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Fachgebiet Nachrichtentechnische Systeme NTS UNIVERSITÄT D U I S B U R G E S S E N Prof. Dr.-Ing. Andreas Czylwik Radio Propagation Channels WS 2009/2010 S. 1 Radio Propagation Channels Prof. Dr.-Ing. Andreas Czylwik
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Radio Propagation Channels

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Page 1: Radio Propagation Channels

FachgebietNachrichtentechnische Systeme

N T SUNIVERSITÄT

D U I S B U R GE S S E N

Prof. Dr.-Ing. Andreas Czylwik Radio Propagation Channels WS 2009/2010

S. 1

Radio Propagation Channels

Prof. Dr.-Ing. Andreas Czylwik

Page 2: Radio Propagation Channels

FachgebietNachrichtentechnische Systeme

N T SUNIVERSITÄT

D U I S B U R GE S S E N

Prof. Dr.-Ing. Andreas Czylwik Radio Propagation Channels WS 2009/2010

S. 2

Radio Propagation ChannelsOrganisational

Lecture 2 hours/week Exercise 1 hour/week Transparencies on web site Written examination

Department for Communication Systems Diploma and Master Theses

Page 3: Radio Propagation Channels

FachgebietNachrichtentechnische Systeme

N T SUNIVERSITÄT

D U I S B U R GE S S E N

Prof. Dr.-Ing. Andreas Czylwik Radio Propagation Channels WS 2009/2010

S. 3

Radio Propagation ChannelsTextbooks

Basic textbooks: T. S. Rappaport: Wireless communications, Prentice Hall G. S. Stüber: Principles of mobile communications, Kluwer

Academic Publishers W. C. Jakes: Microwave mobile communications, John Wiley K. David, T. Benkner: Digitale Mobilfunksysteme, Teubner-

Verlag

Advanced textbooks: J. D. Parsons: The mobile radio propagation channel, John Wiley J. Eberspächer, H.-J. Vögel: GSM - Global system for mobile

communication, Teubner-Verlag H. Holma, A. Toskala: WCDMA for UMTS, John Wiley

Page 4: Radio Propagation Channels

FachgebietNachrichtentechnische Systeme

N T SUNIVERSITÄT

D U I S B U R GE S S E N

Prof. Dr.-Ing. Andreas Czylwik Radio Propagation Channels WS 2009/2010

S. 4

Radio Propagation ChannelsContents

1 Introduction2 Wave propagation in mobile communications3 Linear time-variant systems4 Modulation5 Diversity schemes6 Coding7 Multiple access methods8 Cellular systems9 Methods for capacity enhancement10 Current systems

Page 5: Radio Propagation Channels

FachgebietNachrichtentechnische Systeme

N T SUNIVERSITÄT

D U I S B U R GE S S E N

Prof. Dr.-Ing. Andreas Czylwik Radio Propagation Channels WS 2009/2010

S. 5

Radio Propagation Channels1 Introduction

History of radio transmission 1888 Heinrich Hertz: Proof of propagation of electromagnetic

waves through free space 1895 Gugliemo Marconi: First transmission of messages with a

radio system over a distance of several km‘s 1958-1977 A-Net in Germany 1972-1994 B-Net in Germany 1986-2000 C-Net (1st generation) 1992 D-Net − GSM (2nd generation) 1994 E-Net - DCS 1800 (2nd generation) 2003 UMTS (3rd generation) ?? UMTS LTE

Page 6: Radio Propagation Channels

FachgebietNachrichtentechnische Systeme

N T SUNIVERSITÄT

D U I S B U R GE S S E N

Prof. Dr.-Ing. Andreas Czylwik Radio Propagation Channels WS 2009/2010

S. 6

Radio Propagation Channels1 Introduction Classification of mobile radio systems

Type of mobile station Land radio, marine radio, air radio

Type of base station Terrestrical base stations, satellite base stations

Type of services Broadcast (radio/TV), bidirectional communication (mobile

phone, wireless local area networks –WLANs) Type of communication signals

Speech, pictures, video, data, navigation, location Analog/digital

Structure of the network Cellular net, Ad-hoc net, local net, point-to-point

Page 7: Radio Propagation Channels

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Prof. Dr.-Ing. Andreas Czylwik Radio Propagation Channels WS 2009/2010

S. 7

Radio Propagation Channels1 Introduction

Cellular systems in Germany (2nd and 3rd generation) GSM (Global System for Mobile Communications): public mobile

phone system with world-wide roaming UMTS (Universal Mobile Telecommunication System): higher data

rates (up to 2 Mbit/s)

Page 8: Radio Propagation Channels

FachgebietNachrichtentechnische Systeme

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Prof. Dr.-Ing. Andreas Czylwik Radio Propagation Channels WS 2009/2010

S. 8

Radio Propagation Channels1 Introduction

Local systems in Germany DECT (Digital European Cordless Telephone): Cordless standard

for communication short distances (indoor) Bluetooth: cordless standard for small and smallest distances and

medium data rates WLAN IEEE 802.11: Class of wireless local area networks with

high data rates

Future systems UMTS LTE (long term evolution) Ultra-wideband systems for small distances and highest data rates

Page 9: Radio Propagation Channels

FachgebietNachrichtentechnische Systeme

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D U I S B U R GE S S E N

Prof. Dr.-Ing. Andreas Czylwik Radio Propagation Channels WS 2009/2010

S. 9

Radio Propagation Channels1 Introduction

Mobile radio systems in GermanySystem / Network GSM: D1/D2 GSM: E1/E2 UMTS, W-CDMA UMTS, TD-CDMA DECT Bluetooth WLAN 802.11a

Frequency range 890-915 / 935-960 MHz

1710-1785 / 1805-1880 MHz

1920-1980 / 2110-2170 MHz

1900-1920 / 2010-2025 MHz

1880-1900 MHz 2402-2485 MHz 5150-5350 / 5470-5725 MHz

Bandwidth 25 MHz (× 2) 75 MHz (× 2) 60 MHz (× 2) (20+15) MHz 20 MHz 83 MHz (ISM) 455 MHz

Duplexing method FDD ∆f = 45 MHz

FDD ∆f = 95 MHz

FDD ∆f = 120 MHz

TDD TDD TDD TDD

Multiple access method FDMA / TDMA FDMA / TDMA FDMA / CDMA CDMA FDMA / TDMA FDMA / FDMA/TDMA

Duplex channels 124 × 8 374 × 8 ca. 60 pro Zelle 10 × 12 79 19 ×

Modulation method GMSK GMSK QPSK QPSK GMSK GMSK OFDM

Channel separation 200 kHz 200 kHz 5 MHz 5 MHz (1,6 MHz) 1728 kHz 1 MHz 20 MHz

Data rate 9,6 kbit/s 9,6 kbit/s 16 ... 384 kbit/s (1,92 Mbit/s)

16 ... 384 kbit/s (1,92 Mbit/s)

32 kbit/s max. 721 kbit/s 6 ... 54 Mbit/s

Mobility vmax = 250 km/h vmax = 130 km/h vmax = 300 km/h vmax = 20 km/h vmax = 30 km/h

MS transmit power 13 ... 33 dBm 4 ... 30 dBm 21 ... 33 dBm 21 ... 33 dBm max. 10 dBm 0 dBm / 20 dBm max. 17 dBm

Range ca. 10 km ca. 8 km ca. 10 km Mainly indoor, up to some km’s

200-300 m 10 m / 100 m some 100 m

Network operator T-Mobil

D2 Vodafone

E-Plus

O2

5 Network operators Still open Private networks Private networks Private networks

Page 10: Radio Propagation Channels

FachgebietNachrichtentechnische Systeme

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Prof. Dr.-Ing. Andreas Czylwik Radio Propagation Channels WS 2009/2010

S. 10

Radio Propagation Channels1 Introduction

Basic problems of mobile radio Time variance of the radio channel (fading, Doppler effect) →

Channel coding, diversity schemes

Distance≈λ/2

Rec

eive

d po

wer

[dB

]

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Prof. Dr.-Ing. Andreas Czylwik Radio Propagation Channels WS 2009/2010

S. 11

Radio Propagation Channels1 Introduction

Time dispersion / frequency selectivity → adapted transmission methods / equalizers

Impulse response:

Page 12: Radio Propagation Channels

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Prof. Dr.-Ing. Andreas Czylwik Radio Propagation Channels WS 2009/2010

S. 12

Radio Propagation Channels1 Introduction

Alternative solution: multicarrier transmission

Transfer function:

Page 13: Radio Propagation Channels

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Prof. Dr.-Ing. Andreas Czylwik Radio Propagation Channels WS 2009/2010

S. 13

Radio Propagation Channels1 Introduction

Shared medium → multiple access method necessary Large number of users → cellular systems, since bandwidth is

limited Supporting user mobility:

Handover International roaming

Mobile phone is registered at home location register HLR1 Connecting in a foreign network Information exchange between mobile switching center

MSC2 and MSC1 Entries about absence in the home network and connection

in the foreign network into HLR1 Entry of the new user in the visitor location register VLR2

Page 14: Radio Propagation Channels

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Prof. Dr.-Ing. Andreas Czylwik Radio Propagation Channels WS 2009/2010

S. 14

Radio Propagation Channels1 Introduction

MSC 1

BS 1

HLR 1

PSTN

MSC 2

BS 2

HLR 2PSTN

VLR 2

PSTN

Fixed Network

BS = base stationMS = mobile stationPSTN = public switched telephone networkMSC = mobile switching centerHLR = home location registerVLR = visitor location register

Page 15: Radio Propagation Channels

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Prof. Dr.-Ing. Andreas Czylwik Radio Propagation Channels WS 2009/2010

S. 15

Radio Propagation Channels2 Wave Propagation

Wave propagation Physical effects

Page 16: Radio Propagation Channels

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Prof. Dr.-Ing. Andreas Czylwik Radio Propagation Channels WS 2009/2010

S. 16

Radio Propagation Channels2 Wave Propagation

Maxwell‘s Equations Ampere‘s law:

Faraday‘s law:

Notations:E = electrical field strengthH = magnetic field strengthD = electric displacement or electric flux densityB = magnetic induction or magnetic flux densityJ = electric current density

(2.1)t∂

∂+=

DJHrot

t∂∂

−=BErot (2.2)

Page 17: Radio Propagation Channels

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Prof. Dr.-Ing. Andreas Czylwik Radio Propagation Channels WS 2009/2010

S. 17

Radio Propagation Channels2 Wave Propagation

Material properties: ε = permittivity µ = permeability κ = conductivity

Page 18: Radio Propagation Channels

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Prof. Dr.-Ing. Andreas Czylwik Radio Propagation Channels WS 2009/2010

S. 18

Radio Propagation Channels2 Wave Propagation

Linear media:ε, µ, κ are independent from field amplitudes

Isotropic media:ε, µ, κ are independent from field directions

Homogeneous media:ε, µ, κ are independent from the position

Dispersion-free media:ε, µ, κ are independent from frequency

Loss-free media:κ = 0 and ε, µ are real

Page 19: Radio Propagation Channels

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Prof. Dr.-Ing. Andreas Czylwik Radio Propagation Channels WS 2009/2010

S. 19

Radio Propagation Channels2 Wave Propagation

Material equations for linear homogeneous isotropic lossy dielectric media:

Notations:κ = conductivityε0 = permittivity of vacuumεr = relative permittivityµ0 = magnetic permeability of vacuum

= refraction index

HBED

EJ

0

r0µ

εεκ

=== (2.3)

(2.4)(2.5)

rε=n

Page 20: Radio Propagation Channels

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Prof. Dr.-Ing. Andreas Czylwik Radio Propagation Channels WS 2009/2010

S. 20

Radio Propagation Channels2 Wave Propagation

Wave equation Inserting material equations:

Introducing complex amplitudes:

(2.6)t∂

∂+=

EEH r0rot εεκ

t∂∂

−=HE 0rot µ (2.7)

eRe)(,eRe)( jj tt tt ωω ⋅=⋅= HHEE (2.8)

HEEH

0

r0jrot

)j(rotωµ

εωεκ−=

+= (2.9)(2.10)

Page 21: Radio Propagation Channels

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Prof. Dr.-Ing. Andreas Czylwik Radio Propagation Channels WS 2009/2010

S. 21

Radio Propagation Channels2 Wave Propagation

Wave equation: combining Maxwell‘s equations:

with:

ex, ey, ez = unit vectors of the cartesian coordinate system

(2.11)

(2.12)

(2.13)

HH

EE

)j(

)j(

0r02

0

0r02

0

µεεωκωµ

µεεωκωµ

−=∆

−=∆

(2.14)zzyyxx

zzyyxx

HHH

EEE

eeeH

eeeE

++=

++=

Page 22: Radio Propagation Channels

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Prof. Dr.-Ing. Andreas Czylwik Radio Propagation Channels WS 2009/2010

