Mobile Radio Systems – Small-Scale Channel Modeling · at const. velocity v uniformly distributed scattering around mobile Jakes Doppler spectrum: Sc(ν)= 1 π 1 ν2 max−ν2 (for

GRAZ UNIVERSITY OF TECHNOLOGY

al Processing and Speech Communications Lab

Mobile Radio Systems –Small-Scale Channel Modeling

Klaus Witrisal

[email protected]

Signal Processing and Speech Communication Laboratory

www.spsc.tugraz.at

Graz University of Technology

October 28, 2015

Mobile Radio Systems –Small-Scale Channel Modeling – p. 1/43



Outline

� 3-1 Introduction – Mathematical models forcommunications channels [Molisch 6.2.2; Proakis 1-3]

� 3-2 Stochastic Modeling of Fading Multipath Channels

� Multipath channel [Proakis 14-1]

� Fading amplitude distribution (Rayleigh, Rice)[Molisch 5.4, 5.5]

� Time-selective fading [Molisch 5.6]

� Frequency-selective fading

� WSSUS stochastic channel description [Molisch6.3-6.5, Proakis 14]

� 3-3 Classification of Small-Scale Fading [Molisch 6.5]




References

� A. F. Molisch: Wireless Communications, 2005, Wiley

� J. G. Proakis: Digital Communications, 3rd ed., 1995, 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. Pätzold: Mobile Fading Channels, 2002, Wiley

Figures (partly) extracted from these references




Signal Models

� “Signal processing” channel models can be describedfor different interfaces

� Application/design objective determines choice ofappropriate model

baseband modulation

carrier modulation

011 tream

baseband signal (analog)

radio channel (e.g. c(t) or c(�;t))

propagation channel

baseband equivalent radio channel (e.g. cl(t) or cl(�;t))

sampled (bandlimited) baseband equivalent radio channel (e.g. h or h[k])

carrier de-modulation

baseband signal (analog)

(matched) filter

threshold detector

010011 bit stream sampling

(detection)




Additive Noise Channel

Channel’s frequency response is flat over signal bandwidth

� Simplest model – transmitted (TX) signal corrupted byadditive noise

r(t) = αs(t) + n′(t)

� s(t) ... TX signal

� is a bandpass signal s(t) =√2�{sl(t)ej2πfct}

� r(t) ... received (RX) signal

� for (lowpass equivalent) baseband signals(i.e. complex envelopes of s(t), r(t), n′(t))

rl(t) = hsl(t) + n′l(t), with h ∈ C




Additive Noise Channel (cont’d)

� Noise is usually modeled as white, Gaussian (additivewhite Gaussian noise – AWGN)

φn′(τ) = E{n′(t)n′(t+ τ)} =N0

2δ(τ)

F←→ Sn′(f) =N0

2




Additive Noise Channel (cont’d)

� Sampled AWGN model (lowpass equivalent model)

r[k] = hs[k] + n[k] (all are ∈ C)

� Noise characterization

E{n[k]n∗[l]} = σ2nδ[k − l]

� n[k] is zero-mean circularly symmetric complexGaussian (ZMCSCG)� Real and imaginary components are i.i.d.

(independent, identically distributed)

� σ2n depends on (matched) filter at receiverfront-end

� Real and imaginary components have σ2n/2Mobile Radio Systems –Small-Scale Channel Modeling – p. 7/43



Linear filter channel

Channel’s frequency response is frequency-selective (i.e.non-flat), leading to (linear) signal distortions

� For time-invariant channels

r(t) = s(t) ∗ c(t) + n(t)

=

∫ ∞

−∞

c(τ)s(t− τ)dτ + n(t)

� c(t) ... impulse response of linear filter




Linear filter channel (cont’d)

� Sampled case (lowpass equivalent model)

r[k] =

L−1∑l=0

h[l]s[k − l] + n[k]

� h[k] incorporates� TX pulse shape� RX (matched) filter; ADC filter� (thus bandwidth corresponds to signal bandwidth)� physical channel

� n[k] ... AWGN (ZMCSCG)

� This is actually an equivalent, whitened matched filter(WMF) channel model [Barry/Lee/Messerschmitt]




Linear time-variant filterchannel

� Characterized by time-variant channel impulseresponse (CIR) c(τ ; t)

� response of channel at time t

� to an impulse transmitted at time t− τ

� τ ... “elapsed time”, “age” variable

r(t) = s(t) ∗ c(τ ; t) + n(t)

=

∫ ∞

−∞

c(τ ; t)s(t− τ)dτ + n(t)

� model for multipath propagation

c(τ ; t) =

∞∑i=0

αi(t)δ(τ − τi(t)) (1)




Linear time-variant filter ch.

