+small-scale channel modeling
TRANSCRIPT
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GRAZ UNIVERSITY OF TECHNOLOGY
Signal Processing and Speech Communications Lab
Mobile Radio Systems Small-Scale Channel ModelingKlaus Witrisal
Signal Processing and Speech Communication Laboratory
www.spsc.tugraz.at
Graz University of Technology
November 14, 2006
Mobile Radio Systems Small-Scale Channel Modeling p. 1/41
GRAZ UNIVERSITY OF TECHNOLOGY
Signal Processing and Speech Communications Lab
Outline
3-1 Introduction Mathematical models forcommunications channels
3-2 Stochastic Modeling of Fading Multipath ChannelsMultipath channelFading amplitude distribution (Rayleigh, Rice)WSSUS stochastic channel descriptionJakes Doppler spectrumExponential delay spectrum (power delay prole)
3-3 Classication of Small-Scale Fading
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GRAZ UNIVERSITY OF TECHNOLOGY
Signal Processing and Speech Communications Lab
ReferencesA. F. Molisch: Wireless Communications , 2005, Wiley
J. G. Proakis: Digital Communications , 4th ed., 2000, McGraw Hill
J. R. Barry, E. A. Lee, D. G. Messerschmitt: Digital Communication , 3rd ed., 2004,Kluwer
A. Paulraj, R. Nabar, and D. Gore: Introduction to Space-Time Wireless Communications , 2003, Cambridge
T. S. Rappaport: Wireless Communications Principles and Practice , 2nd ed., 2002,Prentice Hall
M. Ptzold: Mobile Fading Channels , 2002, Wiley
Figures (partly) extracted from these references
Mobile Radio Systems Small-Scale Channel Modeling p. 3/41
GRAZ UNIVERSITY OF TECHNOLOGY
Signal Processing and Speech Communications Lab
Channel Models
Signal processing channel models can be describedfor different interfaces
Application/ design objective determines choice ofappropriate model
basebandmodulation
carriermodulation
011011bit stream baseband
signal(analog)
radio channel
propagationchannel
baseband equivalent radio channel
sampled baseband equivalent radio channel
carrier de-modulation baseband
signal(analog)
(matched)filter
thresholddetector
010011bit streamsampling
(detection)
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GRAZ UNIVERSITY OF TECHNOLOGY
Signal Processing and Speech Communications Lab
Additive Noise Channel
Simplest model transmitted (TX) signal corrupted byadditive noise
r (t) = s (t) + n (t)
s(t) ... TX signalcan be bandpass (passband)
s(t) = 2 {s l(t)e j 2f c t}or (lowpass equivalent) baseband signal s
l(t)
(i.e. complex envelope of s(t))
r (t) ... received (RX) signal
Mobile Radio Systems Small-Scale Channel Modeling p. 5/41
GRAZ UNIVERSITY OF TECHNOLOGY
Signal Processing and Speech Communications Lab
Additive Noise Channel (contd)
Noise is usually modeled as white, Gaussian (additivewhite Gaussian noise AWGN)
n ( ) = N 02( ) F S n (f ) = N 02
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GRAZ UNIVERSITY OF TECHNOLOGY
Signal Processing and Speech Communications Lab
Additive Noise Channel (contd)
Sampled AWGN model
r [k] = s [k] + n[k]
Noise characterization (for complex baseband)
E{n[k]n[l]}= 2n [k l]
n[k] is zero-mean circularly symmetric complexGaussian (ZMCSCG)
Real and imaginary components are i.i.d.(independent, identically distributed)2n depends on (matched) lter at receiverfront-endReal and imaginary components have 2n / 2
Mobile Radio Systems Small-Scale Channel Modeling p. 7/41
GRAZ UNIVERSITY OF TECHNOLOGY
Signal Processing and Speech Communications Lab
Linear lter channel
For time- in variant channels
r (t) = s(t)c(t) + n(t)
= c( )s(t )d + n(t)c(t) ... impulse response of linear lter
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GRAZ UNIVERSITY OF TECHNOLOGY
Signal Processing and Speech Communications Lab
Linear lter channel (contd)
Sampled case
y[k] =L 1
l=0
h[l]s[k l] + n[k]
h[k] incorporatesTX pulse shapeRX (matched) lterphysical channel
n[k] ... AWGN (ZMCSCG)
This is actually an equivalent, whitened matched lter(WMF) channel model [Barry/Lee/Messerschmitt]
Mobile Radio Systems Small-Scale Channel Modeling p. 9/41
GRAZ UNIVERSITY OF TECHNOLOGY
Signal Processing and Speech Communications Lab
Linear time-variant lterchannel
Characterized by time-variant channel impulseresponse (CIR) c( ; t)
response of channel at time tto an impulse transmitted at time t
... elapsed time, age variabler (t) = s(t)c( ; t) + n(t)
=
c( ; t)s(t )d + n(t)
model for multipath propagation
c( ; t) =
i=0 i(t)( i(t)) (1)
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GRAZ UNIVERSITY OF TECHNOLOGY
Signal Processing and Speech Communications Lab
Linear time-variant lter ch.
