multipath interference characterization in wireless ... · multipath interference is spatial...

9/29/00 BYU Wireless Communications 1

Multipath Interference Characterizationin Wireless Communication Systems

Michael Rice

BYU Wireless Communications Lab

166


Multipath Propagation

• Multiple paths between transmitter and receiver

• Constructive/destructive interference

• Dramatic changes in received signal amplitude and phase as a result ofsmall changes (λ/2) in the spatial separation between a receiver andtransmitter.

• For Mobile radio (cellular, PCS, etc) the channel is time-variantbecause motion between the transmitter and receiver results inpropagation path changes.

• Terms: Rayleigh Fading, Rice Fading, Flat Fading, FrequencySelective Fading, Slow Fading, Fast Fading ….

• What do all these mean?

167


LTI System Model

h(t))(ts )(tr

( )

( ) ( )∑

∑−

=

−

=

−+=

−=

1

10

1

0

)(

N

kk

jk

N

kk

jk

teata

teath

k

k

τδδ

τδ

θ

θ

line-of-sightpropagation

multipathpropagation

( )∑−

=

−+=1

10 )()(

N

kk

jk tseatsatr k τθ

line-of-sightcomponent

multipathcomponent

168


Some Important Special Cases

All the delays are so small and we approximate kk allfor 0≈τ

( )

( )

)(

)(

)()(

1

10

1

10

1

10

tseaa

tseatsa

tseatsatr

N

k

jk

N

k

jk

N

kk

jk

k

k

k

+=

+≈

−+=

∑

∑

∑

−

=

−

=

−

=

θ

θ

θ τ

• sum of complex random numbers (random amplitudesand phases)

• if N is large enough, this sum is well approximated bycomplex Gaussian pdf

α

( )( )2

2

,~

,~

aaI

aaR

IR

mN

mN

j

σασα

ααα += { }

{ } { }[ ]ππθ

α

θ

θ

,~ when 0

1

1

1

1

−=

=

==

∑

∑−

=

−

=

U

eEaE

eaEEm

k

N

k

jk

N

k

jka

k

k

169




( )

( )

[ ][ ]

( ) ( ) )(

)(

)(

)(

)(

)()(

220

0

0

1

10

1

10

1

10

tsea

tsja

tsa

tseaa

tseatsa

tseatsatr

jIR

IR

N

k

jk

N

k

jk

N

kk

jk

k

k

k

φ

θ

θ

θ

αα

ααα

τ

++=

++=+=

+=

+≈

−+=

∑

∑

∑

−

=

−

=

−

=

( ) ( ))(

)()(

22

21

220

tseXX

tseatr

j

jIR

φ

φαα

+=

++=

( )201 ,~ aaNX σ ( )2

2 ,0~ aNX σ

170


Important PDF’s

( )( )

( )

=

+=

=

+=

+−

+−

20

02

2

22

21

20

02

2

22

21

22

201

2

220

2

20

2

1)(

,0~

,~

a

ua

aU

a

wa

aW

a

a

uaIe

uup

XXU

awIewp

XXW

NX

aNX

a

a

σσ

σσ

σσ

σ

σ Non-centralChi-square pdf

Rice pdf

( )( )

( ) 2

2

2

22

22

21

22

22

21

22

21

2

1)(

,0~

,0~

a

a

u

aU

w

aW

a

a

eu

up

XXU

ewp

XXW

NX

NX

σ

σ

σ

σ

σσ

−

−

=

+=

=

+=

Chi-square pdf

Rayleigh pdf

171


Back to Some Important Special Cases


( )

( )

( ) ( ) )(

)(

)()(

220

1

10

1

10

tsea

tseatsa

tseatsatr

jIR

N

k

jk

N

kk

jk

k

k

φ

θ

θ

αα

τ

++=

+≈

−+=

∑

∑−

=

−

=

Rice pdf

“Ricean fading”

00 >a

( )

( )

( ) ( ) )(

)(

22

1

1

1

1

tse

tsea

tseatr

jIR

N

k

jk

N

kk

jk

k

k

φ

θ

θ

αα

τ

+=

≈

−=

∑

∑−

=

−

=

Rayleigh pdf

“Rayleigh fading”

00 =a

172



All the delays are small and we approximate kk allfor ττ ≈

( )

( )

( ) ( ) ( )ταα

τ

τ

φ

θ

θ

−++=

−+≈

−+=

∑

∑−

=

−

=

tsetsa

tseatsa

tseatsatr

jIR

N

k

jk

N

kk

jk

k

k

220

1

10

1

10

)(

)(

)()(

Rayleigh pdf

“Line-of-sight with Rayleigh Fading”

00 >a

( )

( )

( ) ( ) ( )ταα

τ

τ

φ

θ

θ

−+=

−≈

−=

∑

∑−

=

−

=

tse

tsea

tseatr

jIR

N

k

jk

N

kk

jk

k

k

22

1

1

1

1

)(

Rayleigh pdf

“Rayleigh fading”

00 =a

173


Multiplicative Fading

In the past two examples, the received signal was of the form

)()( tsFetr jφ=

The fading takes the form of a random attenuation: the transmittedsignal is multiplied by a random value whose envelope is describedby the Rice or Rayleigh pdf.

