Dr.-Ing. Ch. KranzSMS IC CE DlfOctober, 2003Page 1
Communication Networks 1
Infi
neo
n
Principles of Mobile Communications
University Duisburg-EssenWS 2003/2004
N e v e r s t o p t h i n k i n g .
Dr.-Ing. Ch. KranzSMS IC CE DlfOctober, 2003Page 2
Wave PropagationOverview
• Large Scale Path Loss / Large Scale Fading
• Shadowing
• Path Loss Prediction
• Atmospherical Effects
• Channel Simulation
• Linear Time Variant Systems
• Linear Time Variant Channels - statistic description
• Special Channels
• Real mobile channels
Dr.-Ing. Ch. KranzSMS IC CE DlfOctober, 2003Page 3
Wave PropagationLarge Scale Path Loss
n Large scale path loss models
– predict the mean signal strength for an arbitrary TX-RX distance
– average attenuation as a function of distance: averaging over large distances so that the fast fading eliminate itself
– free space propagation (see first lectures):
– Lee‘s empirical approach of the propagation in real environments
2
RTT
Rp4
⋅⋅=
fdc
GGPP
000
0R kff
dd
PPn
⋅
⋅
⋅=
−−γ
Dr.-Ing. Ch. KranzSMS IC CE DlfOctober, 2003Page 4
– d0 and f0 are relative quantities to determine the parameters γ , n , and k0 experimentally
– the parameter γ , n , and k0 depend on the propagation environment
– LOS - line-of-sight: γ = 2
– typical values of the propagation exponent in urban areas: γ = 3 ... 4,5
– corresponds to 30 ... 45 dB attenuation per decade of distance
– numerous other empirical approaches, e.g. the COST-Hata-model for urban areas, that also includes the antenna height as a parameter
Wave PropagationLarge Scale Path Loss
Dr.-Ing. Ch. KranzSMS IC CE DlfOctober, 2003Page 5
Wave PropagationShadowing
n Shadowing
– shadowing effects lead to random fluctuations of the attenuation with respect to its mean, which is determined by the pass loss
– slow fading
– measurement result: attenuation is Gauß-distributed ⇒ log-normal-fading (normal in dB)
– received amplitude a(x) for radial movement away from the BS: (A(x) transmitted amplitude)
– alog-normal is Gauß-distributed with a standard deviation:
)(10)( 20
)(2
00
normal-log
xAdx
axa
xa
⋅⋅
⋅=
−γ
2normal-lognormal-log )(a=σ
Dr.-Ing. Ch. KranzSMS IC CE DlfOctober, 2003Page 6
Wave PropagationShadowing
– the log-normal distribution describes the random shadowing effect which occur over a large number of measurements at the same TX-RX distance
– standard deviation clearly depends on the propagation environment, values: σlog-normal = 4 ... 12 dB, typical value: σlog-normal = 8 dB
– shadowing effects are correlated over larger distances:
– in the literature very different numerical values are documented, e.g.:
n typical suburban area for 900 MHz: σlog-normal = 7,5 dB, Rlog-normal (∆x = 100 m) = 0,82
n microcellular area for 1700 MHz: σlog-normal = 4,3 dB, Rlog-normal (∆x = 10 m) = 0,3
2normal-log
normal-lognormal-lognormal-log
)(
)()()(
σ
xxaxaxR
∆+⋅=∆
Dr.-Ing. Ch. KranzSMS IC CE DlfOctober, 2003Page 7
Wave PropagationTypical Path Loss
– Typical behavior of the received signal power
Slow-Fading
Standard deviation of Slow-Fading
Average path loss
Rec
eive
d si
gnal
pow
er
in d
Bm
d/km
Dr.-Ing. Ch. KranzSMS IC CE DlfOctober, 2003Page 8
Wave PropagationPath Loss Prediction
Path Loss Prediction
Dr.-Ing. Ch. KranzSMS IC CE DlfOctober, 2003Page 9
Wave PropagationPath Loss Prediction
n Predicting the propagation characteristics between two antennas belongs to the most important tasks for the planning of mobile communication systems
