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Fredrik TufvessonDepartment of Electrical and Information Technology
Lund University, [email protected]
Channel Modelling – ETIM10Lecture no:
2011-03-01 Fredrik Tufvesson - ETIM10 1
11Channel modelling
repetition
2011-03-01 Fredrik Tufvesson - ETIM10 2
Why channel modelling?
• The performance of a radio system is ultimately determined by the radio channel
• The channel models basis for – system design– algorithm design– antenna design etc.
• Trend towards more interaction system-channel– MIMO– UWB– radio based positioning
Without reliable channel models, it is hard to design radio systems that work well in real environments.
2011-03-01 Fredrik Tufvesson - ETIM10 3
THE RADIO CHANNELIt is more than just a loss• Some examples:
– behavior in time/place?– behavior in frequency?– directional properties?– bandwidth dependency?– behavior in delay?
2011-03-01 Fredrik Tufvesson - ETIM10 4
THE RADIO CHANNELSome properties
• Path loss– Roughly, received power decays exponentially with distance
• Large-scale fading– Large objects, compared to a wavelength, in the signal path
obstruct the signal
• Small-scale fading– Objects reflecting the signal causes multipath propagation
from transmitter to receiver
exponentn PropagatioDistance power dTransmitte power Received −×∝
2011-03-01 Fredrik Tufvesson - ETIM10 5
Free-space loss
d
ARX
If we assume RX antenna to be isotropic:2
4RX TXP Pdλπ
⎛ ⎞= ⎜ ⎟⎝ ⎠
Attenuation between twoisotropic antennas in freespace is (free-space loss):
( )24⎟⎠
⎞⎜⎝
⎛=λπddLfree
2011-03-01 Fredrik Tufvesson - ETIM10 6
Large-scale fadingLog-normal distribution
Measurements confirm that in many situations, the large-scalefading of the received signal strength has a normal distributionin the dB domain.
”POWER” [dB]
dBTXP |
dBRXP |
|dBL
( ) ( )2| 0|| 2
||
1 exp22dB dB
dBF dBF dB
L Lpdf L
σπσ
⎛ ⎞−⎜ ⎟= −⎜ ⎟⎝ ⎠
Note dB scale
dBDeterministic meanvalue of path loss, L0|dB
( )|dBpdf L
Standard deviation | 4 10 dBF dBσ ≈ K
2011-03-01 Fredrik Tufvesson - ETIM10 7
Small-scale fadingMany incoming waves
1 1,r φ2 2,r φ
3 3,r φ4 4,r φ
,r φ
( ) ( ) ( ) ( ) ( )1 1 2 2 3 3 4 4exp exp exp exp expr j r j r j r j r jφ φ φ φ φ= + + +
1r1φ
2r 2φ
3r
3φ
4r4φ
r
φ
Many incoming waves withindependent amplitudes
and phases
Add them up as phasors
2011-03-01 Fredrik Tufvesson - ETIM10 8
Small-scale fadingRayleigh fading
No dominant component(no line-of-sight)
2D Gaussian(zero mean)
Tap distribution
( )2
2 2exp2
r rpdf rσ σ
⎛ ⎞= −⎜ ⎟
⎝ ⎠
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
2011-03-01 Fredrik Tufvesson - ETIM10 9
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 2
2
2
exp2
Power in LOS componentPower in random components 2
r r A rApdf r I
Ak
σ σ σ
σ
⎛ ⎞+ ⎛ ⎞= −⎜ ⎟ ⎜ ⎟⎝ ⎠⎝ ⎠
= =
Amplitude distribution
Rice
r α=
0 1 2 30
0.5
1
1.5
2
2.5k = 30
k = 10
k = 0
TX RX
( )Im a ( )Re a
2011-03-01 Fredrik Tufvesson - ETIM10 10
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
is the gamma function
with m it is possible to adjust the dominating power
2 1 2
2
2
2 2
2( ) ( ) exp( )( )
( )
( )
m mm mpdf r r rm
m
r
mr
−= −Γ Ω Ω
Γ
Ω =
Ω=
−Ω
2011-03-01 Fredrik Tufvesson - ETIM10 11
Both small-scale and large-scale fading
Large-scale fading - lognormal fading gives a certain mean Small scale fading – Rayleigh distributed given a certain meanThe 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 eσ µπσσπ
σ σσ π
−∞ −−
= ∫log-normal std
log-normal mean
small-scale std for complex components
2011-03-01 Fredrik Tufvesson - ETIM10 12
Small-scale fadingDoppler shifts
c
rvθ
0f f ν= +
Frequency of received signal:
( )0 cosrvfc
ν θ= −
where the doppler shift isReceiving antenna moves withspeed vr at an angle θ relativeto the propagation directionof the incoming wave, whichhas frequency f0.
