Channel Modelling ETI 085

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Channel Modelling – ETI 085

g

Lecture no:

44

Propagation mechanisms

Propagation mechanisms,

Channel characterization

Fredrik Tufvesson

Department of Electrical and Information Technology

Lund University, Swedeny,

[email protected]

2008-11-06 Fredrik Tufvesson - ETI 085 1

Contents

• Propagation mechanisms

– Transmission – Reflection Diffraction – Diffraction – Scattering – Waveguidingg g

• The tap delay line model

• Description of channel properties, introduction

p

p p

,

Complex dielectric constant

conductivity



i

 

i

−

j

_e,_i

2ff_c_c

dielectric constant, permittivity

Describes the dielectric material in one single parameter

Examples Rel. permittivity conductivity

Concrete 6 10-2 Gypsum 6.5 10-2 Wood 23 10-11 Glass 5 10-12 Ai 1 Air 1

Reflection and transmission

Θe  Θr. e

Θ

_r _{reflected angle} 1

ε

₁ transmitted angle 2

ε

t a s tted a g e sinΘ_t sinΘ_e



1 2 sinΘt iΘ  1 . sinΘe 2 t

Θ

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TM and TE waves behave differently

2cosΘ− 1cosΘ 1cosΘ 2cosΘ

R fl ti _ TM  − 2cosΘe− 1cosΘt 2cosΘe 1cosΘt TE  1cosΘe− 2cosΘt 1cosΘe 2cosΘt Reflection coefficient

T

1



2 Transmission coefficient

T



1

−



2

Transmission through walls – layered structures

Total transmission coefficient

T



T1T2e−j

1R1R2e−2j

d

Total transmission coefficient





12e−j2 2j

total reflection coefficient



112e−2j

T1

T2

with the electrical length in the wall





2





1

d

layer

cos



Θ

t



g

Diffraction, Huygen’s principle

Each point of a wavefront can

be considered as a source of a

spherical wave

Bending around corners

and edges

Diffraction

• Single or multiple edges edges • makes it possible to go behind corners • less pronounced

when the wavelength is small compared to is small compared to objects

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Diffraction coefficient

exp−jk0x F   F exp jt2 dt

The Fresnel integral is defined

F F   0

exp −jt

2 dt. with the Fresnel parameter

Total field



F





k 2d1d2 d1d2 E_total  exp−jk0x 1₂ − exp−j/4 2 F F Fresnel integral

Diffraction in real environments

validity region

For real environments we can represent buildings and objects as multiple screens For real environments we can represent buildings and objects as multiple screens

Diffraction – Bullington’s method

tangent

Replace all screens with one equivalent a ge

screen

Height determined by the steepest angle the steepest angle Simple but a bit optimistic optimistic equivalent screen E_total  exp−jk0x 1₂ − exp−j/4 2 F F



F





k 2d1d2 d1d2

Diffraction – Epstein-Petersen Method

L

compute diffraction loss for each screen separately and add the losses L1 L2 L3 Diffraction Ltot=L1+L2+L3 Diffraction –

The same approach is used also for the ITU model, but with an

i i l ti f t

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Scattering

Specular reflection Specular reflection Scattering

Smooth surface Rough surface

Kirchhoff theory – scattering by rough surfaces

calculate distribution of the surface amplitude

“ h d i ” assume no “shadowing” from surface

l l t fl ti

calculate a new reflection coefficient

for Gaussian surface distribution angle of incidence

roughsmoothexp −2 k0hsin 2

t d d d i ti f h i ht

standard deviation of height

Pertubation theory – scattering by rough

surfaces

Include shadowing effects by the surface i l d ti l h 2_W _ __E r h r h r  includes spatial correlation of surface – how fast are the changes i h i ht h r h r in height based on calculation of  based on calculation of an “effective” dielectric constant

More accurate than Krichhoff theory, especially for large angles of incidence and “rougher” surfaces

Waveguiding

Waveguiding effects often result in lower propagation exponents propagation exponents

n=1.5-5

This means lower path loss along certain street corridors

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The WSSUS model

Assumptions

A very common wide-band channel model is the WSSUS-model A very common wide band channel model is the WSSUS model. Recalling that the channel is composed of a number of different contributions (incoming waves) the following is assumed: contributions (incoming waves), the following is assumed:

The channel is Wide-Sense Stationary (WSS), meaning

th t th ti l ti f th h l i i i t ti

that the time correlation of the channel is invariant over time. (Contributions with different Doppler frequency are

uncorrelated.)

