Lecture 6 Multipath Fading

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

)(

)(


25/37

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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Lecture 6 Multipath Fading