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

c

x x'j j

r j f tray

G u t e R G u t er t Re e

x x'

2 2

22 4

l

l

l

x x'

c

l time delay of the ground reflection relative to the LOS ray

a bG G G l

product of the transmit and receive antenna field radiation patterns in the LOS direction

r c dG G G product of the transmit and receive antenna field radiation patterns corresponding to x and x’, respectively

R = Ground reflection coefficient

c

x x'j j

r j f tray

G u t e R G u t er t Re e

x x'

2 2

22 4

l

l

l

c

x x'j j

r j f tray

G u t e R G u t er t Re e

x x'

2 2

22 4

l

l

l

Delay spread = delay between the LOS ray and the reflected ray

x x'

c

l

c

x x'j j

r j f tray

G u t e R G u t er t Re e

x x'

2 2

22 4

l

l

l

If the transmitted signal is narrowband wrt the delay spread

uB u t u t 1

c

x x'j j

r j f tray

G u t e R G u t er t Re e

x x'

2 2

22 4

l

l

l

jr

r t

G R G eP P

x x'

22

4l

l

x x'

2

l phase difference between the two received signal components

d = Antenna separation

h t = Transmitter height

h r = Receiver height

t r t rx x' h h d h h d 2 22 2l

t r t rx x' h h d h h d 2 22 2l

When d is large compared to h t + h r :

x x'

2

lExpand into a Taylor series

t rh hx x'

d

4

2l

The ground reflection coefficient is given by

sin ZR

sin Z

rr

cosZ

2

vertical polarization

rZ cos 2 horizontal polarization

r 15 for ground, pavement, etc...

For very large d:

l rx x' d , , G G , R 0 1l

t rt rr t t

G G h hh hP P P

d d d

2 22

2

4

4l l

r t t rP dBm P dBm log G log h h log d 10 10 1010 20 40l

t r

r t

G h hP P

d

2

2

l

• As d increases, the received power

• Varies inversely with d 4

• Independent of

f = 900 MHz

R = - 1

h t = 50 m

h r = 2 mG l = 1

G r = 1

P t = 0 dBm

The path can be divided into three segments

1. d < h t

• The two rays add constructively

• Path loss is slowly increasing

• Path loss td h

2 2

1

t r

t rt

d h h

for h hd h

22

2 2 2

1 1

l

l

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

t rh hx x'

d

4

2l

To find d c , set

t rc

h hd

4

• In segment 1, d < h t power falls off by

• In segment 2, h t < d < d c power falls off by – 20 db/decade

• In segment 3, d c < d, power falls off by – 40 db/decade

• Cell sizes are typically much less than d c and power falls off by t/ h 21

t/ h 21

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 = 20 m), the receivers at height h r = 3 m, and f c = 2 GHz. Is this a good size for cell radius in a suburban macrocell? Why or why not?

Solution

c. m

f

8

9

3 100 15

2 10

t rc

h hd . km

. 4 4 20 3

1 60 15

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

i

i c

xjj

i x i j f tray

i i

R G u t eG u t er t Re e

x

22

92

1014

l

l

l

x i = path length of the i th reflected ray

ii

x

c

lix

G Product of the transmit and receive antenna gains of the i th ray

Assume a narrowband model such that

iu t u t for all i

i

i

ji x

r ti i

R G eGP P

x

22

9

14l

l

ii

x

2l

• 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 dielectricproperties 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

• 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

Diffraction

Shadowing: Diffraction and Spreading

For h small wrt d and d', the signal must travel an additional distance d

h d d 'd

d d '

2

2

The phase shift is

dv

d d 'v h

d d '

22

2

2

d d 'v h

d d '

2 is called the Fresnel – Kirchhoff

diffraction parameter

Approximations for the path loss relative to LOS are

. v

L v dB log . . v . v

log . e v

log . . . . v v .

.log v .

v

10

0 9510

2

10

10

20 0 5 0 62 0 8 0

20 0 5 0 1

20 0 4 0 1184 0 38 0 1 1 2 4

0 22520 2 4

Scattering

c

s s'j

s j f tG er t Re u t e

ss'

2

2

3

24

r t sP dBm P dBm log G log log

log log s log s'

10 10 10

10 10 10

10 20 10

30 4 20 20

• 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

r t

dP P K

d

0

r t

dP dBm P dBm K dB log

d

10

0

10

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 logd

100

204

General approach:

• Take data at three values of d

• Solve for K, d o , and