Wireless Communication Channels: Small-Scale Fading.

Wireless Communication Channels: Small-Scale Fading

© Tallal Elshabrawy 2

Clarke’s Model for Flat FadingAssumptions: Mobile traveling in x

direction Vertically polarized wave Multiple waves in the x-y

plane arrive at the mobile antenna at the same time

Waves arrive at different angles α

in x-y plane

α

z

x

y

N

z n c nn

E E C f t01

cos 2π θ

For N waves incident at the mobile antennaEach wave arriving at an angle αn will experience a different Doppler shift fn

n n nf t θ 2π φn n

vf cos α

λ

E0 amplitude of the local average E-fieldCn random variable representing the amplitude of individual wavesfc carrier frequencyφn random phase shift due to distance traveled by the nth wave


Clarke’s Model for Flat Fading

Given that: Φn uniformly distributed over 2π N is sufficiently large (i.e., the central limit theorem is

applicable)Therefore:

Both Tc(t) and Ts(t) may be modeled as:Gaussian Random Processes

N N

z n n c n n cn n

E t E C f t E C f t0 01 1

cosθ cos 2π sinθ sin 2π

z c c s cE t T t f t T t f tcos 2π sin 2π

N

c n n nn

T t E C f01

cos 2π φ

N

s n n nn

T t E C f01

sin 2π φ


Clarke’s Model for Flat Fading

z c c s cE t T t f t T t f tcos 2π sin 2π

z c sE t T t T t r t2 2

N

n c sn

If C T T E2 2 2 2 20

1

1 σ 2

c cT t f tcos 2π

s cT t f tsin 2π z cE t r t f t tcos 2π ψ

Power received at mobile antenna zE t r2 2

r r

rp r

r

2

2 2exp 0

σ 2σ

0 0

Rayleigh Distribution


Rayleigh Fading Distribution

r r

rp r

r

2

2 2exp 0

σ 2σ

0 0

α

in x-y plane

x

y

dα

zMain Assumption:- No LOS- All waves at the mobile receiver experience approximately the same attenuation

N

z n c nn

E E C f t01

cos 2π θ

N

nn

C2

1

1

constant p(r)

rσ

0.6065/σ

σ2: Time average received powerσ : rms value of received voltage


Rayleigh Fading Statistics

Probability the received signal does not exceed a value R

R R

r R p r dr2

20

Pr 1 exp2σ

Mean value of the Rayleigh distribution

meanr E r rp r dr0

πσ 1.2533σ

2

Variance of the Rayleigh distribution

r

r

E r E r r p r dr2 2 2 2

0

2 2

πσ σ

2

πσ σ 2 0.4292σ

2

Median of the Rayleigh distribution

medianr

medianp r dr r0

11.177σ

2


Ricean Fading Distribution

r r A Ar

I A rp r

r

2 2

02 2 2exp 0,0

σ 2σ σ

0 0

α

in x-y plane

x

y

dα

zMain Assumption:- LOS- There is a dominant wave component at the mobile receiver in addition to experience multiple waves that experience approximately the same attenuation

A : Peak amplitude of the dominant signalI(.): Modified Bessel function of the first kind and zero-orderσ2: Time average received power of the non-dominant components


Riciean & Rayleigh Fading

p(r)

r

A AK K dB

2 2

2 210 log

2σ 2σ

Define K called the Ricean Factor: The ratio between the deterministic signal power and the power of the non-dominant waves

K=6 dB

K=-∞ dB

Rayleigh Distribution


Level Crossing Rate and Mean fade Duration for Rayleigh Fading Signals

Level Crossing Rate Statistic:The expected rate at which Rayleigh fading envelope normalized to local rms level crosses a specified level in a positive–going direction

R mN f e2ρ2π ρ

ρ:= R/Rrms

fm: Maximum Doppler shift

Mean Fade Duration Statistic:The average period of time for which the received signal is below a specified level R

Mean Fade duration is a very important statistic that helps define the time correlation behavior of BER performance at the receiver

R

R

m

r RN

r R p r dr

f

2

0

2

1τ Pr

Pr 1 exp ρ

exp ρ 1τ

2π ρ


0 10 20 30 40 50 6030

40

50

60

70

80

90

100

0 10 20 30 40 50 60-15

-10

-5

0

5

10

15

0 10 20 30 40 50 60-60

-50

-40

-30

-20

-10

0

10

20

Lognormal ShadowingMobile Speed 3 Km/hrARMA Correlated Shadow Model

Distance PathlossMobile Speed 3 Km/hrPL=137.744+ 35.225log10(DKM)