The maximal Doppler shift is
max 0vfc
ν =
2011-03-01 Fredrik Tufvesson - ETIM10 13
Small-scale fadingThe Doppler spectrum
0 maxf ν− 0 maxf ν+0f
( )0DS fν −
( ) ( ) 2
2 2max
1
jDS e dπν τν ρ τ τ
π ν ν
− Δ= Δ Δ
∝−
∫
Uncorrelated scatteringwith uniform angular distribution
Doppler spectrum by Fouriertransformation of the timecorrelation of the signal:
for max maxν ν ν− < <
Doppler spectrumat center frequency f0.
What does this mean in practice?
2011-03-01 Fredrik Tufvesson - ETIM10 14
Condensed parametersThe time correlation
A property closely related to the Doppler spectrun is the time correlationof the channel. It is in fact the inverse Fourier transform of the Dopplerspectrum:
( ) ( ) ( )exp 2t Bt P j t dρ ν πν ν∞
−∞
Δ = Δ∫
0 0.1 0.2 0.3 0.4 0.5 0.6-90
-85
-80
-75
-70
-65
-60
-55
-50
-45
-40
-35
Position (m)
Fre
quen
cy r
esp
(dB
)
Compare 1/(2*π*vmax)=0.014 s
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1-0.5
0
0.5
1
Tim
e co
rr
Position (m)
measuredtheoretical
2011-03-01 Fredrik Tufvesson - ETIM10 15
Condensed parametersCoherence timeGiven the time correlation of a channel, we can define thecoherence time TC:
( )t tρ Δ
tΔ
( )0tρ
( )02tρ
CT
What does the coherencetime tell us?It shows us over how longtime we can assumethat the channel is fairlyconstant.
E.g. radio systems transmittingdata in frames much shorterthan TC will not experience anyfading within a single frame.
2011-03-01 Fredrik Tufvesson - ETIM10 16
Narrow- versus wide-bandChannel impulse response
The same radio propagation environment is experienceddifferently, depending on the system bandwidth.
“High” BW “Medium” BW “Low” BW( )h τ ( )h τ ( )h τ
τ τ τ
2011-03-01 Fredrik Tufvesson - ETIM10 17
2B
A wide-band system(bandwidth B2) will howeverexperience both frequencyselectivity and delaydispersion.
1B
A narrow-band system(bandwidth B1) will notexperience any significantfrequency selectivity ordelay dispersion.
Narrow- versus wide-bandChannel frequency response
( )|dB
H f
fNote that narrow- or wide-band depends on the
relation between the channel and the system bandwidth. It is not an absolute measure.
2011-03-01 Fredrik Tufvesson - ETIM10 18
Condensed parametersPower-delay profile (cont.)
We can “reduce” the PDP into more compact descriptions of the channel:
( )mP P dτ τ∞
−∞= ∫
( )m
m
P dT
P
τ τ τ∞
−∞= ∫
( )2
mm
P dS T
P
τ τ τ∞
−∞= −∫
2
12
N
m ii
P σ=
=∑
2
12
N
i ii
mm
TP
τ σ==∑
2 2
12
N
i ii
mm
S TP
τ σ== −∑
Total power (time integrated):
Average mean delay:
Average rms delay spread:
For our tapped-delay linechannel:
2 2
2011-03-01 Fredrik Tufvesson - ETIM10 19
Condensed parametersFrequency correlation
A property closely related to the power-delay profile (PDP) is the frequencycorrelation of the channel. It is in fact the Fourier transform of the PDP:
( ) ( ) ( )exp 2f f P j f dρ τ π τ τ∞
−∞Δ = − Δ∫
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
x 108
-90
-85
-80
-75
-70
-65
-60
-55
-50
-45
-40
-35
Frequency (Hz)
Fre
quen
cy r
esp
(dB
)
Compare 1/(2*π*τrms)=9.8 MHz
0 1 2 3 4 5 6
x 107
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Frequency (Hz)
Freq
cor
r
based on PDPbased on H(f)
2011-03-01 Fredrik Tufvesson - ETIM10 20
Condensed parametersCoherence bandwidth
Given the frequency correlation of a channel, we can define thecoherence bandwidth BC:
( )f fρ Δ
fΔ
( )0fρ
( )02fρ
CB
What does the coherencebandwidth tell us?It shows us over how largea bandwidth we can assumethat the channel is fairlyconstant.
