Wireless Communication Channels: SmallScale Fading.

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Transcript of Wireless Communication Channels: SmallScale Fading.
Wireless Communication Channels: SmallScale Fading
© Tallal Elshabrawy 2
Clarke’s Model for Flat FadingAssumptions: Mobile traveling in x
direction Vertically polarized wave Multiple waves in the xy
plane arrive at the mobile antenna at the same time
Waves arrive at different angles α
in xy 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 EfieldCn random variable representing the amplitude of individual wavesfc carrier frequencyφn random phase shift due to distance traveled by the nth wave
© Tallal Elshabrawy 3
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
© Tallal Elshabrawy 4
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
© Tallal Elshabrawy 5
Rayleigh Fading Distribution
r r
rp r
r
2
2 2exp 0
σ 2σ
0 0
α
in xy 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
© Tallal Elshabrawy 6
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
© Tallal Elshabrawy 7
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 xy 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 zeroorderσ2: Time average received power of the nondominant components
© Tallal Elshabrawy 8
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 nondominant waves
K=6 dB
K=∞ dB
Rayleigh Distribution
© Tallal Elshabrawy 9
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
© Tallal Elshabrawy 10
0 10 20 30 40 50 6030
40
50
60
70
80
90
100
0 10 20 30 40 50 6015
10
5
0
5
10
15
0 10 20 30 40 50 6060
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)
SmallScale 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 SmallScale 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
Intracell Interference
Intercell Interference
bk TH : Packets Lost (Outage) bk TH : Packets Received Correctly
12
Diversity Techniques
© Tallal Elshabrawy 14
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
© Tallal Elshabrawy 15
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
© Tallal Elshabrawy 16
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
© Tallal Elshabrawy 17
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
© Tallal Elshabrawy 18
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
© Tallal Elshabrawy 20
Select the strongest signal
Transmitter Receiver
Channel 1
Channel 2
Channel M
SNR Monitor
Select MAXSNR=γmax
Selection Combining
© Tallal Elshabrawy 21
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
© Tallal Elshabrawy 22
Transmitter Receiver
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
© Tallal Elshabrawy 23
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
available at the receiver
© Tallal Elshabrawy 24
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
© Tallal Elshabrawy 25
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)
Transmitter Receiver
Channel 1
Channel 2
Channel M
∑
r1
r2
rM
Equal Ratio Combining
© Tallal Elshabrawy 26
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
available at the receiver