temporal characteristics of fading
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Ghassan Dahman / Fredrik Tufvesson
Department of Electrical and Information Technology
Lund University, Sweden
Channel Modelling โ ETIN10
Lecture no:
Fredrik Tufvesson - ETIN10 1
4
Temporal Characteristics of
Fading
Fredrik Tufvesson - ETIN10 2
Small-scale fading
Doppler shifts
rv
Frequency of received signal:
๐ = ๐0 + ๐
where the Doppler shift is
๐ = โ ๐0
๐ฃ๐
๐cos (๐)
The maximum Doppler shift
๐๐๐๐ฅ = ๐0
๐ฃ๐
๐
Receiving antenna moves with
speed ๐ฃ๐ at an angle ฮธ relative
to the propagation direction
of the incoming wave, which
has frequency ๐0.
k
The relationship ๐๐๐๐ฅ = ๐0 ๐ฃ๐ ๐ is based on several assumptions โ
e.g., static IOs, no double reflections on moving objects, etc.
Fredrik Tufvesson - ETIN10 3
Small-scale fading
Doppler shifts
โข f0=5.2 109 Hz, vr=5 km/h, (1.4 m/s) 24 Hz
โข f0=900 106 Hz, vr=110 km/h, (30.6 m/s) 92 Hz
How large is the maximum Doppler frequency in the
following cases:
โข at pedestrian speeds for 5.2 GHz WLAN,
โข and at highway speeds using GSM 900?
๐๐๐๐ฅ = ๐0
๐ฃ๐
๐
Fredrik Tufvesson - ETIN10 4
โข f0=900 106 Hz, v=40 km/h, (11.1 m/s) 33.33 Hz
Each 30 ms the envelop of the received signal will go to zero.
max 0
vf
c
Consider a GSM 900, and an Rx moving at 40 km/h.
The beating effect
0 0.05 0.1 0.15 0.2 0.25-2
0
2
time (s)
E(t
)
1
๐ฃ๐๐๐ฅ
Superposition of two carriers with different frequencies (beating)
Fredrik Tufvesson - ETIN10 5
Small-scale fading
Example: the time-variant two-path model
Wave 2
Wave 1
v
The two components have different Doppler shifts!
The Doppler shifts will cause a random frequency modulation
๐ธ1 = ๐ธ 1 (๐2๐๐ก๐๐ 1 โ๐ฃ
๐cos ๐1 )
๐1
๐2 v
๐1
๐1 ๐ธ2 = ๐ธ 2 (๐2๐๐ก๐๐ 1 โ
๐ฃ
๐cos ๐2 )
๐01, ๐
02: distances at t=0
๐1, ๐2: wavenumbers
๐1 = ๐2 = ๐0 = 2๐ ๐
๐ธ๐(๐ก) = ๐ธ ๐๐ฅ๐ ๐2๐๐๐๐ก
Fredrik Tufvesson - ETIN10 6
Equivalent cases:
โข Two superimposed incident waves + moving Rx
โข Superimposing two signals with different Doppler shifts at the Rx antenna
Besides the distribution, the temporal behavior is needed to characterize
small-scale fading.
Small-scale fading
Example: the time-variant two-path model
Fredrik Tufvesson - ETIN10 7
Radio Channels
(some properties)
Path loss
Large-scale
fading
Small-scale fading
Attenuation
due to
distance
Variation
around the
mean due to
IOs
Frequency dispersion of the
signal due to the time-varying
nature of the channel
Time dispersion of the signal due
to the multipaths
Fast
fading Slow
fading
Flat
fading
Frequency-
selective
fading
The final effect is a function of the characteristics of
BOTH: the channel, and the transmitted signal
Fredrik Tufvesson - ETIN10 8
0 maxf 0 maxf 0f
Spectrum of received signal
when a f0 Hz signal is transmitted.
RX
RX movement
Incoming waves from several directions
(relative to movement or RX)
All waves of equal strength in
this example, for simplicity.
1
1
2
2
3
3
4
4
The received signal may occupy the ๐0 โ ๐ฃ๐๐๐ฅ to ๐0 + ๐ฃ๐๐๐ฅ frequency band
Doppler spectra
Fredrik Tufvesson - ETIN10 9
Jakes spectrum
Isotropic uncorrelated
scattering
RX
Uniform incoming
power distribution
(isotropic)
Uncorrelated
amplitudes
and phases
RX movement
0 maxf 0 maxf 0f
0DS f
Doppler spectrum
at center frequency f0.
Jakes spectrum
Fredrik Tufvesson - ETIN10 10
Condensed parameters
Coherence time Doppler spectrum vs. the time-autocorrelation function
time-autocorrelation function Doppler power spectrum
0 maxf 0 maxf 0f
๐๐ฃ
Fredrik Tufvesson - ETIN10 11
โข Time correlation โ how static is the channel?