Radio systems using abandwidth much smallerthan BC will not noticethe frequency selectivityof the channel.
2011-03-01 Fredrik Tufvesson - ETIM10 21
Channel measures
2011-03-01 Fredrik Tufvesson - ETIM10 22
Complex dielectric constant
i i j e,i
2 fc
dielectric constant
conductivity
Describes the dielectric material in one single parameter
2011-03-01 Fredrik Tufvesson - ETIM10 23
Reflection and transmission
sin t
sin e
1
2.
eΘ rΘ
tΘ
1ε
2ε
reflected angle
transmitted angle
2011-03-01 Fredrik Tufvesson - ETIM10 24
Diffraction
2011-03-01 Fredrik Tufvesson - ETIM10 25
Diffraction, Huygen’s principle
Each point of a wavefront can be considered as a source of a spherical wave
Bending around corners and edges
2011-03-01 Fredrik Tufvesson - ETIM10 26
Diffraction – Bullington’s method
Replace all screens with one equivalent screen
Height determined by the steepest angle
Simple but a bit optimistic
tangent
equivalent screen
Etotal exp jk0x 12
exp j /42
F F F k2d1d2d1 d2
2011-03-01 Fredrik Tufvesson - ETIM10 27
Scattering
Smooth surface
Specularreflection
Scattering
Rough surface
Specularreflection
2011-03-01 Fredrik Tufvesson - ETIM10 28
Kirchhoff theory – scattering by rough surfaces
rough smoothexp 2 k0 h sin2
calculate distribution of the surface amplitude
assume no “shadowing” from surface
calculate a new reflection coefficient
for Gaussian surface distribution
standard deviation of height
angle of incidence
2011-03-01 Fredrik Tufvesson - ETIM10 29
Waveguiding
Waveguiding effects often result in lower propagation exponents
n=1.5-5
This means lower path loss along certain street corridors
2011-03-01 Fredrik Tufvesson - ETIM10 30
The WSSUS modelAssumptions
A very common wide-band channel model is the WSSUS-model.
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.(Contributions with different Doppler frequency areuncorrelated.)
The channel is built up by Uncorrelated Scatterers (US),meaning that the frequency correlation of the channels isinvariant over frequency. (Contributions with different delaysare uncorrelated.)
2011-03-01 Fredrik Tufvesson - ETIM10 31
Modelling methods
• Stored channel impulse responses– realistic– reproducible– hard to cover all scenarios
• Deterministic channel models– based on Maxwell’s equations– site specific– computationally demanding
• Stochastic channel models– describes the distribution of the field strength etc– mainly used for design and system comparisons
2011-03-01 Fredrik Tufvesson - ETIM10 32
The Okumura-Hata modelHow to calculate prop. loss
( )|logO H kmL A B d C− = + +
Small/medium-size cities
Metropolitanareas
Suburbanenvironments
Rural areas
( )( )( )( )0|
0|
1.1log 0.7
1.56log 0.8
MHz m
MHz
f h
f
− −
−
( )( )( )( )
20
20
8.29 log 1.54 1.1 for 200 MHz
3.2 log 11.75 4.97 for 400 MHz
m
m
h f
h f
− ≤
− ≥
( )ma h =
0
0
( )2
0|2 log / 28 5.4MHzf⎡ ⎤− −⎣ ⎦
( ) ( )2
0| 0|4.78 log 18.33log 40.94MHz MHzf f⎡ ⎤− + −⎣ ⎦
C =
( ) ( ) ( )( )
0|69.55 26.16log 13.82log
44.9 6.55logMHz b m
b
A f h a h
B h
= + − −
= −
hb and hmin meter
2011-03-01 Fredrik Tufvesson - ETIM10 33
The COST 231-Walfish-Ikegami modelHow to calculate prop. loss
0 msd rtsL L L L= + +
BS
MS
d
Free space Roof-topto street
Building multiscreen
Details about calculations can be found in the textbook, Section 7.6.2.
