+Small-Scale Channel Modeling

8/6/2019 +Small-Scale Channel Modeling

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GRAZ UNIVERSITY OF TECHNOLOGY

Signal Processing and Speech Communications Lab

Mobile Radio Systems Small-Scale Channel ModelingKlaus Witrisal

[email protected]

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



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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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




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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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




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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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




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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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]




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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Linear time-variant lter ch.

[g 5-4 Rap]Mobile Radio Systems Small-Scale Channel Modeling p. 11/41



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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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!




Fading of an unmodulatedcarrier (contd)



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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





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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central chi-square PDF Rayleigh PDFof n degrees of freedom characterized by





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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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




Time-selective fading (contd)

Doppler power spectrum: average power output of channelas a function of Doppler frequency



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[Rappaport g. 5-1]Mobile Radio Systems Small-Scale Channel Modeling p. 23/41




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



13/21




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




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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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 )





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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Multipath intensity prole: average power output of channelas a function of delay





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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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




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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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!)




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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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




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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Doppler-delay scattering function

[Paulraj; g 2-9] Mobile Radio Systems Small-Scale Channel Modeling p. 37/41



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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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



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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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


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