Radio Propagation Channels - nts.uni-due.dents.uni-due.de/downloads/rpc/radio_pro_ch_1_122.pdf · Andreas Czylwik Radio Propagation Channels WS ... WCDMA for UMTS, John Wiley Fachgebiet

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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 2005/2006

S. 1

Radio Propagation Channels

Prof. Dr.-Ing. Andreas Czylwik


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Radio Propagation ChannelsOrganisational

Lecture 2 hours/weekExercise 1 hour/weekTransparencies on web siteWritten examination

Department for Communication Systems Diploma and Master ThesesHome page: http://nts.uni-duisburg.de

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Radio Propagation ChannelsTextbooks

Basic textbooks:T. S. Rappaport: Wireless communications, Prentice HallG. S. Stüber: Principles of mobile communications, KluwerAcademic PublishersW. C. Jakes: Microwave mobile communications, John WileyK. David, T. Benkner: Digitale Mobilfunksysteme, Teubner-Verlag

Advanced textbooks:J. D. Parsons: The mobile radio propagation channel, John WileyJ. Eberspächer, H.-J. Vögel: GSM - Global system for mobilecommunication, Teubner-VerlagH. Holma, A. Toskala: WCDMA for UMTS, John Wiley


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Radio Propagation ChannelsContents

1 Introduction2 Wave propagation in mobile communications3 Linear time-variant systems4 Modulation5 Diversity schemes6 Coding7 Multiple access methods8 Cellular systems

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Radio Propagation Channels1 Introduction

History of radio transmission1888 Heinrich Hertz: Proof of propagation of electromagnetic waves through free space1895 Gugliemo Marconi: First transmission of messages with a radio system over a distance of several km‘s1958-1977 A-Net in Germany1972-1994 B-Net in Germany1986-2000 C-Net (1st generation)1992 D-Net − GSM (2nd generation)1994 E-Net - DCS 1800 (2nd generation)2003 UMTS (3rd generation)2010 4th generation ??


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Classification of mobile radio systemsType of mobile station

Land radio, marine radio, air radioType of base station

Terrestrical base stations, satellite base stationsType of services

Broadcast (radio/TV), bidirectional communication (mobile phone, wireless local area networks –WLANs)

Type of communication signalsSpeech, pictures, video, data, navigation, locationAnalog/digital

Structure of the networkCellular net, Ad-Hoc net, local net, point-to-point

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Cellular systems in Germany (2nd and 3rd generation)GSM (Global System for Mobile Communications): public mobile phone system with world-wide roamingUMTS (Universal Mobile Telecommunication System): Standard des zukünftigen breitbandigeren öffentlichen Mobilfunksystems


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Local systems in GermanyDECT (Digital European Cordless Telephone): Cordless standardfor communication short distances (indoor)Bluetooth: cordless standard for small and smallest distances and medium data ratesWLAN IEEE 802.11: Class of wireless local area networks with high data rates

Future systemsUltra-wideband systems for small distances and highest data rates

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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 einige 100 m

Network operator T-Mobil

D2 Vodafone

E-Plus

VIAG Interkom

5 Network operators Still open Private networks Private networks Private networks


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Basic problems of mobile radioTime 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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Time dispersion / frequency selectivity→ adapted transmission methods / equalizers

Impulse response:


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Alternative solution:multicarrier transmission

Transfer function:

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Shared medium → multiple access method necessaryLarge number of users → cellular systems, since bandwidth is limitedSupporting user mobility:

HandoverInternational roaming

Mobile phone is registered at home location register HLR1Connecting in a foreign networkInformation exchange between mobile switching centerMSC2 and MSC1Entries 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


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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

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S. 15

Radio Propagation Channels2 Wave Propagation

Wave propagation

Physical effects


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Maxwell‘s EquationsAmpere‘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)

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Material properties: ε = permittivityµ = permeabilityκ = conductivity


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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

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Material equations for lineare homogeneous isotropic lossy dielectric media:

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

= diffraction index

HBED

EJ

0

r0µ

εεκ

=== (2.3)

(2.4)(2.5)

rε=n


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Wave equationInserting 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)

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S. 21


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

++=

++=


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Solution for cartesian coordinates for κ = 0: homogeneous planewaveExample: 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

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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)


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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

−

−

+

+

−

−

+

+ =−=−==y

x

x

y

y

x

y

xHE

HE

HE

HEZD

Ω≈Ω== 377π1200

00 ε

µZ(2.27)

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PolarizationGeneral approach for a plane wave propagating in z-direction:

Phase difference of waves: ∆ϕ = ϕy − ϕx

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

(2.28)(2.29)

(2.30)

(2.31)

kzyyyxxx

kzyyxx

kzyyxx

tEtE

tEtEt

EE

j

j

j

e])cos(ˆ)cos(ˆ[

e])()([)(

e][

−

−

−

+++=

+=

+=

ee

eeE

eeE

ϕωϕω

yx EE ˆˆ =


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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 =

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S. 27


Planar wave in an arbitrary directionLocation 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)

kr

kr

kr

ee

eeE

eeE

j222111

j2211

j2211

e])cos(ˆ)cos(ˆ[

e])()([)(

e][

−

−

−

+++=

+=

+=

ϕωϕω tEtE

tEtEt

EE

(2.36)(2.37)

(2.38)


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Reflection and diffraction 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>

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S. 29


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)


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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

ααα

nnn

nEE

t−+

==

e22

1221e

22

e21

e

g

sincos

cos2

ααα

nnnn

nnEE

t−+

−==⊥

⊥⊥

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S. 31


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


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Total reflectionFor the roots become imaginary.

Total reflection if:

(only possible if n1 > n2)

(2.49)

0sin e22

122 <− αnn

1|||||| ==⇒= ∗∗ z

zrzzr

1

2esin

nn>α

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S. 33


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


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S. 34


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)

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S. 35


AntennasHertz‘ dipole in free Space

Point-shaped oscillating charges+q and −q

Distance ∆l << λ/4∆l ⋅ I = dipole momentField is symmetric withrespect to rotation

Description in polar coordinates

y

x

z

ϕ

ϑ

r

Er

Eϑ

Hϕ


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S. 36


Complex amplitude of the magnetic field:

Complex amplitude of the electric field:

ϕϕλϑ

λeHH ⋅⋅

+⋅⋅∆== − rkerr

lI jπ2j

1sin2

j

rrk

rkr

errr

lIZ

errr

lIZ

e

eEEE

⋅⋅

+⋅⋅∆+

⋅⋅

++⋅⋅∆=+=

−

−

j2

0

j2

0

π2jπ2jcos2

2j

π2jπ2j1sin

2j

λλϑλ

λλϑλ ϑϑ

(2.53)

(2.54)

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S. 37


Far field approximation:

Wave fronts are spherical surfaces ⇒ spherical waveField strengths do not depend on azimuth angle ϕDependence of field strength with respect elevation: ∼ sin ϑLarge distances:

Curvature of wave fronts is negligibleSperical wave ≈ planar wave

ϑϑ

ϕϕ

ϑλ

ϑλ

eEE

eHH

⋅⋅⋅∆==

⋅⋅⋅∆==

−

−

rk

rk

er

lIZ

er

lI

j0

j

sin2

j

sin2

j (2.55)

(2.56)


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S. 38

Free space propagation

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


2T

isoπ4 dPP =′ (2.57)

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S. 39


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 ⋅⋅⋅=⋅⋅= λ

2

RT

2

RTT

Rπ4π4

⋅⋅=

⋅⋅=fd

cGGd

GGPP λ

(2.58)

(2.59)

(2.60)


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


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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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 ΩRadiated field strength of the transmit antenna:

0

2eff,0

ZE

P =′

dGP

EdGPE

P TTeff,02

TT

0

2eff,0

T30

π4Z⋅⋅

=⇒⋅==′

(2.64)

(2.65)

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S. 43


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


DiffractionWave propagation according to geometrical optics ifλ << object sizeGeometrical optics: tightlight-shadow borderDifference with respect to optics: field strength in the shadow of buildings and other obstacles is not negigibleHuygens‘ principle

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S. 45


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, notationsPlane perpendicular to the line-of-sight,Locations of the same additional time delay: concentrical circles around the line-of-sight axis


d1

h

d2

Transmitter Receiverl1 l2

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S. 47


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


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

2λ⋅=∆ nxn

(2.71)

m2,122

11 =⋅= dr λ

(2.72)nvn ⋅= 2

(2.73)

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S. 49


Fresnel zones: Sum of distances with respect to two points is constant ⇒ ellipseLocations with the same phase difference lie on the Fresnel ellipsoid:

Almost unaffected transmission if no obstacle is within the firstFresnel zone


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S. 50


Model for an obstacle: ideal absorbing half-plane

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

d1

h

d2Transmitter Receiver

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S. 51


Transmission factornormalized to free-space transmission:

∫∞ −+=v

t tEE de

2j1 2

2j

0

π(2.74)


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Diffraction problems in real propagation scenarios are more complex:

Finite dimensions of obstaclesMultiple diffractionsBuildings are not ideal absorbersFinite dimension of the absorbers in propagation directionRough surfacesPropagation 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


Single and multipath propagation, overviewDoppler effectFast fadingTime dispersionPropagation scenariosSpatial correlation


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S. 54


Single path propagationAssumptions: distance x << d , direct line-of-sight, no obstacles, plane waveReceived signal:

with the wave number k = 2π / λ

)cos(j0 10e)( xktAtr θω −= (2.75)

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S. 55


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 HzAmplitude amplitude of the received signals:⇒ no fast fading effect

t

tvt

A

Atr)(j

0

)cos2(j0

D0

10

e

e)(ωω

θλπω

−

−

=

=

konst.)( 0 == Atr

10

1 coscos2

θθλπ

ωcfvvf D

D ===

(2.76)

(2.77)

(2.78)


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S. 56


Two-path propagationReceived 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


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


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


Example: ground reflection, earth curvature neglectedSmall angle of incidence ⇒

Contributions fron two paths:

1hv −≅≅ rr

l1

l2

hT

hR

d

ϕ∆−⋅−⋅+≅+= j0021 )1( eEEEEE (2.84)


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S. 60


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)

31


N T SUNIVERSITÄT



S. 61


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)


N T SUNIVERSITÄT



S. 62


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

32


N T SUNIVERSITÄT



S. 63


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)


N T SUNIVERSITÄT



S. 64


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

33


N T SUNIVERSITÄT



S. 65


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

Squared magnitude (~ received power):

AR and AI are random variablesApproximation: large number of statistically independent propagation paths⇒ central limit theorem is applicable

(2.100)∑=

−=n

i

xki iAtr

1

cosje)( θ

[ ] [ ] 2I

2R

2

1I,

2

1R,

2

)()(

)cossin()coscos()(

xAxA

xkAxkAxrn

iii

n

iii

+=

+

= ∑∑

==θθ (2.101)

(2.102)


N T SUNIVERSITÄT



S. 66


Assumption: AR and AI show a Gaussian distribution and 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)

34


N T SUNIVERSITÄT



S. 67


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

σσπ

+−⋅=

⋅=

(2.108)

(2.109)

(2.110)

(2.111)


N T SUNIVERSITÄT



S. 68


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)

35


N T SUNIVERSITÄT



S. 69



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 σσ

−==

(2.118)


N T SUNIVERSITÄT



S. 70



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

ty

dBin 2 2

A

Pσ(2.120)

36


N T SUNIVERSITÄT



S. 71

Probability density function of the power transfer factor:

P

fP(P)22

1Aσ

22 Aσ



N T SUNIVERSITÄT



S. 72


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)

37


N T SUNIVERSITÄT



S. 73


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 σσ

−==


N T SUNIVERSITÄT



S. 74


Rayleigh probability density function:

0.2

0.4

0.6

0.8

-2 -1 0 1 2 3 4 A/σA

fA(A) ⋅ σA

38


N T SUNIVERSITÄT



S. 75


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)


N T SUNIVERSITÄT



S. 76


Coordinate transformation:

Joint pdf for AR and AI

ϕcosR

2I

2R⋅=

+=AA

AAA (2.131)(2.132)

AR

AI

ϕA

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)

39


N T SUNIVERSITÄT



S. 77


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


N T SUNIVERSITÄT



S. 78


Definition of the modified Bessel function of zeroth order:

Rice' K-factor:

(2.139)

(2.140)

(2.141)

(2.142)

)(Iπ2dedeπ1)(I 0

2π

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

=

40


N T SUNIVERSITÄT



S. 79


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


N T SUNIVERSITÄT



S. 80


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)

41


N T SUNIVERSITÄT



S. 81

-180 -90 0 90 180 ϕ

fϕ(ϕ)

K = 0K = 1

K = 2

K = 8


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


N T SUNIVERSITÄT



S. 82


Doppler spectrumspectral broadening from different Doppler frequencies for eachindiviual path in a multipath propagation environmentThe 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

42


N T SUNIVERSITÄT



S. 83


Calculation of the Doppler spectrum with the following assumptions:

Omnidirectional antenna at the mobile stationMobile stations are moving with constant velocity in any arbitrary directionVery large number of reflectors/scatterers equally distributedaround the mobile stationSame statistical properties for each pathSame average power for each pathAngles of arrival are equally distributedPath amplitudes and angles of arrival are statistically independent


N T SUNIVERSITÄT



S. 84


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

Doppler frequency as a function of the angle of arrival:

Probability density function of the Doppler frequency:

≤≤−=

else0ππfor)( π2

1 ϕϕϕf

)cos()cos()( maxD ϕϕλ

ϕ ⋅== fvf

∑=i i

fi

ff

ff)(

)()(

ddD

DD ϕ

ϕ

ϕ

ϕ

(2.145)

(2.146)

(2.147)

43


N T SUNIVERSITÄT



S. 85


Calculationof the Doppler spectrum

ϕ

fϕ(ϕ)

−π π

fD = fmax⋅cos(ϕ)

−fm

ax

f max

f D

ϕ

f f D(f D

)


N T SUNIVERSITÄT



S. 86


Derivative of the nonlinear characteristic:

Substituting ϕ by fD :

Probability density function of the Doppler frequency

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

D ϕϕλϕ

ϕ −⋅=−= fvd

df (2.148)

2

max

D2

22

1)(cos1)sin(

1)(cos)(sin

−=−=⇒

=+

ffϕϕ

ϕϕ

2D

2max

Dπ

1)(D ff

ff f−

=

(2.149)

(2.150)

(2.151)

44


N T SUNIVERSITÄT



S. 87


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


N T SUNIVERSITÄT



S. 88


2. Approach for calculation of the Doppler spectrum: Analysis of the autocorrelation function of the received signalReceived 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)

45


N T SUNIVERSITÄT



S. 89


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)


N T SUNIVERSITÄT



S. 90

∫∞

∞

−⋅⋅⋅=-

AA PNS ττωω ωτde)(J)( jmax00


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)

46


N T SUNIVERSITÄT



S. 91

<−

⋅⋅=

else 0

for2

)(max22

max0 ωω

ωωωPN

SAA


(2.163)


N T SUNIVERSITÄT



S. 92


Autocorrelation function of the complex amplitude

-3 -2 -1 1 2 3

RAA(τ)

τ⋅fmax

47


N T SUNIVERSITÄT



S. 93


Power spectral density of the complex amplitude

SAA(ω)

