GRAZ UNIVERSITY OF TECHNOLOGY
al Processing and Speech Communications Lab
Mobile Radio Systems –Small-Scale Channel Modeling
Klaus Witrisal
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
GRAZ UNIVERSITY OF TECHNOLOGY
al Processing and Speech Communications Lab
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]
Mobile Radio Systems –Small-Scale Channel Modeling – p. 2/43
GRAZ UNIVERSITY OF TECHNOLOGY
al Processing and Speech Communications Lab
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
Mobile Radio Systems –Small-Scale Channel Modeling – p. 3/43
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al Processing and Speech Communications Lab
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)
Mobile Radio Systems –Small-Scale Channel Modeling – p. 4/43
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al Processing and Speech Communications Lab
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
Mobile Radio Systems –Small-Scale Channel Modeling – p. 5/43
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al Processing and Speech Communications Lab
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
Mobile Radio Systems –Small-Scale Channel Modeling – p. 6/43
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al Processing and Speech Communications Lab
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
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al Processing and Speech Communications Lab
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
Mobile Radio Systems –Small-Scale Channel Modeling – p. 8/43
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al Processing and Speech Communications Lab
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]
Mobile Radio Systems –Small-Scale Channel Modeling – p. 9/43
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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)
Mobile Radio Systems –Small-Scale Channel Modeling – p. 10/43
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al Processing and Speech Communications Lab
Linear time-variant filter ch.
[fig 5-4 Rap]
Mobile Radio Systems –Small-Scale Channel Modeling – p. 11/43
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al Processing and Speech Communications Lab
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
Mobile Radio Systems –Small-Scale Channel Modeling – p. 12/43
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al Processing and Speech Communications Lab
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
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al Processing and Speech Communications Lab
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
Mobile Radio Systems –Small-Scale Channel Modeling – p. 14/43
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al Processing and Speech Communications Lab
Fading of an unmodulatedcarrier (cont’d)
Mobile Radio Systems –Small-Scale Channel Modeling – p. 15/43
GRAZ UNIVERSITY OF TECHNOLOGY
al Processing and Speech Communications Lab
Fading of an unmodulatedcarrier (cont’d)
� 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
Mobile Radio Systems –Small-Scale Channel Modeling – p. 16/43
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al Processing and Speech Communications Lab
Fading of an unmodulatedcarrier (cont’d)
central chi-square PDF Rayleigh PDF
of n degrees of freedom characterized by σ
Mobile Radio Systems –Small-Scale Channel Modeling – p. 17/43
GRAZ UNIVERSITY OF TECHNOLOGY
al Processing and Speech Communications Lab
Fading of an unmodulatedcarrier (cont’d)
� 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
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Fading of an unmodulatedcarrier (cont’d)
� 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
Mobile Radio Systems –Small-Scale Channel Modeling – p. 19/43
GRAZ UNIVERSITY OF TECHNOLOGY
al Processing and Speech Communications Lab
Fading of an unmodulatedcarrier (cont’d)
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
Mobile Radio Systems –Small-Scale Channel Modeling – p. 20/43
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al Processing and Speech Communications Lab
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
Mobile Radio Systems –Small-Scale Channel Modeling – p. 21/43
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Time-selective fading (cont’d)
Doppler power spectrum: average power output ofchannel as a function of Doppler frequency
Mobile Radio Systems –Small-Scale Channel Modeling – p. 22/43
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al Processing and Speech Communications Lab
Time-selective fading (cont’d)
[Rappaport fig. 5-1]Mobile Radio Systems –Small-Scale Channel Modeling – p. 23/43
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Time-selective fading (cont’d)
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
Mobile Radio Systems –Small-Scale Channel Modeling – p. 24/43
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al Processing and Speech Communications Lab
Time-selective fading (cont’d)
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
Mobile Radio Systems –Small-Scale Channel Modeling – p. 25/43
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al Processing and Speech Communications Lab
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)
Mobile Radio Systems –Small-Scale Channel Modeling – p. 26/43
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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 )
Mobile Radio Systems –Small-Scale Channel Modeling – p. 27/43
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Frequency-selective fading(cont’d)
� 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
Mobile Radio Systems –Small-Scale Channel Modeling – p. 28/43
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Frequency-selective fading(cont’d)
Multipath intensity profile: average power output of channelas a function of delay
Mobile Radio Systems –Small-Scale Channel Modeling – p. 29/43
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Frequency-selective fading(cont’d)
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
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Frequency-selective fading(cont’d)
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
Mobile Radio Systems –Small-Scale Channel Modeling – p. 31/43
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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
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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!)
Mobile Radio Systems –Small-Scale Channel Modeling – p. 33/43
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The WSSUS channel (cont’d)
� 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
]
Mobile Radio Systems –Small-Scale Channel Modeling – p. 34/43
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al Processing and Speech Communications Lab
The WSSUS channel (cont’d)
� 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
Mobile Radio Systems –Small-Scale Channel Modeling – p. 35/43
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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)
Mobile Radio Systems –Small-Scale Channel Modeling – p. 36/43
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The WSSUS channel (cont’d)
� Doppler-delay scattering function
[Paulraj; fig 2-9]Mobile Radio Systems –Small-Scale Channel Modeling – p. 37/43
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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)
Mobile Radio Systems –Small-Scale Channel Modeling – p. 38/43
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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
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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
Mobile Radio Systems –Small-Scale Channel Modeling – p. 40/43
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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
Mobile Radio Systems –Small-Scale Channel Modeling – p. 41/43
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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
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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