[fig 5-4 Rap]




Stochastic modeling of fadingmultipath channels

� Motivated by their randomly time-variant nature andlarge number of multipath components

� Derivation of lowpass equivalent CIR from (1)

cl(τ ; t) =

∞∑i=0

αi(t)e−j2πfcτi(t)δ(τ − τi(t))

=

∞∑i=0

αi(t)ejϕi(t)δ(τ − τi(t)) (2)

considering discrete multipath components

� phase term ϕi(t) = −2πfcτi(t) varies dramatically




Fading of an unmodulatedcarrier

� TX signal is unmodulated carrier (CW) sl(t) = 1

� RX signal w/o noise: yl(t) = cl(τ ; t) ∗ 1 = cl(t) · 1

cl(t) =

∞∑i=0

αi(t)ejϕi(t) =

∞∑i=0

αi(t)e−j2πfcτi(t)

sum of vectors (phasors)

� amplitudes αi(t) change slowly

� phases ϕi(t) change by 2π if:� τi(t) changes by 1/fc� i.e.: path length changes by wavelength λ

� large number of multipath components

→ model cl(t) as a random process!Mobile Radio Systems –Small-Scale Channel Modeling – p. 13/43



Fading of an unmodulatedcarrier (cont’d)

Modeling cl(t) as a random process:

� large number of multipath components are added

� by central limit theorem (CLT):

� cl(t) is complex Gaussian

� (CIR cl(τ ; t) is complex Gaussian)

� cl(t) has random phase and amplitude

� in absence of dominant component:cl(t) is zero-mean complex Gaussian

→ its envelope |cl(t)| is Rayleigh distributed

� Rayleigh fading channel









� Rayleigh distribution:

fR(r) =r

σ2e−r2/(2σ2) for r ≥ 0

� characterized by σ2: variance of underlying

Gaussian processes X1, X2 ∼ N (0, σ2), whereX1 = �{cl(t)} and X2 = {cl(t)}

derivation of Rayleigh distribution ...

� Y = X21 +X2

2 ... has χ2-PDF of 2 degrees of freedom

� R =√

X21 +X2

2 ... amplitude |cl(t)| has Rayleigh PDF





central chi-square PDF Rayleigh PDF

of n degrees of freedom characterized by σ





� in presence of a dominant component:cl(t) is non-zero-mean complex Gaussian

→ its envelope |cl(t)| is Ricean distributed

� Ricean fading channel

� Ricean distribution:

fR(r) =r

σ2e−

r2+s

2

2σ2 I0

( rs

σ2

)for r ≥ 0

I0(x) ... zero-order modified Bessel function of first kind

� characterized by

σ2 ... variance of underlying Gaussian processes and

s2 = m21 +m2

2 ... power of mean (i.e. s2 = |E{cl(t)}|2)Mobile Radio Systems –Small-Scale Channel Modeling – p. 18/43




� Shape of Ricean distribution defined by

K =s2

2σ2

K [dB] = 10 logs2

2σ2

Ricean K-factor

� Ratio of deterministic signal power (mean) and varianceof multipath (scattered components)

� For K = 0 = −∞ dB:Ricean distribution equivalent to Rayleigh





0 0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

amplitude r

R

Ricean PDFs with normalized mean power (2nd moment)

K = 0; RayleighK = 2 = 3 dBK = 4 = 6 dBK = 10 = 10 dB

-30 -25 -20 -15 -10 -5 0 5 1010

-3

10-2

10-1

100

amplitude r [dB]

CD

FF

R(r

)=

Pr[

R=

r]

Ricean CDFs with normalized mean power (2nd moment)

K = 0; RayleighK = 2 = 3 dBK = 4 = 6 dBK = 10 = 10 dB

Ricean (and Rayleigh) PDFs Ricean (and Rayleigh) CDFs




Time-selective fading

� Characterization of the time variability

sl(t) = 1 → yl(t) = cl(τ ; t) ∗ 1 = cl(t) =

∞∑i=0

αi(t)ejϕi(t)

� Characterize autocorrelation function of cl(t)

� assume cl(t) is complex Gaussian

� assume cl(t) is wide-sense stationary (WSS)

� Define: spaced-time correlation function

φc(Δt) = E{c∗l (t)cl(t+Δt)} F←→ Sc(ν)

� Doppler power spectrum Sc(ν) =∫∞

−∞φc(Δt)e−j2πνΔtdΔt




Time-selective fading (cont’d)

Doppler power spectrum: average power output ofchannel as a function of Doppler frequency





[Rappaport fig. 5-1]Mobile Radio Systems –Small-Scale Channel Modeling – p. 23/43




Jakes model for Dopplerpower spectrum

� assumes mobile movingat const. velocity v

� uniformly distributedscattering around mobile

� Jakes Doppler spectrum:

Sc(ν) =1

π

1√ν2max − ν2

(for normalized power)

-1 -0.5 0 0.5 10

0.5

1

1.5

2

2.5

normalized Doppler frequency ν/νmax

Dop

pler

pow

ersp

ectr

alde

nsity





Characterization of time-selective fading by parameters

� RMS Doppler spread

νrms =

√ν2 − ν2

second centralized moment of normalized Doppler PSD

� mean and mean squared Doppler spread

ν =

∫νSc(ν)dν∫Sc(ν)dν

ν2 =

∫ν2Sc(ν)dν∫Sc(ν)dν

� Coherence timeTc ≈ 1

νrms




Frequency-selective fading

f a (time-invariant) multipath channel

� Characterization of the time dispersion: CIR cl(τ)

sl(t) = δ(t) → yl(t) = cl(τ ; t) ∗ δ(t) =∞∑i=0

αi(t)ejϕi(t)δ(t− τi(t))

cl(τ) =

∞∑i=0

αiejϕiδ(τ − τi)

� ACF: Uncorrelated scattering assumption:

E{c∗l (τ1)cl(τ2)} = Sc(τ1)δ(τ1 − τ2)

Sc(τ) ... multipath intensity profile (= delay powerspectrum; = average power delay profile)




Frequency-selective fading(cont’d)

� Time-dispersion implies frequency-selectivity

� Equivalent channel characterization by channeltransfer function (TF) Cl(f)

cl(τ)F←→ Cl(f) =

∫ ∞

−∞

cl(τ)e−j2πfτdτ

� ACF of channel TF

Sc(τ)F←→ φC(Δf) = E{C∗

l (f)Cl(f +Δf)}φC(Δf) ... spaced-frequency correlation function

� TF Cl(f) is wide-sense stationary (WSS in f ) if CIRcl(τ) fulfills “uncorrelated scattering” (US in τ )





� Channel IR vs. channel frequency response

-20 0 20 40 60 80 100 120 140 160

15

10

05

00

95

90

85

80

75

70

65

IDFT of transfer function after correction for linear phase shift

excess delay time [ns]0 200 400 600 800 1000

-4

-2

0

2

4

frequency [MHz]

phas

ear

g(H

(f))

[rad

]

phase transfer function (corrected for linear phase shift)

0 200 400 600 800 1000-90

-80

-70

-60

-50

frequency [MHz]

ampl

itude

H(f

)[d

B]

amplitude transfer function





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





Characterization by parameters

� RMS delay spread

τrms =

√τ2 − τ2

second centralized moment of normalized multipathintensity profile

� mean and mean squared delay spread

τ =

∫τSc(τ)dτ∫Sc(τ)dτ

τ2 =

∫τ2Sc(τ)dτ∫Sc(τ)dτ

� Coherence bandwidth

Bc ≈ 1

τrmsMobile Radio Systems –Small-Scale Channel Modeling – p. 30/43




Characterization of mul-tipath intensity profile(simplified; suitable forindoor channels)

� Exponentiallydecaying part

� Line-of-sight (LOS)component

� Defined by channelparameters

Channel parameters:total power P0

K-factor (rel. strength of LOS)RMS delay spread (duration)

Ph(t) [dB]

t




The WSSUS channel

� joint modeling of

� time dispersion (= frequency selectivity)

� and time variability (= Doppler spread)

� Define: ACF of time-variant CIR cl(τ ; t)

E{c∗l (τ1; t)cl(τ2; t+Δt)} = φc(τ1; Δt)δ(τ1 − τ2)

assumes:

� time-variations are wide-sense stationary (WSS)

� attenuation and phase shifts are independent at τ1and τ2: uncorrelated scattering (US)

� for Δt = 0: φc(τ ; Δt) = Sc(τ) multpath intensity profile

� φc(τ ; Δt) ... lagged-time correlation functionMobile Radio Systems –Small-Scale Channel Modeling – p. 32/43



The WSSUS channel (cont’d)

An equivalent representation of the t-var. CIR cl(τ ; t):

� Time-variant channel transfer function (TF) Cl(f ; t)

cl(τ ; t)Fτ←→ Cl(f ; t) =

∫ ∞

−∞

cl(τ ; t)e−j2πfτdτ

� from US property follows WSS in f -domain

� equivalent characterization (ACF)

φC(Δf ; Δt) = E{C∗

l (f ; t)Cl(f +Δf ; t+Δt)}spaced-frequency spaced-time correlation function(WSSWSS!)





