properties of the mobile radio propagation channel
DESCRIPTION
Properties of the Mobile Radio Propagation Channel. Jean-Paul M.G. Linnartz Department Head CoSiNe Nat.Lab., Philips Research. Statistical Description of Multipath Fading. The basic Rayleigh / Ricean model gives the PDF of envelope But: how fast does the signal fade? - PowerPoint PPT PresentationTRANSCRIPT
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Properties of the Mobile Radio Propagation Channel
Jean-Paul M.G. LinnartzDepartment Head CoSiNeNat.Lab., Philips Research
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JPL 2
Statistical Description of Multipath Fading
The basic Rayleigh / Ricean model gives the PDF of envelope But: how fast does the signal fade? How wide in bandwidth are fades?
Typical system engineering questions: What is an appropriate packet duration, to avoid fades? How much ISI will occur? For frequency diversity, how far should one separate carriers? How far should one separate antennas for diversity? What is good a interleaving depth? What bit rates work well? Why can't I connect an ordinary modem to a cellular phone?
The models discussed in the following sheets will provide insight in these issues
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JPL 3
The Mobile Radio Propagation Channel
A wireless channel exhibits severe fluctuations for small displacements of the antenna or small carrier frequency offsets.
Channel Amplitude in dB versus location (= time*velocity) and frequency
TimeFrequency
Amplitude
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JPL 4
Time Dispersion vs Frequency Dispersion
Time Domain Channel variations Delay spreadInterpretation Fast Fading InterSymbol Interference
Correlation Distance Channel equalization
Frequency Doppler spectrum Frequency selective fadingDomain Intercarrier Interf. Coherence bandwidthInterpretation
Time Dispersion Frequency Dispersion
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JPL 5
Two distinct mechanisms
1.) Time dispersion Time variations of the channel are caused by motion of the antenna Channel changes every half a wavelength Moving antenna gives Doppler spread Fast fading requires short packet durations, thus high bit rates Time dispersion poses requirements on synchronization and rate of convergence of
channel estimation Interleaving may help to avoid burst errors
2.) Frequency dispersion Delayed reflections cause intersymbol interference Channel Equalization may be needed. Frequency selective fading Multipath delay spreads require long symbol times Frequency diversity or spread spectrum may help
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JPL 6
Time dispersion of narrowband signal (single frequency)
Transmit: cos(2 fc t)
Receive: I(t) cos(2 fc t) + Q(t) sin(2 fc t)= R(t) cos(2 fc t + )
I-Q phase trajectory
• As a function of time, I(t) and Q(t) follow a random trajectory through the complex plane
• Intuitive conclusion: Deep amplitude fades coincide with large phase rotations
Animate
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JPL 7
Doppler shift and Doppler spread
• All reflected waves arrive from a different angle• All waves have a different Doppler shift
The Doppler shift of a particular wave is cos f c cv
= f 0
Maximum Doppler shift: fD = fc v / c
Joint Signal Model
• Infinite number of waves
• First find distribution of angle of arrival,
then compute distribution of Doppler shifts
• Uniform distribution of angle of arrival : f() = 1/2
• Line spectrum goes into continuous spectrum Calculate
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JPL 8
Doppler Spectrum
If one transmits a sinusoid, …what are the frequency components in the received signal?
Power density spectrum versus received frequency
Probability density of Doppler shift versus received
frequency
The Doppler spectrum has a characteristic U-shape.
Note the similarity with sampling a randomly-phased
sinusoid
No components fall outside interval [fc- fD, fc+ fD]
Components of + fD or -fD appear relatively often
Fades are not entirely “memory-less”
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JPL 9
Derivation of Doppler SpectrumT h e p o w e r s p e c t r u m S ( f ) i s f o u n d f r o m
S ( f ) = p f ( ) G ( ) + f ( - ) G ( - ) d
d f0f 0
w h e r e
f ( ) = 1 / ( 2 ) i s t h e P D F o f a n g l e o f i n c i d e n c e
G ( ) t h e a n t e n n a g a i n i n d i r e c t i o n
p l o c a l - m e a n r e c e i v e d p o w e r , a n d
0 cf = f 1 + v
c c o s
O n e f i n d s d
d f =
f ( f - f )D2
c2
1
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JPL 10
Vertical Dipole
R e ce ive d P o w e r S p e c tru m
)(2 π
3
2f-f-f
1p = S(f)
c2D
• D o p p le r spe c trum is ce n te red a ro u nd fc
• D o p p le r spe c trum ha s w id th 2 fD
A vertical dipole is omni-directional in horizontal plane
G(θ) = 1.5
We assume
• Uniform angle of arrival of reflections
• No dominant wave
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JPL 11
How do systems handle Doppler Spreads?
