Properties of the Mobile Radio Propagation Channel

Jean-Paul M.G. LinnartzDepartment Head CoSiNeNat.Lab., Philips Research

2JPL

Statistical Description of Multipath Fading

The basic Rayleigh / Ricean model gives the PDF of envelope But: how fast does the signal fade? How wide in bandwidth are fades?

Typical system engineering questions: What is an appropriate packet duration, to avoid fades? How much ISI will occur? For frequency diversity, how far should one separate carriers? How far should one separate antennas for diversity? What is good a interleaving depth? What bit rates work well? Why can't I connect an ordinary modem to a cellular phone?

The models discussed in the following sheets will provide insight in these issues

3JPL

The Mobile Radio Propagation Channel

A wireless channel exhibits severe fluctuations for small displacements of the antenna or small carrier frequency offsets.

Channel Amplitude in dB versus location (= time*velocity) and frequency

TimeFrequency

Amplitude

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Time Dispersion vs Frequency Dispersion

Time Domain Channel variations Delay spreadInterpretation Fast Fading InterSymbol Interference

Correlation Distance Channel equalization

Frequency Doppler spectrum Frequency selective fadingDomain Intercarrier Interf. Coherence bandwidthInterpretation

Time Dispersion Frequency Dispersion

5JPL

Two distinct mechanisms

1.) Time dispersion Time variations of the channel are caused by motion of the antenna Channel changes every half a wavelength Moving antenna gives Doppler spread Fast fading requires short packet durations, thus high bit rates Time dispersion poses requirements on synchronization and rate of convergence of

channel estimation Interleaving may help to avoid burst errors

2.) Frequency dispersion Delayed reflections cause intersymbol interference Channel Equalization may be needed. Frequency selective fading Multipath delay spreads require long symbol times Frequency diversity or spread spectrum may help

6JPL

Time dispersion of narrowband signal (single frequency)

Transmit: cos(2 fc t)

Receive: I(t) cos(2 fc t) + Q(t) sin(2 fc t)= R(t) cos(2 fc t + )

I-Q phase trajectory

• As a function of time, I(t) and Q(t) follow a random trajectory through the complex plane

• Intuitive conclusion: Deep amplitude fades coincide with large phase rotations

Animate

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Doppler shift and Doppler spread

• All reflected waves arrive from a different angle• All waves have a different Doppler shift

The Doppler shift of a particular wave is cos f c cv

= f 0

Maximum Doppler shift: fD = fc v / c

Joint Signal Model

• Infinite number of waves

• First find distribution of angle of arrival,

then compute distribution of Doppler shifts

• Uniform distribution of angle of arrival : f() = 1/2

• Line spectrum goes into continuous spectrum Calculate

8JPL

Doppler Spectrum

If one transmits a sinusoid, …what are the frequency components in the received signal?

Power density spectrum versus received frequency

Probability density of Doppler shift versus received

frequency

The Doppler spectrum has a characteristic U-shape.

Note the similarity with sampling a randomly-phased

sinusoid

No components fall outside interval [fc- fD, fc+ fD]

Components of + fD or -fD appear relatively often

Fades are not entirely “memory-less”

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Derivation of Doppler SpectrumT h e p o w e r s p e c t r u m S ( f ) i s f o u n d f r o m

S ( f ) = p f ( ) G ( ) + f ( - ) G ( - ) d

d f0f 0

w h e r e

f ( ) = 1 / ( 2 ) i s t h e P D F o f a n g l e o f i n c i d e n c e

G ( ) t h e a n t e n n a g a i n i n d i r e c t i o n

p l o c a l - m e a n r e c e i v e d p o w e r , a n d

0 cf = f 1 + v

c c o s

O n e f i n d s d

d f =

f ( f - f )D2

c2

1

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Vertical Dipole

R e ce ive d P o w e r S p e c tru m

)(2 π

3

2f-f-f

1p = S(f)

c2D

• D o p p le r spe c trum is ce n te red a ro u nd fc

• D o p p le r spe c trum ha s w id th 2 fD

A vertical dipole is omni-directional in horizontal plane

G(θ) = 1.5

We assume

• Uniform angle of arrival of reflections

• No dominant wave

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How do systems handle Doppler Spreads?

