to depend on parameter g c number of rows of buildings and the
• Comparison with measurements
• Mobile-to-mobile communications
Polytechnic University,
Brooklyn, NY
©2002 by H.L. Bertoni 37

Cylindrical Wave Solutions for Microcells Using
Low Base Station Antennas
Microcell coverage out to about 1 km involves propagation over a limited number of rows.
Must account for the number of rows covered, and hence for the field variation in the plane perpendicular to the rows of buildings.
Therefore use a cylindrical incident wave with axis parallel to the array of absorbing screens to find the field reduction due to propagation past rows of buildings.
Polytechnic University,
Brooklyn, NY
©2002 by H.L. Bertoni 38



Physical Optics Approximations for Reduction of the Rooftop Fields
I. Replace rows of buildings by parallel absorbing screens
E
H a
Incident wave y n
 n
 n y n+ 1 x n= 1 n= 2 n= 3 n n +1
II. For parallel screens, the reduction factor will apply for a spherical wave and for a cylindrical wave. For 2D fields, Kirchhoff integration gives
H ( x n

1
, y n

1
)

Polytechnic University,
Brooklyn, NY



 h n


 cos a n e j

/ 4

 cos
 n

H ( x n
, y n
) jke
 jkr
4
 r dy n dz n
 h
 n
H ( x n
, y n
) e
 jk
 n
 n dy n
, since cos a n
©2002 by H.L. Bertoni
 cos
 n

2
39


Paraxial Approximation for Repeated Kirchhoff
Integration and Screens of Uniform Height
For uniform building height h n

0, and uniform row spacing x n

1
 x n
 d
 n
 d 2
  y n

1
 y n
 2
H ( x
N

1
, y
N

1
)
 e j N

4
(
 d )
N / 2 e
 jkNd
 d

( y n

1
 y n
)
2
2 d
0
  dy
1
  dy
0
 
N



H ( d , y
1
)exp
 j k
2 d
N   n

1 y n

1
 y n

2
 


Let v n
 y n jk
2 d
; dy n
H ( x
N

1
, y
N

1
)
 e
 jkNd

N / 2
 e
 j

/ 4
 d
 dv n
0

 dv
1
 

0
 dv
N


H ( d , y
1
)exp


N  n

1
 v n

1
 v n

2
 
Polytechnic University,
Brooklyn, NY
©2002 by H.L. Bertoni 40

Approximation for Cylindrical Wave of a
Line Source
H ( d , y
1
)
 e
 jk

1

1 where

1
 d
2   y
1
 y
0
 2
In exponent

1
H ( d , y
1
)

 d

 y
1
 y
0

2
2 d e
 jkd e
 jky
2
0
/ 2 d e jky
0 y
1
/ d e
 jky
1
2
/ 2 d d
Define g c
 y
0
 d
and v
1
 y
1
Then
H ( d , y
1
)
 e
 jkd e
 2 jky
0
/ 2 d e
2 g c d jk
2 d j

1 e
  2
1 y
0 y d
1 2 3 4 d
N N+1 x
Polytechnic University,
Brooklyn, NY
©2002 by H.L. Bertoni 41


Integral Representation for Field at the
N +1
Edge
At the roof top of the N +1 row of buildings y
N

1

0
 
N

1

0

H ( x
N

1
,0)
 e
  
1
 d e
 jky
2
0
 N / 2 d
( 2 d ) 

0 dv
1


0 dv
2

  dv
0

N e
2 g c j

1 exp


2 v
1
2

2
N

1  n

1 v n v n

1

2
N  n

2 v 2 n 
Use Taylor series expansion e
2 g c j
 v
1
 q
 

0
1 q !

2 g c j

 q v
1 q
Then
H ( x
N

1
,0)
 e
  
1
 d e
 jky
2
0 d
( 2 d )
  q

0
1 q !

2 g c j

 q
I
N , q
(2) where I
N , q
(2) are Borsma functions
Polytechnic University,
Brooklyn, NY
©2002 by H.L. Bertoni 42


Borsma Functions for Line Source Field
I
N , q
  
1

N / 2


0 d

1


0 d

2
 
0

 d

N



1 q exp


2

1
2

2
N

1 
 n

1
 n

2
N 
 n

1 n

2 n
2
 
Recursion relation for q

2
I
N , q
  
2(
N ( q

1)
N

1)
 
1
I
N , q

2
  
2
1

( N

1)
 
1 n

N

1 
 
1
I n , q

1
N
 n where
I
N ,0
(2)

1

N

1
 3
2
; I
N ,1
(2)

4
1

N  n

1
1 n
2
3

N

1
 n
 3
2
Polytechnic University,
Brooklyn, NY
©2002 by H.L. Bertoni 43


Rooftop Field Reduction Factor for Low
Base Station Antenna
Reduction factor found from cylindrical wave field
Q
N

1
( g c
)

H ( x
N

1
,0) e
 jk

/

where
  
( N

1) d

2  y
0
2

( N

1) d
In terms of Boersma functions
Q
N

1
( g c
)

N

1

 q

0
1 q !