• Analog Carrier frequency is low enough to avoid problems (random FM)
• GSM Channel bit rate well above Doppler spread TDMA during each bit / burst transmission the channel is fairly constant. Receiver training/updating during each transmission burst Feedback frequency correction
• DECT Intended to pedestrian use: only small Doppler spreads are to be anticipated for.
• IS95 Downlink: Pilot signal for synchronization and channel estimation Uplink: Continuous tracking of each signal
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JPL 12
Autocorrelation of the signal
We now know the Doppler spectrum,
but ... how fast does the channel change?
Wiener-Kinchine Theorem:
Power density spectrum of a random signal is the Fourier Transform
of its autocorrelation Inverse Fourier Transform of Doppler spectrum gives autocorrelation
of I(t), or of Q(t)
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JPL 13
Derivation of Autocorrelation of I-Q components
W e f i n d t h e a u t o - c o r r e l a t i o n
g ( ) = I ( t ) I ( t + ) = Q ( t ) Q ( t + )
= S ( f ) 2 ( f - f ) d fc D
c D
f - f
f f
c
E E
c o s
A u t o - c o r r e l a t i o n d e p e n d s o n S ( f ) , t h u s i t d e p e n d s o n t h e d i s t r i b u t i o n o f
a n g l e o f a r r i v a l
• g ( = 0 ) = l o c a l - m e a n p o w e r : g ( 0 ) = E I 2 ( t ) = p
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JPL 14
For uniform angle of arrival
F o r a U - s h a p e d D o p p l e r s p e c t r u m , t h e a u t o - c o r r e l a t i o n f u n c t i o n i s
g ( ) = I ( t ) I ( t + ) = p J 2 f 0 D E
w h e r e
J 0 i s z e r o - o r d e r B e s s e l f u n c t i o n o f f i r s t k i n d : i n t e g r a l o v e r { c o s ( z s i n x ) }
f D i s t h e m a x i m u m D o p p l e r s h i f t
i s t h e t i m e d i f f e r e n c e
N o t e t h a t t h e c o r r e l a t i o n i s a f u n c t i o n o f d i s t a n c e o r t i m e o f f s e t :
D cf = fv
c =
d
w h e r e
d i s t h e a n t e n n a d i s p l a c e m e n t d u r i n g , w i t h d = v
i s t h e c a r r i e r w a v e l e n g t h ( 3 0 c m a t 1 G H z )
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JPL 15
Relation between I and Q phase
S . O . R i c e : C r o s s - c o r r e l a t i o n
h ( ) = I ( t ) Q ( t + ) = - Q ( t ) I ( t + )
= S ( f ) 2 ( f - f ) d fc D
c D
f - f
f f
c
E E
s i n
F o r u n i f o r m l y d i s t r i b u t e d a n g l e o f a r r i v a l
• D o p p l e r s p e c t r u m S ( f ) i s e v e n a r o u n d f c
• C r o s s c o r r e l a t i o n h ( ) i s z e r o f o r a l l
• I ( t 1 ) a n d Q ( t 2 ) a r e i n d e p e n d e n t
F o r a n y d i s t r i b u t i o n o f a n g l e s o f a r r i v a l
• I ( t 1 ) a n d Q ( t 1 ) a r e i n d e p e n d e n t { = 0 : h ( ) = 0 }
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JPL 16
PDF of the real amplitude R
),,,()φ,φ,,( 21212121 QQIIfJrrf
For the amplitude r1 and r2
212121π2
21 φφ)φ,φ,,(),( ddrrfrrf
2221
022
22
21
2421
21ρ1σ
ρ
ρ12σexp
ρ1σ),(
rrI
rrrrrrf
NB: )()(
),()( 2
1
2112 rRician
rf
rrfrrf
2
20
)Δω(1
τωρ
rms
c
T
J
Correlation:
R2 = I2 + Q2
Note: get more elegant expressions for
complex amplitude H
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JPL 17
Autocorrelation of amplitude R2 = I2 + Q2
E
ER(t)R(t + ) =
2 p 1 +
1
4
I(t)I(t+ )
p2
2
210 0
2121 ),()τ()(E drdrrrfrrtRtR
221
212 ρ;1;,σ
2
π)τ()(E FtRtR
The solution is known as the hypergeometric function F(a,b;c;z)
or, in good approximation, ..