• Analog Carrier frequency is low enough to avoid problems (random FM)

• GSM Channel bit rate well above Doppler spread TDMA during each bit / burst transmission the channel is fairly constant. Receiver training/updating during each transmission burst Feedback frequency correction

• DECT Intended to pedestrian use: only small Doppler spreads are to be anticipated for.

• IS95 Downlink: Pilot signal for synchronization and channel estimation Uplink: Continuous tracking of each signal

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Autocorrelation of the signal

We now know the Doppler spectrum,

but ... how fast does the channel change?

Wiener-Kinchine Theorem:

Power density spectrum of a random signal is the Fourier Transform

of its autocorrelation Inverse Fourier Transform of Doppler spectrum gives autocorrelation

of I(t), or of Q(t)

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Derivation of Autocorrelation of I-Q components

W e f i n d t h e a u t o - c o r r e l a t i o n

g ( ) = I ( t ) I ( t + ) = Q ( t ) Q ( t + )

= S ( f ) 2 ( f - f ) d fc D

c D

f - f

f f

c

E E

c o s

A u t o - c o r r e l a t i o n d e p e n d s o n S ( f ) , t h u s i t d e p e n d s o n t h e d i s t r i b u t i o n o f

a n g l e o f a r r i v a l

• g ( = 0 ) = l o c a l - m e a n p o w e r : g ( 0 ) = E I 2 ( t ) = p

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For uniform angle of arrival

F o r a U - s h a p e d D o p p l e r s p e c t r u m , t h e a u t o - c o r r e l a t i o n f u n c t i o n i s

g ( ) = I ( t ) I ( t + ) = p J 2 f 0 D E

w h e r e

J 0 i s z e r o - o r d e r B e s s e l f u n c t i o n o f f i r s t k i n d : i n t e g r a l o v e r { c o s ( z s i n x ) }

f D i s t h e m a x i m u m D o p p l e r s h i f t

i s t h e t i m e d i f f e r e n c e

N o t e t h a t t h e c o r r e l a t i o n i s a f u n c t i o n o f d i s t a n c e o r t i m e o f f s e t :

D cf = fv

c =

d

w h e r e

d i s t h e a n t e n n a d i s p l a c e m e n t d u r i n g , w i t h d = v

i s t h e c a r r i e r w a v e l e n g t h ( 3 0 c m a t 1 G H z )

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Relation between I and Q phase

S . O . R i c e : C r o s s - c o r r e l a t i o n

h ( ) = I ( t ) Q ( t + ) = - Q ( t ) I ( t + )

= S ( f ) 2 ( f - f ) d fc D

c D

f - f

f f

c

E E

s i n

F o r u n i f o r m l y d i s t r i b u t e d a n g l e o f a r r i v a l

• D o p p l e r s p e c t r u m S ( f ) i s e v e n a r o u n d f c

• C r o s s c o r r e l a t i o n h ( ) i s z e r o f o r a l l

• I ( t 1 ) a n d Q ( t 2 ) a r e i n d e p e n d e n t

F o r a n y d i s t r i b u t i o n o f a n g l e s o f a r r i v a l

• I ( t 1 ) a n d Q ( t 1 ) a r e i n d e p e n d e n t { = 0 : h ( ) = 0 }

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PDF of the real amplitude R

),,,()φ,φ,,( 21212121 QQIIfJrrf

For the amplitude r1 and r2

212121π2

21 φφ)φ,φ,,(),( ddrrfrrf

2221

022

22

21

2421

21ρ1σ

ρ

ρ12σexp

ρ1σ),(

rrI

rrrrrrf

NB: )()(

),()( 2

1

2112 rRician

rf

rrfrrf

2

20

)Δω(1

τωρ

rms

c

T

J

Correlation:

R2 = I2 + Q2

Note: get more elegant expressions for

complex amplitude H

17JPL

Autocorrelation of amplitude R2 = I2 + Q2

E

ER(t)R(t + ) =

2 p 1 +

1

4

I(t)I(t+ )

p2

2

210 0

2121 ),()τ()(E drdrrrfrrtRtR

221

212 ρ;1;,σ

2

π)τ()(E FtRtR

The solution is known as the hypergeometric function F(a,b;c;z)

or, in good approximation, ..