S. 22

Radio Propagation Channels2 Wave Propagation

Solution for cartesian coordinates for κ = 0: homogeneous plane wave

Example: propagation in z direction Field equations for

(2.15)

(2.16)

(2.17)0,0

jj

jj

r00

r00

==

=∂

∂=

−=∂

∂−=

∂∂

zz

yx

xy

xy

yx

HE

Ez

HHz

E

Ez

HH

zE

εωεωµ

εωεωµ

0=∂∂

=∂∂

yx

Page 23: Radio Propagation Channels

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Prof. Dr.-Ing. Andreas Czylwik Radio Propagation Channels WS 2009/2010

S. 23

Radio Propagation Channels2 Wave Propagation

Independent wave equations per component:

with k2 = ω2ε0εrµ0 k = 2π/λn = ω n/c0

Solution for the electrical field:

(2.18)

(2.19)

(2.20)

0

0

22

2

22

2

=+∂

=+∂

yy

xx

Ekz

E

EkzE

kzy

kzyy

kzx

kzxx

eEeEE

eEeEEjj

jj

+−

−+

+−

−+

+=

+= (2.21)

(2.22)

Page 24: Radio Propagation Channels

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Prof. Dr.-Ing. Andreas Czylwik Radio Propagation Channels WS 2009/2010

S. 24

Radio Propagation Channels2 Wave Propagation

Solution for the magnetic field:

with the characteristic impedance of the dielectric:

Characteristic impedance of vacuum:

(2.23)

(2.24)

(2.25)

( )( )kz

ykz

yx

kzx

kzxy

eEeEZ

H

eEeEZ

H

jj

D

jj

D1

1

+−

−+

+−

−+

−−=

−=

(2.26)

nZZ 0

r0

0D ==

εεµ

+

+

+

+ =−=−==x

y

x

y

y

x

y

x

HE

HE

HE

HEZD

Ω≈Ω== 377π1200

00 ε

µZ(2.27)

Page 25: Radio Propagation Channels

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Prof. Dr.-Ing. Andreas Czylwik Radio Propagation Channels WS 2009/2010

S. 25

Radio Propagation Channels2 Wave Propagation

Polarization General approach for a plane wave propagating in z-direction:

Phase difference of waves: ∆ϕ = ϕy − ϕx

∆ϕ = 0 (or ∆ϕ = π) ⇒ linearly polarized wave∆ϕ = ±π/2 and ⇒ circularly polarized wave∆ϕ = ϕ0 ⇒ elliptically polarized wave

(2.28)(2.29)

(2.30)

(2.31)

kz

yyxxkz

yyxx

ty

kzyx

kzx

yyyxxx

yyxx

EEEE

EE

kztEkztE

tEtEt

yx

yx

jjjj

j)(j)(j

e][e]eˆeˆ[

e]eˆeˆ[Re

)cos(ˆ)cos(ˆ)()()(

−−

−−

+=+=

⋅+=

−++−+=

+=

eeeeE

ee

ee

eeE

ϕϕ

ωϕϕ

ϕωϕω

yx EE ˆˆ =

Page 26: Radio Propagation Channels

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Prof. Dr.-Ing. Andreas Czylwik Radio Propagation Channels WS 2009/2010

S. 26

Radio Propagation Channels2 Wave PropagationElectrical field

∆ϕ = 0 ∆ϕ = π/3 ∆ϕ = π/2 and

xExE−

yE−

yEyE

xE xExE−

yE−

yEyE

xE xExE−

yE−

yE

yE

xE

yx EE ˆˆ =

Page 27: Radio Propagation Channels

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Prof. Dr.-Ing. Andreas Czylwik Radio Propagation Channels WS 2009/2010

S. 27

Radio Propagation Channels2 Wave Propagation

Planar wave in an arbitrary direction Location vector: r = xex + yey + zez

Vector wave number: k = kxex + kyey + kzez

Relation to scalar wave numbers:

Generalized planar harmonic wave:

with e1⋅k = 0, e2⋅k = 0, e1⋅e2 = 0

2222zyx kkkkk ++=⇔=⋅kk

(2.32)(2.33)

(2.34)

(2.35)

222111

2211

j2211

)cos(ˆ)cos(ˆ)()()(

e][

ekrekr

eeEeeE kr

−++−+=

+=+= −

ϕωϕω tEtE

tEtEtEE

(2.36)(2.37)

(2.38)

Page 28: Radio Propagation Channels

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Prof. Dr.-Ing. Andreas Czylwik Radio Propagation Channels WS 2009/2010

S. 28

Radio Propagation Channels2 Wave Propagation

Reflection and refraction at the boundary surface z = 0 (x-y-plane) between two lossless dielectrica

⊥eE

||eE⊥rE

||rE

||gE

⊥gE

x

y z

ek

rk

gk

eα rα

gα)( 12

1nn

n>

Page 29: Radio Propagation Channels

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Prof. Dr.-Ing. Andreas Czylwik Radio Propagation Channels WS 2009/2010

S. 29

Radio Propagation Channels2 Wave Propagation

Incident wave:

Reflected wave:

Refracted wave:

Continuity conditions at the boundary surface: Et,1 = Et,2, Ht,1 = Ht,2

Law of reflection: αe = αr

Law of refraction: n1 sin αe = n2 sin αg

rkrk eeEEE ee jee||e||e

je||ee e][e][ −

⊥⊥−

⊥ +=+= EE

rkrk eeEEE rr jrr||r||r

jr||rr e][e][ −

⊥⊥−

⊥ +=+= EE

rkrk eeEEE gg jgg||g||g

jg||gg e][e][ −

⊥⊥−

⊥ +=+= EE

(2.39)

(2.40)

(2.41)

(2.42)

(2.43)

Page 30: Radio Propagation Channels

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Prof. Dr.-Ing. Andreas Czylwik Radio Propagation Channels WS 2009/2010

S. 30

Radio Propagation Channels2 Wave Propagation

Reflection and transmission factors (Fresnel equations):

(2.44)

(2.45)

(2.46)

(2.47)

e22

122e1

e22

122e1

||e

||r||

sincos

sincos

αα

αα

nnn

nnnEE

r−+

−−==

e22

1221e

22

e22

1221e

22

e

r

sincos

sincos

αα

αα

nnnn

nnnnEEr

−+

−−−==

⊥⊥

e22

122e1

e1

||e

||g||

sincos

cos2

αα

α

nnn

nEE

t−+

==

e22

1221e

22

e21

e

g

sincos

cos2

αα

α

nnnn

nnEE

t−+

−==⊥

⊥⊥

Page 31: Radio Propagation Channels

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Prof. Dr.-Ing. Andreas Czylwik Radio Propagation Channels WS 2009/2010

S. 31

Radio Propagation Channels2 Wave Propagation

Can reflection factors become zero?

vanishes only if no boundary surface exists.

αB = Brewster angle

(2.48)

21e22

122e

221

e22

122e1||

sincos

0sincos0

nnnnn

nnnr

=⇒−=

=−−⇒=

αα

αα

22

21

22

B2

e22

122

21e

242

e22

1221e

22

sin

)sin(cos

0sincos0

nnn

nnnn

nnnnr

+=

−=

=−−⇒=⊥

α

αα

αα

||r

Page 32: Radio Propagation Channels

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Prof. Dr.-Ing. Andreas Czylwik Radio Propagation Channels WS 2009/2010

S. 32

Radio Propagation Channels2 Wave Propagation

Total reflection For the roots become imaginary.

Total reflection if:

(only possible if n1 > n2)

(2.49)

0sin e22

122 <− αnn

1|||||| ==⇒= ∗∗ z

zrzzr

1

2esin

nn

Page 33: Radio Propagation Channels

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Prof. Dr.-Ing. Andreas Czylwik Radio Propagation Channels WS 2009/2010

S. 33

Radio Propagation Channels2 Wave Propagation

Reflexion factors for different angles of incidence: αe = 0 αe = π/2n1 > n2

n1 < n2

||r ⊥r

||r ⊥r

1

1

11

1−1

−1

−1

−1 −1

21

21nnnn

+−

21

21nnnn

+−

21

21nnnn

+−

21

21nnnn

+−

ReRe

ReRe

Im

Im

Im

Im

Page 34: Radio Propagation Channels

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Prof. Dr.-Ing. Andreas Czylwik Radio Propagation Channels WS 2009/2010

S. 34

Radio Propagation Channels2 Wave Propagation

Reflection factors for a radio channel with reflection at a lossy dielectric medium Horizontal polarization:

Vertical polarization:

Limit for very flat incidence αe → π/2:

(2.50)

(2.51)

e2

0re

e2

0re

he,

hr,h

sin)/j(cos

sin)/j(cos

αωεκεα

αωεκεα

−−+

−−−==

EE

r

e2

0re0r

e2

0re0r

ve,

vr,v

sin)/j(cos)/j(

sin)/j(cos)/j(

αωεκεαωεκε

αωεκεαωεκε

−−+−

−−−−==

EE

r

1limlim h2/π

v2/π ee

−==→→

rrαα

(2.52)

Page 35: Radio Propagation Channels

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Prof. Dr.-Ing. Andreas Czylwik Radio Propagation Channels WS 2009/2010

S. 35

Radio Propagation Channels2 Wave Propagation

Antennas Hertz‘ dipole in free space

Point-shaped oscillating charges+q and −q

Distance ∆l << λ/4 ∆l ⋅ I = dipole moment Field is symmetric with

respect to rotation Description in polar coordinates

y

x

z

ϕ

ϑ

r

Er

Page 36: Radio Propagation Channels

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Prof. Dr.-Ing. Andreas Czylwik Radio Propagation Channels WS 2009/2010

S. 36

Radio Propagation Channels2 Wave Propagation

Complex amplitude of the magnetic field:

Complex amplitude of the electric field:

ϕϕλϑ

λeHH ⋅⋅

+⋅⋅

∆== − rk

rrlI je

π2j1sin

2j

rrk

rkr

rrrlIZ

rrrlIZ

e

eEEE

⋅⋅

+⋅⋅

∆+

⋅⋅

++⋅⋅

∆=+=

j2

0

j2

0

eπ2jπ2j

cos22

j

eπ2jπ2j

1sin2

j

λλϑλ

λλϑλ ϑϑ

(2.53)

(2.54)

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S. 37

Radio Propagation Channels2 Wave Propagation

Far field approximation:

Wave fronts are spherical surfaces ⇒ spherical wave Field strengths do not depend on azimuth angle ϕ Dependence of field strength with respect elevation: ∼ sin ϑ Large distances:

Curvature of wave fronts is negligible Spherical wave ≈ planar wave

ϑϑ

ϕϕ

ϑλ

ϑλ

eEE

eHH

⋅⋅⋅∆

==

⋅⋅⋅∆

==

rk

rk

rlIZ

rlI

j0

j

esin2

j

esin2

j (2.55)

(2.56)

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S. 38

Free space propagation

Power considerations: spherical radiation of power Power density of an isotropical radiator (power per m2):

Radio Propagation Channels2 Wave Propagation

2T

isoπ4 dPP =′ (2.57)

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S. 39

Radio Propagation Channels2 Wave Propagation

Power density of a transmit antenna (power per m2):

Available power at the receive antenna:

Power transfer factor:

2TT

Tπ4 d

GPP ⋅=′

π4π4π4R

2

2TT

R2TT

RG

dGPA

dGPP ⋅

⋅⋅

=⋅⋅

2

RT

2

RTT

Rπ4π4

⋅⋅=

⋅⋅=

fdcGG

dGG

PP λ

(2.58)

(2.59)

(2.60)

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S. 40

Radio Propagation Channels2 Wave Propagation

Antenna gain: gain factor of the power density relative to the (not realizable) isotropic radiator

Relation between antenna gain and effective antenna surface:

Notations:PR = received powerPT = transmit powerGR = gain of the receive antennaGT = gain of the transmit antennaAR = effective surface of the receive antennaλ = carrier wavelength, f = carrier frequency

GA ⋅=π4

2eff

λ(2.61)

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S. 41

Radio Propagation Channels2 Wave Propagation

Path loss:

Free space attenuation:

RT

2

RTT

RP

log10log10

π4log10log10

GGL

dGG

PPL

F −−=

⋅⋅−=

−=

λ

+

+=

=

=

kmlog20

GHzlog20dB44,92

π4log20π4log20F

dfc

fddLλ

(2.62)

(2.63)

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S. 42

Radio Propagation Channels2 Wave Propagation

Relation between power density (magnitude of the Pointing vector) and the electric field strength:

with Z0 = characteristic impedance of free space: Z0 = 120 π Ω≈ 377 Ω

Radiated field strength of the transmit antenna:

0

2eff,0

ZE

P =′

dGPE

dGPE

P TTeff,02

TT

0

2eff,0

T30

π4Z⋅⋅

=⇒⋅

==′

(2.64)

(2.65)