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JPL 18
Autocorrelation of amplitude R2 = I2 + Q2
E
ER(t)R(t + ) =
2 p 1 +
1
4
I(t)I(t+ )
p2
2
Result for Autocovariance of Amplitude
Remove mean-value and normalize
Autocovariance
C J 2 f D 02
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JPL 19
Amplitude r(t0) and Derivative r’(t0) are uncorrelated
J0()
τ
τπ2)τ()(E
τ)(')(E
20
d
fdJtrtr
d
dtrtr D
Correlation is 0 for = 0
τ
τπ2)τ()(E
τ)(')(E 0
d
fdJthth
d
dthth D
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JPL 20
Simulation of multipath channels
Jakes' Simulator for narrowband channel
generate the two bandpass “noise” components by adding many sinusoidal signals. Their frequencies are non-uniformly distributed to approximate the typical U-shaped Doppler spectrum.
N Frequency components are taken at
2fi = fm cos -------- 2(2N+1)
with i = 1, 2, .., N
All amplitudes are taken equal to unity. One component at the maximum Doppler shift is also added, but at amplitude of 1/sqrt(2), i.e., at about 0.707 . Jakes suggests to use 8 sinusoidal signals. Approximation (orange)
of the U-Doppler spectrum (Black)
?
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JPL 21
How to handle fast multipath fading?
Analog User must speak slowly
GSM Error correction and interleaving to avoid burst errors Error detection and speech decoding Fade margins in cell planning
DECT Diversity reception at base station
IS95 Wideband transmission averages channel behaviour This avoids burst errors and deep fades
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Frequency Dispersion
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JPL 23
Frequency Dispersion
• Frequency dispersion is caused by
the delay spread of the channel
• Frequency dispersion has no relation
to the velocity of the antenna
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JPL 24
Frequency Dispersion: Delay Profile
Typical sample of impulse response h(t)
If we transmit a pulse (t) we receive h(t)
Delay profile: PDF of received power: "average h2(t)"
Local-mean power in delay bin is p f()
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JPL 25
RMS Delay Spread and Maximum delay spread
D e f i n i t i o n s
n - t h m o m e n t o f d e l a y s p r e a d n0
n = f ( ) d
R M S v a l u e
R M S
02 0 2 1
2T =
1 -
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JPL 26
Typical Values of Delay Spreads
Macrocells TRMS < 8 sec
• GSM (256 kbit/s) uses an equalizer
• IS-54 (48 kbit/s): no equalizer
• In mountanous regions delays of 8 sec and more occur
GSM has some problems in Switzerland
Microcells TRMS < 2 sec
• Low antennas (below tops of buildings)
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JPL 27
Typical values of delay spreadPicocells 1 .. 2 GHz:
TRMS < 50 nsec - 300 nsec
• Home 50 nsec
• Shopping mall 100 - 200 nsec
• Railway station 200 - 450 nsec
• Office block 100 - 400 nsec
• Indoor: often 50 nsec is assumed
• DECT (1 Mbit/s) works well up to 90 nsec
Outdoors, DECT has problem if range > 200 .. 500
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JPL 28
Typical Delay Profiles
1) Exponential
2) Uniform Delay Profile
Experienced on some indoorchannels
Often approximated by N-RayChannel
3) Bad Urban
For a bandlimited signal, one may sample the delay profile. This give a tapped delay line model for the channel, with constant delay times
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JPL 29
COST 207 Typical Urban Reception (TU6)
COST 207 describes typical channel characteristics for over transmit bandwidths of 10 to 20 MHz around 900MHz. TU-6 models the terrestrial propagation in an urban area. It uses 6 resolvable paths
COST 207 profiles were adapted to mobile DVB-T reception in the E.U. Motivate project.
