Ray Tracing
A radio signal will typically encounter multiple objects and will be reflected, diffracted, or scattered
These are called multipath signal components

• Represent wavefronts as simple particle
• Geometry determines received signal from each signal component
• Typically includes reflected rays, can also include scattered and diffracted rays
• Requires site parameters
• Geometry
• Dielectric properties
• Error is smallest when the receiver is many wavelengths from the nearest scatterer and when all the scatterers are large relative to a wavelength

• Accurate model under these conditions
• Rural areas
• City streets when the TX and RX are close to the ground
• Indoor environments with adjusted diffraction coefficients
• If the TX, RX, and reflectors are all immobile, characteristics are fixed
• Otherwise, statistical models must be used

Two – Ray Model
Used when a single ground reflection dominates the multipath effects.
Approach:
• Use the free – space propagation model on each ray
• Apply superposition to find the result

 r
2
 ray

  
Re





4
 x
 x'
 l
 c



 G u t e l
   j 2
 l
 l

R G u t r
    x
 x' e
 j 2

 x
 x '





 e j 2
 f t


 time delay of the ground reflection relative to the LOS ray
G l

G G a b
 product of the transmit and receive antenna field radiation patterns in the
LOS direction

r
2
 ray
G r
 


Re





4

G G c d





G u t e l l
 j 2
 l


R G u t r
    x
 x' e
 j 2

 x
 x '





 product of the transmit and receive e j 2
 f t


 antenna field radiation patterns corresponding to x and x’, respectively
R = Ground reflection coefficient

r
2
 ray
  
Re





4




 G u t e l
   j 2
 l
 l

R G u t r
    x
 x' e
 j 2

 x
 x '





 e j 2
 f t



Delay spread = delay between the LOS ray and the reflected ray x
 x'
 l c

r
2
 ray
  
Re





4




 G u t e l
   j 2
 l
 l

R G u t r
    x
 x' e
 j 2

 x
 x '





 e j 2
 f t



If the transmitted signal is narrowband wrt the delay spread
 
B u

1        

r
2
 ray
  
Re





4





P r

P t
G u t e l
  


4




2 l j 2
 l


R G u t r
    x
 x' e
 j 2

 x
 x '





 e j 2
 f t



2 l
G l 
R G e x
 x'
 j

 
2
 x
 x'

 l
 phase difference between the two received signal components

d = Antenna separation h t
= Transmitter height h r
= Receiver height x
 x' l
 h t
 h r

2  d 2 
 h t
 h r

2  d 2

x
 x' l
 h t
 h r

2  d 2 
 h t
 h r

2  d 2
When d is large compared to h t
+ h r
:
 
2
 x
 x'

 l
 
2
 x
 x'

 l
Expand into a Taylor series

4

 h h r d

The ground reflection coefficient is given by sin
 
Z
R
 sin
 
Z
Z
  r
 cos
 r
2  vertical polarization
Z
  r
 cos
2  horizontal polarization
 
15 for ground, pavement, etc...
r

For very large d: x
 x' l d ,
 
0 , G l

G r
, R
 
1
P r




4

G l d


2


4

 h h t r d


2
P t



G h h l r


2 d 2
P t
P dBm

P dBm

10 log
10
 

20 log
10
 h h r


40 log
10
 

P r



G h h l r


2 d 2
P t
• As d increases, the received power
• Varies inversely with d 4
• Independent of


G l
= 1
G r
= 1
P t
= 0 dBm f = 900 MHz
R = - 1 h t
= 50 m h r
= 2 m

The path can be divided into three segments
• Path loss

1 d 2  h t
2 l
 d 2 
 h t
 h r

2
1 l
2

1 d 2  h t
2 for h t
 h r
1. d < h t
• The two rays add constructively
• Path loss is slowly increasing

2. h t
< d c
• Wave experiences constructive and destructive interference
• Small – scale (Multipath) fading
• If power is averaged in this area, the result is a piecewise linear approximation
3. d c
< d
• Signal power falls off by d
– 4
• Signal components only combine destructively