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S. 43

Radio Propagation Channels2 Wave Propagation

Received power for a given field strength E0,eff :

Formulas for free space transmission can be directly used for point-to-point transmissions (fixed radio systems)

Reciprocity: the antenna gain is the same for transmit and receive usage

Ω

⋅=

⋅⋅

Ω=⋅=

120π2π4π120R

2eff,0R

22eff,0

R0

2eff,0

RGEGE

AZ

EP

λλ (2.66)

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S. 44

Radio Propagation Channels2 Wave Propagation

Diffraction Wave propagation

according to geometrical optics ifλ << object size

Geometrical optics: tight light-shadow border

Difference with respect to optics: field strength in the shadow of buildings and other obstacles is not negigible

Huygens‘ principle

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S. 45

Radio Propagation Channels2 Wave Propagation

Light: Wave fronts are modeled by point sources with spherical waves that combine to planar wave fronts

Shadow: spherical waves of point sources combine to diffracted radiation

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S. 46

Geometry, notations Plane perpendicular to the line-of-sight,

Locations of the same additional time delay: concentrical circles around the line-of-sight axis

Radio Propagation Channels2 Wave Propagation

d1

h

d2

Transmitter Receiverl1 l2

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S. 47

Radio Propagation Channels2 Wave Propagation

Additional path length:

Corresponding phase difference

with the Fresnel-Kirchhoff diffraction parameter:

hdddd

h

ddhdhdddllx

>>

+≈

−−+++=−−+=∆

2121

221

222

2212121

,for112 (2.67)

2

21

2

2π11

2π2π2 v

ddhx

⋅=

+⋅=

∆=∆

λλϕ

+⋅=

21

112dd

hvλ

(2.69)

(2.68)

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S. 48

Radio Propagation Channels2 Wave Propagation

Definition of Fresnel zones: path difference

Radii of Fresnel zones depend on the location between the antennas:

Numerical example: f = 1 GHz, d1 = d2 = 1 km

(2.70)

21

21dd

ddnrh n +⋅⋅== λ

⋅=∆ nxn

(2.71)

m2,122

11 =

⋅=

dr λ

(2.72)nvn ⋅= 2

(2.73)

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S. 49

Radio Propagation Channels2 Wave Propagation

Fresnel zones: Sum of distances with respect to two points is constant ⇒ ellipse

Locations with the same phase difference lie on the Fresnel ellipsoid:

Almost unaffected transmission if no obstacle is within the first Fresnel zone

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S. 50

Radio Propagation Channels2 Wave Propagation

Model for an obstacle: ideal absorbing half-plane

h, v > 0 ⇒ shadowing h, v < 0 ⇒ no shadowing

d1

h

d2Transmitter Receiver

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S. 51

Radio Propagation Channels2 Wave Propagation

Transmission factor normalized to free-space transmission:

∫∞ −+

=v

t tEE de

2j1 2

2j

0

π(2.74)

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S. 52

Radio Propagation Channels2 Wave Propagation

Diffraction problems in real propagation scenarios are more complex: Finite dimensions of obstacles Multiple diffractions Buildings are not ideal absorbers Finite dimension of the absorbers in propagation direction Rough surfaces Propagation over a long distance: earth curvature is not

negligible Solution: empirical formulas for the attenuation in specific

propagation scenarios

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S. 53

Radio Propagation Channels2 Wave Propagation

Single and multipath propagation, overview Doppler effect Fast fading Time dispersion Propagation scenarios Spatial correlation

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S. 54

Radio Propagation Channels2 Wave Propagation

Single path propagation Assumptions: distance x << d , direct line-of-sight, no

obstacles, plane wave Received signal:

with the wave number k = 2π / λ

)cos(j0 10e)( xktAtr θω −= (2.75)

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S. 55

Radio Propagation Channels2 Wave Propagation

Received signal:

With velocity v ⇒ x = v ⋅ t andDoppler frequency

Numerical example: f0 = 1 GHz, v = 30 m/s = 108 km/h, θ1= 0° ⇒ f D = 100 Hz

Amplitude of the received signals: ⇒ no fast fading effect

t

tvt

A

Atr)(j

0

)cos2(j0

D0

10

e

e)(ωω

θλπω

=

=

const.)( 0 == Atr

10

1 coscos2

θθλπ

ωcfvvf D

D ===

(2.76)

(2.77)

(2.78)

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S. 56

Radio Propagation Channels2 Wave Propagation

Two-path propagation Received signal:

)cos(j2

)cos(j1 2010 ee)( xktxkt AAtr θωθω −− +=

[ ][ ]

)()cossin()cossin(

)coscos()coscos()(2

2211

22211

2

xfxkAxkA

xkAxkAtr

=++

+=

θθ

θθ

(2.79)

(2.80)

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S. 57

Radio Propagation Channels2 Wave Propagation

Special case: A1 = A2= A0

. . .

)cos(j0

)cos(j0 2010 ee)( xktxkt AAtr θωθω −− +=

2)cos(coscos2)( 21

0θθ −

=xkAxr

(2.81)

(2.82)

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S. 58

Radio Propagation Channels2 Wave Propagation

Example: θ1 = 0, θ2 = π

)2cos(2)( 0 λπ xAxr =

0.0

0.2

0.4

0.6

0.8

1.0

0.00 0.25 0.50 0.75 1.00 1.25 1.50 x/λ

02)(

Axr

(2.83)

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S. 59

Radio Propagation Channels2 Wave Propagation

Example: ground reflection, earth curvature neglected Small angle of incidence ⇒

Contributions from two paths:

1hv −≅≅ rr

l1

l2

hT

hR

d

ϕ∆−⋅−⋅+≅+= j0021 e)1(EEEEE (2.84)

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S. 60

Radio Propagation Channels2 Wave Propagation

Magnitude of the complex amplitude:

Phase difference:

)sinjcos1(0 ϕϕ ∆+∆−= EE (2.85)

2sin2

2sin22cos22

sin)cos1(

0

200

220

ϕ

ϕϕ

ϕϕ

∆⋅⋅=

∆⋅=∆−=

∆+∆−=

E

EE

EE

)(21212 lllklk −=⋅−⋅=∆

λπϕ

2RT

22

2RT

21 )(and)(with hhdlhhdl ++=−+=

(2.86)

(2.87)

(2.88)

(2.89)

(2.90)

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S. 61

Radio Propagation Channels2 Wave Propagation

Phase difference:

(2.91)

dhh

dhhd

dhh

dhhd

dhh

dhhd

hhdhhd

λλ

λ

λ

λϕ

RT2

RT

2

2RT

2

2RT

2

2RT

2

2RT

2RT

22RT

2

π4222π2

2)(1

2)(1π2

)(1)(1π2

)()(π2

=⋅

⋅⋅=

−−−

++⋅⋅≈

−+−

++⋅⋅=

−+−++=∆

(2.92)

(2.93)

(2.94)

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S. 62

Radio Propagation Channels2 Wave Propagation

Transmission factor because of ground reflection:

Attenuation because of ground reflection:

Example: hT = 100 λ, hR = 5 λ, λ = 0,3 m

(2.95)

(2.96)

(2.97)

dhh

EE

λRT

0

π2sin2 ⋅=

λλ

λ

RTRT

RT

0ground

forπ22lg20

π2sin2lg20lg20

hhddhh

dhh

EE

a

>>⋅−≈

⋅−=−=−

m150500RT ==>>⇒ λλhhd

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S. 63

Radio Propagation Channels2 Wave Propagation

Additional attenuation because of ground reflection: −aground

-50

-40

-30

-20

-10

0

10

0 1 2 3 4

RTlg

hhd

⋅⋅ λ

−aground[dB]

(2.97)

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S. 64

Radio Propagation Channels2 Wave Propagation

Total transmission factor including free-space attenuation:

Approximation for long distances

(2.98)

(2.99)

λRThhd >>

dhh

dGG

PP

λλ RT2

2

RTT

R π2sin4π4

⋅⋅

⋅⋅=

2

2RT

RTT

R

⋅⋅=

dhhGG

PP

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S. 65

Radio Propagation Channels2 Wave Propagation

n-path propagation for unmodulated carrier signals Complex amplitude of the received signal:

Squared magnitude (~ received power):

AR and AI are random variables Approximation: large number of statistically independent

propagation paths⇒ central limit theorem is applicable

(2.100)∑=

−=n

i

xki iAtr

1

cosje)( θ

[ ] [ ]2I2

R

2

1I,

2

1R,

2

)()(

)cossin()coscos()(

xAxA

xkAxkAxrn

iii

n

iii

+=

+

= ∑∑

==θθ (2.101)

(2.102)

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S. 66

Radio Propagation Channels2 Wave Propagation

Assumption: AR and AI show a Gaussian distribution and are statistically independent

Probability density functions:

with

(2.103)∑=

−=n

i

xki iAxr

1

cosje)( θ

2I

2I

II

2R

2R

RR

2I

2R

e2

1)(

e2

1)(

A

A

A

AA

A

AA

Af

Af

σ

σ

σπ

σπ

⋅=

⋅=

222IR AAA σσσ ==

(2.104)

(2.105)

(2.106)

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S. 67

Radio Propagation Channels2 Wave Propagation

Variance of the complex random variable r :

Joint probability density function:

(2.107)

22I

2R

IRIR2

IR

2E

)j)(j(EE

j

AAEA

AAAAr

AAr

σ=+=

−+=

+=

2

2I

2R

IRIR

22

IRIR

e2

1

)()(),(

A

AA

A

AAAA AfAfAAf

σ

σπ

+−⋅=

⋅=

(2.108)

(2.109)

(2.110)

(2.111)

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S. 68

Radio Propagation Channels2 Wave Propagation

Statistical properties of the power transfer factor

Cumulative distribution function of the power transfer factor:

Coordinate transformation

(2.112)

(2.113)

2I

2R

2 AAPr +==

AR

AI

P

∫ ∫=

≤=

IRIR

00

dd),(

)()(

IRAAAAf

PPpPF

AA

P

ϕ

ϕ

ddddej

IR

jIR

AAAAAAAr

⋅=⇒⋅=+=

(2.114)

(2.115)

(2.116)

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S. 69

Radio Propagation Channels2 Wave Propagation

Cumulative distribution function of the power transfer factor:

Probability density function of the power transfer factor:

(2.117)

(2.119)

20

0 2

2

2

0

π2

0

220

e1

dde2

1)(

A

A

P

P

A

A

AP AAPF

σ

ϕ

σ ϕπσ

= =

−=

⋅⋅⋅= ∫ ∫

222 e

21

d)(d)( A

P

A

PP P

PFPf σ

σ

−==

(2.118)

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S. 70

Radio Propagation Channels2 Wave Propagation

Cumulative distribution function of the power transfer factor:

PPP

PF

A

P

P

P

A

A

==

−−≈

−=−

2

2

2

2

11

e1)(

2

2

σ

σ

σ

-40 -30 -20 -10 0 1010-4

10-3

10-2

10-1

100

Out

age

prob

abili

tydBin

2 2A

Pσ(2.120)

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S. 71

Probability density function of the power transfer factor:

P

fP(P)22

1Aσ

22 Aσ

Radio Propagation Channels2 Wave Propagation

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S. 72

Radio Propagation Channels2 Wave Propagation

Amplitude transfer factor A

Statistical properties of the amplitude transfer factor

Coordinate transformation (see Eqns. (2.115) and (2.116))

(2.121)2I

2R AAAr +==

∫ ∫=

≤=

IRIR

00

dd),(

)()(

IRAAAAf

AApAF

AA

A (2.122)

(2.123)

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S. 73

Radio Propagation Channels2 Wave Propagation

Cumulative distribution function of the amplitude transfer factor:

Probability density function of the amplitude transfer factor:

(2.124)

(2.125)

(2.126)

2

20

0 2

2

2

0

π2

0

220

e1

dde2

1)(

A

A

A

A

A

A

AA AAAF

σ

ϕ

σ ϕπσ

= =

−=

⋅⋅⋅= ∫ ∫

2

2

22 e

d)(d)( A

A

A

AA

AA

AFAf σ

σ

−==

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S. 74

Radio Propagation Channels2 Wave Propagation

Rayleigh probability density function:

0.2

0.4

0.6

0.8

-2 -1 0 1 2 3 4 A/σA

fA(A) ⋅ σA

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S. 75

Radio Propagation Channels2 Wave Propagation

n-path propagation with a dominant path: EAR = S

Pdf´s

IR jAAr +=

2

2I

2R

IR

2

2I

I

2

2R

R

2)(

2IR

2I

2)(

R

e2

1),(

e2

1)(

e2

1)(

A

A

A

ASA

AAA

A

AA

SA

AA

AAf

Af

Af

σ

σ

σ

σπ

σπ

σπ

+−−

−−

⋅=

⋅=

⋅= (2.128)

(2.127)

(2.130)

(2.129)

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S. 76

Radio Propagation Channels2 Wave Propagation

Coordinate transformation:

Joint pdf for AR and AI

ϕcosR

2I

2R⋅=

+=

AAAAA (2.131)

(2.132)

AR

AI

ϕ

A

2

22

2R

22

IR

2cos2

2

22

2IR

e2

1

e2

1),(

A

A

SASA

A

SASA

AAA AAf

σϕ

σ

σπ

σπ

−+−

−+−

⋅=

⋅= (2.133)

(2.134)

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S. 77

Radio Propagation Channels2 Wave Propagation

Cumulative distribution function:

(2.135)

(2.136)

(2.137)

(2.138)

∫ ∫=

≤=

IRIR

00

dd),(

)()(

IRAAAAf

AApAF

AA

A

AA

AAAF

A

A

SASA

A

A

A

SASA

AA

AA

A

ddee2

1

dde2

1)(

0 22

22

0 2

22

0

π2

0

2cos2

22

0

π2

0

2cos2

20

∫ ∫

∫ ∫

= =

+−

= =

−+−

⋅⋅⋅⋅=

⋅⋅⋅=

ϕ

σϕ

σ

ϕ

σϕ

ϕπσ

ϕπσ

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S. 78

Radio Propagation Channels2 Wave Propagation

Definition of the modified Bessel function of zeroth order:

Rice' K-factor:

(2.139)

(2.140)

(2.141)

(2.142)

)(Iπ2dedeπ1)(I 0

0

cosπ

0

cos0 xttx txtx =⇒= ∫∫

AASAAFA

A A

SA

AA A dIe)(

0 2

22

020

220 ∫

=

+−

⋅⋅=

σσσ

⋅⋅=

+−

202

2 Ie)(2

22

A

SA

AA

ASAAf A

σσσ

2

2

2 A

SKσ

=

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S. 79

Radio Propagation Channels2 Wave Propagation

Ricean pdf for different K-factors:

0.2

0.4

0.6

-2 -1 0 1 2 3 4 5 6 7 A/σA

fA(A) ⋅ σAK = 0

K = 1

K = 2K = 8

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S. 80

Radio Propagation Channels2 Wave Propagation

Pdf of the phase:

with the error function erf(x):

+⋅

⋅⋅+⋅⋅=−

A

S

A

SSSf AA

σϕ

σϕϕ σ

ϕσ

ϕ 2coserf1ecos

2π1e

π21)(

2

22

2

2

2cos

2

(2.143)

∫ −⋅=x

t tx0

deπ

2)(erf2

(2.144)

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S. 81

-180 -90 0 90 180 ϕ

fϕ(ϕ)

K = 0K = 1

K = 2

K = 8

Radio Propagation Channels2 Wave Propagation

Rice´ pdf for the phase and different K-factors:

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S. 82

Radio Propagation Channels2 Wave Propagation

Doppler spectrum spectral broadening from different Doppler frequencies for each

indiviual path in a multipath propagation environment The number and location of the scatterers depends on the scenario. Special case: large number of scatterers and reflectors in the

vicinity of the mobile station

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S. 83

Radio Propagation Channels2 Wave Propagation

Calculation of the Doppler spectrum with the following assumptions: Omnidirectional antenna at the mobile station Mobile stations are moving with constant velocity in any

arbitrary direction Very large number of reflectors/scatterers equally distributed

around the mobile station Same statistical properties for each path Same average power for each path Angles of arrival are equally distributed Path amplitudes and angles of arrival are statistically

independent

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S. 84

Radio Propagation Channels2 Wave Propagation

1. Approach for calculation of the Doppler spectrum: Transformation of angles of arrival into Doppler frequencies Probability density function of the angles of arrival:

Doppler frequency as a function of the angle of arrival:

Probability density function of the Doppler frequency:

≤≤−

=else0

ππfor)( π21 ϕϕϕf

)cos()cos()( maxD ϕϕλ

ϕ ⋅== fvf

∑=i i

fi

ff

ff)(

)()(

ddD

DD ϕ

ϕ

ϕ

ϕ

(2.145)

(2.146)

(2.147)

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S. 85

Radio Propagation Channels2 Wave Propagation

Calculation of the Doppler spectrum

ϕ

fϕ(ϕ)

−π π

fD = fmax⋅cos(ϕ)

−fm

ax

f max

f D

ϕ

f f D(f D

)

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S. 86

Radio Propagation Channels2 Wave Propagation

Derivative of the nonlinear characteristic:

Substituting ϕ by fD :

Probability density function of the Doppler frequency

))sin(())sin(()(max

D ϕϕλϕ

ϕ−⋅=−= fv

ddf (2.148)

2

max

D2

22

1)(cos1)sin(

1)(cos)(sin

−=−=⇒

=+

ffϕϕ

ϕϕ

2D

2max

1)(D ff

ff f−

=

(2.149)

(2.150)

(2.151)

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S. 87

Radio Propagation Channels2 Wave Propagation

Equal power for all paths → received spectrum is proportional to the pdf of the Doppler frequency

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S. 88

Radio Propagation Channels2 Wave Propagation

2. Approach for calculation of the Doppler spectrum: Analysis of the autocorrelation function of the received signal Received signal:

with

Autocorrelation function:

)j(])(j[ 000,D0 e)(ReeRe)( ϕωϕωω +++ =

= ∑ t

i

ti tAAtr i

∑=i

tji

iAtA ,De)( ω

⋅=+= ∑∑ +−

j

tj

i

tiAA

ji AAtAtAR )(j*j* ,D,D eeE)()(E)( τωωττ

(2.152)

(2.153)

(2.154)

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S. 89

Radio Propagation Channels2 Wave Propagation

Autocorrelation function:

Assumption: Ai, Aj and ωD,i, ωD,j are statistically independent

∑∑ −−=i j

tjiAA

jjiAAR ])j[(* ,D,D,DeE)( τωωωτ

[ ]))cos(sin(Ej))cos(cos(E

eE

eEEeE)(

maxmax0

)cos(j0

j2j2

max

,D,D

τϕωτϕω

τ

τϕω

τωτω

ii

ii

iiAA

PN

PN

AAR

i

ii

−⋅⋅=

⋅⋅=

⋅==

−− ∑∑

(2.155)

(2.156)

(2.157)

(2.158)

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Prof. Dr.-Ing. Andreas Czylwik Radio Propagation Channels WS 2009/2010

S. 90

∫∞

−⋅⋅⋅=-

AA PNS ττωω ωτ de)(J)( jmax00

Radio Propagation Channels2 Wave Propagation

Power spectral density: SAA(ω) RAA(τ)

)(J)(

))cos(sin(2π1j

))cos(cos(2π1)(

max00

π

πmax

π

πmax0

τωτ

ϕτϕω

ϕτϕωτ

⋅⋅=

⋅⋅=

PNR

d

dPNR

AA

-ii

-iiAA

(2.159)

(2.160)

(2.161)

(2.162)

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S. 91

<−

⋅⋅=

else 0

for2

)(max22

max0 ωω

ωωωPN

SAA

Radio Propagation Channels2 Wave Propagation

(2.163)

Page 92: Radio Propagation Channels

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S. 92

Radio Propagation Channels2 Wave Propagation

Autocorrelation function of the complex amplitude

-3 -2 -1 1 2 3

RAA(τ)

τ⋅fmax

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S. 93

Radio Propagation Channels2 Wave Propagation

Power spectral density of the complex amplitude (Jakes spectrum)

SAA(ω)

ωωmax−ωmax

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S. 94

Radio Propagation Channels2 Wave Propagation

Received RF signal:

Autocorrelation function of the received RF signal:

(2.164)

(2.165)

( ))j(-*)j(

)j(

0000

00

e)(e)(21

e)(Re)(

ϕωϕω

ϕω

++

+

⋅+⋅=

⋅=

tt

t

tAtA

tAtr

( )( )

⋅++⋅+⋅

⋅+⋅=+⋅

++++

++

))(j(-*))(j(

)j(-*)j(

0000

0000

e)(e)(

e)(e)(41E)()(E

ϕτωϕτω

ϕωϕω

ττ

τ

tt

tt

tAtA

tAtAtrtr

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Prof. Dr.-Ing. Andreas Czylwik Radio Propagation Channels WS 2009/2010

S. 95

Radio Propagation Channels2 Wave Propagation

Expectation with respect to ϕ0:

Power spectral density of the received RF signal:

(2.166)

(2.167)

( )

( )τωτω

τωτω

τω

τω

τ

τ

τ

ττ

00

00

0

0

jj

jj*

j*

j*

ee)(41

ee)()(E41

e)()(E41

e)()(E41)()(E

+−

+−

+

+⋅=

+⋅+=

⋅++

⋅+=+⋅

AAR

tAtA

tAtA

tAtAtrtr

[ ])()(41)()()()(E 00 ωωωωωττ −++==+⋅ AAAArrrr SSSRtrtr

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S. 96

Radio Propagation Channels2 Wave Propagation

Power spectral density of the received RF signal r(t)

Srr(ω)

−ω0−ωmax −ω0+ωmax ω0−ωmax ω0+ωmax

−ω0 ω0 ω

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S. 97

Radio Propagation Channels2 Wave Propagation

Temporal dispersion Description of a radio channel in the time domain:

Idealized representation of the impulse response:

Impulse response taking into account the band limitation:

Average time delay:

∑=

−⋅=N

iii tAth

1)(δ)( τ

∑=

−⋅=N

iii thAth

1BP )()( τ

∫∞

∞⋅

=

0

20

2

d)(

d)(

tth

tthtt (2.170)

(2.169)

(2.168)

Page 98: Radio Propagation Channels

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S. 98

Radio Propagation Channels2 Wave Propagation

Standard deviation of the impulse spreading (delay spread):

∫∞

∞⋅−

=∆

0

20

22

d)(

d)()(

tth

tthttt (2.171)

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S. 99

Radio Propagation Channels2 Wave Propagation

Average received power per time delay (power delay profile):

Frequently applicable (especially in case of indoor communications): negative-exponential power delay profile

P0 = average received power∆τ = time constant

ττττ

ττ d)d...(

)(+

=P

P (2.172)

ττ

τττ ∆−

∆= e)( 0PP log(Pτ(τ))

τ

log(P0/∆τ)(2.173)

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S. 100

Radio Propagation Channels2 Wave Propagation

Power delay profiles for testing GSM systems rural (non-hilly) area

≤≤⋅

=⋅−

else0μs7,00fore(0))(

μs/2,9 τττ

ττ

PP

-35

-30

-25

-20

-15

-10

-5

0

0 2 4 6 8 10 12 14 16 18 20

10 log(Pτ(τ)/Pτ(0))

τ/µs

20 log(|h(τ)|/hmax)

τ/µs

(2.174)

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S. 101

Radio Propagation Channels2 Wave Propagation

typical urban (non-hilly) area

≤≤⋅

=−

else0μs70fore(0))(

μs/ τττ

ττ

PP

-35

-30

-25

-20

-15

-10

-5

0

0 2 4 6 8 10 12 14 16 18 20

10 log(Pτ(τ)/Pτ(0))

τ/µs

20 log(|h(τ)|/hmax)

τ/µs

(2.175)

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S. 102

Radio Propagation Channels2 Wave Propagation

bad case for a hilly urban area

≤≤⋅⋅≤≤⋅

= −−

else 0μs10μs5fore(0)5,0

μs50for e(0))( μs/)μs5(

μs/

ττ

τ ττ

ττ

τ PP

P

-35

-30

-25

-20

-15

-10

-5

0

0 2 4 6 8 10 12 14 16 18 20

10 log(Pτ(τ)/Pτ(0))

τ/µs

20 log(|h(τ)|/hmax)

τ/µs

(2.176)

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S. 103

Radio Propagation Channels2 Wave Propagation

hilly terrain

≤≤⋅⋅≤≤⋅

= −−

⋅−

else 0μs20μs15fore(0)1,0

μs20for e(0))( μs/)μs15(

μs/5,3

ττ

τ ττ

ττ

τ PP

P

-35

-30

-25

-20

-15

-10

-5

0

0 2 4 6 8 10 12 14 16 18 20

10 log(Pτ(τ)/Pτ(0))

τ/µs

20 log(|h(τ)|/hmax)

τ/µs

(2.177)

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S. 104

Radio Propagation Channels2 Wave Propagation

Short term fading Rayleigh fading with Jakes Doppler spectrum (non-frequency-

selective) Logarithmic representation

-40

-30

-20

-10

0

10

0 2 4 6 8 10 12-40

-30

-20

-10

0

10

7.0 7.5 8.0 8.5 9.0

10 log(|A(t)|2/⟨|A(t)|2⟩)

t⋅fmaxx/λ

t⋅fmaxx/λ

10 log(|A(t)|2/⟨|A(t)|2⟩)

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S. 105

Radio Propagation Channels2 Wave Propagation

Linear representation

Design of digital communication systems: Frequency and duration of signal fades

0.0

0.5

1.0

1.5

2.0

0 2 4 6 8 10 120.0

0.5

1.0

1.5

2.0

7.0 7.5 8.0 8.5 9.0t⋅fmaxx/λ

t⋅fmaxx/λ

2|)(|

|)(|

tA

tA2|)(|

|)(|

tA

tA

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S. 106

Radio Propagation Channels2 Wave Propagation

Power fluctuations: The probability that the power level falls below a specific value decreases with this value.