โข For a uniform scattering environment the time correlation of the Re-
and Im-components is described by a โzeroth-order Bessel function of
the first kindโ
โข The time correlation for the amplitude is
*
0 max2t E a t a t t J t
2
0 max2t J t
The channel time correlation
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
0.5
1
d/
J 02(2
d/
)
Fredrik Tufvesson - ETIN10 12
The channel time correlation (example)
Example: Assume that an MS is located in a fading dip. On average, what
distance should the MS move so that it is no longer influenced by this
fading dip?
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
d/
J 02(2
d/
)0.38๐
0.18๐
๐ = 0 (๐๐๐๐๐๐๐ก๐ ๐๐๐๐๐๐๐๐๐๐ก๐๐๐)
๐ = 0.5
Fredrik Tufvesson - ETIN10 13
*r
*r
Time
Received amplitude [dB]
rmsr|dBM
What about the length and the frequency
of fading dips ?
Important for system design
block length
code design
choice of modulation
Temporal Dependence of small-scale fading
Fredrik Tufvesson - ETIN10 14
Frequency of the fading dips Length of fading dips
These curves are for Rayleigh fading & isotropic uncorrelated scattering (Jakesโ doppler spectrum).
Temporal Dependence of small-scale fading (cont.)
Example: assume a multipath environment where the received signal has a Rayleigh
distribution and the Doppler spectrum has Jakes shape. Compute the LCR and the
ADF for a maximum Doppler frequency=50Hz, and r/rmin=0.1.
a) 0.25 = LCR/50 => LCR=12.5
b) 0.04 = ADFร50 => ADF=0.8 msec
Fredrik Tufvesson - ETIN10 15
โข Caused by the motion of the Tx, Rx, or IOs.
โข Characterizes the time-varying nature of the channel in a small-
scale region (How fast does the channel change?) The
Coherence Time ๐๐(defined as the time during which the channel is
time invariant) is used as a measure.
โข Characterizes the frequency-dispersion of the channel ( How does
the Doppler shifts spread the frequency components of the
transmitted signal?) The Doppler spread ๐ท๐ (defined as the
Maximum Doppler Shift) is used as a measure.
โข As a general approximation :
Tc =1
Ds
Temporal Nature of the Channel
Fredrik Tufvesson - ETIN10 16
Condensed parameters
Coherence time
Given the time correlation of a channel, we can define the
Coherence Time TC.
t t
t
0t
0
2
t
CT
TC shows us over how long time
we can assume that the channel is
fairly constant.
If the Symbol Time ๐๐ is much
shorter than TC , it will
experience the same channel
(i.e., it will get attenuated by the
channel, however, it will not get
distorted due to fading).
The coherence time
Fredrik Tufvesson - ETIN10 17
Condensed parameters
Coherence time
If the bandwidth of the base band
signal W is much greater than
๐ท๐, the effect of the Doppler spread
on the received signal is negligible (no
distortion).
Bandwidth vs. Doppler spread
Doppler power spectrum
0 maxf 0 maxf 0f
๐๐ฃ
Fredrik Tufvesson - ETIN10 18
Condensed parameters
Coherence time Slow fading vs. Fast fading
๐๐ โซ ๐๐ OR ๐ท๐ โช ๐
Slow Fading (i.e., the channel fading rate is less than the symbol
rate which results only in SNR degradation)
๐๐ < ๐๐ OR ๐ท๐ > ๐ (or comparable)
Fast Fading (i.e., the channel fading rate is greater than the symbol
rate which results in distortion degradation)
Mitigation Techniques:
โข Robust modulation (schemes that donโt require phase tracking, and
reduces the detectorโs integration time)
โข Increasing the symbol rate
โข Coding and Interleaving
Fredrik Tufvesson - ETIN10 19
Radio Channels
(some properties)
Path loss
Large-scale
fading
Small-scale fading
Attenuation
due to
distance
Variation
around the
mean due to
IOs The final effect is a function of the characteristics of
BOTH: the channel, and the transmitted signal
or comparable values
Time dispersion of the signal due
to the multipaths
Flat
fading Frequency-
selective
fading
Fredrik Tufvesson - ETIN10 20
Measurement example
โข Measurement in the lab
โข Center frequency 3.2 GHz
โข Measurement bandwidth 200 MHz, 201 frequency points
โข 60 measurement positions, spaced 1 cm apart
โข Measured with a vector network analyzer
Fredrik Tufvesson - ETIN10 21
Coherence time, measured
0 0.1 0.2 0.3 0.4 0.5 0.6-90
-85
-80