Tap number Delay (us) Power (dB) Fading model
1 0.0 -3 Rayleigh
2 0.2 0 Rayleigh
3 0.5 -2 Rayleigh
4 1.6 -6 Rayleigh
5 2.3 -8 Rayleigh
6 5.0 -10 Rayleigh
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JPL 30
COST 207 A sample fixed channel
Tap number Delay ( (s) Amplitude r Level (dB) Phase ( (rad)
1 0.050 0.36 -8.88 -2.875
2 0.479 1 0 0
3 0.621 0.787 -2.09 2.182
4 1.907 0.587 -4.63 -0.460
5 2.764 0.482 -6.34 -2.616
6 3.193 0.451 -6.92 2.863
• COST 207 Digital land mobile radio Communications, final report, September 1988.
• European Project AC 318 Motivate: Deliverable 06: Reference Receiver Conditions for Mobile Reception, January 2000
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JPL 31
How do systems handle delay spreads?
Analog Narrowband transmission
GSM Adaptive channel equalization Channel estimation training sequence
DECT Use the handset only in small cells with small delay spreads Diversity and channel selection can help a little bit
“pick a channel where late reflections are in a fade”
IS95 Rake receiver separately recovers signals over paths with excessive
delays
DTV and Digital Audio Broacasting OFDM multi-carrier modulationThe radio channel is split into many narrowband (ISI-free) subchannels
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JPL 32
Correlation of Fading vs. Frequency Separation
W h e n d o w e e x p e r i e n c e f r e q u e n c y - s e l e c t i v e f a d i n g ?
H o w t o c h o o s e a g o o d b i t r a t e ?
W h e r e i s f r e q u e n c y d i v e r s i t y e f f e c t i v e ?
I n t h e n e x t s l i d e s , w e w i l l . . . g i v e a m o d e l f o r I a n d Q , f o r t w o s i n u s o i d s w i t h t i m e a n d f r e q u e n c y o f f s e t , d e r i v e t h e c o v a r i a n c e m a t r i x f o r I a n d Q , d e r i v e t h e c o r r e l a t i o n o f e n v e l o p e R , g i v e t h e r e s u l t f o r t h e a u t o c o v a r i a n c e o f R , a n d d e f i n e t h e c o h e r e n c e b a n d w i d t h .
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JPL 33
Inphase and Quadrature-Phase Components
Consider two (random) sinusoidal signals
Sample 1 at frequency f1 at time t1 Sample 2 at frequency f2 = f1 + f at time t2
Effect of displacement on each phasor:Spatial or temporal displacement: Phase difference due to DopplerSpectral displacement Phase difference due to excess delay
Mathematical Treatment:• I(t) and Q(t) are jointly Gaussian random
processes• (I1, Q1, I2, Q2) is a jointly Gaussian random
vectorConsider complex H1 = I1+jQ1, H2 =I2+jQ2 as a
jointly Gaussian random vector
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JPL 34
Multipath Channel
• Transmit signal s(t)
• Received signal r(t) = an gn (t) + n(t),
Channel model:
Iw reflected waves have
the following properties:
• Di is the amplitude
• Ti is the delay
Signal parameters:
•an is the datac is the carrier frequency
)()}(ωωexp{)(1
0tnTtnjDatr
wI
iiscin
}exp{)( tjats cn
TO DO make math consistent
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JPL 35
Doppler Multipath Channel
Correlation between p-th and q-th derivative
1
0
1
0}exp{E
)(*)(E
w wI
i
I
kkickip
TTjDDdt
tdg
td
tdg
TO DO make math consistent
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JPL 36
Doppler Multipath Channel
ddTT
DDI rmsrmsAi
iiw
exp2
1E
lim *
)dθθcos(2ωθcos2::τ ffidtTiA ii
0
2
0
)(*)(
exp
cos2
)1(E
dT
dT
j
c
vgg
rms
qp
rms
qpq
qp
qps
qpcq
mp
n
TO DO make math consistent
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JPL 37
Random Complex-Gaussian Amplitude
Special case
• This defines the covariance matrix of subcarrier amplitudes at different frequencies
• This is used in OFDM for cahnnel estimation
srmsmn Tmnj
vvω)(1
1E
TO DO make math consistent
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JPL 38
Coherence BandwidthDefinition: Coherence. Bandwidth is the frequency separation for which the