To find d c
, set
 
2
 x
 x'

 l

4

 h h r d
  d c

4 h h t r

• In segment 1, d < h t power falls off by
• In segment 2, h t db/decade
< d < d c
1 / h t
2 power falls off by – 20
• In segment 3, d c db/decade
< d, power falls off by – 40
• Cell sizes are typically much less than d c power falls off by 1 / h t
2 and

Problem 2 – 5
Find the critical distance, d c
, under the two – ray model for a large macrocell in a suburban area with the base station mounted on a tower or building (h t receivers at height h r
= 3 m, and f c
= 20 m), the
= 2 GHz. Is this a good size for cell radius in a suburban macrocell? Why or why not?
Solution
  c
 f
3

10
8
2

10
9

.
m d c

4 h h r

 

Ten – Ray Model (Dielectric Canyon)
• Assumptions:
• Rectilinear streets
• Buildings along both sides of the street
• Transmitter and receiver heights close to street level
• 10 rays incorporate all paths with 1, 2, or 3 reflections
• LOS (line of sight)
• GR (ground reflected)
• SW (single wall reflected)
• DW (double wall reflected
• TW (triple wall reflected)
• WG (wall – ground reflected)
• GW (ground – wall reflected)

Overhead view of 10 – ray model r
10
 ray
  
Re





4





G u t e l
   j 2
 l
 l

9 i


1
R i
G x i

  i
 x i e
 j 2
 x
 i



 e j 2
 f t




 i x i
= path length of the i th reflected ray
 x i c
 l
G x i
 Product of the transmit and receive antenna gains of the i th ray

Assume a narrowband model such that
  

  i
 for all i
P r

P t


4




2
G l l

9 i


1
R G e i x i x i
 j
 i
2
 i

2
 x i

 l
• Power falloff is proportional to d - 2
• Multipath rays dominate over the ground reflected rays that decay proportional to d - 4

General Ray Tracing
• Models all signal components
– Reflections
– Scattering
– Diffraction
• Requires detailed geometry and dielectric properties of site
– Site specific
• Similar to Maxwell, but easier math
• Computer packages often used
• The GRT method uses geometrical optics to trace the propagation of the LOS and reflected signal components

Shadowing: Diffraction and Spreading
Diffraction
• Diffraction occurs when the transmitted signal
"bends around" an object in its path
• Most common model uses a wedge which is asymptotically thin
• Fresnel knife – edge diffraction model

For h small wrt d and d', the signal must travel an additional distance
 d
 d
 h 2 d
 d '
2
The phase shift is
 d d '
 v
 h
2
 
 d


2
2
 d
 d '
 d d '
 v
2

v
 h
2
 d
 d '
 d d '
 is called the Fresnel – Kirchhoff diffraction parameter
Approximations for the path loss relative to LOS are

20 log
10

.

.
v
 
0 8 v 0

20 log
10
 0 5

.
v 0 1

20 log
10

.

.
 
.

. v

2



20 log
10

 v


1 v

2 4 v 2 4

Scattering
  


Re u t

   

G s
 e
 j 2

3
2 s s' s
 s'
 e j 2
 f t 




P dBm r

P dBm t

30 log
10

10 log
10
 

20 log
10

20 log
10
 

10 log
10

20 log
10
 

• Okumura model
• Empirically based (site/freq specific)
• Awkward (uses graphs)
• Hata model
• Analytical approximation to Okumura model
• Cost 136 Model:
• Extends Hata model to higher frequency (2 GHz)
• Walfish/Bertoni:
• Cost 136 extension to include diffraction from rooftops

Simplified Path – Loss Model
P r

P K t
 d
0


 d 
P dBm r

P dBm t

K dB

10
 log
10

 
 d d
0


K = dimensionless constant that depends on the antenna characteristics and the average channel attenuation d
0
= reference distance for the antenna far field

= path – loss exponent
LOS, 2 – ray model, Hata model, and the COST extension all have this basic form

Generally valid where d > d
0 d
0
= 1 – 10 m indoors
= 10 – 100 m outdoors
K dB

20 log
10
4

 d
0
General approach:
• Take data at three values of d
• Solve for K, d o
, and
