EE542 452 Class10 Fade

8/6/2019 EE542 452 Class10 Fade

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EE 552/452, Spring, 2008Wireless Communications

(and Networks)

Zhu Han

Department of Electrical and Computer Engineering

Class 10

Feb. 21st, 2008

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EE 542/452 Spring 2008EE 552/452 Spring 2007

OutlineOutline

R eview: Four types of Fading ± Slow Fading

± Fast Fading

± Flat Fading

± Frequency Selective Fading

R ayleigh and R icean Distributions

Statistical Models

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EE 542/452 Spring 2008

Types of Small Types of Small- -scale Fading scale Fading

Small-scale Fading(Based on Multipath Tme Delay Spread)

Flat Fading

1. BW Signal < BW of Channel

2. Delay Spread < Symbol Period

Frequency Selective Fading

1. BW Signal > Bw of Channel

2. Delay Spread > Symbol Period

Small-scale Fading(Based on Doppler Spread)

Fast Fading

1. High Doppler Spread2. Coherence Time < Symbol Period3. Channel variations faster than baseband

signal variations

Slow Fading

1. Low Doppler Spread2. Coherence Time > Symbol Period3. Channel variations smaller than baseband

signal variations

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Fading DistributionsFading Distributions

Describes how the received signal amplitude changes with time. ± R emember that the received signal is combination of multiple signals

arriving from different directions, phases and amplitudes.

± With the received signal we mean the baseband signal, namely the

envelope of the received signal (i.e. r(t)).

It is a statistical characterization of the multipath fading.

Two distributions

± R ayleigh Fading

± R icean Fading

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R ay leigh DistributionsR ay leigh Distributions

Describes the received signal envelope distribution for channels, where all

the components are non-LOS:

± i.e. there is no line-of±sight (LOS) component.

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R icean DistributionsR icean Distributions

Describes the received signal envelope distribution for channels where one

of the multipath components is LOS component.

± i.e. there is one LOS component.

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R ay leigh Fading R ay leigh Fading

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R ay leigh Fading R ay leigh Fading

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Rayleigh Fading DistributionRayleigh Fading Distribution

The R ayleigh distribution is commonly used to describe thestatistical time varying nature of the received envelope of a flatfading signal, or the envelope of an individual multipath

component.

The envelope of the sum of two quadrature Gaussian noise

signals obeys a R ayleigh distribution.

W is the rms value of the received voltage before envelopedetection, and W2 is the time-average power of the received

signal before envelope detection.

p r

r r r

r

( )exp( )

! e e g

®

¯±

°±

2

2

22

0

0 0

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Rayleigh Fading DistributionRayleigh Fading Distribution

The probability that the envelope of the received signal doesnot exceed a specified value of R is given by the CDF:

r peak =W and p(W)=0.6065/W

´ R R

r edr r p Rr P R P 0

2 2

2

1)()()( W

W

W

W

T

W

2

)(2

1177.1

2533.12)(][

0

0

!

!!

!!!!

´

´

¡

rms

r

median

mean

r

dr r pr

dr r rpr E r

median

s l if

W W T

W r E r E r r p r dr 2 2 2 22

0

2

20 4292! ! !

g

´[ ] [ ] ( ) .

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R ay leigh PDF R ay leigh PDF

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 1 2 3 4 50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 1 2 3 4 5W W W W W

W

mean = 1.2533W

median = 1.177W

variance = 0.4292W

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A typical Rayleigh fading envelope at 900MHz. A typical Rayleigh fading envelope at 900MHz.

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R icean DistributionR icean Distribution

When there is a stationary (non-fading) LOS signal present, then theenvelope distribution is R icean.

The R icean distribution degenerates to R ayleigh when the dominantcomponent fades away.

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Ricean Fading DistributionRicean Fading Distribution

When there is a dominant stationary signal component present, the small-scale fading envelope distribution is R icean. The effect of a dominant signalarriving with many weaker multipath signals gives rise to the R iceandistribution.

The R icean distribution degenerates to a R ayleigh distribution when thedominant component fades away.

The R icean distribution is often described in terms of a parameter K which isdefined as the ratio between the deterministic signal power and the varianceof the multipath.

K is known as the R icean factor

As Ap0, K p -g dB, R icean distribution degenerates to R ayleighdistribution.

p r

r r A

I

A r

r A

r

( ) exp[

( )

] ( ) ,!

e e g u

®

±̄

°±W W W 2

2 2

2 0 22 0 0

0 0

K A !2

22W

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C DF C DF

Cumulative distribution for three small-scale fading measurements and their fit to R ayleigh, R icean, and log-normal distributions.

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PDF PDF

Probability density function of R icean distributions: K=-�dB(R ayleigh) and K=6dB. For K>>1, the R icean pdf isapproximately Gaussian about the mean.

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Rice time seriesRice time series

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N akagami Model N akagami Model

Nakagami Model

r: envelope amplitude

=<r2>: time-averaged power of received signal

m: the inverse of normalized variance of r2

± Get R ayleigh when m=1

m

mm

m

r m

r m

r p;+

;

!