[g 5-4 Rap]Mobile Radio Systems Small-Scale Channel Modeling p. 11/41
GRAZ UNIVERSITY OF TECHNOLOGY
Signal Processing and Speech Communications Lab
Stochastic modeling of fadingmultipath channels
Motivated by their randomly time-variant nature andlarge number of multipath components
Skip noise
Derivation of lowpass equivalent CIR from (1)
cl( ; t) =
i=0 i(t)e
j 2f c i (t)( i(t))
=
i=0
i(t)e j i (t)(
i(t)) (2)
considering discrete multipath componentsphase term i(t) varies dramatically
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GRAZ UNIVERSITY OF TECHNOLOGY
Signal Processing and Speech Communications Lab
Fading of an unmodulatedcarrier
TX signal is unmodulated carrier (CW) s l(t) = 1
RX signalr l(t) =
i=0 i(t)e j i (t)
sum of vectors (phasors)amplitudes i(t) change slowlyphases i (t) change by 2 if:
i changes by 1/f ci.e.: path length changes by wavelength
model r l(t) as a random process!
Mobile Radio Systems Small-Scale Channel Modeling p. 13/41
GRAZ UNIVERSITY OF TECHNOLOGY
Signal Processing and Speech Communications Lab
Fading of an unmodulatedcarrier (contd)
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GRAZ UNIVERSITY OF TECHNOLOGY
Signal Processing and Speech Communications Lab
Fading of an unmodulatedcarrier (contd)
Modeling r l(t) as a random process:
large number of multipath components are addedby central limit theorem (CLT):
r l(t) is complex Gaussian(CIR cl( ; t) is complex Gaussian in t-variable)
r l(t) has random phase and amplitude
in absence of dominant component:r l(t) is zero-mean complex Gaussian
its envelope |r l(t)| is Rayleigh distributedRayleigh fading channel
Mobile Radio Systems Small-Scale Channel Modeling p. 15/41
GRAZ UNIVERSITY OF TECHNOLOGY
Signal Processing and Speech Communications Lab
Fading of an unmodulatedcarrier (contd)
Rayleigh distribution:
f R (r ) =r
2 e
r 2 / (2 2 ) for r
0
characterized by 2: variance of underlying Gaussianprocesses
derivation of Rayleigh distribution ...
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GRAZ UNIVERSITY OF TECHNOLOGY
Signal Processing and Speech Communications Lab
Fading of an unmodulatedcarrier (contd)
central chi-square PDF Rayleigh PDFof n degrees of freedom characterized by
Mobile Radio Systems Small-Scale Channel Modeling p. 17/41
GRAZ UNIVERSITY OF TECHNOLOGY
Signal Processing and Speech Communications Lab
Fading of an unmodulatedcarrier (contd)
in presence of a dominant component:r l(t) is non-zero-mean complex Gaussian
its envelope
|r
l(t)
|is Ricean distributed
Ricean fading channel
Ricean distribution:
f R (r ) =r
2e
r 2 + s 2
2 2 I 0rs2
for r 0I 0(x) ... zero-order modied Bessel function of rst kindcharacterized by2 ... variance of underlying Gaussian processes ands2 = m21 + m22 ... power of mean (i.e. s2 = |E{r l(t)}|2)
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GRAZ UNIVERSITY OF TECHNOLOGY
Signal Processing and Speech Communications Lab
Time-selective fading
Characterization of the time variability
s l(t) = 1 r l(t) = cl( ; t)1 = cl(t) =
i=0 i(t)e j i (t)
Characterize autocorrelation function of cl(t)assume cl(t) is complex Gaussianassume cl(t) is wide-sense stationary (WSS)
Dene: spaced-time correlation function
c( t) = E {c
l (t)cl(t + t)}F
S c( )Doppler power spectrum S c( ) =
c( t)e j 2 td t
Mobile Radio Systems Small-Scale Channel Modeling p. 21/41
GRAZ UNIVERSITY OF TECHNOLOGY
Signal Processing and Speech Communications Lab
Time-selective fading (contd)
Doppler power spectrum: average power output of channelas a function of Doppler frequency
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GRAZ UNIVERSITY OF TECHNOLOGY
Signal Processing and Speech Communications Lab
Time-selective fading (contd)
[Rappaport g. 5-1]Mobile Radio Systems Small-Scale Channel Modeling p. 23/41
GRAZ UNIVERSITY OF TECHNOLOGY
Signal Processing and Speech Communications Lab
Time-selective fading (contd)
Jakes model for Dopplerpower spectrum
assumes mobile movingat const. velocity vuniformly distributedscattering around mobile
Jakes Doppler spectrum:
S c( ) =1
1
2max
2
(for normalized power)1 0.5 0 0.5 10
0.5
1
1.5
2
2.5
normalized Doppler frequency / max
D o p p
l e r p o w e r s p e c t r a
l d e n s i
t y
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GRAZ UNIVERSITY OF TECHNOLOGY