n Number of subscribers rises -> size of cells reducesfrom macro cells (> some km) to micro cells (few hundreds of meters) in urban areas
n With decreasing size the importance of path loss prediction increases
n Path Loss Prediction will provide only mean or median values
n Small scale fading is adequately represented by Rayleigh- or Rice-distributions
Dr.-Ing. Ch. KranzSMS IC CE DlfOctober, 2003Page 10
Wave PropagationPath Loss Prediction
n path loss prediction
– empirical path loss models
– ray-tracing methods
n Tracing the rays for a prescribed number of reflections or some maximum attenuation
n precise collection of measurement data required (regarding geometry, material properties)
– diffraction theory
– combination of different approaches
Dr.-Ing. Ch. KranzSMS IC CE DlfOctober, 2003Page 11
Wave PropagationChannel Simulation
– geometry based models- most time consuming part is the determination of all relevant paths
from transmitter and receiver- the computations are performed with the help of the theory of
diffraction, reflection, scattering
BS
Dr.-Ing. Ch. KranzSMS IC CE DlfOctober, 2003Page 12
Wave PropagationChannel Simulation
Example:
n Ray-Tracing Examples of the city center of Helsinki @2125MHz 5Mchips/s
n a) low antenna 5m below the rooftop of a 30-m building- the street canyon effect became more important
n b) high antenna 1.5m above the rooftop- the over rooftop propagation plays an important role
n surrounding buildings height is app. 20-25 meters
n BS directional antenna GT=11.5dB, downtilted by 7degree
n Path Losses are calculated by adding the mean absolute tap powers of the complex impulse response
n Delay Spread is calculated out of the impulse response
Free (limited) evaluation software: www.awe-communications.com
(What’s that for a communication system?)
Dr.-Ing. Ch. KranzSMS IC CE DlfOctober, 2003Page 13
Wave PropagationPath Loss Prediction
Source: Verifying path loss and delay spread predictions of a 3D ray tracing propagation model in urban environment, Terhi Rautiainen1, Gerd Wölfle2, Reiner Hoppe3
Dr.-Ing. Ch. KranzSMS IC CE DlfOctober, 2003Page 14
Wave PropagationPath Loss Prediction
Source: Verifying path loss and delay spread predictions of a 3D ray tracing propagation model in urban environment, Terhi Rautiainen1, Gerd Wölfle2, Reiner Hoppe3
Dr.-Ing. Ch. KranzSMS IC CE DlfOctober, 2003Page 15
Wave PropagationPath Loss Prediction
Source: Verifying path loss and delay spread predictions of a 3D ray tracing propagation model in urban environment, Terhi Rautiainen1, Gerd Wölfle2, Reiner Hoppe3
Dr.-Ing. Ch. KranzSMS IC CE DlfOctober, 2003Page 16
Wave PropagationAtmospheric Effects
n atmospheric effects
– Attenuation due to resonance's of water and oxygen molecules for large frequencies (GHz-area)
– Additional attenuation due to rain, fog and snow
– Interpretation as layered wave guides
f/GHz
atm
osph
eric
alat
tenu
atio
n in
dB
/km H2O
O2
Dr.-Ing. Ch. KranzSMS IC CE DlfOctober, 2003Page 17
Wave PropagationChannel Simulation
n Channel simulation: model of the propagation channel which includes all major effects
– Propagation channel without
time dispersion
τ (x)
Filter
αlog-normal
realGauß-
process
path loss
shadowing
complexGauß-
process
Rice fading
2010x
y =2
0
γ−
⋅
dx
k
Filter
(delay)
(large scale path loss only)
IR j)( AAtA +=
Dr.-Ing. Ch. KranzSMS IC CE DlfOctober, 2003Page 18
Wave PropagationChannel Simulation
– Propagationchannel withtime dispersion
τN
path loss
power delay profile
2
0
γ−
⋅
dx
k
. . .
large scale fading
. . .
small scale fading
. . .
. . .