Level crossing rate = average number of crossing a specific level (undershooting or overshooting) per time interval

Magnitude of the complex amplitude:

Time derivative of the amplitude:

22)( IR AAtA +=

t

A(t)

A + dA A

dt

AAt

tAA

dd

dd

=⇒=

(2.178)

(2.179)

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S. 107

Radio Propagation Channels2 Wave Propagation

Probability that the amplitude and its derivative are in the range:

Average time duration that the amplitude and its derivative can be found in D during a time interval of length T :

Average number of level crossings (over- or undershootings):

dd AAAAAAD +∩+=

(2.180)AAAAfAAP AA dd),(),(d ,=

(2.182)

(2.181)AAAAfTAAPTAAT AA dd),(),(d),(d ,⋅=⋅=

AAAAfT

AA

AAAAfTt

AATAAN AAAA

T

d),(d

dd),(d

),(d),(d ,, ⋅=

⋅==

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S. 108

Radio Propagation Channels2 Wave Propagation

Number of level crossings per time interval T for the interval of time derivatives :

Number of all level crossings per time:

(2.183)

(2.184)

AAAAfT

AANAAN AAT

d),(),(d),(d ,==

Ad

∫∫∞∞

==0

,0

d),(d),(d)( AAAAfAAANAN AA

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S. 109

Radio Propagation Channels2 Wave Propagation

Joint probability density function of a complex Gaussian process with „Jakes“ Doppler spectrum:

with

and (2.187)

(2.185)

(2.186)

),(, AAf AA

)()(eeπ21),(

2

22

2

22

2, AfAfAAAf AA

A

A

A

AAA

AA ⋅=⋅=−−

σσ

σσ

2I

2R

2 AAA ==σ

2dd

dd 2

max22

I2

R2 ωσσ ⋅=

=

= AA t

At

A

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S. 110

Radio Propagation Channels2 Wave Propagation

Number of all over- and undershootings per time interval:

(2.190)

(2.188)

(2.189)

2

2

2

2

2

2

2

2

2

2

2

2

2max22

0

2222

2

0

22

0,

eππ2

e

eπ21e

deπ21e

d),()(

AA

AA

AA

A

AA

A

A

A

AA

A

A

A

A

A

A

AA

AfA

A

AAA

AAAAfAN

σσ

σσ

σσ

σσ

σ

σσσ

σσ

−−

∞−−

−∞−

=⋅=

⋅⋅=

⋅⋅=

=

(2.191)

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S. 111

Radio Propagation Channels2 Wave Propagation

Substitution:

Number of over- and undershootings per wavelength:

(2.192)

(2.193)2

eπ2)( maxRRfRN −⋅⋅⋅=

2

2

2 A

ARσ

=

2eπ2

max

RRfN

vNTNN −⋅⋅==⋅=∆⋅=∆

λ

λλ (2.194)

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S. 112

Radio Propagation Channels2 Wave Propagation

level crossing rate (average number of crossings of the level Rper wavelength)

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

-60 -50 -40 -30 -20 -10 0 10

0.0

0.2

0.4

0.6

0.8

1.0

-30 -20 -10 0 1020 lg R

maxfNN

=∆ λ

=∆

maxlg

fNN

λ

20 lg R

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S. 113

Radio Propagation Channels2 Wave Propagation

Average time interval between two fades:

Average fade duration:

)()()(

)(1

)()(0

00F

0

0F0 RN

RRPRT

RN

RTRRP

<=⇒=<

)(1RN

)(F RT

2

2

e2π

e1)(max

F R

R

RfRT

⋅⋅⋅

−=

−⋅⋅

⋅= 1e1

2π1)(

2

maxF

RRf

RT

(2.195)

(2.196)

(2.197)

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S. 114

Radio Propagation Channels2 Wave Propagation

Average fade length:

Average fade duration:

−⋅⋅=⋅=∆ 1e1

2π)(

2FF

RR

vRTx λ (2.198)

-3.5

-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

-60 -50 -40 -30 -20 -10 0 10

0.0

0.2

0.4

0.6

0.8

1.0

-30 -20 -10 0 10

=⋅λ

FmaxFlg xfT

λF

maxFxfT ∆

=⋅

20 lg R20 lg R

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S. 115

Radio Propagation Channels2 Wave Propagation

Average fade length and number of fades per wavelength as a function of the fade depth

fade depth in dB: −20 lg R

average fade length in wave lengths ∆x/λ

average number of fades per wave length ∆Nλ

0 0,479 1,043 10 0,108 0,615 20 0,033 0,207 30 0,010 0,066

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S. 116

Radio Propagation Channels2 Wave Propagation

Spatial correlation

Correlation at the mobile station has already been treated:

Transformation of coordinates:)π2(J)()(E)( max00

* τττ fPNtAtARAA ⋅⋅=+⋅=

)π2(J)()(E)( 00*

λxPNxxAxAxRAA

∆⋅⋅=∆+⋅=∆

λτ

τx

vxffxv ∆

=∆

⋅=⋅⇒∆

= maxmax

(2.199)

(2.200)

(2.201)

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S. 117

-3 -2 -1 1 2 3

RAA(∆x)

∆x/λ

Radio Propagation Channels2 Wave Propagation

The spatial correlation does not depend on the direction

The correlation decreases fast within small distances

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S. 118

Radio Propagation Channels2 Wave Propagation

Correlation at the base station

Calculation procedure similar as for the calculation at the mobile station: base station moves with velocity v.

Differences:

No local scatterers close to the base station.

All waves arrive from a narrow angular range.

Direction of movement of the base station plays an important role

Assumptions

All (micro-)paths exhibit the same average power.

Path amplitudes are statistically independent from angles of arrival.

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S. 119

Radio Propagation Channels2 Wave Propagation

Approximation: angles of arrival are equally distributed within a small angular range

(2.202)

BS

∆ϕ

ϕ0MS

+≤≤−

∆=∆∆

else0

for1)( 2020

ϕϕ

ϕϕϕϕ

ϕϕf

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S. 120

Radio Propagation Channels2 Wave Propagation

Calculation of the autocorrelation function corresponding to Eq. (2.157):

No analytic solution ⇒ numerical evaluation

(2.203)

∫−

∆−

∆−

⋅⋅=

⋅⋅=∆⇔

⋅⋅=

π

π

)cos(π2j0

)cos(π2j0

)cos(j0

d)(e

eE)(

eE)( max

ϕϕ

τ

ϕϕ

λ

ϕλ

τϕω

fPN

PNxR

PNR

x

x

AA

AA

(2.204)

(2.205)

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S. 121

Radio Propagation Channels2 Wave Propagation

Autocorrelation function at the base station

ACF for different average angles of arrival ϕ0 and angular spread ∆ϕ = 5°

ACF for different angular spreads ∆ϕfor a fixed angle of arrival ϕ0 = 60°

0.0

0.2

0.4

0.6

0.8

1.0

0 10 20 30 40 500.0

0.2

0.4

0.6

0.8

1.0

0 10 20 30 40 50

RAA(∆x)/RAA(0)

∆x/λ ∆x/λ

ϕ0 = 0°ϕ0 = 30°ϕ0 = 60°ϕ0 = 90°

∆ϕ = 2,5°∆ϕ = 5°∆ϕ = 10°∆ϕ = 20°

RAA(∆x)/RAA(0)

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S. 122

Radio Propagation Channels2 Wave Propagation

Path loss models Average attenuation as a function of distance: averaging across

medium distances (hundreds of wavelengths) so that fast fading and shadowing effects cancel out

Free-space propagation:

Lee‘s empirical approach for propagation in real environments:

2

RTT

Rπ4

⋅⋅=

fdcGG

PP

000

0R kff

ddPP

n⋅

⋅=

−−γ

(2.206)

(2.207)

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S. 123

Radio Propagation Channels2 Wave Propagation

d0 and f0 are reference quantities to determine experimentally the parameters γ , n , and k0

Parameters γ , n , and k0 depend on the propagation scenario

LOS - line-of-sight: γ = 2

Typical values for the propagation exponent in built-up areas: γ = 3 ... 4,5

Corresponds to 30 ... 45 dB loss per decade of distance

Different empirical approaches for the path loss, e.g. the COST-Hata model, which includes also the dependence on antenna heights

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S. 124

Radio Propagation Channels2 Wave Propagation

Shadowing Shadowing leads to random fluctuations of the attenuation with

respect to its average value which is determined by the path loss (slow fading, shadowing)

Measurements: attenuation is Gaussian distributed ⇒ log-normal fading

Received amplitude when moving away from the base station:

alog-normal is Gaussian distributed with the standard deviation:

)(10)( 20)(

2

00

normal-log

xAdxaxa

xa

⋅⋅

⋅=

−γ(2.208)

2normal-lognormal-log )(a=σ (2.209)

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S. 125

Radio Propagation Channels2 Wave Propagation

Standard deviation depends highly on the propagation environ-ment, values: σlog-normal = 4 ... 12 dB, typical value: σlog-normal = 8 dB

Shadowing effects show a correlation for short distances:

In literature very different values are reported, measurement examples:

typical suburban area at 900 MHz: σlog-normal = 7,5 dB, Rlog-normal (∆x = 100 m) = 0,82

micro cellular area at 1700 MHz: σlog-normal = 4,3 dB, Rlog-normal (∆x = 10 m) = 0,3

(2.210)2normal-log

normal-lognormal-lognormal-log

)(

)()()(

σ

xxaxaxR

∆+⋅=∆

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S. 126

Radio Propagation Channels2 Wave Propagation

Typical characteristic of the received power level

large-scale fading

standard deviation of the large-scale fading

average path loss

Rec

eive

d po

wer

leve

l in

dBm

d/km

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S. 127

Radio Propagation Channels2 Wave Propagation

Path loss prediction Empirical path loss models

Ray-launching, ray-tracing

Rays are observed for a given number of reflections or until a maximum attenuation is reached

Precise data base of the propagation environment required (geometry of objects, material properties)

Diffraction theory

Combination of prediction methods

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S. 128

Radio Propagation Channels2 Wave Propagation

Atmospherical effects Attenuation by water and

oxygen resonances in the upper GHz range

Additional attenuation by rainfall and snowfall

film waveguide

f/GHz

atm

osph

eric

al a

ttenu

atio

n in

dB

/km H2O

O2

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S. 129

Radio Propagation Channels2 Wave Propagation

Channel simulation: modell for a radio channel that includes all significat effects:

Radio channel without temporal dispersion

τ (x)

filter

αlog-normal

realGaussianprocess

path loss

shadowing

complexGaussianprocess

Rayleigh fading

tfsS π2je

IR j)( AAtA +=

2010x

y =2

0

γ−

dxk

filter

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S. 130

Radio Propagation Channels2 Wave Propagation

Radio channel with temporal dispersion

τN

path loss

power delay profile

2

0

γ−

dxk

. . .

large scale fading

. . .

small scale fading

. . .

. . .Dopplerprocess 1

Σ

shadowingprocess 1

Dopplerprocess 2

shadowingprocess 2

Dopplerprocess N

shadowingprocess N

τ2

τ1

h1

h2

hN

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S. 131

Radio Propagation Channels2 Wave Propagation

Geometry-based models

BS

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S. 132

Radio Propagation Channels3 Linear Time-Variant Stochastic Systems

Linear time-variant Systems Mobile radio channel = linear time-variant system

Input-output-representation in the complex baseband:

h(t,τ) = time-variant impulse response = input delay-spread function

Linear time-variant

system

e)(Re)( 0jHF

ttxtx ω= e)(Re)( 0jHF

ttyty ω=

∫∞

∞−−= τττ d),()()( thtxty (3.1)

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S. 133

Radio Propagation Channels3 Linear Time-Variant Stochastic Systems

Response due to a single impulse:

Causality: h(t,t − t0) = 0 for t < t0

Substitution: t − t0 = τ ⇒ h(t,τ) = 0 for t < t − τ

⇒ h(t,τ) = 0 for τ < 0

h(t,τ))()( 0tttx −= δ ),()( 0ttthty −=

∫∞

∞−−=−−= ),(d),()()( 00 ttththttty τττδ (3.2)

(3.3)

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S. 134

Radio Propagation Channels3 Linear Time-Variant Stochastic Systems

Time-variant impulse response − response to a single impulse

h(t,τ) = 0

τ

t

t0

t0 = 0

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S. 135

Radio Propagation Channels3 Linear Time-Variant Stochastic Systems

Discrete time representation:

Transversal filter with time-variant coefficients

∑=

∆∆∆−=n

mmthmtxty

0),()()( τττ

x(t). . . .

