properties of the mobile radio propagation channel jean-paul m.g. linnartz department head cosine...
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Properties of the Mobile Radio Propagation Channel
Jean-Paul M.G. LinnartzDepartment Head CoSiNeNat.Lab., Philips Research
2JPL
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
3JPL
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
4JPL
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
5JPL
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
6JPL
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
7JPL
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
8JPL
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”
9JPL
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
10JPL
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
11JPL
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
12JPL
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)
13JPL
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
14JPL
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 )
15JPL
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 }
16JPL
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
17JPL
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, ..
18JPL
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
19JPL
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
20JPL
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
2ifi = 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)
?
21JPL
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
Frequency Dispersion
23JPL
Frequency Dispersion
• Frequency dispersion is caused by
the delay spread of the channel
• Frequency dispersion has no relation
to the velocity of the antenna
24JPL
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()
25JPL
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 -
26JPL
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)
27JPL
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
28JPL
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
29JPL
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
30JPL
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
31JPL
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
32JPL
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 .
33JPL
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
34JPL
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
35JPL
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
36JPL
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
37JPL
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
38JPL
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
39JPL
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
40JPL
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
41JPL
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
Effects of fading on modulated radio signals
JPL
Effects of Multipath (I)
Frequency
Tim
e
Narrowband
Frequency
Tim
e
OFDM
Frequency
Tim
e
Wideband
JPL
Effects of Multipath (II)
Frequency
Tim
e
+-+--+-+
DS-CDMA
Frequency
Tim
e +
-
-
FrequencyHopping
Tim
e
Frequency
+ - + -
+ - +-
+ - +-
MC-CDMA
45JPL
BER
• BER for Ricean fading
calculate
Time Dispersion Revisited
The duration of fades
47JPL
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
48JPL
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
49JPL
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
50JPL
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
+
51JPL
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
52JPL
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
53JPL
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
54JPL
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
55JPL
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
56JPL
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
57JPL
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.
58JPL
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
59JPL
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
60JPL
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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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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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 r Rayleigh
r(t2)givenr(t1) Ricean
r’(t1)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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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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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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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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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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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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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 )*()(