� 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

rece

ive

dp

ow

er

[dB

]





� Equivalent representations: time-variant systemfunctions – Bello functions [Bello63]

� cl(τ ; t) and Cl(f ; t) and two more Fouriertransformed functions w.r.t. t ↔ ν and f ↔ τ

� Equivalent (2-nd order) characterizations: correlationfunctions of Bello functions

� φc(τ ; Δt) and φC(Δf ; Δt) and two more Fouriertransformed functions w.r.t. Δt ↔ ν and Δf ↔ τ

overview shown on next slide




The WSSUS channel (cont’d)φc(τ)

multipath intensity profile

delay power spectrum

τmax max. delay spread

φC(Δf) spaced-frequency

correlation function

Bc coherence bandwidth

φc(τ;Δt) delay cross-power spectral

density

φC(Δf;Δt) spaced-frequency, spaced-

time correlation function

φC(Δt) spaced-time correlation

function

Tc … coherence time

SC(ν) Doppler power spectrum

fm … max. Doppler freq.; Doppler spread

SC(Δf;ν) Doppler cross-power

spectral density

S(τ;ν)

Scattering function

cl(τ;t) equivalent lowpass

time-variant

impulse response

Cl(f;t) time-variant

transfer function FT

(τ ↔ f)

FT (τ ↔ Δf)

ACF (WSSUS)

ACF (WSSWSS)

FT (τ ↔ Δf)

FT (Δt ↔ ν)

FT (Δt ↔ ν)

FT (Δt ↔ ν)

FT (τ ↔ Δf)

Δt = 0 Δt = 0

Δf = 0

Δf = 0

wide-band characterization (time-invariant channel)

characterization of time variations (flat fading)





� Doppler-delay scattering function

[Paulraj; fig 2-9]Mobile Radio Systems –Small-Scale Channel Modeling – p. 37/43



Channel as a space-timerandom field

� Homogenous (HO) channel is (locally) stationary inspace

� characterization:

E{c∗l (τ ; t;d)cl(τ ; t;d+Δd)} = φd(τ ; t; Δd)

� agrees with discrete scattering model: eachscatterer has discrete ToA τi and AoA θi

� space-angle transform: assume d lies on x-axis;parameterized by x (and dropping t)

cl(τ ;x) =

∫ ∞

−∞

cl(τ ; θ)e−j2π sin(θ) x

λdθ

� we may define the angle-delay scattering functionSc(τ ; θ)




Channel as a space-timerandom field (cont’d)

� Angle-delay scattering function

[Paulraj; fig 2-10]Mobile Radio Systems –Small-Scale Channel Modeling – p. 39/43



Channel as a space-timerandom field (cont’d)

Sc(θ) ... angle power spectrum: average power vs. angleof arrival

� Characterization by parameters:

� RMS angle spread: θrms =

√θ2 − θ

2

second centralized moment of normalized angle powerspectrum

� mean and mean squared angle spread

θ =

∫θSc(θ)dθ∫Sc(θ)dθ

θ2 =

∫θ2Sc(θ)dθ∫Sc(θ)dθ

� Coherence distance Dc ∝ 1θrms




3-3 Classification ofSmall-Scale Fading

� Compares system and channel parameters

Classification w.r.t. symbol period w.r.t. bandwidth

Ts Bs ∝ 1/Ts

dispersiveness

flat fading Ts � τrms Bs � Bc

frequency selective Ts < τrms Bs > Bc

time variations

slow fading Ts � Tc Bs � νrms

fast fading Ts > Tc Bs < νrms




Classification example – GSM

� Key air-interface parameters:

� Carrier frequency ... 900 MHz, 1.8 GHz

� Bandwidth ... 200 kHz

� Frame; slot length ... ∼4.6 ms; ∼0.6 ms

� Time dispersiveness

� τrms (typical urban and suburban) ... 100–800 ns

� corresponds to Bc ≈ 1.2–10 MHz

� flat fading

� Time variability

� assume v = 50 m/s at fc = 1 GHz → νmax = 167 Hz

� corresponds to Tc ≈ 6 ms

� Time-invariant during slotMobile Radio Systems –Small-Scale Channel Modeling – p. 42/43



Classification example – WLAN

� Key air-interface parameters:

� Carrier frequency ... 2.4; 5 GHz

� Bandwidth ... 17 MHz (sampling f: fs = 20 MHz)

� OFDM symbol length ... 4 μs

� Time dispersiveness

� τrms (indoor) ... 10–300 ns

� corresponds to Bc ≈ 3–100 MHz

� frequency selective

� Time variability

� assume v = 2 m/s at fc = 5 GHz → νmax = 33 Hz

� corresponds to Tc ≈ 30 ms; several 1000 symbols

� Time-invariant during packetMobile Radio Systems –Small-Scale Channel Modeling – p. 43/43

Documents

Mobile Radio Systems – Small-Scale Channel Modeling · at const. velocity v uniformly distributed scattering around mobile Jakes Doppler spectrum: Sc(ν)= 1 π 1 ν2 max−ν2 (for