. . . .y(t)

∆τ ∆τ ∆τ

h(t,0)⋅∆τ h(t,∆τ)⋅∆τ h(t,2∆τ)⋅∆τ h(t,n∆τ)⋅∆τ

(3.4)

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S. 136

Radio Propagation Channels3 Linear Time-Variant Stochastic Systems

Representation in the frequency domain:

H(ν,ω) = Doppler-variant transfer function= output Doppler-spread function

∫∞

∞−−−= uuuHuXY d),()(

21)( ωωπ

ω (3.5)

x(t)

X(ω)

y(t)

Y(ω)

h(t,τ)

H(ν,ω)

∫ ∫∞

∞−

∞−

−−= ττων ωτν ddee),(),( jj tthH t (3.6)

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S. 137

Radio Propagation Channels3 Linear Time-Variant Stochastic Systems

Discrete frequency representation:

Bank of coefficients with subsequent Doppler shifts

∑−=

∆∆−∆∆−=

n

nmmmHmXY

π2),()()( ωωωωωωω

X(ω)

. . . .

. . . .

Y(ω)

−n∆ω

H(−n∆ω,ω)⋅∆f

−∆ω

H(−∆ω,ω)⋅∆f

0⋅∆ω

H(0,ω)⋅∆f

∆ω

H(∆ω,ω)⋅∆f

. . . .

. . . .

n∆ω

H(n∆ω,ω)⋅∆f

(3.7)

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S. 138

Radio Propagation Channels3 Linear Time-Variant Stochastic Systems

Time-variant transfer function T(t,ω)

Delay Doppler-spread function S(ν,τ)

(3.8)

∫∞

∞−= ωωω ω de),()(

π21)( j ttTXty

∫∞

∞−

−= ττω ωτ de),(),( jthtT

(3.9)

(3.10)

(3.11)

∫∞

∞−

−= tthS t de),(),( jνττν

∫ ∫∞

∞−

∞−

−= τντντ ν dde),()(π2

1)( j tStxty

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S. 139

Radio Propagation Channels3 Linear Time-Variant Stochastic Systems

Relations between the four possible representations of a linear time-variant system

Variables: t = observation time, τ = delay time, ω = (angular) frequency, ν = Doppler (angular) frequency

many real radio channels: slow temporal fluctuations

h(t,τ)

S(ν,τ) T(t,ω)

H(ν,ω)

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S. 140

Radio Propagation Channels3 Linear Time-Variant Stochastic Systems

Measurement 1: time-variant impulse response:f0 = 1,8 GHz, LOS − line-of sight, omnidirectional fixed antennas, distance = 95 m, industrial area

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S. 141

Radio Propagation Channels3 Linear Time-Variant Stochastic Systems

Measurement example 1: time-variant transfer function:f0 = 1,8 GHz,LOS − line-of sight, omnidirectional fixed antennas, distance = 95 m, industrial area

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S. 142

Radio Propagation Channels3 Linear Time-Variant Stochastic Systems

Measurement example 2: time-variant impulse response:f0 = 1,8 GHz, NLOS − non-line-of sight, omnidirectional fixed antennas, distance = 230 m, industrial area

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S. 143

Radio Propagation Channels3 Linear Time-Variant Stochastic Systems

Measurement example 2: time-variant transfer function f0 = 1,8 GHz, NLOS − non-line-of sight, omnidirectional fixed antennas, distance = 230 m, industrial area

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S. 144

Radio Propagation Channels3 Linear Time-Variant Stochastic Systems

Measurement example 3: time-variant impulse response:f0 = 1,8 GHz, LOS − line-of sight, omnidirectionale antennas, run length = 1 m, distance = 95 m, industrial area

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S. 145

Radio Propagation Channels3 Linear Time-Variant Stochastic Systems

Measurement example 3: time-variant transfer function:f0 = 1,8 GHz, LOS − line-of sight, omnidirectional antennas, run length = 1 m, distance = 95 m, industrial area

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S. 146

Radio Propagation Channels3 Linear Time-Variant Stochastic Systems

Mobile radio channels are linear time-variant stochastic systems ⇒ description with stochastic methods

Restriction to second order statistical averages

because of complexity reasons

Gaussian processes are described completely by second order statistics

realistic approach: correlation functions

Autocorrelation function of the received signal

[ ][ ])j(

2*)j(

221

)j(1

*)j(12

12HF1HF

020020

010010

e)(e)(

e)(e)(E)()(Eϕωϕω

ϕωϕω

+−+

+−+

+

⋅+=⋅tt

tt

tyty

tytytyty

(3.12)

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S. 147

Radio Propagation Channels3 Linear Time-Variant Stochastic Systems

Averaging with respect to the phase ϕ0:

)2)(j(2

*1

*

)(j21

*

)(j2

*1

)2)(j(214

12HF1HF

0210

210

210

0210

e)()(

e)()(

e)()(

e)()(E)()(E

ϕω

ω

ω

ϕω

++−

−−

++

+

+

+

=⋅

tt

tt

tt

tt

tyty

tyty

tyty

tytytyty

)(j

2121

)(j2

*12

12HF1HF

210

210

e),(Re

e)()(ERe)()(Ett

yy

tt

ttR

tytytyty−

=

=⋅

ω

ω

(3.13)

(3.14)

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S. 148

Radio Propagation Channels3 Linear Time-Variant Stochastic Systems

Autocorrelation function of the complex amplitude of the output signal:

∫ ∫

∫ ∫

∞−

∞−

∞−

∞−

−−=

−−=

⋅=

2122*

1122*

11

2122*

22*

1111

2*

121

dd),(),(E)()(E

dd),()(),()(E

)()(E),(

ττττττ

ττττττ

ththtxtx

thtxthtx

tytyttRyy

(3.15)

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S. 149

Radio Propagation Channels3 Linear Time-Variant Stochastic Systems

Autocorrelation functions of the system functions

Eh(t1,τ1)⋅h*(t2,τ2) = Rhh(t1,t2;τ1,τ2)

EH(ν1,ω1)⋅H*(ν2,ω2) = RHH(ν1,ν2;ω1,ω2)

ET(t1,ω1)⋅T*(t2,ω2) = RTT(t1,t2;ω1,ω2)

ES(ν1,τ1)⋅S*(ν2,τ2) = RSS(ν1,ν2;τ1,τ2)

(3.15)

(3.16)

(3.17)

(3.18)

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S. 150

Radio Propagation Channels3 Linear Time-Variant Stochastic Systems

Relations between the autocorrelation functions:

Rhh(t1,t2;τ1,τ2)

RSS(ν1,ν2;τ1,τ2) RTT(t1,t2;ω1,ω2)

RHH(ν1,ν2;ω1,ω2)

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S. 151

Radio Propagation Channels3 Linear Time-Variant Stochastic Systems

Special channels:

WSS − wide-sense stationary channel: During short time intervals, the autocorrelation function Rhh(t1,t2;τ1,τ2) depends only on the difference between observation times ∆t = t2 − t1

Rhh(t1,t1+∆t;τ1,τ2) = Rhh(∆t;τ1,τ2)

RTT(t1,t1+∆t ;ω1,ω2) = RTT(∆t;ω1,ω2)

Consequence of the WSS property:

),(),(E),;,( 22*

112121 τντνττνν SSRSS ⋅=

(3.19)

(3.20)

(3.21)

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S. 152

Radio Propagation Channels3 Linear Time-Variant Stochastic Systems

)(π2),;(

)(π2de),;(

dde),;(

dde),(),(E

de),(de),(E),;,(

21212

21)j(

21

1)j(

21

21)j(

22*

11

*

2j

221j

112121

2

21211

2211

2211

ννδττν

ννδττ

ττ

ττ

ττττνν

ν

ννν

νν

νν

−⋅−=

−⋅∆∆=

∆∆=

=

⋅=

∫ ∫

∫ ∫

∫∫

∞−

∞−

∞−

∆−−−

∞−

∞−

−−

∞−

−∞

∞−

SS

thh

ttthh

tt

ttSS

P

ttR

tttR

ttthth

tthtthR

(3.22)

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S. 153

Radio Propagation Channels3 Linear Time-Variant Stochastic Systems

Consequence of the WSS property:

Scattering contributions with different Doppler frequencies are uncorrelated.

Channel with uncorrelated scatterers (US − uncorrelated scattering channel): In the frequency domain the autocorrelation function (ACF) RHH(ν1,ν2;ω1,ω2) depends only on the frequency difference ∆ω = ω2 − ω1.

RHH(ν1,ν2;ω1,ω1+∆ω) = RHH(ν1,ν2;∆ω)

RTT(t1,t2;ω1,ω1+∆ω) = RTT(t1,t2;∆ω)

Consequence of the US property:),(),(E),;,( 22

*112121 ττττ ththttRhh ⋅=

(3.23)

(3.24)

(3.25)

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S. 154

Radio Propagation Channels3 Linear Time-Variant Stochastic Systems

)();,(

)(de);,(

dde);,(

dde),(),(E

de),(de),(E),;,(

21221

21)j(

21π21

1)j(

21π)2(1

21)j(

22*

11π)2(1

*

2j

22π21

1j

11π21

2121

2

221112

22112

2211

ττδτ

ττδωω

ωωω

ωωωω

ωωωωττ

τω

τωτωτω

τωτω

τωτω

−⋅−=

−⋅∆∆=

∆∆=

=

⋅=

∫ ∫

∫ ∫

∫∫

∞−

∆−

∞−

∞−

∆−−

∞−

∞−

∞−

∞−

ttP

ttR

ttR

tTtT

tTtTttR

hh

TT

TT

hh

(3.26)

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S. 155

Radio Propagation Channels3 Linear Time-Variant Stochastic Systems

Consequence of the US property:

Scattering contributions of elementary scatterers with different time delays are uncorrelated.

Weakly stationary channel with uncorrelated scattering (WSSUS − wide-sense stationary uncorrelated scattering channel): important class of practical mobile radio channels:

Stationarity with respect to the observation time (small scale fading)

Uncorrelated contributions of scatterers with different time delay

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S. 156

Radio Propagation Channels3 Linear Time-Variant Stochastic Systems

Autocorrelation functions of the WSSUS channel:

Rhh(t1,t1+∆t;τ1,τ2) = Phh(∆t;τ2) ⋅ δ(τ1−τ2)

RHH(ν1,ν2;ω1,ω1+∆ω) = PHH(ν2;∆ω) ⋅ 2π δ(ν1−ν2)

RTT(t1,t1+∆t ;ω1,ω1+∆ω) = PTT(∆t;∆ω)

RSS(ν1,ν2;τ1,τ2) = PSS(ν2;τ2) ⋅ 2π δ(ν1−ν2) ⋅ δ(τ1−τ2)

Phh(∆t,τ)

PSS(ν,τ) PTT(∆t,∆ω)

PHH(ν,∆ω)

(3.28)(3.27)

(3.30)

(3.29)

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S. 157

Radio Propagation Channels3 Linear Time-Variant Stochastic Systems

Physical model for a WSSUS channel:

single scattering at a large number of scatterers

each scatterer ist described by its time delay, Doppler shift, and its scattering coefficient

independent scatterers

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S. 158

Radio Propagation Channels3 Linear Time-Variant Stochastic Systems

PSS(ν;τ) is proportional to the scattering function σ(ν,τ)

σ(ν,τ) describes the distribution of power with respect to the Doppler frequency and time delay = delay-Doppler spectrum

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S. 159

Radio Propagation Channels3 Linear Time-Variant Stochastic Systems

Real mobile radio channels

in general non-stationary − but stationarity is often found for short runlengths within small areas

assumption: stationarity within small areas − significant scattering centers do not change within these areas (small scale fading): WSSUS approach is valid

for larger runlengths shadowing effects have to be considered, so that significant scattering areas change (shadowing − large scale fading)

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S. 160

Radio Propagation Channels3 Linear Time-Variant Stochastic Systems

Properties in the time domain

no difference in the observation time: ∆t = 0 ⇒ power delay profile = averaging the delay-Doppler spectrum with respect to all Doppler frequencies:

average time delay:

∫∞

∞−== ντνττ ν de);();0()( 0j

π21

SShhhh PPP

∫∞

∞⋅

=

0

0

d)(

d)(

ττ

ττττ

hh

hh

P

P

(3.31)

(3.32)

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S. 161

Radio Propagation Channels3 Linear Time-Variant Stochastic Systems

delay spread:

∫∞

∞⋅−

=∆

0

0

2

)d(

)d()(

ττ

τττττ

hh

hh

P

P

(3.33)

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S. 162

Radio Propagation Channels3 Linear Time-Variant Stochastic Systems

Properties in the frequency domain

The correlation of the transfer function between different frequencies decreases with increasing frequency difference.

