Channel modelling repetition...Fredrik Tufvesson Department of Electrical and Information Technology...

Fredrik TufvessonDepartment of Electrical and Information Technology

Lund University, SwedenFredrik.Tufvesson@eit.lth.se

Channel Modelling – ETIM10Lecture no:

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11Channel modelling

repetition

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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.

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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?

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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 −×∝

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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

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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

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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

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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 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

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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 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

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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

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

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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 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

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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 =

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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?

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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

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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.

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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

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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.

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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

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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)

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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.

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Channel measures

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Complex dielectric constant

i i j e,i

2 fc

dielectric constant

conductivity

Describes the dielectric material in one single parameter

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Reflection and transmission

sin t

sin e

1

2.

eΘ rΘ

tΘ

1ε

2ε

reflected angle

transmitted angle

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Diffraction

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Diffraction, Huygen’s principle

Each point of a wavefront can be considered as a source of a spherical wave

Bending around corners and edges

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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

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Scattering

Smooth surface

Specularreflection

Scattering

Rough surface

Specularreflection

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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

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Waveguiding

Waveguiding effects often result in lower propagation exponents

n=1.5-5

This means lower path loss along certain street corridors

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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.)

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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

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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

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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.

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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

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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

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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

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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)

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Saleh-Valenzuela Model (cont’d)

Typical inter-cluster decay: 10-30 nsTypical intra-cluster decay: 1-60 ns

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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

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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

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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!!

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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

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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

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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

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Goals of MIMO

• increase power• beamforming

• multiply data rates• spatially orthogonal channels

• mitigate fading• space-time coding

Array gain

Spatial multiplexing

Diversity

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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

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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+=

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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

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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

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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

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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)

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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

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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

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Mobile unit

Fixed unitFixed unit

τ1 τ2