2011-03-01 Fredrik Tufvesson - ETIM10 34
Motley-Keenan indoor model
PL PL0 10n log d/d0 Fwall Ffloor
For indoor environments, the attenuation is heavily affected by the building structure, walls and floors play an important rule
distance dependentpath loss sum of attenuations
from walls, 1-20 dB/wall
sum of attenuation from the floors (often larger than wall attenuation)
site specific, since it is valid for a particular case
2011-03-01 Fredrik Tufvesson - ETIM10 35
Power delay profile
• Often described by a single exponential decay
• though often there is more than one “cluster”
( / ) 0( )
0sc
exp SP
otherwiseττ τ
τ− ≥⎧
= ⎨⎩
0,,
( ) 0( )
0
cck
sc kck k
P PSP
otherwiseτ
τ τ ττ
⎧− ≥⎪
= ⎨⎪⎩
∑
log( ( ))scP τ
τ
log( ( ))scP τ
τ
delay spread
2011-03-01 Fredrik Tufvesson - ETIM10 36
arrival time
• If the bandwidth is high, the time resolution is large so we might resolve the different multipath components
• Need to model arrival time– The Saleh-Valenzuela model:
– The Δ-K-model:
S1 S2
arrival rate: 0 ( )tλ
0 ( )K tλ
2011-03-01 Fredrik Tufvesson - ETIM10 37
Saleh-Valenzuela Model
• Originally not for UWB [A.M. Saleh, R.A. Valenzuela, 1987]• MPCs arrive in clusters• Impulse responses given by
• Path interarrival times given by Poisson-distributed arrival process
• Different occurance rates for clusters (L) and rays (l)
2011-03-01 Fredrik Tufvesson - ETIM10 38
Saleh-Valenzuela Model (cont’d)
Typical inter-cluster decay: 10-30 nsTypical intra-cluster decay: 1-60 ns
2011-03-01 Fredrik Tufvesson - ETIM10 39
Wideband modelsCOST 207 model for GSM
[ ]sτ µ
[ ]P dB010−
20−
30−0 1 [ ]sτ µ
[ ]P dB010−
20−
30−0 1 2 3 4 5 6 7
[ ]sτ µ
[ ]P dB010−
20−
30−0 5 10 [ ]sτ µ
[ ]P dB010−
20−
30−100 20
Four specified power-delay profiles
RURAL AREA TYPICAL URBAN
BAD URBAN HILLY TERRAIN
2011-03-01 Fredrik Tufvesson - ETIM10 40
Wideband modelsCOST 207 model for GSM
Four specified Doppler spectra
maxν− maxν+0
( ),s iP ν τ
maxν− maxν+0
maxν− maxν+0 maxν− maxν+0
CLASS GAUS1
GAUS2 RICE
0.5 siτ µ≤ 0.5 s 2 siµ τ µ< ≤
2 siτ µ> Shortestpath in
rural areas
( ),s iP ν τ
( ),s iP ν τ( ),s iP ν τ
2011-03-01 Fredrik Tufvesson - ETIM10 41
Narrowband vs. UWB Channel Models
• Assumptions about standard wireless channels:– “Narrowband” in the RF sense (bandwidth much smaller than carrier
frequency– WSSUS assumption– Complex Gaussian fading (Rayleigh or Rice) in each delay tap
• Specialties of UWB channel:– Bandwidth comparable to carrier frequency– Different frequency components can “see” different reflection/
diffraction coefficients of obstacles– Few components per delay bin ® central limit theorem (Gaussian
fading) not valid anymoreð New channel models are needed!!
2011-03-01 Fredrik Tufvesson - ETIM10 42
Why directional channel models?
• The spatial domain can be used to increase the spectral efficiency of the system– Smart antennas– MIMO systems
• Need to know directional properties– How many significant reflection points?– Which directions?