Dopplerprocess 1
Σ
Shadingprocess 1
Dopplerprocess 2
Shadingprocess 2
Dopplerprocess N
Shadingprocess N
τ2
τ1
h1
h2
hN
Dr.-Ing. Ch. KranzSMS IC CE DlfOctober, 2003Page 19
Wave PropagationPath Loss Prediction
Linear Time Variant Systems
Dr.-Ing. Ch. KranzSMS IC CE DlfOctober, 2003Page 20
Wave PropagationLinear Timevariant Systems
n Linear timevariant systems
– mobile channel = linear timevariant systeme.g. time variation due to receiver motion in space
– input-output description in the complex baseband:
n h(t,τ) = timevariant impulse response = input delay-spread function
Linear Timevariant
System
}e)(Re{)( 0jHF
ttxtx ω= }e)(Re{)( 0jHF
ttyty ω=
∫∞
∞−−= τττ d),()()( thtxty
Dr.-Ing. Ch. KranzSMS IC CE DlfOctober, 2003Page 21
Wave PropagationLinear Timevariant Systems
n Response to a single impulse
n causality: h(t,t − t0) = 0 if 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
Dr.-Ing. Ch. KranzSMS IC CE DlfOctober, 2003Page 22
Wave PropagationLinear Timevariant Systems
h(t,τ) = 0
τ
t
t0
t0 = 0
n timevariant impulse response - response to a single impulse
τ<0
t − t0 = τ
h(t0,τ)
h(t1,τ)
h(t2,τ)
h(t,τ) = 0 for t < t − τ
Dr.-Ing. Ch. KranzSMS IC CE DlfOctober, 2003Page 23
Wave PropagationLinear Timevariant Systems
n „discrete time“ representation:
n FIR filter with timevariant 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
Dr.-Ing. Ch. KranzSMS IC CE DlfOctober, 2003Page 24
Wave PropagationLinear Timevariant Systems
– transfer to the frequency domain
n H(ν,ω) = output Doppler-spread function
∫∞
∞−−−= uuuHuXY d),()(
21
)( ωωπ
ω
x(t)
X(ω)
y(t)
Y(ω)
h(t,τ)
H(ν,ω)
∫ ∫∞
∞−
∞
∞−
−−= ττων ωτν ddee),(),( jj tthH t
Dr.-Ing. Ch. KranzSMS IC CE DlfOctober, 2003Page 25
Wave PropagationLinear Timevariant Systems
– transfer to the frequency domain
Dr.-Ing. Ch. KranzSMS IC CE DlfOctober, 2003Page 26
Wave PropagationLinear Timevariant Systems
n „discrete frequency“ representation
∑−=
∆∆−∆∆−=
n
nmmmHmXY
p2),()()(
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
Dr.-Ing. Ch. KranzSMS IC CE DlfOctober, 2003Page 27
Wave PropagationLinear Timevariant Systems
– time variant transfer function T(t,ω)
– delay Doppler spread function S(ν,τ) (e.g. time variance because of a moving receiver)
∫∞
∞−= ωωω ω de),()(
p21
)( j ttTXty
∫∞
∞−
−= ττω ωτ de),(),( jthtT
∫∞
∞−
−= tthS t de),(),( jνττν
∫ ∫∞
∞−
∞
∞−
−= τντντ ν dde),()(p2
1)( j tStxty
(Convolution integral with inverse transf. doppler spread function)
Dr.-Ing. Ch. KranzSMS IC CE DlfOctober, 2003Page 28
Wave PropagationLinear Timevariant Systems
– Relation between the four presentation of a time variant linear system
– variables: t = observation time, τ = delay, ω = frequency, ν = Doppler frequency
– often true for real channels: slowly timevariant
h(t,τ)
S(ν,τ) T(t,ω)
H(ν,ω)
delay Doppler spread function
time variant transfer function
frequency Doppler-spread function
time variant impulse response
Dr.-Ing. Ch. KranzSMS IC CE DlfOctober, 2003Page 29
Wave PropagationLinear Timevariant Systems
measurement no. 1:timevariant impulse response: f0 = 1,8 GHz, LOS - line-of sight, omnidirectional non-moving antennas, distance = 95 m, industrial area
Slowly time variant
Dr.-Ing. Ch. KranzSMS IC CE DlfOctober, 2003Page 30
Wave PropagationLinear Timevariant Systems
measurement no. 1:timevariant transfer function: f0 = 1,8 GHz, LOS - line-of sight, omnidirectional non-moving antennas, distance = 95 m, industrial area
