Radio Propagation Channels
Transcript of 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
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
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
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
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
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
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. 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)
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. 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
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. 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
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. 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
]
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. 11
Radio Propagation Channels1 Introduction
Time dispersion / frequency selectivity → adapted transmission methods / equalizers
Impulse response:
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. 12
Radio Propagation Channels1 Introduction
Alternative solution: multicarrier transmission
Transfer function:
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. 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
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. 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
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. 15
Radio Propagation Channels2 Wave Propagation
Wave propagation Physical effects
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. 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)
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. 17
Radio Propagation Channels2 Wave Propagation
Material properties: ε = permittivity µ = permeability κ = conductivity
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. 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
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. 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
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. 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)
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. 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
++=
++=
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. 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
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. 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)
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. 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)
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. 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 =
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. 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 =
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. 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)
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. 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>
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. 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)
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. 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−+
−==⊥
⊥⊥
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. 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
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. 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
>α
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. 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
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. 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)
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. 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
Eϑ
Hϕ
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. 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)
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. 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)
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. 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)
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. 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 ⋅
⋅⋅
=⋅⋅
=λ
2
RT
2
RTT
Rπ4π4
⋅⋅=
⋅⋅=
fdcGG
dGG
PP λ
(2.58)
(2.59)
(2.60)
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. 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)
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. 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)
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. 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)
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. 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)
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. 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
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. 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
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. 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
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. 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)
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. 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
2λ
⋅=∆ nxn
(2.71)
m2,122
11 =
⋅=
dr λ
(2.72)nvn ⋅= 2
(2.73)
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. 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
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. 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
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. 51
Radio Propagation Channels2 Wave Propagation
Transmission factor normalized to free-space transmission:
∫∞ −+
=v
t tEE de
2j1 2
2j
0
π(2.74)
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. 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
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. 53
Radio Propagation Channels2 Wave Propagation
Single and multipath propagation, overview Doppler effect Fast fading Time dispersion Propagation scenarios Spatial correlation
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. 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)
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. 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)
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. 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)
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. 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)
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. 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)
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. 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)
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. 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)
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. 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)
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. 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
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. 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)
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. 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
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. 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)
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. 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)
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. 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)
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. 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)
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. 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)
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. 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)
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. 71
Probability density function of the power transfer factor:
P
fP(P)22
1Aσ
22 Aσ
Radio Propagation Channels2 Wave Propagation
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. 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)
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. 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
−==
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. 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
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. 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)
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. 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)
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. 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
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. 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
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
=
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. 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
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. 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)
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. 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:
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. 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
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. 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
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. 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)
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. 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
)
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. 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
Dπ
1)(D ff
ff f−
=
(2.149)
(2.150)
(2.151)
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. 87
Radio Propagation Channels2 Wave Propagation
Equal power for all paths → received spectrum is proportional to the pdf of the Doppler frequency
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. 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)
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. 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)
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. 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)
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. 91
<−
⋅⋅=
else 0
for2
)(max22
max0 ωω
ωωωPN
SAA
Radio Propagation Channels2 Wave Propagation
(2.163)
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. 92
Radio Propagation Channels2 Wave Propagation
Autocorrelation function of the complex amplitude
-3 -2 -1 1 2 3
RAA(τ)
τ⋅fmax
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. 93
Radio Propagation Channels2 Wave Propagation
Power spectral density of the complex amplitude (Jakes spectrum)
SAA(ω)
ωωmax−ωmax
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. 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
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. 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
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. 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
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. 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)
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. 98
Radio Propagation Channels2 Wave Propagation
Standard deviation of the impulse spreading (delay spread):
∫
∫∞
∞⋅−
=∆
0
20
22
d)(
d)()(
tth
tthttt (2.171)
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. 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)
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. 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)
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. 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)
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. 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)
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. 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)
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. 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⟩)
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. 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
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. 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)
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. 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 ,, ⋅=
⋅==
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. 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
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. 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
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. 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)
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. 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)
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. 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
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. 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)
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. 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
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. 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
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. 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)
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. 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
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. 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.