This is sometimes called multiplicative fading for the obviousreason. It is also called flat fading since all spectral components ins(t) are attenuated by the same value.

174


An Example

τ1θ

f

2)( fH

( )21 a+

( )21 a−

( ) ( )( )

( ) ( )θτπ

τδδτπθ

θ

−++=

+=−+=

−

faafH

aefH

taetthfj

j

2cos21

1

)(

22

)2(

τ1θ

f

(dB) )(2

fH

( )210 1log10 a+

( )210 1log10 a−

h(t))(ts )(tr

( )fS ( )fR

( )fS

fWW−

( )fR

fWW−

????

?

??

?

175


Example (continued)

f

2)( fH( )fS

fWW−

( )fR

fWW−

h(t))(ts )(tr

( )fS ( )fRτ is very small

f

2)( fH( )fS

fWW−

( )fR

fWW−

τ is very large

attenuation is even across thesignal band (i.e. channeltransfer function is “flat” in thesignal band)

attenuation is uneven acrossthe signal band -- this causes“frequency selective fading”

176


Another important special case

The delays are all different:

( )∑−

=

−+=1

10 )()(

N

kk

jk tseatsatr k τθ

121 −<<< Nτττ L

intersymbol interference

if the delays are “long enough”, the multipath reflections are resolvable.

177


Two common models for non-multiplicative fading

∆ ∆ ∆L

× × × × ×

+ + + +

1α 2α 3α 2−Nα 1−Nα

)(ts

)(tr

Taped delay-line withrandom weights

Additive complexGaussian random process ( )

)()(

)()(

0

1

10

ttsa

tseatsatrN

kk

jk

k

ξ

τθ

+≈

−+= ∑−

=

central limit theorem:approximately a Gaussian RP

178


Multipath Intensity Profile

The characterization of multipath fading as either flat (multiplicative) or frequency selective(non-multiplicative) is governed by the delays:

small delays ⇒ flat fading (multiplicative fading)large delays ⇒ frequency selective fading (non-multiplicative fading)

The values of the delay are quantified by the multipath intensity profile S(τ)

( )τS

τ2τ1τ 1−Nτ

1. “maximum excess delay” or “multipath spread”

1−= NmT τ

∑−

=−=

1

11

1 N

kkN

ττ∑

∑−

=

−

== 1

1

1

1N

kk

N

kkk

a

a ττ

2. average delay

or

3. delay spread

21

1

2

1

1 ττστ −−

= ∑−

=

N

kkN

21

1

2

1

1

22

ττ

στ −=∑

∑−

=

−

=N

kk

N

kkk

a

aorpo

wer

( ) ( ) ( ){ }( ) ( )

( ) ( ){ }2

211

21*

21,

ττ

ττδτττττ

hES

S

hhERhh

=

−==

uncorrelatedscattering (US)assumption

179


Characterization using the multipath intensity profile

( )τS

τ2τ1τ 1−Nτ

1. “maximum excess delay” or “multipath spread”

1−= NmT τ

∑−

=−=

1

11

1 N

kkN

ττ∑

∑−

=

−

== 1

1

1

1N

kk

N

kkk

a

a ττ

2. average delay

or

3. delay spread

21

1

2

1

1 ττστ −−

= ∑−

=

N

kkN

21

1

2

1

1

22

ττ

στ −=∑

∑−

=

−

=N

kk

N

kkk

a

aor

Compare multipath spread Tm with symboltime Ts:

Tm < Ts ⇒ flat fading (frequency non-selective fading)

Tm > Ts ⇒ frequency selective fading

Compare multipath spread Tm with symboltime Ts:

Tm < Ts ⇒ flat fading (frequency non-selective fading)

Tm > Ts ⇒ frequency selective fading

pow

er

180


Spaced Frequency Correlation Function

( )τS

τ2τ1τ 1−Nτ

( )fR ∆

f

0f

Fourier Xformpo

wer

R(∆f) is the “correlation between the channelresponse to two signals as a function of thefrequency difference between the two signals.”

“What is the correlation between receivedsignals that are spaced in frequency ∆f = f1-f2?”

Coherence bandwidth f0 = a statistical measureof the range of frequencies over which thechannel passes all spectral components withapproximately equal gain and linear phase.