)(

)exp(2

)(

212

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Small Small- -scale fading mechanismscale fading mechanism

Assume signals arrive from allangles in the horizontal plane0<<360

Signal amplitudes are equal,independent of

Assume further that there is no

multipath delay: (flat fading

assumption)

Doppler shifts

nn av

f cosP

!

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SmallSmall--scale fading: effect of Doppler in ascale fading: effect of Doppler in amultipath environmentmultipath environment

f m, the largest Doppler shift

2

21

8

1)( ¹¹

º

¸©©ª

¨!

mm

bbEz f

f k

f f S

T

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C arrier Do ppler s pectrumC arrier Do ppler s pectrum

Spectrum Empirical investigations show results that deviate

from this model Power Model Power goes to infinity at fc+/-fm

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Baseband S pectrum Do ppler Faded Signal Baseband S pectrum Do ppler Faded Signal

Cause baseband spectrum has a maximum frequency of 2fm

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Simulating Do ppler/Small Simulating Do ppler/Small- -scale fading scale fading

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Simulating Do ppler fading Simulating Do ppler fading

Procedure in page 222

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Level C rossing R ate ( LCR)Level C rossing R ate ( LCR)

Threshold (R)

LCR is defined as the expected rate at which the Rayleigh fadingenvelope, normalized to the local rms signal level, crosses a specifiedthreshold level R in a positive going directionpositive going direction. It is given by:

secondper crossings

rms)tonormalizedvalueenvelope(specfied

where

:

/

22

R

r ¢ s

¢ R

N

r R

e f N

!

!

V

VT V

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Average Fade Duration Average Fade Duration

Defined as the average period of time for which the received signal isbelow a specified level R.

For Rayleigh distributed fading signal, it is given by:

r s

R R

r

R

f

e

e N

Rr N

!

!

!e!

VT V

X

X

V

V

,2

1

11]Pr[1

2

2

Example 5.7, 5.8, 5.9

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Fading Model: Gilbert Fading Model: Gilbert- -Elliot Model Elliot Model

Fade Period

Time t

Signal Amplitude

Threshold

Good(Non-fade)

Bad(Fade)

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Gilbert Gilbert- -Elliot Model Elliot Model

Good(Non-fade)

Bad(Fade)

1/ANFD

1/AFD

The channel is modeled as a Two-State Markov Chain.Each state duration is memory-less and exponentially distributed.

The rate going from Good to Bad state is: 1/AFD (AFD: Avg Fade Duration)The rate going from Bad to Good state is: 1/ANFD (ANFD: Avg Non-Fade

Duration)

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Simulating 2 Simulating 2- -ray multi pathray multi path

a1 and a2 are independent R ayleigh fading

J1 and J2 are uniformly distributed over [0,2T )

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Simulating multi path with Do ppler Simulating multi path with Do ppler- -induced R ay leigh fading induced R ay leigh fading

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Saleh and Valenzuela Indoor Model Saleh and Valenzuela Indoor Model

Measured same-floor indoor characteristics

± Found that, with a fixed receiver, indoor channel is very slowly time-varying

± R MS delay spread: mean 25ns, max 50ns

± Maximal delay spread 100ns-200ns

± With no LOS, path loss varied over 60dB range and obeyed log distance

power law, 3 > n > 4

Model assumes a structure and models correlated multipath components.

Multipath model ± Multipath components arrive in clusters, follow Poisson distribution.

Clusters relate to building structures.

± Within cluster, individual components also follow Poisson distribution.Cluster components relate to reflecting objects near the TX or R X.

± Amplitudes of components are independent R ayleigh variables, decayexponentially with cluster delay and with intra-cluster delay

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SI RC IM and SM RC IM indoor/outdoor ModelsSI RC IM and SM RC IM indoor/outdoor Models

These models were developed byR

appaport and seidel SIR

CIM is acomputer program , that generates small scale indoor channel responsemeasurements.

The most salient feature of the model is that it produces multipath channel conditionsthat are very realistic since they are based on real world measurements and may thus be used for meaningful system design in factories and office buildings

These programs are very useful and poplar and are used in over 100institutions.

Model can measure individual multipath fading and small scale receiver spacing.

Multipath delay inside the building was found to be 40ns to 800ns.

Mean multipath delay ranged from 30-300 ns.

Arriving multipath component has a Gaussian distribution.

Average number of multipath components range from 9 to 36

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SI RC IM and SM RC IM indoor/outdoor ModelsSI RC IM and SM RC IM indoor/outdoor Models

SIR

CIM Model ± Based on measurements at 1300MHz in 5 factory and other

buildings

± Model power-delay profile as a piecewise function

±±±

°

±±±

¯

®

!

ns500Tns2001360

200T-0.22

ns200ns110360

11065.0

ns110367

1

),(

K K

1 K

K

K K

K R T T

T T

S T P

±°

±¯

!

ns500ns100)75

100T0.62exp(0.08

ns100667

55.0),( 2

K

K

K

K R

T

T T

S T P

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EE542 452 Class10 Fade