Dr.-Ing. Ch. KranzSMS IC CE DlfOctober, 2003Page 31
Wave PropagationLinear Timevariant Systems
measurement no. 2:time variant impulse response: f0 = 1,8 GHz, NLOS - non-line-of sight, omnidirectional non-moving antennas, distance = 230 m, industrial area
Dr.-Ing. Ch. KranzSMS IC CE DlfOctober, 2003Page 32
Wave PropagationLinear Timevariant Systems
measurement no. 2:time variant transfer response: f0 = 1,8 GHz, NLOS - non-line-of sight, omnidirectional non-moving antennas, distance = 230 m, industrial area
Dr.-Ing. Ch. KranzSMS IC CE DlfOctober, 2003Page 33
Wave PropagationLinear Timevariant Systems
measurement no. 3:time variant impulse response: f0 = 1,8 GHz, LOS - line-of sight, omnidirectional antennas, driven distance = 1m,distance to base = 95 m, industrial area
Dr.-Ing. Ch. KranzSMS IC CE DlfOctober, 2003Page 34
Wave PropagationLinear Timevariant Systems
measurement no. 3:time variant transfer response: f0 = 1,8 GHz, LOS - line-of sight, omnidirectional antennas, driven distance = 1m,distance to base = 95 m, industrial area
Dr.-Ing. Ch. KranzSMS IC CE DlfOctober, 2003Page 35
Wave PropagationPath Loss Prediction
Linear Time Variant Channels
Statistic Description
Dr.-Ing. Ch. KranzSMS IC CE DlfOctober, 2003Page 36
Wave PropagationLinear Timevariant Systems
– mobile channels are timevariant random systems ⇒ description with random methods
– restriction to (squared) means
n for computational costs
n Gauß-processes are completely described by (squared) means
n realistic approach: correlation
– auto correlation function of the received signal
[ ]{[ ]})j(
2*)j(
221
)j(1
*)j(12
12HF1HF
020020
010010
)()(
)()(E)}()({E
ϕωϕω
ϕωϕω
+−+
+−+
+
⋅+=⋅
tt
tt
etyety
etyetytyty
Dr.-Ing. Ch. KranzSMS IC CE DlfOctober, 2003Page 37
Wave PropagationLinear Timevariant Systems
n averaging over the carrier 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)}()({E
ttyy
tt
ttR
tytytyty
−
−
=
=⋅
ω
ω
Dr.-Ing. Ch. KranzSMS IC CE DlfOctober, 2003Page 38
Wave PropagationLinear Timevariant Systems
– auto correlation function of the complex amplitude of the received signal:
∫ ∫
∫ ∫
∞
∞−
∞
∞−
∞
∞−
∞
∞−
−−=
−−=
⋅=
2122*
1122*
11
2122*
22*
1111
2*
121
dd}),(),(}E{)()(E{
dd),()(),()(E
)}()({E),(
ττττττ
ττττττ
ththtxtx
thtxthtx
tytyttRyy
Dr.-Ing. Ch. KranzSMS IC CE DlfOctober, 2003Page 39
Wave PropagationLinear Timevariant Systems
n auto correlation of the system functions
E{h(t1,τ1)⋅h*(t2,τ2)} = Rhh(t1,t2;τ1,τ2)
E{H(ν1,ω1)⋅H*(ν2,ω2)} = RHH(ν1,ν2;ω1,ω2)
E{T(t1,ω1)⋅T*(t2,ω2)} = RTT(t1,t2;ω1,ω2)
E{S(ν1,τ1)⋅S*(ν2,τ2)} = RSS(ν1,ν2;τ1,τ2) ACF of delay Doppler spread function
ACF of time variant transfer function
ACF of freq. Doppler-spread function
ACF of time variant impulse response
Dr.-Ing. Ch. KranzSMS IC CE DlfOctober, 2003Page 40
Wave PropagationLinear Timevariant Systems
– relations between the auto correlation functions:
Rhh(t1,t2;τ1,τ2)
RSS(ν1,ν2;τ1,τ2) RTT(t1,t2;ω1,ω2)
RHH(ν1,ν2;ω1,ω2)
Dr.-Ing. Ch. KranzSMS IC CE DlfOctober, 2003Page 41
Wave PropagationPath Loss Prediction
Special Channels
Dr.-Ing. Ch. KranzSMS IC CE DlfOctober, 2003Page 42
Wave PropagationLinear Timevariant Systems
– special channels:
n WSS − wide-sense stationary channel: in between short time intervals the ACF Rhh(t1,t2;τ1,τ2) depends only on the time difference ∆t = t2 − t1.