Signal Processing and Speech Communications Lab
Time-selective fading (contd)
Characterization of time-selective fading by parameters
RMS Doppler spread rms = 2 2
second centralized moment of normalized Doppler PSD
mean and mean squared Doppler spread
= S c( )d
S c( )d 2
= 2S c( )d
S c( )d Coherence time
T c 1
rms
Mobile Radio Systems Small-Scale Channel Modeling p. 25/41
GRAZ UNIVERSITY OF TECHNOLOGY
Signal Processing and Speech Communications Lab
Frequency-selective fading
of a (time-invariant) multipath channel
Characterization of the time dispersion
s l(t) = (t) r l(t) = cl( ; t)(t) =
i=0 i(t)e j i (t )(t i(t))
=
i=0 ie j i (t i)
ACF: Uncorrelated scattering assumption:
E{cl ( 1)cl( 2)}= S c( 1)( 1 2)S c( ) ... multipath intensity prole (= delay powerspectrum; = average power delay prole)
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GRAZ UNIVERSITY OF TECHNOLOGY
Signal Processing and Speech Communications Lab
Frequency-selective fading(contd)
Time-dispersion implies frequency-selectivityEquivalent channel characterization by channeltransfer function (TF) C l(f )
cl( )F
C l(f ) =
cl( )e
j 2f d
ACF of channel TF
S c( )F
C ( f ) = E {C
l (f )C l(f + f )}C ( f ) ... spaced-frequency correlation function
TF C l(f ) is wide-sense stationary (WSS in f ) if CIRcl( ) fullls uncorrelated scattering (US in )
Mobile Radio Systems Small-Scale Channel Modeling p. 27/41
GRAZ UNIVERSITY OF TECHNOLOGY
Signal Processing and Speech Communications Lab
Frequency-selective fading(contd)
Channel IR vs. channel TF
20 0 20 40 60 80 100 120 140 160
115
110
105
100
95
90
85
80
75
70
65
IDFT of transfer function after correction for linear phase shift
excess delay time [ns]
a m p
l i t u
d e
[ d B ]
0 200 400 600 800 10004
2
0
2
4
frequency [MHz]
p h a s
e a r g
( H ( f ) ) [ r a
d ]
phase transfer function (corrected for linear phase shift)
0 200 400 600 800 100090
80
70
60
50
frequency [MHz]
a m p
l i t u
d e
| H ( f ) | [ d B ]
amplitude transfer function
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GRAZ UNIVERSITY OF TECHNOLOGY
Signal Processing and Speech Communications Lab
Frequency-selective fading(contd)
Multipath intensity prole: average power output of channelas a function of delay
Mobile Radio Systems Small-Scale Channel Modeling p. 29/41
GRAZ UNIVERSITY OF TECHNOLOGY
Signal Processing and Speech Communications Lab
Frequency-selective fading(contd)
Characterization by parameters
RMS delay spread
rms = 2 2second centralized moment of normalized multipathintensity prole
mean and mean squared delay spread
= S c( )d
S c( )d
2 = 2S c( )d
S c( )d
Coherence bandwidth
Bc 1
rmsMobile Radio Systems Small-Scale Channel Modeling p. 30/41
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GRAZ UNIVERSITY OF TECHNOLOGY
Signal Processing and Speech Communications Lab
Frequency-selective fading(contd)
Characterization of mul-tipath intensity prole(simplied; suitable forindoor channels)
Exponentiallydecaying part
Line-of-sight (LOS)componentDened by channelparameters
Channel parameters: total power P 0 K-factor (rel. strength of LOS) RMS delay spread (duration)
P h (t ) [dB]
t
Mobile Radio Systems Small-Scale Channel Modeling p. 31/41
GRAZ UNIVERSITY OF TECHNOLOGY
Signal Processing and Speech Communications Lab
The WSSUS channel
joint modeling oftime dispersion (= frequency selectivity)and time variability (= Doppler spread)
Dene: ACF of time-variant CIR cl( ; t)
E{c
l ( 1; t)cl( 2; t + t)}= c( 1; t)( 1 2)assumes:
time-variations are wide-sense stationary (WSS)attenuation and phase shifts are independent at 1and 2: uncorrelated scattering (US)
for t = 0 : c( ; t) = c( ) multpath intensity prole
c( ; t) ... lagged-time correlation functionMobile Radio Systems Small-Scale Channel Modeling p. 32/41
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GRAZ UNIVERSITY OF TECHNOLOGY
Signal Processing and Speech Communications Lab
The WSSUS channel (contd)
An equivalent representation of the t-var. CIR cl( ; t):
Time-variant channel transfer function (TF) C l(f ; t)
cl( ; t)F
C l(f ; t) =
cl( ; t)e
j 2f d
from US property follows WSS in f -domain
equivalent characterization (ACF)
C ( f ; t) = E {C l (f ; t)C l(f + f ; t + t)}spaced-frequency spaced-time correlation function(WSSWSS!)
