
2/22


BS

Pt

Pr1

Pt

Pr2

Radio wave propagation - fading

Line-of-Sight (LOS)

Reflection ()

Scattering (>>object)

= signalwavelength


3/22

Too complicated to use in practice

Other, simpler solutions are used

Maxwells equations



Usually separated in to three groups

Path loss

Shadow fading

Multipath fading

Propagation effects


4/22


Path loss

Shadow fading

Multipath fading

PrxPtr

d=vt

v

Prx/Ptr

d=vt

Very slow

Slow

Fast

Ptr= transmitted powerPrx = received power


shadow fading gainsg

distance dependent

average path gain

pg


5/22


BS

Pt

Pr1

Pt

Pr2


Mobile unit is distance raway from antenna

Distance dependence:

Distance dependent path loss (gp)

Shadow fading gain (gs) The multipath gain (gm)

RECEIVED POWER

rx tx p s mP P g g g = + + +

in dB scale

rx tr p s m tr P gP g g g P = =


6/22


Radio wave propagation (dB)

RECEIVED POWER in dB scale (No bars)

Definition of decibel (dB), dbW and dBm

1010log

gain in dB

= input power

= output power

outdB

in

dB

in

out

PG

P

G

P

P

=

=

rx tx p s mP P g g g = + + +

10

dBW = decibel-Watt

10 log1 W

WdBW

PowerPower =

10

dBm = decibel-milliWatt

10 log

1 mW

mWdBm

PowerPower =


7/22


Separated in to three groups

Path loss

Shadow fading

Multipath fading

Propagation effects

Distance power loss

Average path loss

10 2log ,

in dB

p pg C r =

2

2;

4p tr rx

Cg C G G

r

= =

2

2

4

rx tr tr rx

tr p tr

P P G G

rC

P g P r

=

= =

Carrier wavelength =r = propagation distance


8/22


9/22

Shadowing

Shadowing

Shadowing

A terminal moving behind a hill, etc.

At or above 300 MHz, the amount of diffracted energy is

low shadows will be distinct

Signal will fluctuate shadow fading

Shadowing gain can be estimated

Included in Geographical Information Systems (GIS)planning tools

Finer details not known

maps have resolution of 50-100 m


10/22

Shadowing

Path loss

Shadow fading

PrxPtr

d=vt

v

Prx/Ptr

d=vt

Very slow

Slow

Ptr= transmitted powerPrx = received power

log-normal

Shadowing

Shadowing

A common model log-normal, probability density

function

log-standard deviation 8-12 dB

2

2

( )

21( )

2

i avg

i

g g

i

i

p g e

=


11/22

Shadowing

ShadowingA common model log-normal, probabilitydensity function

Model parametersobained

empirically.

dB power normally

distributed

Shadowing

Shadowing

log-normal

rx tr p tr

C

P g P r= =

Prx/Ptr

d=vt

Very slow

Slow

log-normal


12/22


13/22

Shadowing

Autocorrelation between dB-samples of the signal level

taken at sample rate 1/T in a mobile moving at speed v,

given by

where = correlation of two shadow fading gains,

X(r) andX(r+D)

2( )k

XR k a= /T Da =

D

Shadowing

hb =

Antenna

height

hm =

Mobile

height


14/22

Shadowing

Okumura-Hata

Multipath fading


15/22

Multipath fading

Multipath fading

Looking more microscopic model:

Fast multipath fading dependent on phase differencesbetween wave components

Shadow fading is slow (wavelengths 10-100 m),multipath fading very rapid (wavelengths 0.5-1 m)

Multipath fading is narrowband phenomenon, when thedelay of the multipath components < the symbol durationof the transmitted signals

For wideband signals, the received power fluctuationshave considerably lower amplitude.

Multipath fading

Line-of-Sight (LOS)

Reflection ()

Scattering (>>object)

= signalwavelength


16/22

Effects of multipath fading

Intersymbol interference

Multipath model

Random number of multipathcomponents, each with

Random amplitude Random phase

Random Doppler shift

Random delay

Each component varies with time

Model is time-varying impulseresponse


17/22

Multipath model

Response of a channel at time ttoimpulse at t-

Each component varies with time

Model is time-varying impulse

response

Multipath model

Response of a channel at time tto impulse at t-

t = time when impulse is observed

t-= time when impulse was put into the channel

= how long ago impulse was put into channel

( ) ( )1

, ( ) ( ( ))nN

j tn n

n

c t t e t

==


18/22

Multipath model

Response of a channel at time tto impulse at t-

n = changes slowly

n = changes quickly

Amplitude fading

( ) ( )1

, ( ) ( ( ))nN

j t

n n

n

c t t e t

=

=

Multipath fading

Narrowband signals central limit theorem used

Received signal hasNcomponents

Herex(t) andy(t) uncorrelated Gaussian processes

(N large and incoming components have the same

statistical properties).

{ } ( )( )1 1 1

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )N N N

j t

n n n n

n n n

z t t e x t jy t x t jy t x t jy t

= = =

= = + = + = +


19/22

Multipath fading

Amplitude

Phase

2 2( ) ( ) ( )t x t y t = +

( )( ) arctan

( )

x tt

y t =

Multipath fading

The joint probability density function

2-D Gaussian

Joint probability density function of

has uniform distribution over the interval

independent of

( ) and (t)t

2 22

2( , )

2

Marginal distribution of

p e

=

2 2- 2

2Rayleigh dist

rp() = e ibu

tion

( )t

( )t [0,2 ]


20/22

Multipath fading

First two moments of

Compute the density of the power of

Then

[ ] 2

( )2

2 14

E

Var

=

=

( )t

2( ), (t)= ( )t t

0

0

1( )p e

=

2 2

0 ( ) 2E E = = =

ll signalcomponents have

the same energy.

The Resource Management Problem

ll signalcomponents have

the same energy

If there is a strong

component

one can compute( ) ( ) ( )z t x t jy t = + +

2 2 2( ) 2

0 2 2

0

( )

zero-order Bessel function

amplitude of the dominant, constant signal

(line-of-sight)

p I e

I

+ =

Nakami-Rice or just Rice distribution


21/22

Multipath fading time series models

Jakes model in fading

Multipath fading Jakes model

N0 low-frequency oscillators with frequency

= Doppler shifts

m=cos(2n/N), n=1,2,...,N0, +

m are used to generate signals with frequency-shifted from a carrier frequency c using modu-lation methods. The amplitudes are =1 except mwhich is = .

n chosen so that the probability distribution ofthe resultant phase is close to uniform didtri-

bution (1/2).N0=8.

1/ 2


22/22

Multipath fading time series models

and others

Slow correlation in fading

{ }

[ ] [ ]

1 2 1

2 2

2

1 2 1

( ) ( 1) ( 2) ( ) ( 1),

( ) ,

, , 1.8384,0.8395, 0.9634 , 2.01,

Sample instanst are with respect to spatial sample interval ,

0.1 m

s s s s s

s e

T T

e

s

s

g k a g k a g k e k b e k

Var e k

a a b

k x

x

= + +

=

= =

=

1 channel models 2004

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