2 g c j
 q 
I
N , q
(2)
For y
0

0, g c
 y
0
 d

0 and
Q
N

1
( g c
)

N

1
1
( N

1)
3/ 2

1
N

1
Polytechnic University,
Brooklyn, NY
©2002 by H.L. Bertoni 44

Field Reduction Past Rows of Buildings
10
1 y
0
= 0m
0.1
0.01
y
0
= +11.25m
1/M
0.001
y
0
= –11.25m
0.0001
1 y
0

10
Number of Screens M = N+1 h
BS

H
B
, g c
 y
0 d

100
Field after multiple diffraction over absorbing screens. Values of y
0 a frequency of 900MHz and d=50 m.
are for
Polytechnic University,
Brooklyn, NY
©2002 by H.L. Bertoni 45


Slope of field H ( M ) vs. Number of Screens for different Tx heights at 1800MHz
1.2
1.1
1
0.9
0.8
0.7
0.6
y
0
< 0 y
0
> 0
0.5
0 10 20 30 40 50 60 70 80 90 100
Number of Screens
M s
  log Q
M

1
 log Q
M log( M

1)
 log M
  log Q log
 

M
M

1

1

Q
M
M


Polytechnic University,
Brooklyn, NY
©2002 by H.L. Bertoni 46


Modifications for Propagation Oblique to the
Street Grid
Base Station
 mobile x= 0 x = Md + d /2
Radio propagation with oblique incidence
PG
2

10log



 kr

1 cos

1
 


1
2
  

 2



 g c
 y
0 x cos

 d
Polytechnic University,
Brooklyn, NY
©2002 by H.L. Bertoni 47


0
-5
-10
Comparison of Base Station Height Gain with
Har/Xia Measurement Model f

1.8 GHz d

50 m y
0
 h
B S
R

1 km

H
B
-15
-20
-25
-30
Q
10
Q exp
For perpendicular propagation
 
0
N

1

R d

20
-35
-40
-45
-50
-8 -6
Q
20
-4 -2 0 y
0
2 4 6 8
For oblique propagation
 
60

N

1

R d /cos


10
Polytechnic University,
Brooklyn, NY
©2002 by H.L. Bertoni 48


Experimentally Based Expression for
Q exp
We can compare the theoretical Q with the Har/Xia measurements using
L
 
PG
 
PG
0

20log Q

PG
2
The Har/Xia formulas for path loss on staircase and transverse paths give
L , so that
20log Q exp
 
L

PG
0

PG
2
Substituing their expression for L gives
20log Q exp
 
 
138.3

38.9log
f
G

 
13.7

4.6log
f


40.1

4.4 sgn( y
G
0
 sgn( y
0

 y
0


 y
0
  log R k


10log


4

R k


10 3
 2


10log

 2


2 ( H
B
 h m
) 2

Polytechnic University,
Brooklyn, NY
©2002 by H.L. Bertoni 49

Comparison of Range Index n with Har/Xia
Measurement Model n =2+2 s
4.4
4.3
4.2
4.1
4
3.9
3.8
3.7
3.6
3.5
3.4
-8 theory
-6 exp
-4 -2 8 d f

1 .
8 GHz

50 m
R

1 / 2 km
Polytechnic University,
Brooklyn, NY
0 y
0
2 4
©2002 by H.L. Bertoni
6
50

Q for Mobile to Mobile Communications

0


0

1


1
H
B h
0 n= 1 2
R
M h
1
Peak of first building acts as line source of strength
D
  
0
Propagation past remaining peaks gives factor 1/( M -1)
Effective reduction factor
Q e


0
D
 
( M

1)
Polytechnic University,
Brooklyn, NY
©2002 by H.L. Bertoni 51


Comparison of Q Factors for Plane Waves,
Cylindrical Waves and Mobile-to-Mobile
0
-10
-20
-30 Q
20
( g c
)
Q ( g p
)
 
1/3 m, d

50 m, M

20 g c
 y
0
 d g p
 sin a d

 h
B S

Md
H
B d


1
M y
0
 d

1
M g c
-40
-50
-60
-15 -10 -5
  d
Polytechnic University,
Brooklyn, NY
Q e
0 y
0
 d
5 10 15
©2002 by H.L. Bertoni
Use plane wave factor for y
0
  d use mobile - to - mobile factor for y
0
   d
52

Path Loss for Mobile-to-Mobile Communication
L
 
20 log
 

4

R



20log Q e
Since R

Md

10 log
D
1

1
2
L
 
20 log


4




20log
 dM ( M

1)
 
10 log
D
0

0
2

10 log
D
1

1
2
If both mobile are at same height and in the middle of the street, using
D

2

1
2
 k
 
1
2

1
2
 k ( d / 2) d / 2

H
B
 2
 h m

 d

8

2 ( H
B
 h m
) 2 gives
L

20 log(16

3 )

20 log

M ( M

1)
 
40 log


H
B
 h m



Polytechnic University,
Brooklyn, NY
©2002 by H.L. Bertoni 53

Walk About Range for Low Buildings

0

0


1


1
H
B h
0 n= 1 2
R
M h
1
PG dB
 


3


20 log

M ( M

1)
 

H
B
 h m
)
 
For
 
2 / 3 m, H
B
PG dB

10 m, h m

2 m
 
53.9

20log

M ( M

1)
 
43.2
 
138
Thus 20log

M ( M

1)
 
40.9 or M ( M

1)

111 or M

11
For d

50 m, R

Md

550 m = 0.55 km
Polytechnic University,
Brooklyn, NY
©2002 by H.L. Bertoni 54

Summary of Solution for Low Base
Station Antennas
• Reduction factor found using an incident cylindrical wave
• Q
M depends on parameter g c and the number of rows of buildings M over which the signal passes
• Theory gives the correct trends for base station height gain and slope index, but is pessimistic for antennas below the rooftops
• Theory give simple expressions for path gain in the case of Mobile-to-mobile communications
Polytechnic University,
Brooklyn, NY
©2002 by H.L. Bertoni 55