correlation coefficient is down from 1 to 0.5 Thus 1 = 2(f1 - f2) TRMS
so Coherence Bandwidth BW = 1 (2 TRMS)
We derived this for an exponential delay profile
Another rule of thumb:
ISI affects BER if Tb > 0.1TRMS
Conclusion: Either keep transmission bandwidth much smaller than the coherence
bandwidth of the channel, or use signal processing to overcome ISI, e.g. Equalization DS-CDMA with rake OFDM
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JPL 39
Frequency and Time Dispersion
M o d e l : E a c h w a v e h a s i t s o w n a n g l e a n d e x c e s s d e l a y
A n t e n n a m o t i o n c h a n g e s p h a s e
c h a n g i n g c a r r i e r f r e q u e n c y c h a n g e s p h a s e
T h e s c a t t e r i n g e n v i r o n m e n t i s d e f i n e d b y
a n g l e s o f a r r i v a l
e x c e s s d e l a y s i n e a c h p a t h
p o w e r o f e a c h p a t h
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JPL 40
Scatter Function of a Multipath Mobile Channel
Gives power as function of
Doppler Shift (derived from angle of arrival )
Excess Delay
Example of a scatter plot
Horizontal axes: x-axis: Excess delay time y-axis:Doppler shiftVertical axis z-axis: received power
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JPL 41
Special casesZero displacement / motion: = 0Zero frequency separation: f = 0
1 2R ,R2
02
21 2
2 2RMS
C = J (2
v)
1 + 4 ( f - f ) T
Freq. and time selective channels
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Effects of fading on modulated radio signals
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JPL
Effects of Multipath (I)
Frequency
Tim
e
Narrowband
Frequency
Tim
e
OFDM
Frequency
Tim
e
Wideband
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JPL
Effects of Multipath (II)
Frequency
Tim
e
+-+--+-+
DS-CDMA
Frequency
Tim
e +
-
-
FrequencyHopping
Tim
e
Frequency
+ - + -
+ - +-
+ - +-
MC-CDMA
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JPL 45
BER
• BER for Ricean fading
calculate
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Time Dispersion Revisited
The duration of fades
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JPL 47
Time Dispersion Revisited: Duration of Fades
In the next slides we study the temporal behavior of fades.
Outline:
# of level crossings per second
Model for level crossings
Derivation
Model for I, Q and derivatives
Model for amplitude and derivative
Discussion of result
Average non-fade duration
Effective throughput and optimum packet length
Average fade-duration
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JPL 48
Two state model
Very simple mode: channel has two-states Good state: Signal above “threshold”, BER is virtually zero Poor state: “Signal outage”, BER is 1/2, receiver falls out of sync, etc
Markov model approach: Memory-less transistions Exponential distribution of sojourn time
This model may be sufficiently realistic for many investigations
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JPL 49
Level crossings per second
• Av. number of crossings per sec = [av. interfade time]-1
Number of level crossing per sec is proportional to
• speed r' of crossing R (derivative r' = dr/dt)
• probability of r being in [R, R + dR]. This Prob = fR(r) dr
)(rfdt
drN R
)()(1
rfdt
drdrRrRP
TN R
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JPL 50
Derivation of Level Crossings per Second
Random process r’ is derivative of the envelope r w.r.t. time Note: here we need the joint PDF;
not the conditional PDF f(r’r=R)
• We derive f(r’r=R) from f(r’r) and f(i1,i2,q1,q2)
''' dr R),f(r r = N0
+
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JPL 51
Covariance matrix of (I, Q, I’, Q’)
In-phase, quadrature and the derivatives are Jointly Gaussian withcovariance matrix:
Here bn is n-th moment of Dopplerspectral power density
nn
f - f
f + f
cn
b = (2 ) S(f) (f - f ) dfc D
c D
For uniform angles of arrival
n
Dn
b =
p (2 f )(n -1)!!
n!