Rhh(t1,t1+∆t;τ1,τ2) = Rhh(∆t;τ1,τ2)
RTT(t1,t1+∆t ;ω1,ω2) = RTT(∆t;ω1,ω2)
Ø consequences of the WSS-property:
)},(),({E),;,( 22*
112121 τντνττνν SSRSS ⋅=(ACF of delay Doppler spread function)
Dr.-Ing. Ch. KranzSMS IC CE DlfOctober, 2003Page 43
Wave PropagationLinear Timevariant Systems
)(p2),;(
)(p2de)},;(
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
Dr.-Ing. Ch. KranzSMS IC CE DlfOctober, 2003Page 44
Wave PropagationLinear Timevariant Systems
Ø consequence of the WSS-property:
scattering contributions with different Doppler frequencies are uncorrelated
n channel with uncorrelated scatterers (US − uncorrelated scattering channel): in the frequency domain the ACF RHH(ν1,ν2;ω1,ω2) depends only on the frequency difference ∆ω = ω2 − ω1, wide sense stationary in the frequency domain. (the mean and correlation do not depend on the choice offrequency)
RHH(ν1,ν2;ω1,ω1+∆ω) = RHH(ν1,ν2;∆ω)
RTT(t1,t2;ω1,ω1+∆ω) = RTT(t1,t2;∆ω)
Ø consequence of the wide sense stationary in the frequency property:
)},(),({E),;,( 22*
112121 ττττ ththttRhh ⋅=
Dr.-Ing. Ch. KranzSMS IC CE DlfOctober, 2003Page 45
Wave PropagationLinear Timevariant Systems
)();,(
)(de)};,(
dde)};,(
dde)},(),({E
de),(de),(E),;,(
21221
21)j(
21p21
1)j(
21p)2(1
21)j(
22*
11p)2(1
*
2j
22p21
1j
11p21
2121
2
221112
22112
2211
ττδτ
ττδωω
ωωω
ωωωω
ωωωωττ
τω
τωτωτω
τωτω
τωτω
−⋅−=
−⋅∆∆=
∆∆=
=
⋅=
∫
∫ ∫
∫ ∫
∫∫
∞
∞−
∆−
∞
∞−
∞
∞−
∆−−
∞
∞−
∞
∞−
−
∞
∞−
∞
∞−
ttP
ttR
ttR
tTtT
tTtTttR
hh
TT
TT
hh
∆ω = ω2 − ω1
Dr.-Ing. Ch. KranzSMS IC CE DlfOctober, 2003Page 46
Wave PropagationLinear Timevariant Systems
n consequence of the wide sense frequency stationarity:
– scattering of elementary scatters with different delays are uncorrelated -> US (uncorrelated scattering)
n WSSUS − wide-sense stationary uncorrelated scattering channel:
WSSUS is an important class of mobile channels with practical relevance:
– stationarity with respect to the observation time (small scale fading)e.g. GSM900, 150km/h -> 7.2ms/wavelength
– uncorrelated contributions due to scatterers with different time delay
Dr.-Ing. Ch. KranzSMS IC CE DlfOctober, 2003Page 47
Wave PropagationLinear Timevariant Systems
n autocorrelation 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)
ACF of delay Doppler spread function
ACF of time variant transfer function
ACF of freq. Doppler-spread function
ACF of time variant impulse response
Dr.-Ing. Ch. KranzSMS IC CE DlfOctober, 2003Page 48
Wave PropagationLinear Timevariant Systems
– physical model of the WSSUS-channel:
n single scattering with a large number of scatterers
n each scatterer is assigned a time delay, a Doppler shift as well as a scattering coefficient
independent scatterers
Dr.-Ing. Ch. KranzSMS IC CE DlfOctober, 2003Page 49
Wave PropagationLinear Timevariant Systems
n PSS(ν;τ) is proportional to the scattering function σ(ν,τ)
n σ(ν,τ) describes the
power distribution as
a function of the
Doppler frequency and
the time delay à
Doppler delay spectrum
Dr.-Ing. Ch. KranzSMS IC CE DlfOctober, 2003Page 50
Wave PropagationPath Loss Prediction
Real Mobile Channels
Dr.-Ing. Ch. KranzSMS IC CE DlfOctober, 2003Page 51
Wave PropagationReal Mobile Channels
– real mobile channels
n are in general non-stationary – however, stationarity is often a reasonable assumption for short distances in areas of small size (mobile phone <-> DVB)
