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8/9/2019 Lecture 6 Multipath Fading
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MULTIPATH FADING
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Various Features in a Wireless Channel
Multipath fading
Shadowing
Interference
Path Loss
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Excess Delay
The propagation delay relative to that of the
shortest path
t=0
t=8ms
t=47ms
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Strength VariationAs the vehicle moves, the strength of each path varies
because the surfaces are complex
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The Channel is a Filter The multipath channel can be represented as linear, time-
varying bandpass filter
),( !th
TransmittedSignal
x(t)
ReceivedSignal
y(t)
!!! dthtxty "+#
#$
$= ),()()(
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Measured Data from Darmstadt,
Germany [Molisch, 01]
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Baseband Impulse Response More convenient to work with baseband signals
{ }{ }tj
tj
tj
b
c
c
c
etrty
etctx
ethth
!
!
!
""
)(Re)(
)(Re)(
),(Re),(
=
=
=
The factor of !ensures that baseband
average powerequals passband
average power!!! dthtctr
b"+#
#$
$= ),()(2
1)(
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Path Model
The channel is assumed to comprise N
discrete paths of propagation (rays)
Each path has an amplitude !(t), a phase "(t)
and a propagation delay #
)(0
0)( tjet !")(
11)( tjet
!" )(
22)( tjet
!" )(
33)(
tjet !
" )(4
4)( tj
et !
"
1!0
!2
! 3!
4!
delay
),( !thb 5=N
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Probing the ChannelThe channel may be probed or soundedby
transmitting a pulsep(t)and recording the
response at the receiver
The response is the convolution of p(t)with the
channel impulse response
)()(21)(
1
0
)(
i
N
i
tj
i tpettr i !"
# $= %$
=
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Suppose a pulse much wider than the length
of the impulse response is transmitted at time
t=0
1!
0!
2!
3!
4! delay
),( !thb
t
p(t)
g
Tp
Pulse Width >>#
max
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Instantaneous Power
1!
0!
2!
3!
4! delay
),( !thb
t
r(t)
The magnitude squaredof any sample in the interval t4and
t0 + Tpwill equal 21
0
)(2
2
)(4)( !
"
=
=
N
i
tj
i
i
ettr
#
$
%
We say, the multipath is not resolved
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Time Variation of the Probe Response
If one or both of the terminals moves, the path phaseschange because the path lengths change
The path amplitudes do not change much These changes yield large changes in the magnitude
of the received waveform
t
|r(t)| in dBm [not real data]
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Narrowband Fading This same type of fading happens to a digital waveform if
the symbol period is much larger than (>10 times) the
channel length
Such long symbol periods correspond tonarrowband
signals
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Average Power for Narrowband Signals
Assuming the channel is ergodic, the ensemble
average may be approximated by a time average:
where the interval [t-T/2,t+T/2] corresponds to a local
area
{ } dsesT
trE
Tt
Tt
N
i
sj
ii! "
+
# =
$=%2
2
2
0
)(2
2)(
4
1)(
&
'& '(
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Uncorrelated Scattering
Assume that the phases of different paths are
uncorrelated and that the energy of the pulse
is one Then the time average simplifies to
where
{ } !=
"=#N
i
itrE
0
22)( $$%
dssT
Tt
Tt
ii !
+
"
#2
2
22)(
1$$
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Pulse Width
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Multipath Resolved
Pulses do not overlap
1!0! 2
! 3! 4!
),( !tr)(
00)(
tjet !
"#)(
11)( tjet
!"# )(
22)(
tjet !
"# )(3
3)( tj
et !
"# )(4
4)( tj
et !
"#
t
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Wideband Signal
The Fourier Transform of such a narrow pulse has a widespectrum
t
p(t)
g
Tp
F.T.
P(f)
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Power Delay Profile (PDP)
The PDP is a time-average of |r(t,#)|2over asmall interval (assuming the terminal is moving)
dssrT
P
Tt
Tt
!
+
"
#2
2
2),(
1)( $$
1!0! 2
! 3! 4!
t
)(!P2
0
2!"
2
1
2!" 2
2
2!" 2
3
2!" 2
4
2!"