Coherence bandwidth = frequency difference for a significant correlation

Frequency correlation without observation time delay: ∆t = 0 ⇒ Frequency correlation spectrum = averaging the frequency-Doppler spectrum versus all Doppler frequencies:

∫∞

∞−∆=∆=∆ νωνωω ν de);();0()( 0j

π21

HHTTTT PPP (3.34)

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S. 163

Radio Propagation Channels3 Linear Time-Variant Stochastic Systems

Relation between frequency-correlation spectrum and power-delay profile:

Phh(0,τ) = Phh(τ) PTT(0,∆ω) = PTT(∆ω)

Example for the frequency-correlation function:

(3.35)

PTT(∆ω) / PTT(0)

∆ω / 2π MHz

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S. 164

Radio Propagation Channels4 Modulation

Block diagram for a digital mobile radio link

Passivefilter

Receivefilter

Up-conversion

Down-conversion

Transmitfilter

Assignmentof complex

symbols

EqualizerDetection

Rad

io c

hann

el

Synchronization

Data

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Prof. Dr.-Ing. Andreas Czylwik Radio Propagation Channels WS 2009/2010

S. 165

Radio Propagation Channels4 Modulation Model of the transmitter

hT(t) = impulse response of the transmit filter (often a rectangular function)

aν = complex symbols

linear modulation: only linear operations after the assignment of complex symbols

ReAssignmentof complex

symbols

Data( )thT

νa

t0je ω( )∑∞

−∞=−

ννδ Tt

⋅−= ∑∞

−∞=

tTthatx 0jT e)(Re)( ω

νν ν

x(t)

(4.1)

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S. 166

-1 1 2 3 4 5 6 7

Radio Propagation Channels4 Modulation

ASK − amplitude shift keying

most simple case: binary ASK = OOK (on-off-keying)

binary ASK = most simple type of digital modulation

Ra

Ia

t/T

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S. 167

-1 1 2 3 4 5 6 7

Radio Propagation Channels4 Modulation

PSK − phase shift keying

most simple case: binary PSK = BPSK (binary phase shift keying)

t/T

Ra

Ia

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S. 168

Radio Propagation Channels4 Modulation

4-PSK = QPSK 8-PSK

Ra

Ia

Ra

Ia

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S. 169

Radio Propagation Channels4 Modulation

modulation requirements for mobile radio applications:

bandwidth efficiency: low adjacent channel interference

applicability of nonlinear amplifiers ⇒ low fluctuations of the magnitude of the transmit signal

OQPSK = offset quadriphase shift keying

ReAssignmentof complex

symbols

Data( )thT

νa

t0je ω( )∑∞

−∞=−

ννδ Tt

Re

Im T/2

aν ∈ 1+j,1−j,−1+j,−1−j

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S. 170

Radio Propagation Channels4 Modulation

OQPSK π/4-DQPSK

Ra

Ia

Ra

Ia

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S. 171

Radio Propagation Channels4 Modulation

QAM - quadrature amplitude modulation

2m states, m = 2k

Ra

Ia

4-QAM

64-QAM

16-QAM

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S. 172

Radio Propagation Channels4 Modulation

QAM - quadrature amplitude modulation

2m states, m = 2k + 1

Ra

Ia

32-QAM

128-QAM

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S. 173

Radio Propagation Channels4 Modulation

8-QAM 8-QΑΜ

Ra

Ia

Ra

Ia

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S. 174

Radio Propagation Channels4 Modulation

FSK - frequency shift keying

CPM (continuous phase modulation)

Phase of the transmit signal:

-1 1 2 3 4 5 6 7 t/T

))((coseRe)( 00))(j(

0CPM 0 ttxxtx tt ϕωϕω +⋅=

⋅= +

∫ ∑∞

=−=

t

ifi iTgd

Tht

0 0d)()( ττπϕ

(4.2)

(4.3)

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S. 175

Radio Propagation Channels4 Modulation

Notations:

gf(t) = frequency impulse

gf(t) = 0 outside of 0 ≤ t ≤ T ⇒ full response CPM

gf(t) ≠ 0 outside of 0 ≤ t ≤ T ⇒ partial response CPM

Normalization:

di = data signal: di ∈ −1,1

h = modulation index

instantaneous frequency difference:

∫∞

∞−= Tg f ττ d)( (4.4)

tddϕω =∆ (4.5)

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S. 176

0.5

1.0

1.5

2.0

-0.5 0.0 0.5 1.0 1.5

gf (t)gϕ(t)

t/T

Radio Propagation Channels4 Modulation

Phase impulse

gϕ(t) = phase impulse

Normalization:

Phase variation during one symbol: ∆ϕ = h ⋅ π

Example: cos2-impulse (raised cosine)

∫∞

∞−==∞= 1d)(1)( ττϕ fg

Ttg (4.6)

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S. 177

Radio Propagation Channels4 Modulation

Block diagram of a CPM transmitter

with frequency modulator

with phase modulator

gf (t)

( )∑∞

−∞=−

νν νδπ Ttd

Th

∆ω(t) Frequencymodulator

(VCO)

xCPM(t)

∫∞−

ttd gf (t)

( )∑∞

−∞=−

νν νδπ Ttd

Th

ϕ(t) Phasemodulator

xCPM(t)

gϕ(t)

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S. 178

Radio Propagation Channels4 Modulation

Advantage of CPM methods: constant amplitude, rapidly decreasing spectrum ⇒ low adjacent channel interference

CPFSK (continuous phase frequency shift keying)

rectangular frequency impulse ⇒ CPFSK

(maximum) frequency deviation

orthogonal time functions for:

smallest modulation index, for which orthogonality between transmit waveforms is fulfilled: h = 0,5 ⇒ MSK (minimum shift keying)

Th

dd

==∆ϕω

,...3,2,1for2π

dd

=⋅==∆ iT

itϕω

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S. 179

Radio Propagation Channels4 Modulation

Instantaneous frequency and phase for MSK

ω (t)

t/T

ω0+∆ω

ω0−∆ωω0 1 2 3 4 5 6 7 8 9

ϕ (t)

t/T0

ππ/2

−π/2−π

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S. 180

Radio Propagation Channels4 Modulation

GMSK

Kinks of the phase function cause a broadened spectrum

Smoothing filter with Gaussian shaped impulse response:

rT (t)

( )∑∞

−∞=−

νν νδπ Ttd

Th

∆ω(t) Frequencymodulator

(VCO)

xGMSK(t)hGauss (t)

222

2lnπ2

Gauss e2lnπ2)(

tB

Bth⋅−

⋅⋅= (4.7)

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S. 181

Radio Propagation Channels4 Modulation

resulting frequency impulse:

with

with

−⋅⋅−

⋅⋅=

TTtB

TtBtg f ππ

2lnπ2erf

2lnπ2erf

21)(

≤≤

= else0

0for1)(

TttrT

)()()( Gauss trthtg Tf ∗= (4.8)

(4.9)

(4.10)

∫ −=x

t tx0

deπ2)(erf

2(4.11)

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S. 182

Radio Propagation Channels4 Modulation

Frequency impulse for GMSK

0.2

0.4

0.6

0.8

1.0

-2 -1 0 1 2 3

gf (t) BT = 10

t/T

BT = 0,5BT = 0,3 (GSM)

BT = 0,15

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S. 183

Radio Propagation Channels4 Modulation

TFM (tamed frequency modulation) CPM method with defined partial response behavior Partial response behavior with a transversal filter with three

coefficients additional Nyquist filter for smoothing the phase function

( )∑∞

−∞=−

νν νδπ Ttd

Th

∆ω(t) Frequencymodulator

(VCO)

xTFM(t)hNyquist (t)

T

−T

1/2

1/2

1

gf (t)

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S. 184

Radio Propagation Channels4 Modulation

OFDM (orthogonal frequency division multiplexing)

Multicarrier transmission scheme which is based on the discrete Fourier transform (DFT)

High importance since powerful FFT processors exist

Channels with large temporal dispersion: no complex equalizer as for single carrier transmission is needed

Problem: amplitude distribution corresponding to a complex Gaussian signal

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S. 185

Radio Propagation Channels4 Modulation

Transmitter for multicarrier transmission

g(t)

I/Q-Modulator

DataInput

tω∆je

g(t)

g(t)

Σ

Mod

ulat

ortω∆2je

tN ω∆je

X1

X2

XN

...

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S. 186

Radio Propagation Channels4 Modulation

Receiver for multicarrier transmission

h(t)

I/QDemodulator

DataOutput

tω∆− je

h(t)

tω∆− 2je

h(t)

tN ω∆− je

Dem

odul

ator

...

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S. 187

Radio Propagation Channels4 Modulation

Realization of multicarrier transmission using the discrete Fourier transform (DFT) ⇒ OFDM

Multicarrier signal:

DFT:

inverse DFT:

tnN

nn tXty ω∆

=⋅= ∑ j

1e)()(

tknN

k

NknN

ktkxtkxnX ∆∆−

=

=⋅∆=⋅∆=∆ ∑∑ ωπ

ω j-1

0

2j-1

0e)(e)()(

tknN

n

NknN

nnX

NnX

Ntkx ∆∆−

=

=⋅∆=⋅∆=∆ ∑∑ ωπ

ωω j1

0

2j1

0e)(1e)(1)(

(4.14)

(4.13)

(4.12)

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S. 188

Radio Propagation Channels4 Modulation

Signals for DFT x(t = k∆t)

X(ω = n∆ω)

−ωa/2 ∆ω ωa/2 ω

−T/2 ∆t T/2 t(4.15)

(4.16)

∆t ⋅ ωa = 2π

∆ω ⋅ T = 2π

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S. 189

Radio Propagation Channels4 Modulation

Periodic repetition of the transmit signal during a guard interval in order to avoid interblock interference

Duration of guard interval > duration of impulse response

Loss in bandwidth efficiency due to the guard interval:

−T/2 ∆t T/2 t

x(t = k∆t)

Tg T

Guard interval g

gTT

T+

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S. 190

Radio Propagation Channels4 Modulation

Orthogonal transmit waveforms

Rectangular envelope ⇒overlap in the frequency domain

Optimum reception also by DFT

T t

T t

T t

T t

T t

1

1

1

1

1

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S. 191

Radio Propagation Channels5 Diversity methods

Overview Problem of mobile radio channels: fading, especially Rayleigh

fading

Solution: simultaneous reception and evaluation of different signals containing the same information via channels which are as independent as possible = diversity methods

Different methods for combining the signals

Requirement: uncorrelated channels

Diversity concepts at transmitter and receiver

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S. 192

Radio Propagation Channels5 Diversity methods

Classification

spatial diversity: signals via different antennas

micro diversity: distance between antennas: a few wavelengths⇒ method to reduce the fast Rayleigh fading

macro diversity: distance of antennas in the range of kilometers ⇒ method against shadowing effects

angular diversity: reception of signals from different directions

polarisation diversity: reception of signals with orthogonal polarisations

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S. 193

Radio Propagation Channels5 Diversity methods

frequency diversity: transmitting and receiving signals at different carrier frequencies

disadvantage: loss of bandwith efficiency

temporal diversity: multiple transmissions at different time intervals

disadvantage: loss of bandwidth efficiency, additional time delay

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S. 194

Radio Propagation Channels5 Diversity methods

Simplified model for a baseband representation of a single mobile radio channel

power of the transmit signal: ⟨|x(t)|2⟩ = Ps

power of the additive noise: ⟨|ni(t)|2⟩ = Pn

power of the desired received signal averaged with respect to the desired information:

Si = ⟨|x(t)|2⟩ ⋅ |ai(t)|2 = Ps ⋅ |ai(t)|2

x(t) yi(t)ni(t)ai(t)

(5.1)

(5.2)

(5.3)

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S. 195

Radio Propagation Channels5 Diversity methods

signal-to-noise power ratio (SNR):

|ai| is Rayleigh distributed and γi is negative-exponentially distributed:

with

iii

ii aPP

tn

atx

NS γ=⋅=

⋅= 2

n

s2

22

)(

)(

Γ=Γ

else0

0fore1)( i

iii

i

if γγ

γ

γ

2

n

siii a

PP

⋅==Γ γ (5.6)

(5.5)

(5.4)

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S. 196

Radio Propagation Channels5 Diversity methods

Selection combining: selection of the strongest signal

selectionlogic

antennas

receiver...y1 y2 y3 yM

1 2 3 M

ySC

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S. 197

Radio Propagation Channels5 Diversity methods

assumption: equal average SNR in all branches: Γi = Γ

probability that the instantaneous SNR falls below the threshold value γs:

instantaneous SNR at the output of a selection combiner: maximum with respect to all branches:

probability that the instantaneous SNR falls below the threshold value γs simultaneously in all branches:

Γ−

−==<s

e1)()( ss

γ

γ γγγi

FP i

MM

FP

Γ≈

−==< Γ

− sssSC

s

SCe1)()( γγγγ

γ

γ

),...,,,max( 321SC Mγγγγγ =

(5.9)

(5.8)

(5.7)

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S. 198

Radio Propagation Channels5 Diversity methods

outage probability:

-40 -30 -20 -10 0 1010-4

10-3

10-2

10-1

100

dBin sΓγ

)( sSCγγF

M = 2

M = 1

M = 4 M = 8

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S. 199

Radio Propagation Channels5 Diversity methods

Switched Combining

disadvantage of selection combining: M receivers necessary

principle: switching to the next branch if the SNR γi of the current branch falls below the threshold γT

higher outage probability than selection combining

switchingselection receiver

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S. 200

Radio Propagation Channels5 Diversity methods

Maximum Ratio Combining (MRC)

Calculation ofcompensation

phases andamplitude

factors

receiver...y1 y2 y3 yM

1ϕ∆ 2ϕ∆ 3ϕ∆ Mϕ∆

c1c2 c3 cM

antennas

yMRC

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S. 201

Radio Propagation Channels5 Diversity methods

MRC determines the optimum linear combination of the antenna signals

first step: phase compensation

second step: linear combination

SNR at the output of the linear combiner:

∑∑∑=

=

=

∆ +==M

iii

M

iii

M

iii ncxacycy iii

1

j

1

j

1

jMRC eee ϕϕϕ (5.10)

=

=

=

=

=⋅

⋅⋅

= M

ii

M

iii

M

iii

M

iii

c

ca

PP

ctn

catx

1

2

2

1

n

s

1

22

2

1

2

MRC

||

)(

||)(γ (5.11)

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S. 202

Radio Propagation Channels5 Diversity methods

maximum search:

solution: cj = k ⋅ |aj|

(5.12)02||||||2

dd !