-75
-70
-65
-60
-55
-50
-45
-40
-35
Position (m)
Fre
quency r
esp (
dB
)
Assume 1 m/s, max 0
vf
c =10.7 Hz Compare 1/(2**vmax)=0.014 s
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1-0.5
0
0.5
1
Tim
e c
orr
Position (m)
measured
theoretical
Pr(
dB
)
Fredrik Tufvesson - ETIN10 22
Probability density function
0 0.002 0.004 0.006 0.008 0.01 0.0120
20
40
60
80
100
120
140
160
180
amplitude
measured
theoretical
Vehicle-to-vehicle communication
Fredrik Tufvesson - ETIN10 23
This channel has highly dynamic scatterers, both Tx and Rx are moving
Vehicle-to-Vehicle channel characteristics
24 / 12
Time-delay characteristics:
0 100 200 300 400Propagation distance [m]
Po
wer
t = 0 s
0 100 200 300 400Propagation distance [m]
Po
wer
t = 0.2 s
0 100 200 300 400Propagation distance [m]
Po
wer
t = 0.4 s
0 100 200 300 400Propagation distance [m]
Po
wer
t = 0.6 s
0 100 200 300 400Propagation distance [m]
Po
wer
t = 0.8 s
0 100 200 300 400Propagation distance [m]
Po
wer
t = 1 s
0 100 200 300 400Propagation distance [m]
Po
wer
t = 1.3 s
0 100 200 300 400Propagation distance [m]
Po
wer
t = 1.5 s
0 100 200 300 400Propagation distance [m]
Po
wer
t = 1.7 s
0 100 200 300 400Propagation distance [m]
Po
wer
t = 1.9 s
0 100 200 300 400Propagation distance [m]
Po
wer
t = 2.1 s
0 100 200 300 400Propagation distance [m]
Po
wer
t = 2.3 s
0 100 200 300 400Propagation distance [m]
Po
wer
t = 2.5 s
0 100 200 300 400Propagation distance [m]
Po
wer
t = 2.8 s
0 100 200 300 400Propagation distance [m]
Po
wer
t = 3 s
0 100 200 300 400Propagation distance [m]
Po
wer
t = 3.2 s
0 100 200 300 400Propagation distance [m]
Po
wer
t = 3.4 s
0 100 200 300 400Propagation distance [m]
Po
wer
t = 3.6 s
0 100 200 300 400Propagation distance [m]
Po
wer
t = 3.8 s
0 100 200 300 400Propagation distance [m]
Po
wer
t = 4.1 s
0 100 200 300 400Propagation distance [m]
Po
wer
t = 4.3 s
0 100 200 300 400Propagation distance [m]
Po
wer
t = 4.5 s
0 100 200 300 400Propagation distance [m]
Po
wer
t = 4.7 s
0 100 200 300 400Propagation distance [m]
Po
wer
t = 4.9 s
0 100 200 300 400Propagation distance [m]
Po
wer
t = 5.1 s
0 100 200 300 400Propagation distance [m]
Po
wer
t = 5.3 s
0 100 200 300 400Propagation distance [m]
Po
wer
t = 5.6 sRX
TX
0 100 200 300 400Propagation distance [m]
Po
wer
t = 5.8 s
0 100 200 300 400Propagation distance [m]
Po
wer
t = 6 s
0 100 200 300 400Propagation distance [m]
Po
wer
t = 6.2 sLOS
0 100 200 300 400Propagation distance [m]
Po
wer
t = 6.4 s
0 100 200 300 400Propagation distance [m]
Po
wer
t = 6.6 s
0 100 200 300 400Propagation distance [m]
Po
wer
t = 6.9 s
0 100 200 300 400Propagation distance [m]
Po
wer
t = 7.1 s
0 100 200 300 400Propagation distance [m]
Po
wer
t = 7.3 s
0 100 200 300 400Propagation distance [m]
Po
wer
t = 7.5 s
0 100 200 300 400Propagation distance [m]
Po
wer
t = 7.7 s
0 100 200 300 400Propagation distance [m]
Po
wer
t = 7.9 s
0 100 200 300 400Propagation distance [m]
Po
wer
t = 8.1 sDiscrete comp.
Diffuse
comp. Other vehicles
Houses, road
signs etc.
โข Rapidly varying channel
โข Discrete components carry significant energy and change delay bin with time
โข Diffuse components following LOS
Fredrik Tufvesson - ETIN10
Fredrik Tufvesson - ETIN10 25
Time variant impulse response
Letโs take a closer
look at the Doppler
shifts here
Fredrik Tufvesson - ETIN10 26
Scattering function, t=8.5-8.65 s
27 / 12
Doppler-delay characteristics
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Delay [ns]
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Doppler-delay characteristics:
โข Discrete components: small Doppler spread, but can change delay bin rapidly
โข Diffuse components: large delay and Doppler spread
โข Time-variant Doppler spectrum โ WSSUS conditions are violated
Fredrik Tufvesson - ETIN10
Impulse response, fading of scatterer
Fredrik Tufvesson - ETIN10 28
300 350 400 450 500-105
-100
-95
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-75
Propagation distance [m]
Rec
eiv
ed p
ow
er [
dB
]Impulse response,
two cars, same direction
90 km/h
Fading of a single scatterer
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