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Autocorrelation of amplitude R2 = I2 + Q2

E

ER(t)R(t + ) =

2 p 1 +

1

4

I(t)I(t+ )

p2

2

Result for Autocovariance of Amplitude

Remove mean-value and normalize

Autocovariance

C J 2 f D 02

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Amplitude r(t0) and Derivative r’(t0) are uncorrelated

J0()

τ

τπ2)τ()(E

τ)(')(E

20

d

fdJtrtr

d

dtrtr D

Correlation is 0 for = 0

τ

τπ2)τ()(E

τ)(')(E 0

d

fdJthth

d

dthth D

20JPL

Simulation of multipath channels

Jakes' Simulator for narrowband channel

generate the two bandpass “noise” components by adding many sinusoidal signals. Their frequencies are non-uniformly distributed to approximate the typical U-shaped Doppler spectrum.

N Frequency components are taken at

2ifi = fm cos -------- 2(2N+1)

with i = 1, 2, .., N

All amplitudes are taken equal to unity. One component at the maximum Doppler shift is also added, but at amplitude of 1/sqrt(2), i.e., at about 0.707 . Jakes suggests to use 8 sinusoidal signals. Approximation (orange)

of the U-Doppler spectrum (Black)

?

21JPL

How to handle fast multipath fading?

Analog User must speak slowly

GSM Error correction and interleaving to avoid burst errors Error detection and speech decoding Fade margins in cell planning

DECT Diversity reception at base station

IS95 Wideband transmission averages channel behaviour This avoids burst errors and deep fades

Frequency Dispersion

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Frequency Dispersion

• Frequency dispersion is caused by

the delay spread of the channel

• Frequency dispersion has no relation

to the velocity of the antenna

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Frequency Dispersion: Delay Profile

Typical sample of impulse response h(t)

If we transmit a pulse (t) we receive h(t)

Delay profile: PDF of received power: "average h2(t)"

Local-mean power in delay bin is p f()

25JPL

RMS Delay Spread and Maximum delay spread

D e f i n i t i o n s

n - t h m o m e n t o f d e l a y s p r e a d n0

n = f ( ) d

R M S v a l u e

R M S

02 0 2 1

2T =

1 -

26JPL

Typical Values of Delay Spreads

Macrocells TRMS < 8 sec

• GSM (256 kbit/s) uses an equalizer

• IS-54 (48 kbit/s): no equalizer

• In mountanous regions delays of 8 sec and more occur

GSM has some problems in Switzerland

Microcells TRMS < 2 sec

• Low antennas (below tops of buildings)

27JPL

Typical values of delay spreadPicocells 1 .. 2 GHz:

TRMS < 50 nsec - 300 nsec

• Home 50 nsec

• Shopping mall 100 - 200 nsec

• Railway station 200 - 450 nsec

• Office block 100 - 400 nsec

• Indoor: often 50 nsec is assumed

• DECT (1 Mbit/s) works well up to 90 nsec

Outdoors, DECT has problem if range > 200 .. 500

28JPL

Typical Delay Profiles

1) Exponential

2) Uniform Delay Profile

Experienced on some indoorchannels

Often approximated by N-RayChannel

3) Bad Urban

For a bandlimited signal, one may sample the delay profile. This give a tapped delay line model for the channel, with constant delay times

29JPL

COST 207 Typical Urban Reception (TU6)

COST 207 describes typical channel characteristics for over transmit bandwidths of 10 to 20 MHz around 900MHz. TU-6 models the terrestrial propagation in an urban area. It uses 6 resolvable paths

COST 207 profiles were adapted to mobile DVB-T reception in the E.U. Motivate project.