Compare coherence bandwidth f0 withtransmitted signal bandwidth W:

f0 > W ⇒ flat fading (frequency non-selective fading)

f0 < W ⇒ frequency selective fading

Compare coherence bandwidth f0 withtransmitted signal bandwidth W:

f0 > W ⇒ flat fading (frequency non-selective fading)

f0 < W ⇒ frequency selective fading

equations (8) - (13) are commonly used relationships between delay spread and coherence bandwidth

181


Time Variations

Important Assumption

Multipath interference is spatial phenomenon. Spatial geometry is assumed fixed. Allscatterers making up the channel are stationary -- whenever motion ceases, the amplitude andphase of the receive signal remains constant (the channel appears to be time-invariant).Changes in multipath propagation occur due to changes in the spatial location x of thetransmitter and/or receiver. The faster the transmitter and/or receiver change spatial location,the faster the time variations in the multipath propagation properties.

Important Assumption

Multipath interference is spatial phenomenon. Spatial geometry is assumed fixed. Allscatterers making up the channel are stationary -- whenever motion ceases, the amplitude andphase of the receive signal remains constant (the channel appears to be time-invariant).Changes in multipath propagation occur due to changes in the spatial location x of thetransmitter and/or receiver. The faster the transmitter and/or receiver change spatial location,the faster the time variations in the multipath propagation properties.

( )

( ) ( )∑

∑−

=

−

=

−+=

−=

1

1

)(0

1

0

)(

)()()(

)()();(

N

kk

xjk

N

kk

xjk

xtexatxa

xtexaxth

k

k

τδδ

τδ

θ

θ

line-of-sightpropagation

multipathpropagation

complex gains andphase shifts are afunction of spatiallocation x.

182


Spatially Varying Channel Impulse Response

τ

x

);( xth

• channel impulse response changeswith spatial location x

• generalize impulse response toinclude spatial information

• Transmitter/receiver motion causechange in spatial location x

• The larger , the faster the rate ofchange in the channel.

• Assuming a constant velocity v, theposition axis x could be changed to atime axis t using t = x/v.

);()( xthth →

x&

183


Generalize the Multipath Intensity Profile

( ) ( ) ( ){ }( ) ( )

( ) ( ){ }2

211

21*

21,

ττ

ττδτττττ

hES

S

hhERhh

=

−== ( ) ( ) ( ){ }

( ) ( )

( ) ( ) ( ){ }

( ) ( ) ( ){ }

( ) ( ) ( ){ }tththEtS

ththEttS

xhxhExxS

xxS

xhxhExxRhh

∆+=∆

=

=

−==

;;;

;;,;

;;,;

,;

;;,;,

*

21*

21

21*

21

21211

2211*

2121

τττ

τττ

τττ

ττδτττττ

From before… The generalization …

US assumption

US assumption

vxt /=

WSS assumption

this function is the keyto the WSSUS channel

184


A look at S(τ;∆t)

τ

x∆

);( xS ∆τ

τ

t∆

);( tS ∆τ

( ) ( )0;ττ SS =

S(τ)S(τ)

185


Time Variations of the Channel:The Spaced-Time Correlation Function

t∆

);( tS ∆τ

integrate along delay axis

t∆)( tR ∆

( ) ∫ ∆=∆ ττ dtStR );(

186


Time Variations of the Channel:The Spaced-Time Correlation Function

0 0T t∆

)( tR ∆

R(∆t) specifies the extent to which there is correlation between the channelresponse to a sinusoid sent at time t and the response to a similar sinusoid attime t+∆t.

Coherence Time T0 is a measure of the expected time duration over whichthe channel response is essentially invariant. Slowly varying channels have alarge T0 and rapidly varying channels have a small T0.

187


Re-examination of special cases

( )

( )

[ ] )(

)(

)(

)()(

0

1

10

1

10

1

10

tsa

tseaa

tseatsa

tseatsatr

N

k

jk

N

k

jk

N

kk

jk

k

k

k

α

τ

θ

θ

θ

+=

+=

+≈

−+=

∑

∑

∑

−

=

−

=

−

=

( )

( )

[ ]

[ ] )()(

)()(

)()(

)()(

)()()()(

0

0

1

1

)(0

1

1

)(0

1

1

)(0

tsta

tsxa

tsexaa

tsexatsa

xtsexatsatr

N

k

xjk

N

k

xjk

N

kk

xjk

k

k

k

α

α

τ

θ

θ

θ

+=

+=

+=

+≈

−+=

∑

∑

∑

−

=

−

=

−

=

complex Gaussian RV

vxt /=

complex Gaussian RandomProcess with autocorrelation

( ) ( ) ( ){ }( )tR

tttEtR

∆=∆+=∆ ααα

*

From before… The generalization …

188


Commonly Used Spaced-Time Correlation Functions

( ) ∞<∆<∞−=∆ ttR a22σ

( ) tv

a etR∆−

=∆ λπ

σ2

22

( )

