Fading Statistical description of the wireless channel · Ghassan Dahman \ Fredrik Tufvesson...
Transcript of Fading Statistical description of the wireless channel · Ghassan Dahman \ Fredrik Tufvesson...
Ghassan Dahman \ Fredrik TufvessonDepartment of Electrical and Information Technology
Lund University, Sweden
Channel Modelling – ETIM10Lecture no:
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FadingStatistical description of the
wireless channel
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Contents
• Why statistical description• Large scale fading• Fading margin• Small scale fading
– without dominant component– with dominant component
• Statistical models• Measurement example
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Why statistical description
• Unknown environment• Complicated environment• Can not describe everything in detail• Large fluctuations
• Need a statistical measure since we can not describe every point everywhere
“There is a x% probability that the amplitude/power will be above the level y”
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The WSSUS modelAssumptions
A very common wide-band channel model is the WSSUS-model. Roughly speaking it means that the statistical properties remain the same over the considered time (or area)
Recalling that the channel is composed of a number of differentcontributions (incoming waves), the following is assumed:
The channel is Wide-Sense Stationary (WSS), meaningthat the time correlation of the channel is invariant over time.
The channel is built up by Uncorrelated Scatterers (US), meaningthat contributions with different delays are uncorrelated.
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What is large scale and small scale?
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Large-scale fadingBasic principle
d
Received power
PositionA B C C
A
B
C
D
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Large-scale fadingLog-normal distribution
A normal distributionin the dB domain.
2| 0|
| 2||
1 exp22
dB dBdB
F dBF dB
L Lpdf L
dBDeterministic meanvalue of path loss, L0|dB
|dBpdf L
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Large-scale fadingExample• What is the probability that the small scale averaged amplitude
will be 10 dB above the mean if the large scale fading can be described as log-normal with a standard deviation of 5 dB?
dB
|dBpdf L
What if the standard deviation is 10 dB instead?
02.021010Pr 0 QQLLPdB
dBout
0L
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The Q(.)-functionUpper-tail probabilities
4.265 0.000014.107 0.000024.013 0.000033.944 0.000043.891 0.000053.846 0.000063.808 0.000073.775 0.000083.746 0.000093.719 0.000103.540 0.000203.432 0.000303.353 0.000403.291 0.000503.239 0.000603.195 0.000703.156 0.000803.121 0.00090
3.090 0.001002.878 0.002002.748 0.003002.652 0.004002.576 0.005002.512 0.006002.457 0.007002.409 0.008002.366 0.009002.326 0.010002.054 0.020001.881 0.030001.751 0.040001.645 0.050001.555 0.060001.476 0.070001.405 0.080001.341 0.09000
1.282 0.100000.842 0.200000.524 0.300000.253 0.400000.000 0.50000
x Q(x) x Q(x) x Q(x)
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If these are consideredrandom and independent,we should get a normaldistribution in thedB domain.
Large-scale fading, why log-normal?
12
N
Many diffraction points adding extra attenuation to the pathloss.
Total pathloss:
1 2tot NL L d| 1| 2| ||tot dB dB dB N dBdB
L L d
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Small-scale fadingTwo waves
Wave 2
Wave 1
ETX(t)=A cos(2 fct)
ERX1(t)=A1 cos(2 fct-2 / *d1)k0=2 /
E=E1 exp(-jkod1)
ERX2(t)=A2 cos(2 fct-2 / *d2)E=E2 exp(-jkod2) E1
E2
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Small-scale fadingTwo waves
Wave 1 + Wave 2
Wave 2
Wave 1
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Small-scale fadingDoppler shifts
c
rv
0f f
Frequency of received signal:
0 cosrvfc
where the doppler shift isReceiving antenna moves withspeed vr at an angle relativeto the propagation directionof the incoming wave, whichhas frequency f0.
The maximal Doppler shift is
max 0vfc
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Small-scale fadingDoppler shifts
• f0=5.2 109 Hz, v=5 km/h, (1.4 m/s) 24 Hz• f0=900 106 Hz, v=110 km/h, (30.6 m/s) 92 Hz
max 0vfc
How large is the maximum Doppler frequency at pedestrian speeds for 5.2 GHz WLAN and at highway speeds using GSM 900?