Tap number Delay (us) Power (dB) Fading model

1 0.0 -3 Rayleigh

2 0.2 0 Rayleigh

3 0.5 -2 Rayleigh

4 1.6 -6 Rayleigh

5 2.3 -8 Rayleigh

6 5.0 -10 Rayleigh

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COST 207 A sample fixed channel

Tap number Delay ( (s) Amplitude r Level (dB) Phase ( (rad)

1 0.050 0.36 -8.88 -2.875

2 0.479 1 0 0

3 0.621 0.787 -2.09 2.182

4 1.907 0.587 -4.63 -0.460

5 2.764 0.482 -6.34 -2.616

6 3.193 0.451 -6.92 2.863

• COST 207 Digital land mobile radio Communications, final report, September 1988.

• European Project AC 318 Motivate: Deliverable 06: Reference Receiver Conditions for Mobile Reception, January 2000

31JPL

How do systems handle delay spreads?

Analog Narrowband transmission

GSM Adaptive channel equalization Channel estimation training sequence

DECT Use the handset only in small cells with small delay spreads Diversity and channel selection can help a little bit

“pick a channel where late reflections are in a fade”

IS95 Rake receiver separately recovers signals over paths with excessive

delays

DTV and Digital Audio Broacasting OFDM multi-carrier modulationThe radio channel is split into many narrowband (ISI-free) subchannels

32JPL

Correlation of Fading vs. Frequency Separation

W h e n d o w e e x p e r i e n c e f r e q u e n c y - s e l e c t i v e f a d i n g ?

H o w t o c h o o s e a g o o d b i t r a t e ?

W h e r e i s f r e q u e n c y d i v e r s i t y e f f e c t i v e ?

I n t h e n e x t s l i d e s , w e w i l l . . . g i v e a m o d e l f o r I a n d Q , f o r t w o s i n u s o i d s w i t h t i m e a n d f r e q u e n c y o f f s e t , d e r i v e t h e c o v a r i a n c e m a t r i x f o r I a n d Q , d e r i v e t h e c o r r e l a t i o n o f e n v e l o p e R , g i v e t h e r e s u l t f o r t h e a u t o c o v a r i a n c e o f R , a n d d e f i n e t h e c o h e r e n c e b a n d w i d t h .

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Inphase and Quadrature-Phase Components

Consider two (random) sinusoidal signals

Sample 1 at frequency f1 at time t1 Sample 2 at frequency f2 = f1 + f at time t2

Effect of displacement on each phasor:Spatial or temporal displacement: Phase difference due to DopplerSpectral displacement Phase difference due to excess delay

Mathematical Treatment:• I(t) and Q(t) are jointly Gaussian random

processes• (I1, Q1, I2, Q2) is a jointly Gaussian random

vectorConsider complex H1 = I1+jQ1, H2 =I2+jQ2 as a

jointly Gaussian random vector

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Multipath Channel

• Transmit signal s(t)

• Received signal r(t) = an gn (t) + n(t),

Channel model:

Iw reflected waves have

the following properties:

• Di is the amplitude

• Ti is the delay

Signal parameters:

•an is the datac is the carrier frequency

)()}(ωωexp{)(1

0tnTtnjDatr

wI

iiscin

}exp{)( tjats cn

TO DO make math consistent

35JPL

Doppler Multipath Channel

Correlation between p-th and q-th derivative

1

0

1

0}exp{E

)(*)(E

w wI

i

I

kkickip

TTjDDdt

tdg

td

tdg

TO DO make math consistent

36JPL

Doppler Multipath Channel

ddTT

DDI rmsrmsAi

iiw

exp2

1E

lim *

)dθθcos(2ωθcos2::τ ffidtTiA ii

0

2

0

)(*)(

exp

cos2

)1(E

dT

dT

j

c

vgg

rms

qp

rms

qpq

qp

qps

qpcq

mp

n

TO DO make math consistent

37JPL

Random Complex-Gaussian Amplitude

Special case

• This defines the covariance matrix of subcarrier amplitudes at different frequencies

• This is used in OFDM for cahnnel estimation

srmsmn Tmnj

vvω)(1

1E

TO DO make math consistent

38JPL

Coherence BandwidthDefinition: Coherence. Bandwidth is the frequency separation for which the

correlation coefficient is down from 1 to 0.5 Thus 1 = 2(f1 - f2) TRMS

so Coherence Bandwidth BW = 1 (2 TRMS)