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. 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
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. 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)
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. 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)
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. 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)
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. 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
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. 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)
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. 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
∆+⋅=∆
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. 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
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. 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
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. 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
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. 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
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. 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
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. 131
Radio Propagation Channels2 Wave Propagation
Geometry-based models
BS
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. 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)
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. 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)
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. 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
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. 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)
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. 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)
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. 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)
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. 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
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. 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)
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. 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
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. 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
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. 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
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. 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
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. 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
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. 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
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. 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)
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. 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)
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. 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)
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. 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)
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. 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)
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. 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)
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. 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)
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. 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)
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. 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)
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. 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
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. 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)
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. 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
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. 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
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. 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)
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. 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)
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. 161
Radio Propagation Channels3 Linear Time-Variant Stochastic Systems
delay spread:
∫
∫∞
∞⋅−
=∆
0
0
2
)d(
)d()(
ττ
τττττ
hh
hh
P
P
(3.33)
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. 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)
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. 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
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. 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
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. 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)
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. 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
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. 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
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. 168
Radio Propagation Channels4 Modulation
4-PSK = QPSK 8-PSK
Ra
Ia
Ra
Ia
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. 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
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. 170
Radio Propagation Channels4 Modulation
OQPSK π/4-DQPSK
Ra
Ia
Ra
Ia
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. 171
Radio Propagation Channels4 Modulation
QAM - quadrature amplitude modulation
2m states, m = 2k
Ra
Ia
4-QAM
64-QAM
16-QAM
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. 172
Radio Propagation Channels4 Modulation
QAM - quadrature amplitude modulation
2m states, m = 2k + 1
Ra
Ia
32-QAM
128-QAM
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. 173
Radio Propagation Channels4 Modulation
8-QAM 8-QΑΜ
Ra
Ia
Ra
Ia
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. 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)
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. 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)
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. 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)
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. 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)
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. 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
tπ
dd
==∆ϕω
,...3,2,1for2π
dd
=⋅==∆ iT
itϕω
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. 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
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. 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)
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. 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)
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. 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
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. 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)
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. 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
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. 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
...
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. 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
...
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. 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)
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. 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
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. 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+
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. 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
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. 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
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. 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
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. 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
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. 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)
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. 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)
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. 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
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. 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)
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. 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
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. 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
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. 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
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. 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)
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. 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
cγ
(5.13)0||||11
2 =⋅⋅−⋅⇒ ∑∑==
jM
iiij
M
ii ccaac
(5.14)
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. 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
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. 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)
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. 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)
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. 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
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. 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
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. 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
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. 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
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. 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)
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. 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)
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. 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
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. 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
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. 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)
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. 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
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. 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
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. 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
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. 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
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. 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)
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. 220
Radio Propagation Channels6 Coding
Block-
interleaver
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. 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
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. 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
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. 223
Radio Propagation Channels7 Multiple Access Schemes
TDMA (time division multiple access)
FDMA (frequency division multiple access)
CDMA (code division multiple access)
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. 224
Radio Propagation Channels7 Multiple Access Schemes
Combination − TDMA/FDMA
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. 225
Radio Propagation Channels7 Multiple Access Schemes
Frequency hopping
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. 226
Radio Propagation Channels7 Multiple Access Schemes
Space division multiple access – SDMA
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. 227
Radio Propagation Channels7 Multiple Access Schemes
Example: transmission frame for GSM
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. 228
Radio Propagation Channels7 Multiple Access Schemes
TDMA frame for GSM
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. 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)
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. 230
Radio Propagation Channels8 Cellular Systems
Frequency reuse Cluster = group of k cells, on which the frequency channels are distributed (k = cluster size)
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. 231
Radio Propagation Channels8 Cellular Systems
Frequency reuse (k = cluster size = 7)
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. 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)
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. 233
Radio Propagation Channels8 Cellular Systems
Interference in the uplink
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. 234
Radio Propagation Channels8 Cellular Systems
Interference in the downlink
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. 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)
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. 236
Radio Propagation Channels9 Methods for Capacity Enhancement
Sectorization Smart antennas Multiuser detection Interference cancellation Adaptation to the radio channel
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!