W. Mohr, Modeling of wideband mobile radio channels based on propagation measurements,
in Proc. 16th Int. Symp. Personal, Indoor, Mobile Radio Communications, vol. 2, pp.397-401, 1995
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Average Power for Wideband Signals
The average power is the integral of the PDP
!==
=
"
#$
=
+%
1
0
2
0
)(
N
i
i
AVG dPP
&
''
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Local Average Powers Are The Same
Narrowband and Wideband averaged powers are equal
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Moments of the PDP
Channels are often described by their rms delay spread
To compute rms delay spread, normalize the PDP tomake it like a PDF for a random variable (unit area) andthen find its standard deviation
Must you use excess delay to compute rms delay spread?
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Mean Delay
Must first compute the mean delay
For this to be mean excess delay, the origin of
the #axis needs to be the time of the firstarriving path
!!
!!!
!
dP
dP
"
"#+
+#
=
0
0
)(
)(
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Second Moment
Next need the second moment of this PDF
!!
!!!
!
dP
dP
"
"#+
+#
=
0
0
2
2
)(
)(
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RMS Delay Spread
Recall that standard deviation is the square root ofvariance and variance is the second moment minus the
first moment squared
( )2
22!!"
! #=
Variance
( )22 !!"!
#=rms delay spread
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Example Data[Rappaport, 02]
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How RMS Delay Spread Can Be Used
If !"
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The Frequency Domain View
!"
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Narrowband Case
The channel appears flat to the signal
),( ftH
f900MHz
Transmitted
Signal Spectrum
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Flat Fading
When!"
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Wideband Case
!">> symbol period implies that the frequency
response of the channel varies significantly with
frequency over the bandwidth of the transmitted
signal
f900MHz
TransmittedSignal Spectrum
),( ftH
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Frequency Selective Fading
When!">> symbol period, we say the signal undergoes
frequency selective fading
The channel frequency response is strong for some
frequencies and not for others within the signal bandwidth
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Complete Narrowband Statistical
Fading Model is the signal envelope. Recall
represents the small-scale or multipath fading, and has eitherthe Rayleigh or Rician distribution with unit mean square value, i.e.
Therefore,
represents the large-scale or shadow fading, and has alognormal distribution. Recall is local averagepower
Let . Then is (the zs are thepoints on the scatter plot)
is the path loss, with the model
x
xytr == )(!
12=xE
y
210log10 yz = 2),( !dpN
)( dp
!!
"
#$$
%
&'=
o
o
d
dndpdp 10log10)()(
z
!=2
yE
22yEE =!
{ } !"
=
=#=1
0
22
N
i
iE $$
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Complete Wideband Statistical
Fading Model Recall . Each path of the wideband model has
independent small-scale fading:
is the small-scale or multipath fading on each path, and has
either the Rayleigh or Rician distribution with mean square value
represents the large-scale or shadow fading, as before, andhas a lognormal distribution.
Let . Then is (the zs are thepoints on the scatter plot)
is the path loss, with the model
ix
!= /22
ii ExE "
y
2
10log10 yz =2),( !dpN
)( dp
!!
"
#$$
%
&'=
o
o
d
dndpdp 10log10)()(
z
! "
=
=# 1
0
2N
i i$
yxii
=!
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Fade Margin
Extra power (i.e. a fademargin) is required, to keep
above the decoding
thresholdt
One channel withmultipath fading
RXPower
DecodingThreshold
Fade Margin
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Summary
The multipath channel model has a discrete number ofpropagation paths
Each path has amplitude, phase and delay
The PDP is the local average of the magnitude squared of
the impulse response of the channelAverage power of the channel is the integral of the PDP
Average power is same for narrowband and widebandchannels
The fading is flator frequency selectivedepending on
the comparison between rms delay spread and thesymbol period
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References
[Rapp, 02] T.S. Rappaport, WirelessCommunications, Prentice Hall, 2002
[Molisch, 01] Andreas F. Molisch (ed), Wideband
Wireless Digital Communications, Prentice HallPTR, 2001.
W. Mohr, Modeling of wideband mobile radiochannels based on propagation measurements,in Proc. 16th Int. Symp. Personal, Indoor, MobileRadio Communications, vol. 2, pp. 397-401, 1995
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