2

1

2

2

111

2

n

sMRC =

⋅−⋅⋅⋅

=

∑∑∑

=

===

M

ii

jM

iiij

M

iii

M

ii

jc

ccaacac

PP

(5.13)0||||11

2 =⋅⋅−⋅⇒ ∑∑==

jM

iiij

M

ii ccaac

(5.14)

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S. 203

Radio Propagation Channels5 Diversity methods

SNR after the optimum linear combination:

sum of statistically independent random variables: pdf = convolution of individual pdf‘s

(5.15)

(5.16)

(5.17)

∑∑

=

=

= =

=M

iiM

ii

M

ii

aPP

ak

ak

PP

1

2

n

s

1

22

2

1

2

n

sMRC ||

||

||γ

∑=

=M

ii

1MRC γγ

Mfffff γγγγγ γ ∗∗∗∗= ...)(

321MRC MRC

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S. 204

Radio Propagation Channels5 Diversity methods

Fourier transform ⇒ characteristic function:

characteristic function of the sum of statistically independent random variables:

Assumption: all branches show the same average SNR:

(5.18)

(5.19)

(5.20)

Mγγγγγ Φ⋅⋅Φ⋅Φ⋅Φ=ΩΦ ...)(321MRC

ii ii

fΩΓ−

=ΩΦj11)()( γγ γ

Γ=Γ= iiγ

M)j1(1)(

MRC ΩΓ−=ΩΦγ (5.21)

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S. 205

Radio Propagation Channels5 Diversity methods

backward Fourier transform yields the χ2 pdf:

cumulative distribution function:

alternative approach: representation by real and imaginary part of the transfer factor:

⇒ γMRC is the sum of 2M squared Gaussian distributed random variables ⇒ χ2 pdf

(5.22)

(5.23)

Γ−− ⋅⋅

Γ−=

MRC

MRCe)(

)!1(1)( 1

MRCMRC

γ

γ γγ MMM

f

1MRC

1MRC )!1(

1e1)(MRC

MRC

=

Γ−

Γ⋅

−−= ∑

iM

i iF γγ

γ

γ

∑=

+=M

iii aa

PP

1

2I

2R

n

sMRCγ (5.24)

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S. 206

Radio Propagation Channels5 Diversity methods

outage probability:

-40 -30 -20 -10 0 1010-4

10-3

10-2

10-1

100

dBin sΓγ

)( sMRCγγF

M = 2

M = 1

M = 4 M = 8

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S. 207

Radio Propagation Channels5 Diversity methods

Equal Gain Combining (EGC)

simple to be implemented since coefficients need not to be calculated

the same performance as with maximum ratio combining if the branches show the same instantaneous SNR

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S. 208

Radio Propagation Channels6 Coding

Block diagram of a mobile radio link

Interleaver I Kanal-Codierer II Interleaver II Modulator

Mobilfunkkanal

DemodulatorDeinterleaver II

Kanal-Decodierer II

DeinterleaverI

Daten-quelle

Quellen-Codierer

Kanal-Codierer I

Kanal-Decodierer I

Quellen-Decodierer

Daten-senke

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S. 209

Radio Propagation Channels6 Coding

Classification of coding schemes source coding, goal: reduction of redundancy

channel coding

block codes

convolutional codes

concatenated codes

parameters of channel coding

code rate

coding gain

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S. 210

Radio Propagation Channels6 Coding

Design of code words of block codes

Code C = set of all code words

Code word c = (c0, c1, ... , cn-1) with c ∈ C

Coding is a memory-less assignment:

information word code word

u = (u0, u1, ... , uk-1) → c = (c0, c1, ... , cn-1)

k information bits n code word bits n ≥ k

General notation: (n,k,dmin)q-block code

q = number of symbols (size of the symbol alphabet)

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S. 211

Radio Propagation Channels6 Coding

Code rate:

Number of code words: N = 2k

Systematic (separable) codes: code word c = (u, p)

m = n-kparity-check bits

1C ≤=nkR

u0 u1 u2 u3 ... ... uk-1

↓ ↓ ↓ ↓ ↓ ↓ ↓

c0 c1 c2 c3 ... ... ck-1 ck ck+1 ... ... cn-1

(6.1)

(6.2)

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S. 212

Radio Propagation Channels6 Coding

Example: (7,4) Hamming code Error correction capability: single errors within code words

parity check bits:c4 = c0 + c1 + c2

c5 = c0 + c1 + c3

c6 = c0 + c2 + c3

u0 u1 u2 u3

↓ ↓ ↓ ↓

c0 c1 c2 c3 c4 c5 c6

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S. 213

Radio Propagation Channels6 Coding

Error correction by evaluation of parity check equations:s0 = y0 + y1 + y2 + y4

s1 = y0 + y1 + y3 + y5

s2 = y0 + y2 + y3 + y6

Syndrome : s = (s0 s1 s2)

Syndrome does not depend on the code word, only on the error

Table for the assignment of error positions

syndrome error position s0 s1 s2

no error 0 0 0 error at bit No. 0 1 1 1 error at bit No. 1 1 1 0 error at bit No. 2 1 0 1 error at bit No. 3 0 1 1 error at bit No. 4 1 0 0 error at bit No. 5 0 1 0 error at bit No. 6 0 0 1

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S. 214

Radio Propagation Channels6 Coding

Error correction and error detection

Number of detectable errors: te = dmin − 1

Number of correctable errors: dmin is even: t = (dmin − 2) / 2 dmin is odd: t = (dmin − 1) / 2

dmin = 3 dmin = 4

c1 c2 c3

t

te

c1 c2 c3

t

te

(6.3)

(6.4)(6.5)

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S. 215

Radio Propagation Channels6 Coding

Convolutional codes, properties and definition no blockwise generation of code words, but convolution of a whole

sequence with a set of generator coefficients

no analytical methods for construction of code words ⇒computer search

simple processing of reliability information of the demodulators (soft decision input)

sensitive with respect to burst errors

usually binary codes

ML decoding with the Viterbi algorithm

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S. 216

Radio Propagation Channels6 Coding

Block diagram of a general (n,k,m) convolutional encoder (m = L − 1)

+ + +

k. . . .

ar

1 2 3 . . . .ur

n-1. . . . . . . . . . 1 2 3 n

++

L

k. . . .1 2 3 k. . . .1 2 3

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S. 217

Radio Propagation Channels6 Coding

Example:

n = 2, k = 1, m = 2

(2,1,2) convolutional encoder

RC = 1/2

ar,1 = ur + ur−1 + ur−2

ar,2 = ur + ur−2

u = (1,1,0,1,0,0,...)

a = (11,01,01,00,10,11,...)

+

ar,2ar,1

+

ur ur-1 ur-2ur

ar

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S. 218

Radio Propagation Channels6 Coding

General behaviour of bit error probabilities, coding gain

1

10−1

10−2

10−3

10−4

10−5

Perr

−4 −2 0 2 4 6 8 10Eb/N0 in dB

with channel coding

without channel coding

coding gain

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S. 219

Radio Propagation Channels6 Coding

Interleaver: Reordering of bits so that at the output of the deinterleaver quasi-

single errors are obtained Transparent transmission Problem: delay Interleaving for bits or blocks

Delay in coded transmission systems Delay because of the block structure of encoder and decoder Delay because of interleavers Delay from the decoding process (fundamental problem for

convolutional codes)

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S. 220

Radio Propagation Channels6 Coding

Block-

interleaver

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S. 221

Radio Propagation Channels6 Coding

Coding in GSM, example 1:TCH/F9.6 - traffic channel, full-rate 9,6 kbit/s

punctured convolutional

codeMUX

12 kBit/s(9,6 kBit/s)

11461

C =R

240 Bitdata

zeros4 Bit

244 Bit 456 Bit22,8 kBit/s

data per block with ∆t = 20 ms

data rate

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S. 222

Radio Propagation Channels6 Coding

Coding in GSM, example 2:TCH/FS - traffic channel, full-rate speech

convolu-tionalcodeMUX64 kBit/s

21

C =Rzeros

50 Bit

1280 Bit456 Bit

22,8 kBit/sspeechencoder

Ia

Ib

II

cyclic redun-dancy checkcode (CRC)

3 parity check bits

132 Bit

78 Bit

4 Bit

132 Bit

53 Bit

189 Bit

MUX378 Bit

78 Bit

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S. 223

Radio Propagation Channels7 Multiple Access Schemes

TDMA (time division multiple access)

FDMA (frequency division multiple access)

CDMA (code division multiple access)

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S. 224

Radio Propagation Channels7 Multiple Access Schemes

Combination − TDMA/FDMA

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S. 225

Radio Propagation Channels7 Multiple Access Schemes

Frequency hopping

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S. 226

Radio Propagation Channels7 Multiple Access Schemes

Space division multiple access – SDMA

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S. 227

Radio Propagation Channels7 Multiple Access Schemes

Example: transmission frame for GSM

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S. 228

Radio Propagation Channels7 Multiple Access Schemes

TDMA frame for GSM

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S. 229

Principle serving a large number of mobile stations ⇒ subdividing the area

to be served into radio cells limiting effects in cellular radio systems:

interference from other mobile and base stations radio wave attenuation ⇒ limited range

relation for the average received power: control of cell radius by adjusting the transmit power cell shape of idealized systems: circle – may be modelled by a

regular hexagon frequency can be reused behind a certain distance

Radio Propagation Channels8 Cellular Systems

γ−dP ~E (8.1)

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S. 230

Radio Propagation Channels8 Cellular Systems

Frequency reuse Cluster = group of k cells, on which the frequency channels are distributed (k = cluster size)

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S. 231

Radio Propagation Channels8 Cellular Systems

Frequency reuse (k = cluster size = 7)

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S. 232

Radio Propagation Channels8 Cellular Systems

Cellular network with frequency reuse

Definition of geometry R = cell radius D = reuse distance normalized reuse distance:

kRDq 3≈= (8.2)

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S. 233

Radio Propagation Channels8 Cellular Systems

Interference in the uplink

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S. 234

Radio Propagation Channels8 Cellular Systems

Interference in the downlink

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S. 235

Radio Propagation Channels8 Cellular Systems

Coarse estimate for the SIR in the downlink:

numerical evaluation for γ = 4:

γ

γ

γ

γ

=

−≥≈

∑ DR

d

rIS

ii

66

1

( ) 2/361

61 γγ kq

IS

≈≈

k 3 4 7 9 12 13 10 lg S/I 11,3 dB 13,8 dB 18,7 dB 20,8 dB 23,3 dB 24,0 dB

(8.3)

(8.4)

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Prof. Dr.-Ing. Andreas Czylwik Radio Propagation Channels WS 2009/2010

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Radio Propagation Channels9 Methods for Capacity Enhancement

Sectorization Smart antennas Multiuser detection Interference cancellation Adaptation to the radio channel

Page 237: Radio Propagation Channels

FachgebietNachrichtentechnische Systeme

N T SUNIVERSITÄT

D U I S B U R GE S S E N

Prof. Dr.-Ing. Andreas Czylwik Radio Propagation Channels WS 2009/2010

S. 237

Radio Propagation Channels

Thank you for your attention!