7. Channel Models

Signal Losses due to three Effects:

1. Large Scale Fading: due to

distance

2. Medium Scale Fading: due to shadowing and

obstacles3. Small Scale Fading: due to

multipath

Wireless Channel

Several Effects:• Path Loss due to dissipation of energy: it depends on distance only• Shadowing due to obstacles such as buildings, trees, walls. Is caused by

absorption, reflection, scattering …• Self-Interference due to Multipath.

transm

rec

P

P10log10

distancelog10

Frequencies of Interest: in the UHF (.3GHz – 3GHz) and SHF (3GHz – 30 GHz) bands;

Path Loss due to Free Space Propagation:

Transmit antenna

Receive antenna

2

4rec transmP Pd

wavelength c

F

d

Path Loss in dB:

10 10 1010log 20log ( ( )) 20log ( ( )) 32.45transm

rec

PL F MHz d km

P

1.1. Large Scale Fading: Free Space

For isotropic antennas:

2. Medium Scale Fading: Losses due to Buildings, Trees, Hills, Walls …

pp LEL

The Power Loss in dB is random:

approximately gaussian with

dB126

expected value

random, zero mean

00

10log10}{ Ld

dLE p

Path loss exponent

Reference distance• indoor 1-10m• outdoor 10-100m

Free space loss at reference distance

dB

Average Loss

10 0log ( / )d d

0pE L L

10110 010210

20dB

10 Values for Exponent :

Free Space 2

Urban 2.7-3.5

Indoors (LOS) 1.6-1.8

Indoors(NLOS) 4-6

• Okumura: urban macrocells 1-100km, frequencies 0.15-1.5GHz, BS antenna 30-100m high;

• Hata: similar to Okumura, but simplified

• COST 231: Hata model extended by European study to 2GHz

Empirical Models for Propagation Losses to Environment

3. Small Scale Fading due to Multipath.

a. Spreading in Time: different paths have different lengths;

time

Transmit Receive

0( ) ( )x t t t

0t

0( ) ( ) ...k ky t h t t

1 2 30t

2138

100 10sec

3 10c

Example for 100m path difference we have a time delay

Typical values channel time spread:

channel

0( ) ( )x t t t

1 2 MAX0t

0t

1

Indoor 10 50 sec

Suburbs 2 10 2 sec

Urban 1 3 sec

Hilly 3-10 sec

n

b. Spreading in Frequency: motion causes frequency shift (Doppler)

time

time

Transmit Receive

Frequency (Hz)

Doppler Shift

v

cf

2( ) cj F tTx t X e

2( ) cj F F tRy t Y e

for each path

cF F

time

Transmit Receive

v

Put everything together

time

)(tx )(ty

Re{.}

tFj Ce 2 tFj Ce 2

)(th

)(tw

)(tgT

LPF

)(tgR

LPF

( )x t( )y t

2 ( )( )( )( ) Re ( ) cj F tFy t x t ea t

Each path has … …shift in time …

…shift in frequency …

… attenuation…

(this causes small scale time variations)

paths

channel

2.1 Statistical Models of Fading Channels

Several Reflectors:

Transmit

v

( )x t

t ( )y t

t

1

2

For each path with NO Line Of Sight (NOLOS):

2 ( )( )( ) Re ( )c kj F tk

kk

Fy t a e x t

v

( )y t average time delay

• each time delay

• each doppler shift

k

DF F

cos( )v t

t

)2 ( )( 22( ) Re ( )c k cFF j F j F tj t

kkky t e e x t ea

2 ( )2( ) ( )c kj F Fj F tk

k

r t a e e x t

Assume: bandwidth of signal <<

( ) ( )kx t x t … leading to this:

Some mathematical manipulation …

k/1

2( ) Re ( ) cj F ty t r t e

( ) ( ) ( )r t c t x t

with 2 ( )2( ) c kj F Fj F t

kk

c t a e e random, time varying

Statistical Model for the time varying coefficients

2 ( )2

1

( ) c k

Mj F Fj F t

kk

c t a e e

randomBy the CLT is gaussian, zero mean, with:( )c t

*0( ) ( ) (2 )DE c t c t t P J F t

D C

v vF F

c with the Doppler frequency shift.

Each coefficient is complex, gaussian, WSS with autocorrelation

*0( ) ( ) (2 )DE c t c t t P J F t

( )c t

and PSD

20

2 1 if | |

( ) (2 ) 1 ( / )

0 otherwise

DDD D

F FFS F FT J F t F F

with maximum Doppler frequency.DF

( )S F

DF F

This is called Jakes spectrum.