∆=∆ t

vJtR a λ

πσ 22 02

( )2

22

∆−

=∆t

v

a etR λπ

σ

( )t

v

tv

tR a

∆

∆

=∆

λπ

λπ

σ2

2sin2 2

Time Invariant

Land Mobile(Jakes)

Exponential

Gaussian

“Rectangular”

t∆

( )tR ∆

t∆

( )tR ∆

t∆

( )tR ∆

t∆

( )tR ∆

t∆

( )tR ∆

189


Characterization of time variations using the spaced-timecorrelation function

0 0T t∆

)( tR ∆

• Fast Fading– T0 < Ts

– correlated channel behavior lasts less than a symbol ⇒ fading characteristicschange multiple times during a symbol ⇒ pulse shape distortion

• Slow Fading– T0 > Ts

– correlated channel behavior lasts more than a symbol ⇒ fading characteristicsconstant during a symbol ⇒ no pulse shape distortion ⇒ error bursts…

190


Doppler Power SpectrumFrequency Domain View of Time-Variations

0 0T t∆

)( tR ∆

Fourier Xform

( )νS

νdf

Time variations on the channel are evidenced as a Doppler broadening and perhaps, in addition as aDoppler shift of a spectral line.

Doppler power spectrum S(ν) yields knowledge about the spectral spreading of a sinusoid (impulsein frequency) in the Doppler shift domain. It also allows us to glean how much spectral broadeningis imposed on the transmitted signal as a function of the rate of change in the channel state.

Doppler Spread of the channel fd is the range of values of ν over which the Doppler powerspectrum is essentially non zero.

191


Doppler Power Spectrum and Doppler Spread

( )νS

νdf

Compare Doppler Spread fd withtransmitted signal bandwidth W:

fd > W ⇒ fast fading

fd < W ⇒ slow fading

Compare Doppler Spread fd withtransmitted signal bandwidth W:

fd > W ⇒ fast fading

fd < W ⇒ slow fading

equations (18) - (21) are commonly used relationships between Doppler spread and coherence time

192


Common Doppler Power Spectra

Time Invariant

Land Mobile(Jakes)

Exponential(1st order Butterworth)

Gaussian

“Rectangular”

ν

( )νS

( ) ( )νδσν 22 aS =

( )( )22

2

/

2

λνσν

vS a

−=

( ) ( )( )22

2

/

/2

λνπλσν

v

vS a

+=

( )( )

( )2

2

/

2

2

/

2 λν

λπσν va ev

S−

=

( )

<<−=otherwise0

///

2

λνλλ

σν vv

vSa

ν

( )νS

ν

( )νS

ν

( )νS

ν

( )νS

193


Putting it all together…

( )fR ∆

f

0f

( )τS

τ2τ1τ 1−Nτ

pow

er

0 0T t∆

)( tR ∆

( )νS

νdf

);( tfS ∆∆

0=∆f0=∆t

Fourier transform Fourier transform

);( ντS

∫ νντ dS );( ∫ τντ dS );(

);( tS ∆τ Fourier transform

f∆↔τ

spaced-frequencycorrelation function

multipath intensityprofile

spaced-timecorrelation function

Doppler powerspectrum

spaced-frequency, spaced-time correlation function

scattering function

194


Scattering Function

delay τ

frequency ν

);( ντS

195


References

• John Proakis, Digital Communications, Third Edition. McGraw-Hill. Chapter 10.

• William Jakes, Editor, Microwave Mobile Communications. John Wiley & Sons.Chapter 1.

• William Y. C. Lee, Mobile Cellular Communications, McGraw-Hill.

• Parsons, J. D., The Mobile Radio Propagation Channel, John Wiley & Sons.

• Bernard Sklar, “Rayleigh Fading Channels in Digital Communication systems Part I:Characterization,” IEEE Communication Magazine, July 1997, pp. 90 - 100.

• Peter Bello, “Characterization of Randomly Time Variant Linear Channels,” IEEETransactions on Communication Systems, vol. 11, no. 4, December 1963, pp. 360-393.

• R. H. Clarke, “A Statistical Theory of Mobile Radio Reception,” Bell Systems TechnicalJournal, vol. 47, no. 6, July-August 1968, pp. 957-1000.

• M. J. Gans, “A Power-Spectral Theory of Propagatio9in in the Mobile RadioEnvironment,” IEEE Transactions on Vehicular Technology, vol. VT 21, February1972, pp. 27-38.,

196

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