ωωmax−ωmax


N T SUNIVERSITÄT



S. 94


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

48


N T SUNIVERSITÄT



S. 95


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


N T SUNIVERSITÄT



S. 96


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

Srr(ω)

−ω0−ωmax −ω0+ωmax ω0−ωmax ω0+ωmax

−ω0 ω0 ω

49


N T SUNIVERSITÄT



S. 97


Temporal dispersionDescription 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)


N T SUNIVERSITÄT



S. 98


Standard deviation of the impulse spreading (delay spread):

∫

∫∞

∞⋅−

=∆

0

20

22

d)(

d)()(

tth

tthttt (2.171)

50


N T SUNIVERSITÄT



S. 99


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

ττττττ d

)d...()(

+=

PP (2.172)

ττ

τττ ∆−

∆= e)( 0PP log(Pτ(τ))

τ

log(P0/∆τ)(2.173)


N T SUNIVERSITÄT



S. 100


Power delay profiles for testing GSM systemsrural (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)

51


N T SUNIVERSITÄT



S. 101


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


τ/µs


τ/µs

(2.175)


N T SUNIVERSITÄT



S. 102


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


τ/µs


τ/µs

(2.176)

52


N T SUNIVERSITÄT



S. 103


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


τ/µs


τ/µs

(2.177)


N T SUNIVERSITÄT



S. 104


Short term fadingRayleigh 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⟩)

53


N T SUNIVERSITÄT



S. 105


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


N T SUNIVERSITÄT



S. 106


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 thecomplex amplitude:

Time derivative ofthe amplitude:

22)( IR AAtA +=

t

A(t)

A + dA A

dt

AAt

tAA

&& dd

dd =⇒=

(2.178)

(2.179)

54


N T SUNIVERSITÄT



S. 107


Probability that the amplitude and ist 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 &&K&K +∩+=

(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 ,, ⋅=

⋅==


N T SUNIVERSITÄT



S. 108


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 ,==

A&d

∫∫∞∞

==0

,0

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

55


N T SUNIVERSITÄT



S. 109


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&


N T SUNIVERSITÄT



S. 110


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

A

A

A

A

A

AA

A

A

A

A

A

A

AA

AfA

A

AAA

AAAAfAN

σσ

σσ

σσ

σσ

σ

σσσ

σσ

−−

∞−−

−∞−

∞

=⋅=

−

⋅⋅=

⋅⋅=

=

∫

∫

&

&

&&

&

&

&

&

& &&

&&&&

(2.191)

56


N T SUNIVERSITÄT



S. 111


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


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

57


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S. 113


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


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

∆=⋅

λF

maxFlg xfT

λF

maxFxfT ∆=⋅

20 lg R20 lg R

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S. 115


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


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


The spatial correlation doesnot depend on the direction

The correlation decreasesfast within small distances


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S. 118


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 power.

Path amplitudes are statistically independent from angles of arrival.

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S. 119


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

(2.202)

BS

∆ϕ

ϕ0MS

+≤≤−

∆=∆∆

else0

for1)( 2020

ϕϕ

ϕϕτϕ

ϕϕf


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S. 120


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

No analytical 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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Autocorrelation function at the base station

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(∆x)

∆x/λ ∆x/λ

ϕ0 = 0°ϕ0 = 30°ϕ0 = 60°ϕ0 = 90°

∆ϕ = 2,5°∆ϕ = 5°∆ϕ = 10°∆ϕ = 20°

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°


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S. 122


Path loss modelsAverage 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

⋅⋅=fd

cGGPP

000

0R kff

ddPP

n⋅

⋅

⋅=

−−γ

(2.206)

(2.207)

Radio Propagation Channels - nts.uni-due.dents.uni-due.de/downloads/rpc/radio_pro_ch_1_122.pdf · Andreas Czylwik Radio Propagation Channels WS ... WCDMA for UMTS, John Wiley Fachgebiet

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