Bottom Line. This:

time

v

time

)(tx )(ty

11( )c t

( )c t

N( )Nc t

( )y t)(tx

… can be modeled as:

delays

1

N

time time

time

For each path

( ) ( )c t P c t

• unit power• time varying (from

autocorrelation)

• time invariant• from power distribution

Parameters for a Multipath Channel (No Line of Sight):

Time delays: L 21 sec

Power Attenuations: LPPP 21 dB

Doppler Shift: DF Hz

)()()( txtcty

( ) ( )c t P c t

)(tc WSS with Jakes PSD

Summary of Channel Model:

Non Line of Sight (NOLOS) and Line of Sight (LOS) Fading Channels

1. Rayleigh (No Line of Sight).

Specified by:

Time delays

Power distribution

],...,,[ 21 NT

],...,,[ 21 NPPPP

Maximum Doppler DF

0)}({ tcE

2. Ricean (Line of Sight) 0)}({ tcE

Same as Rayleigh, plus Ricean Factor

Power through LOS

Power through NOLOS

TotalLOS PK

KP

1

TotalNOLOS PK

P

1

1

K

Simulink Example

-K-

TransmitterGain

B-FFT

SpectrumScope

RectangularQAM

Rectangular QAMModulatorBaseband

-K-

Receiver Gain

RayleighFading

Multipath RayleighFading Channel

-K-ChannelAttenuation

BernoulliBinary

Bernoulli BinaryGenerator

Rayleigh Fading Channel Parameters

M-QAM Modulation

Bit Rate

Set Numerical Values:

modulation

power

channel

CD Fc

vF Recall the Doppler Frequency:

carrier freq.

sec/103 8 m

velocity

Easy to show that: GHzChkmHzD FvF /

Channel Parameterization

1. Time Spread and Frequency Coherence Bandwidth

2. Flat Fading vs Frequency Selective Fading

3. Doppler Frequency Spread and Time Coherence

4. Slow Fading vs Fast Fading

1. Time Spread and Frequency Coherence Bandwidth

Try a number of experiments transmitting a narrow pulse at different random times

)()( ittptx

)(tp

We obtain a number of received pulses

( ) ( ) ( ) ( ) ( )i i i iy t c t p t t c t p t t

1tt 1 2

it t1 2

0

0

Nt t1 2 0

)( 11 itc2 2( )ic t

( )ic t

transmitted

Take the average received power at time it t

1 2 0

1P2P P

2|)(| tcEP

MEAN

RMS

0

10

20

Received Power

time

More realistically:

This defines the Coherence Bandwidth.

Take a complex exponential signal with frequency . The response of the channel is:

)(2)()( MEANtFjetcty

If

)(tx F

1|| RMSF 2 ( )( ) ( ) MEANj F ty t c t e

then

i.e. the attenuation is not frequency dependent

Define the Frequency Coherence Bandwidth as

1

5cRMS

B

1

5cRMS

B

frequencyCoherence Bandwidth

Channel “Flat” up to the Coherence Bandwidth

This means that the frequency response of the channel is “flat” within the coherence bandwidth:

Frequency CoherenceSignal Bandwidth<>

Frequency Selective Fading

Flat Fading Just attenuation, no distortion

Distortion!!!

Example: Flat Fading

Channel : Delays T=[0 10e-6 15e-6] sec

Power P=[0, -3, -8] dB

Symbol Rate Fs=10kHz

Doppler Fd=0.1Hz

Modulation QPSK

Spectrum: fairly uniform

Very low Inter Symbol Interference (ISI)

Example: Frequency Selective Fading

Channel : Delays T=[0 10e-6 15e-6] sec

Power P=[0, -3, -8] dB

Symbol Rate Fs=1MHz

Doppler Fd=0.1Hz

Modulation QPSK

Spectrum with deep variations

Very high ISI

3. Doppler Frequency Spread and Time Coherence

Back to the experiment of sending pulses. Take autocorrelations:

)()()( * ttctcEtR

Where:

1tt 1 2

it t1 2

0

0

Nt t1 2 0

)( 11 itc2 2( )ic t

( )ic t

1( )R t2 ( )R t

( )R t

transmitted

Take the FT of each one:

( )S F

DF F

This shows how the multipath characteristics change with time.

It defines the Time Coherence:

)(tc

9

16CD

TF

Within the Time Coherence the channel can be considered Time Invariant.

Summary of Time/Frequency spread of the channel

Time Spread

Frequency Spread ),( FtS

F

t

RMS

DF

Frequency Coherence

1

5cRMS

B

Time Coherence

9

16CD

TF

mean

Stanford University Interim (SUI) Channel Models

Extension of Work done at AT&T Wireless and Erceg etal.

Three terrain types:• Category A: Hilly/Moderate to Heavy Tree density;• Category B: Hilly/ Light Tree density or Flat/Moderate to Heavy Tree density• Category C: Flat/Light Tree density

Six different Scenarios (SUI-1 – SUI-6).

Found in

IEEE 802.16.3c-01/29r4, “Channel Models for Wireless Applications,” http://wirelessman.org/tg3/contrib/802163c-01_29r4.pdf

V. Erceg etal, “An Empirical Based Path Loss Model for Wireless Channels in Suburban Environments,” IEEE Selected Areas in Communications, Vol 17, no 7, July 1999