• Model independent on specific antenna pattern
2011-03-01 Fredrik Tufvesson - ETIM10 43
Double directional impulse response
h t, r TX, r RX, , ,1
N r
h t, r TX, r RX, , ,
TX position RX position
delay direction-of-departuredirection-of-arrival
h t, r TX, r RX, , , |a |ej
number of multipath components for these positions
2011-03-01 Fredrik Tufvesson - ETIM10 44
Angular spread
lτ
double directional delay power spectrum
angular delay power spectrum
angular power spectrum
power
E s , , , s , , , Ps , , ,
DDDPS , , Ps , , , d
ADPS , DDDPS , , GMS d
APS APDS , d
P APS d
2011-03-01 Fredrik Tufvesson - ETIM10 45
Goals of MIMO
• increase power• beamforming
• multiply data rates• spatially orthogonal channels
• mitigate fading• space-time coding
Array gain
Spatial multiplexing
Diversity
2011-03-01 Fredrik Tufvesson - ETIM10 46
H 1,1 H 2,1
H n,1 T
Transmitter Power P
Receiver
Antenna 1
Antenna 1
Antenna 2
Antenna 2
Antenna n T
Antenna n R
H 1, n R H 2, n R
H n, T n R
γ ...SNR at each receiver branch
Signal model
H...transfer function
TXRX
2011-03-01 Fredrik Tufvesson - ETIM10 47
Capacity formula
• Instantaneous channel characterized by matrix H• Shannon’s formula (for two-dimensional symbols):
• Foschini’s formula:
Hzsbitsn
C H
TnR
//detlog2 ⎟⎟⎠
⎞⎜⎜⎝
⎛⎥⎦
⎤⎢⎣
⎡+= HHI γ
HzsbitsHC //)||1(log 22 γ+=
2011-03-01 Fredrik Tufvesson - ETIM10 48
Channel measurements
In order to model the channel behavior we need to measureits properties
– Time domain measurements• impulse sounder• correlative sounder
– Frequency domain measurements• Vector network analyzer
• Directional measurements– directional antennas– real antenna arrays– multiplexed arrays– virtual arrays
2011-03-01 Fredrik Tufvesson - ETIM10 49
Real, multiplexed, and virtual arrays
• Real array: simultaneous measurement at all antenna elements
• Multiplexed array: short time intervals between measurements at different elements
• Virtual array: long delayno problem with mutual coupling
RX RX
RX
RX
RX
Digital Signal Processing
Digital Signal Processing
Digital Signal Processing
2011-03-01 Fredrik Tufvesson - ETIM10 50
Directional analysis
• The DoA can, e.g., be estimated by correlating the received signals with steering vectors.
• An element spacing of d=5.8 cm and an angle of arrival of φ =20 degrees gives a time delay of 6.6·10-11 s between neighboring elements
d
φ
d sin φ
a
1exp jk0dcosexp j2k0dcos
exp j M 1 k0dcos
2011-03-01 Fredrik Tufvesson - ETIM10 51
Important antenna parameters
• Directivitiy– Total power in a certain direction compared to total transmitted power
• Efficiency
• Q-factor– Stored energy compared to dissipated energy
• Mean effective gain– Include influence of random channel – Average received power compared to average received power by isotropic
antenna in real environment• Polarization• Bandwidth
rad
rad ohmic match
RR R R
η =+ +
Techniques for wireless positioning
Three main measurement principles:• Angle-of-arrival (AOA)• Received signal strength (RSS)• Propagation-time:
– Time-of-arrival (TOA)– Roundtrip-time-of-flight (RTOF)– Time-difference-of-arrival (TDOA)
2011-03-01 Fredrik Tufvesson - ETIM10 52
These differ both in terms of system requirements and in accuracy
Angle-of-arrival (AOA) based positioning
• Based on bearing estimation followed by intersection of different direction pointers
• Requires antenna arrays or directive antennas at measuring side: requires complex hardware
• Accuracy limited by size of antenna array or directivity• No requirements on synchronization2011-03-01 Fredrik Tufvesson - ETIM10 53
Mobile unit
Fixed unit
Fixed unit
α β
Received signal strength (RSS) based positioning
• Based on propagation-loss equations• Propagation-loss is often more complex than free-space
(1/d^2) loss, e.g., indoors:– Advanced models required– Fingerprinting (learn actual field strength from measuerements)
• Feasible implementation: Most radio modules already provide an RSS indicator
2011-03-01 Fredrik Tufvesson - ETIM10 54
Mobile unit
Fixed unitFixed unit
P(d1) P(d2)
Time-based positioning:Time-of-arrival (TOA)
• Based on one-way propagation time• Requires precise synchronization of all involved units (time
synchronization directly affects accuracy)– Ex. A 1 ns clock drift implies a distance error of 0.3 m
• Bandwidth dependent (accuracy inversely proportional to bandwidth)• Can provide higher accuracy than AOA and RSS methods• In practice, very expensive or less inaccurate
2011-03-01 Fredrik Tufvesson - ETIM10 55
Mobile unit
Fixed unitFixed unit
τ1 τ2