The channel is built up by Uncorrelated Scatterers (US), meaning that the frequency correlation of the channels is invariant over frequency. (Contributions with different delays are uncorrelated.)

The WSSUS model

A “prototype” used in the following

A prototype used in the following

In the following slides, we will use the following “prototype” tappedg g p yp pp

delay-line model of the channel as an example:

( )

exp

(

( )

)

(

)

N

h t

( )

τ

∑

α

( )

t

(

j

θ

( )

t

)

δ τ τ

(

)

1

,

_i

exp

_i _i i

h t

τ

α

t

j

θ

t

δ τ τ

=

∑

−

where the attenuations are described by the uncorrelated isotropicy p

scattering.

( )

t

exp

(

j

( )

t

)

α

_i

( )

t

exp

(

j

θ

_i

( )

t

)

2D complex zero-mean independent

α

θ

2D complex zero mean independent

Gaussian processes with

Time correlation:

( )

2

(

)

2 2 t J t ρ Δ σ πν Δ Time correlation: Variance: ρi t,

( )

0 =2σi2

( )

(

)

, 2 0 2 ,max i t t i J i t ρ Δ = σ πν Δ

Measured data used in the following

Measurement in the lab with a vector network analyzer -60 -50 -40 e sp ( d B ) •Center frequency 3.2 GHz •Measurement bandwidth 200 MHz, 201 frequency points -80 -70 -60 F requenc y r e 201 frequency points

•60 measurement positions, spaced 1 cm apart 0 4 1 1.5 2 8 -90 0 0.2 0.4 0 0.5 1 x 108 Position (m) Frequency (Hz)

Condensed parameters

Power-delay profile

One interesting channel property is the power-delay profile (PDP), which is the expected value of the received power at a certain delay:

( )

2

E

t

,

P

τ

=

⎡

⎣

h t

τ

⎤

⎦

( )

E

( )

(

( )

)

(

)

2 N

P

⎡

⎢

∑

j

θ

δ

⎤

⎥

For our tapped-delay line we get:

( )

(

( )

)

(

)

1

E

_t _i

exp

_i _i i N N

P

τ

α

t

j

θ

t

δ τ τ

=

⎢

−

⎥

⎢

⎥

⎣

∑

⎦

( ) (

)

(

)

2 2 1 1

E

2

N N t i i i i i i

t

α

δ τ τ

σ δ τ τ

= =

⎡

⎤

=

∑

⎣

⎦

−

=

∑

−

(6)

Power delay profile, meassured

70 -60 -50 60 -55 -50 -90 -80 -70 PD P -70 -65 -60 PD P -120 -110 -100 -85 -80 -75 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 x 10-7 -130 120 Delay 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x 10-7 -90 Delay

We often have an exponential decay, i.e. linear in the dB-domain

Decay constant 80 dB/0.22μs

Power-delay profile (cont )

Power-delay profile (cont.)