Small-Scale FadingMobile Speed 3 Km/hrJakes’s Rayleigh Fading Model

d

d

d

How Wireless Channels Components Fit Together

© Tallal Elshabrawy

How Wireless Channels Components Fit Together

11

Wireless Channel

PTGT

GR

PR=PTGTGR x Distance Pathloss x Shadowing Parameters x Small-Scale Fading Power

© Tallal Elshabrawy

System Modeling of Wireless Networks: Example

b bkb kbH ,

b bkb kbH ,

b bkb' kb'H ,

k

bkP b'P

b bkP P

b k

b' ONOFF ONOFF ONOFF

ONOFF ONOFF ONOFF

bSK Active Sessions

ONOFF ONOFF ONOFF

bkP , bP

b'P

1

B

b b bb k kb kb

k Nb b b b b b' b b

k k kb kb kb' kb'b'b' b

GAP H

P P H P H N

Target Signal

Intra-cell Interference

Inter-cell Interference

bk TH : Packets Lost (Outage) bk TH : Packets Received Correctly

12

Diversity Techniques


What is Diversity? Diversity techniques offer two or more inputs at the receiver

such that the fading phenomena among these inputs are uncorrelated

If one radio path undergoes deep fade at a particular point in time, another independent (or at least highly uncorrelated) path may have a strong signal at that input

By having more than one path to select from, both the instantaneous and average SNR at the receiver may be improved


Transmitter Receiver

0

1

2

M

Diversity Techniques: Space Diversity

Receiver Space Diversity M different antennas appropriately separated

deployed at the receiver to combine uncorrelated fading signals


ReceiverTransmitter

0

1

2

M

Diversity Techniques: Space Diversity Transmitter Space Diversity

M different antennas appropriately separated deployed at the transmitter to obtain uncorrelated fading signals at the receiver

The total transmitted power is split among the antennas


t

f

Δf>Bc

Diversity Techniques: Frequency Diversity

Modulate the signal through M different carriers The separation between the carriers should be at least

the coherent bandwidth Bc

Different copies undergo independent fading Only one antenna is needed The total transmitted power is split among the

carriers


f

t

Δt>Tc

Diversity Techniques: Time Diversity

Transmit the desired signal in M different periods of time i.e., each symbol is transmitted M times

The interval between transmission of same symbol should be at least the coherence time Tc

Different copies undergo independent fading

Diversity Combining Techniques


Select the strongest signal


Channel 1

Channel 2

Channel M

SNR Monitor

Select MAXSNR=γmax

Selection Combining


Outage Probability of a Single Branch

Selection Combining Consider M independent Rayleigh fading channels

available at the receiver

iγ

iγ Γ1

f eΓ

Average SNR at all Diversity BranchesSNR = Γ

Instantaneous SNR at Diversity Branch iSNR = γi

Rayleigh Fading Voltage means Exponentially Distributed Power

iγ γ γ

i i γ γ γΓ Γ

0

1Pr e d 1 e

Γ

Outage Probability of of Selection Diversity Combining

γ

max γ γ

M

ΓPr 1 e



Channel 1

Channel 2

Channel M

G1

G2

GM

∑

r1

r2

rM

Maximal Ratio Combining

Selection Combining does not benefit from power received across all diversity branches

Maximal Ratio Combining conducts a weighted sum across all branches with the objective of maximizing SNR


Envelope applied to receiver detector

M

MRC i ii 1

r rG

Total Noise Power applied to detector

M2

MRC ii 1

N N G

SNR at the receiver detector

γ

2 2M MMRC 2

MRC i i ii 1 i 1MRC

rrG N G

N

Cauchy’s Inequality 2 2 2i i i iab a b

iγ γ

2 2M M M 2i i

i i M M2i 1 i 1 i 1

MRC iM M2 2 i 1 i 1i i

i 1 i 1

r rNG NG

1N N rN

N G N G

γMRC is maximized when Gi=ri (MRC requires channel measurements)

Maximal Ratio Combining Consider M independent Rayleigh fading channels



γMRC is maximized when Gi=ri (MRC requires channel measurements)

iγ

iγ Γ1

f eΓ

Rayleigh Fading Voltage means

Exponentially Distributed Power

iγ γM

MRCi 1

SNR γMRC is Gamma distributed (sum of M exponential random variables)

Outage Probability of of Maximal Ratio Diversity Combining

γ

γ γ γ γ γ

MRCM 1Γ

MRC

MRC MRCM0

ePr d

Γ M 1 !

γ

γγ

MRCM 1Γ

MRC

MRC M

ef

Γ M 1 !

Maximal Ratio Combining


Maximal Ratio Combining requires estimation of the channel across all diversity branches

Equal Gain Combining conducts a sum across all branches (i.e. Gi=1 for all i)


Channel 1

Channel 2

Channel M

∑

r1

r2

rM

Equal Ratio Combining


EGC is a special case of MRC with Gi=1SNR and outage probability performance in EGC is inferior to that of MRC

Envelope applied to receiver detector

Total Noise Power applied to detector

SNR at the receiver detector

M

EGC ii 1

r r

EGCN MN

γ

2 2MEGC

EGC ii 1EGC

rr MN

N

Equal Gain Combining Consider M independent Rayleigh fading channels


Wireless Communication Channels: Small-Scale Fading.

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