n
0 n
for even
for odd
where (n-1)!! = 1 3 5 .....(n-1)
I,Q,I,Q1
1
1 2
1 2
= p 0 0 b
0 p - b 0
0 - b b 0
b 0 0 b
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JPL 52
Joint PDF of R, R’, , ’
. . . . A f t e r s o m e a l g e b r a . . .
b
r +b
r +
p
r2
1-
bp4
r=),,rf(r,2
22
2
2
22
2
'2'
expˆˆ
S o
r a n d r ’ a r e i n d e p e n d e n t
p h a s e i s u n i f o r m
F a s t p h a s e c h a n g e o n l y o c c u r w h e n a m p l i t u d e i s s m a l l
A v e r a g i n g o v e r a n d ’ g i v e s
r i s R a y l e i g h
r ^ i s z e r o m e a n G a u s s i a n w i t h v a r i a n c e b 2
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JPL 53
Level Crossings per Second
W e i n s e r t t h i s p d f i n
''' dr )R=R,f(r r = N 0
0
+
W e f i n d
+ D -1
N = 2 f
e
w h e r e
i s t h e f a d e m a r g i n = 2 p / ( R 02 )
L e v e l c r o s s i n g r a t e h a s a m a x i m u m f o r t h r e s h o l d s R 0 c l o s e t o t h e m e a n v a l u e o f
a m p l i t u d e . S i m i l a r l y f o r R i c e a n f a d i n g
+D R 0N = p f f ( R )
Calculate
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JPL 54
Average Fade / Nonfade Duration
F a d e d u r a t i o n s a r e r e l e v a n t t o c h o o s e p a c k e t d u r a t i o n
D u r a t i o n d e p e n d o n
F a d e m a r g i n = 2 p / R 02
D o p p l e r s p r e a d
A v e r a g e f a d e d u r a t i o n T F N T = ( R R )F 0P r
A v e r a g e n o n f a d e d u r a t i o n T N F+
N F 0N T = ( R R )P r
Calculate
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JPL 55
Average nonfade duration
N FD
T = = 2 f
1 P r ( o u t a g e )
L e v e l C r o s s R a t e
A v e r a g e n o n f a d e d u r a t i o n i s i n v e r s e l y p r o p o r t i o n a l t o D o p p l e r s p r e a d
A v e r a g e n o n f a d e d u r a t i o n i s p r o p o r t i o n a l t o f a d e m a r g i n
Calculate
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JPL 56
How to handle long fades when the user is stationary?
Analog Disconnect user
GSM Slow frequency hopping Handover, if appropriate Power control
DECT Diversity at base station Best channel selection by handset
IS95 Wide band transmission avoids most deep fades (at least in macro-cells) Power control
Wireless LANs Frequency Hopping, Antenna Diversity
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JPL 57
Optimal Packet length
We want to optimize
#User bitsEffective throughput = ---------------- Prob(success)
#Packet bits
This requires a trade off between
Short packets: much overhead (headers, sync. words etc).
Long packets:may experience fade before end of packet.
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JPL 58
Optimal Packet length
At 1200 bit/s, throughput is virtually zero:Almost all packets run into a fade. Packets are too long
At high bit rates, many packets can be exchanged during non-fadeperiods. Intuition:
Packet duration < < Av. nonfade duration
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JPL 59
Derivation of Optimal Packet length
A s s u m e e x p o n e n t i a l , m e m o r y l e s s n o n f a d e d u r a t i o n s( T h i s i s a n a p p r o x i m a t i o n : I n r e a l i t y m a n y n o n f a d e p e r i o d s h a v e
d u r a t i o n o f / 2 , d u e t o U - s h a p e d D o p p l e r s p e c t r u m )
S u c c e s s f u l r e c e p t i o n i f1 ) A b o v e t h r e s h o l d a t s t a r t o f p a c k e t2 ) N o f a d e s t a r t s b e f o r e p a c k e t e n d s
F o r m u l a :
P s u c c e x p e x p( ) = -1
- T
T = -
1-
2 f TL
N F
D L
w i t h T L p a c k e t d u r a t i o n
L a r g e f a d e m a r g i n : s e c o n d t e r m d o m i n a t e s : p e r f o r m a n c e i m p r o v e s o n l y s l o w l y w i t h i n c r e a s i n g O u t a g e p r o b a b i l i t y i s t o o o p t i m i s t i c Calculate