We can “reduce” the PDP into more compact descriptions of the channel: We can reduce the PDP into more compact descriptions of the channel:

Total power (time integrated): For our tapped-delay line

( )

m P ∞ P

τ τ

d −∞ =

∫

2 1 2 N m i i P

σ

= =

∑

Total power (time integrated):

channel:

( )

P d T

τ τ τ

∞ ∞

∫

2 2 N i i

τ σ

∑

Average mean delay:

( )

m m T P −∞ =

∫

i1 m m T P = = Average rms delay spread:

( )

2 m P d S T P

τ

τ τ

∞ −∞ =

∫

− 2 2 1 2 N i i i m S T P

τ σ

= =

∑

− 2 2

m m

P Pm m

Delay spread, measured

-55 -50 M d l 14 -65 -60 P Mean delay 14 ns RMS delay spread 16 ns -80 -75 -70 PD P 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 10-7 -90 -85 Delay x 107 Delay

Frequency correlation

A property closely related to the power-delay profile (PDP) is the frequencyp p y y p y p ( ) q y correlation of the channel. It is in fact the Fourier transform of the PDP:

( )

f

P

( )

exp

(

j

2

f

)

d

ρ

_f

( )

Δ =

f

∫

∞

P

( )

τ

exp

(

−

j

2

π τ τ

Δ

f

)

d

ρ

τ

π τ τ

−∞

Δ =

∫

Δ

For our tapped delay-line channel we get:

( )

2

(

)

(

)

2

exp

2

N f

f

i i

j

f

d

ρ

Δ =

∞

⎛

⎜

σ δ τ τ

−

⎞

⎟

−

π τ τ

Δ

⎝

∑

⎠

∫

pp y g

( )

(

)

(

)

(

)

1 2

p

2

f i i i N

f

j

f

j

f

ρ

−∞ =

⎜

⎟

⎝

⎠

Δ

∑

∫

∑

(

)

1

2

i

exp

2

i i

j

f

σ

π τ

=

∑

−

Δ

(7)

Condensed parameters

Coherence bandwidth

Given the frequency correlation of a channel, we can define theq y

coherence bandwidthBC:

( )

f f ρ Δ Wh t d th h

( )

0 f

ρ What does the coherence

bandwidth tell us?

It shows us over how large

( )

0 2

f

ρ

It shows us over how large a bandwidth we can assume that the channel is fairly constant

2 constant.

Radio systems using a bandwidth much smaller

f

Δ

C

B

bandwidth much smaller

than BCwill not notice

the frequency selectivity of the channel

C of the channel.

Condensed parameters

Coherence time

Given the time correlation of a channel, we can define the

h ti T coherence timeTC:

( )

t t ρ Δ

( )

0 ρ_t

( )

0 _{Wh t d} _th _h

ρ _{What does the coherence}

time tell us?

It shows us over how long

( )

0 2

t

ρ

It shows us over how long time we can assume that the channel is fairly constant

constant.

E.g. radio systems transmitting data in frames much shorter

t

Δ

C

T

data in frames much shorter

than TCwill not experience any

fading within a single frame.

Coherence time and bandwidth?

-60 -50 -40 e sp ( d B ) -80 -70 -60 F requenc y r e 0 4 1 1.5 2 8 -90 0 0.2 0.4 0 0.5 1 x 108 Position (m) Frequency (Hz)

Coherence bandwidth, measured

-40 -35 0.9 1 based on PDP based on H(f) -55 -50 -45 p (d B ) 0.6 0.7 0.8 rr -75 -70 -65 -60 F req uen c y r e s p 0.3 0.4 0.5 F req c o r 0 0 2 0 4 0 6 0 8 1 1 2 1 4 1 6 1 8 2 -90 -85 -80 -75 0 1 2 3 4 5 6 0 0.1 0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 x 108 Frequency (Hz) Compare 1/(2*π*τrms)=9.8 MHz 0 1 2 3 4 5 6 x 107 Frequency (Hz) p ( rms)

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Coherence time, measured

-45 -40 -35 1 measured theoretical -60 -55 -50 y res p (d B ) 0.5 co rr 80 -75 -70 -65 F req uen c y 0 Ti m e 0 0.1 0.2 0.3 0.4 0.5 0.6 -90 -85 -80 Position (m) 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 -0.5 Position (m) Assume 1 m/s, max 0

v

f

c

ν

=

=10.7 Hz Compare 1/(2*π*vmax)=0.014 s