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Small-scale fadingTwo waves with Doppler
0 2 20
2exp( cos( ) )cvE E jk d f tc
Wave 2
Wave 1
ETX(t)=A cos(2 fct)
0 1 10
1exp( cos( ) )cvE E jk d f tc
v
E1E2
The two components have different Doppler shifts!The Doppler shifts will cause a random frequency modulation
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Small-scale fadingMany incoming waves
1 1,r 2 2,r
3 3,r4 4,r
,r
1 1 2 2 3 3 4 4exp exp exp exp expr j r j r j r j r j
1r1
2r 2
3r
3
4r4
r
Many incoming waves withindependent amplitudes
and phases
Add them up as phasors
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Small-scale fadingMany incoming waves
1r1
2r 2
3r
3
4r4
r
Re and Im components aresums of many independentequally distributed components
Re(r) and Im(r) are independent
The phase of r has a uniform distribution
2Re( ) (0, )r N
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Small-scale fadingRayleigh fading
No dominant component(no line-of-sight)
2D Gaussian(zero mean)
Tap distribution
2
2 2exp2
r rpdf r
Amplitude distribution
Rayleigh
0 1 2 30
0.2
0.4
0.6
0.8
r a
No line-of-sightcomponent
TX RXX
Im a Re a
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Small-scale fadingRayleigh fading
minr
min 2min
min 20
Pr 1 expr
rms
rr r pdf r drr
Probability that the amplitudeis below some threshold rmin:
0 r
Rayleigh distribution
2rmsr
2
2 2exp2
r rpdf r
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Small-scale fadingRayleigh fading – outage probability
• What is the probability that we will receive an amplitude 20 dB below the rrms?
• What is the probability that we will receive an amplitude below rrms?
2min
min 2Pr 1 exp 1 exp( 0.01) 0.01rms
rr rr
2min
min 2Pr 1 exp 1 exp( 1) 0.63rms
rr rr
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Small-scale fadingRayleigh fading – fading margin• To Ensure that we in most cases receive enough power we
transmit extra power – fading margin
2
2min
rmsrMr
2
| 10 2min
10log rmsdB
rMr
minr0 r2rmsr
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Small-scale fadingRayleigh fading – fading margin
How many dB fading margin, against Rayleigh fading, do we need toobtain an outage probability of 1%?
2min
min 2Pr 1 exprms
rr rr
1% 0.01
Some manipulation gives2
min21 0.01 exp
rms
rr
2min
2ln 0.99rms
rr
2min
2 ln 0.99 0.01rms
rr
2
2min
1/ 0.01 100rmsrMr
| 20dBM
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Small-scale fadingRayleigh fading – signal and interference• Both the desired signal and the
interference undergo fading• For a single user inteferer and
Rayleigh fading:
• where is the mean signal to interference radio
2
2 2 2
2( )( )
SIRrpdf rr
2
2 2( ) 1
( )SIR
rcdf rr
22 221
-20 -15 -10 -5 0 5 10 15 200
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
SIR(dB)
pdf for 10 dB mean signal to interference ratio
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Small-scale fadingRayleigh fading – signal and interference• What is the probability that the instantaneous SIR will be below 0 dB
if the mean SIR is 10 dB when both the desired signal and the interferer experience Rayleigh fading?
2
minmin 2 2
min
10Pr 1 1 0.09(10 1)( )
rr rr
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Small-scale fadingone dominating componentIn case of Line-of-Sight (LOS) one component dominates.• Assume it is aligned with the real axis
• The recieved amplitude has now a Ricean distribution instead of a Rayleigh– The fluctuations are smaller– The phase is dominated by the LOS component– In real cases the mean propagation loss is often smaller due
to the LOS• The ratio between the power of the LOS component and
the diffuse components is called Ricean K-factor
2Re( ) ( , )r N A 2Im( ) (0, )r N
2
2
Power in LOS componentPower in random components 2
Ak
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Small-scale fadingRice fading
A dominant component(line of sight)
2D Gaussian(non-zero mean)
Tap distribution
A
Line-of-sight (LOS)component with
amplitude A.
2 2
02 2 2exp2
r r A rApdf r I
Amplitude distribution
Rice
r
0 1 2 30
0.5
1
1.5
2
2.5k = 30
k = 10
k = 0
TX RX
Im a Re a
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Small-scale fadingRice fading, phase distribution
The distribution of the phase is dependent on the K-factor
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Small-scale fadingNakagami distribution• In many cases the received signal can not be described as a pure
LOS + diffuse components• The Nakagami distribution is often used in such cases
with m it is possible to adjust the dominating power
increasing m{1 2 3 4 5}
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Both small-scale and large-scale fading
Large-scale fading - lognormal fading gives a certain mean Small scale fading – Rayleigh distributed given a certain mean
The two fading processes adds up in a dB-scale
Suzuki distribution:
2
22
20log( )24
20
20 1( )2 ln(10) 2
F
r
F
rpdf r e e
log-normal sdt
log-normal mean
small-scale std for complex components
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Both small-scale and large-scale fading
• An alternative is to add fading terms (fading margins) independently
• Pessimistic approach, but used quite often• The communication link can be OK if there is a fading dip
for the small scale fading but the log-normal fading have a rather small attenuation.
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Some special cases
• Rayleigh fading
• Rice fading, K=0
Rice fading with K=0 becomes Rayleigh
• Nakagami, m=1
Nakagami with m=1 becomes Rayleigh
2
2 2exp2
r rpdf r
Example, shadowing from people
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Two persons communicating with each other using PDAs, signal sometimes blocked by persons moving randomly
Example, shadowing from people
0 2 4 6 8 10-40
-30
-20
-10
0
10
Time [s]
Rec
eive
d po
wer
[dB
]
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line-of-sight
obstructed LOS