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JPL 60
Average fade duration
1
η
1exp
π
η-
f 2 = T
D
F
A F D i s i n v e r s e l y p r o p o r t i o n a l t o D o p p l e r s p r e a d
F a d e d u r a t i o n s r a p i d l y r e d u c e w i t h i n c r e a s i n g m a r g i n , b u t t i m e b e t w e e n f a d e s
i n c r e a s e s m u c h s l o w e r
E x p e r i m e n t s : F o r l a r g e f a d e m a r g i n s : e x p o n e n t i a l l y d i s t r i b u t e d f a d e d u r a t i o n s
R e l e v a n t t o f i n d l e n g t h o f e r r o r b u r s t s a n d d e s i g n o f i n t e r l e a v i n g
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JPL 61
Conclusion
• The multipath channel is characterized by two effects: Time and Frequency Dispersion
• Time Dispersion effects are proportional to speed and carrier frequency
• System designer need to anticipate for channel anomalies
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JPL 62
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JPL 63
Statistical properties of a Rayleigh signal
Parameter Probability Distribution
I(t1) and Q(t1) i.i.d. Gaussian, Zero mean
I(t1) and Q(t2) Correlated jointly Gaussian
I(t2) given I(t1) Gaussian
Amplitude rRayleigh
rgivenrRicean
r’derivative r Gaussian, Independent of amplitude
Phase Uniform
Derivative of Phase Gaussian, Dependent on amplitude
Power Exponential
The nice thing about jointly Gaussian r.v.s is that the covariance matrix fully describes the behavior
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JPL 64
Taylor Expansion of Amplitude
• Rewrite the Channel Model as follows
• Tayler expansion of the amplitude
• gn(t) = gn(0)+ gn
(1) (t-t) + gn(2) (t-t)2/2 + .. .
• gn(q) : the q-th derivative of amplitude wrt time, at instant t = t.
• gn(p) is a complex Gaussian random variable
• To address frequency separation ns (later):
)(}exp{)()( tntjtgatr cnn
1
0
)( })(exp{wI
itiisci
qi
qn jTnjDjg
)(}))((exp{)(1
0tntjTtjDatr
wI
iiicin
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JPL 65
Doppler Multipath Channel
Simplified baseband model for narrowband linear modulation
Received signal r(t) = an g(t) + n(t), with time-varying channel amplitude g(t)
Channel model:
Iw reflected waves, each has the following properties:
• Di is the amplitude
I is the Doppler shift
• Ti is the path delay
Note g(t) is not an impulse response;
It is the channel amplification and phase
Signal parameters:• linear modulation • an is the user data• c is the carrier frequency
• n(t) is the noise
1
0
)( })(exp{)(wI
iiisci
qi
qn tjTnjDjtg
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JPL 66
Doppler Multipath Channel
p-th Derivative gp(t) of the channel w.r.t. to time t
All derivatives are gaussian rv’s if Iw and Di i.i.d.
The covariance matrix fully describes the statistical properties
1
0}exp{
)( wI
iiic
pii
pp
p
tjTjDjdt
tgd
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JPL 67
Doppler Multipath Channel
Correlation between p-th and q-th derivative
1
0
1
0
)*()(
}exp{1E
)()(E
w wI
i
I
kkikicki
qk
pi
qpq
q
p
p
tjTTjDDj
dt
tdg
td
tdg
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JPL 68
Doppler Multipath Channel
ddTT
DDI rmsrmsAi
iiw
exp2
1E
lim *
)dθθcos(2ωθcos2::τ ffidtTiA ii
0
2
0
)(*)(
exp
cos2
)1(E
dT
dT
j
c
vgg
rms
qp
rms
qpq
qp
qps
qpcq
mp
n
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JPL 69
Random Complex-Gaussian Amplitude
It can be shown that for p + q is even
and 0 for p + q is odd.
• This defines the covariance matrix of subcarrier amplitudes and derivatives,
• OFDM: allows system modeling and simulation between the input of the transmit I-FFT and output of the receive FFT.
srms
qpqqp
Dq
mp
n Tmnj
j
qp
qpfvv
)(1
)1(
!)!(
!)!1(2E )*()(