n assumption: stationarity in small areas, significant scattering regions do not vary inside these areas (small scale fading) àWSSUS-approach is valid
n for larger distances shadowing becomes remarkable, so that the significant scattering regions will vary (large scale fading)
Dr.-Ing. Ch. KranzSMS IC CE DlfOctober, 2003Page 52
Wave PropagationReal Mobile Channels
– properties in the time domain (stationarity)
n no time delay for the observation time: ∆t = 0 ⇒ power delay profile = averaging of the delay Doppler spectrum over all Doppler frequencies:
n average time delay:
∫∞
∞−== ντνττ ν de);();0()( 0j
p21
SShhhh PPP
∫
∫∞
∞⋅
=
0
0
d)(
d)(
ττ
τττ
τ
hh
hh
P
P
(time invariant)
Dr.-Ing. Ch. KranzSMS IC CE DlfOctober, 2003Page 53
Wave PropagationReal Mobile Channels
n standard deviation of the delay spread:
∫
∫∞
∞⋅−
=∆
0
0
2
)d(
)d()(
ττ
ττττ
τ
hh
hh
P
P
Dr.-Ing. Ch. KranzSMS IC CE DlfOctober, 2003Page 54
Wave PropagationReal Mobile Channels
– properties in the frequency domain
n the correlation of the transfer function depends for different frequencies on the increasing frequency difference.
n coherence bandwidth = frequency shift for which there is still a significant correlation. (statistical measure over which the channel can be considered “flat”)
n frequency correlation without time delay of the observation time: ∆t = 0 ⇒ frequency-correlation spectrum = averaging of the frequency Doppler spectrum over all Doppler frequencies:
∫∞
∞−∆=∆=∆ νωνωω ν de);();0()( 0j
p21
HHTTTT PPP
Dr.-Ing. Ch. KranzSMS IC CE DlfOctober, 2003Page 55
Wave PropagationReal Mobile Channels
n relation between the frequency correlation spectrum and the power delay profile:
Phh(0,τ) = Phh(τ) PTT(0,∆ω) = PTT(∆ω)
n example for the frequency-correlation: PTT(∆ω) / PTT(0)
∆ω / 2π MHz
Dr.-Ing. Ch. KranzSMS IC CE DlfOctober, 2003Page 56
Wave PropagationReal Mobile Channels
Definition of coherence time Tc and coherence bandwidth Bc by using the autocorrelation function of the transfer function.
Dr.-Ing. Ch. KranzSMS IC CE DlfOctober, 2003Page 57
Wave PropagationReal Mobile Channels
Coherence Bandwidth:
- It can be assumed that the transfer function is almost constant within the coherence bandwidth
Coherence Time:
- The transfer function changes only slightly during the coherence time
Both have considerable effect on the design of a transmission system which is adapted to the channel characteristics.
Symbol Duration
Channel varies during the transmission of a single symbol-> large distortions result -> high bit error probability -> strong time variant behavior.
Channel can be considered constant during symbol transmission. Time invariant behavior of received signal
Dr.-Ing. Ch. KranzSMS IC CE DlfOctober, 2003Page 58
Wave PropagationReal Mobile Channels
Bandwidth B:
• The channel transfer function changes over the bandwidth B of a signal
• Multi-path spread is larger than symbol duration. IntersymbolInterference (ISI) occurs
• Frequency selective behavior of the radio channel
• Broadband Channel (frequency selective fading)
• Transfer function constant over the signal bandwidth B
• Only small ISI
• Narrowband channel (non frequency selective fading)
In practical mobile applications single Narrowband Channels are not applicable.
1. The velocity of the mobile is not small
2. Due to fading effects either very little or very strong attenuation occurs in a narrowband channel -> no technical possibility to improve the receive situation withsingle receivers