Mobile Radio Systems Small-Scale Channel Modeling p. 33/41
GRAZ UNIVERSITY OF TECHNOLOGY
Signal Processing and Speech Communications Lab
The WSSUS channel (contd)time- and frequency-selective transfer function
0
0.2
0.4
0.6
0.8
1
01
23
4
-20
-10
0
10
20
30
frequency
time
r e c e
i v e
d p o w e r
[ d B ]
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GRAZ UNIVERSITY OF TECHNOLOGY
Signal Processing and Speech Communications Lab
The WSSUS channel (contd)
Equivalent representations : time-variant systemfunctions Bello functions [Bello63]
cl( ; t) and C l(f ; t) and two more Fouriertransformed functions w.r.t. t and f
Equivalent (2-nd order) characterizations : correlationfunctions of Bello functions
c( ; t) and C ( f ; t) and two more Fouriertransformed functions w.r.t. t and f
overview shown on next slide
Mobile Radio Systems Small-Scale Channel Modeling p. 35/41
GRAZ UNIVERSITY OF TECHNOLOGY
Signal Processing and Speech Communications Lab
The WSSUS channel (contd) h( )
multipath intensity profiledelay power spectrum
max max. delay spread
H ( f )spaced-frequencycorrelation function
( f )c coherence bandwidth
h( ; t )delay cross-power spectraldensity
H ( f ; t )spaced-frequency, spaced-time correlation function
H ( t )spaced-time correlationunction
( t )c coherence time
S H ( ) Doppler power spectrum
m max. Doppler freq.;Doppler spread
S H ( f ; ) Doppler cross-power spectral density
S( ; )Scattering function
h( ;t )equivalent lowpasstime-variant impulse response
H ( f ;t )time-variant
transfer functionFT
( f )
FT( f )
ACF(WSSUS)
ACF(WSSWSS)
FT( f )
FT( t )
FT( t )
FT( t )
FT( f )
t = 0 t = 0
f = 0
f = 0
wide-band characterization(time-invariant channel)
characterization of timevariations (narrow-band)
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GRAZ UNIVERSITY OF TECHNOLOGY
Signal Processing and Speech Communications Lab
The WSSUS channel (contd)
Doppler-delay scattering function
[Paulraj; g 2-9] Mobile Radio Systems Small-Scale Channel Modeling p. 37/41
GRAZ UNIVERSITY OF TECHNOLOGY
Signal Processing and Speech Communications Lab
Channel as a space-timerandom eld
Homogenous (HO) channel is (locally) stationary inspace
characterization:E{cl ( ; t; d )cl( ; t; d + d )}= d ( ; t; d )
agrees with discrete scattering model: eachscatterer has discrete ToA i and AoA i
space-angle transform: assume d lies on x-axis;parameterized by x (and dropping t)
cl( ; x) =
cl( ; )e j 2 sin( ) x
d we may dene the angle-delay scattering functionS c( ; )
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GRAZ UNIVERSITY OF TECHNOLOGY
Signal Processing and Speech Communications Lab
Channel as a space-timerandom eld (contd)
Angle-delay scattering function
[Paulraj; g 2-10]Mobile Radio Systems Small-Scale Channel Modeling p. 39/41
GRAZ UNIVERSITY OF TECHNOLOGY
Signal Processing and Speech Communications Lab
Channel as a space-timerandom eld (contd)
S c( ) ... angle power spectrum : average power vs. angleof arrival
Characterization by parameters:
RMS angle spread: rms = 2 2second centralized moment of normalized angle powerspectrum
mean and mean squared angle spread
= S c( )d
S c( )d 2
= 2S c( )d
S c( )d Coherence distance Dc
1 rms
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Signal Processing and Speech Communications Lab
3-3 Classication ofSmall-Scale Fading
Compares system and channel parameters
Classication w.r.t. symbol period w.r.t. bandwidthT s Bs 1/T s
dispersivenessat fading T s rms Bs Bcfrequency selective T s < rms Bs > B ctime variationsslow fading T s T c Bs rmsfast fading T s > T c Bs < rms
Mobile Radio Systems Small-Scale Channel Modeling p. 41/41