We derived this for an exponential delay profile

Another rule of thumb:

ISI affects BER if Tb > 0.1TRMS

Conclusion: Either keep transmission bandwidth much smaller than the coherence

bandwidth of the channel, or use signal processing to overcome ISI, e.g. Equalization DS-CDMA with rake OFDM

39JPL

Frequency and Time Dispersion

M o d e l : E a c h w a v e h a s i t s o w n a n g l e a n d e x c e s s d e l a y

A n t e n n a m o t i o n c h a n g e s p h a s e

c h a n g i n g c a r r i e r f r e q u e n c y c h a n g e s p h a s e

T h e s c a t t e r i n g e n v i r o n m e n t i s d e f i n e d b y

a n g l e s o f a r r i v a l

e x c e s s d e l a y s i n e a c h p a t h

p o w e r o f e a c h p a t h

40JPL

Scatter Function of a Multipath Mobile Channel

Gives power as function of

Doppler Shift (derived from angle of arrival )

Excess Delay

Example of a scatter plot

Horizontal axes: x-axis: Excess delay time y-axis:Doppler shiftVertical axis z-axis: received power

41JPL

Special casesZero displacement / motion: = 0Zero frequency separation: f = 0

1 2R ,R2

02

21 2

2 2RMS

C = J (2

v)

1 + 4 ( f - f ) T

Freq. and time selective channels

Effects of fading on modulated radio signals

JPL

Effects of Multipath (I)

Frequency

Tim

e

Narrowband

Frequency

Tim

e

OFDM

Frequency

Tim

e

Wideband

JPL

Effects of Multipath (II)

Frequency

Tim

e

+-+--+-+

DS-CDMA

Frequency

Tim

e +

-

-

FrequencyHopping

Tim

e

Frequency

+ - + -

+ - +-

+ - +-

MC-CDMA

45JPL

BER

• BER for Ricean fading

calculate

Time Dispersion Revisited

The duration of fades

47JPL

Time Dispersion Revisited: Duration of Fades

In the next slides we study the temporal behavior of fades.

Outline:

# of level crossings per second

Model for level crossings

Derivation

Model for I, Q and derivatives

Model for amplitude and derivative

Discussion of result

Average non-fade duration

Effective throughput and optimum packet length

Average fade-duration

48JPL

Two state model

Very simple mode: channel has two-states Good state: Signal above “threshold”, BER is virtually zero Poor state: “Signal outage”, BER is 1/2, receiver falls out of sync, etc

Markov model approach: Memory-less transistions Exponential distribution of sojourn time

This model may be sufficiently realistic for many investigations

49JPL

Level crossings per second

• Av. number of crossings per sec = [av. interfade time]-1

Number of level crossing per sec is proportional to

• speed r' of crossing R (derivative r' = dr/dt)

• probability of r being in [R, R + dR]. This Prob = fR(r) dr

)(rfdt

drN R

)()(1

rfdt

drdrRrRP

TN R

50JPL

Derivation of Level Crossings per Second

Random process r’ is derivative of the envelope r w.r.t. time Note: here we need the joint PDF;

not the conditional PDF f(r’r=R)

• We derive f(r’r=R) from f(r’r) and f(i1,i2,q1,q2)

''' dr R),f(r r = N0

+

51JPL

Covariance matrix of (I, Q, I’, Q’)

In-phase, quadrature and the derivatives are Jointly Gaussian withcovariance matrix:

Here bn is n-th moment of Dopplerspectral power density

nn

f - f

f + f

cn

b = (2 ) S(f) (f - f ) dfc D

c D

For uniform angles of arrival

n

Dn

b =

p (2 f )(n -1)!!

n!

n

0 n

for even

for odd

where (n-1)!! = 1 3 5 .....(n-1)

I,Q,I,Q1

1

1 2

1 2

= p 0 0 b

0 p - b 0

0 - b b 0

b 0 0 b

52JPL

Joint PDF of R, R’, , ’

. . . . A f t e r s o m e a l g e b r a . . .

b

r +b

r +

p

r2

1-

bp4

r=),,rf(r,2

22

2

2

22

2

'2'

expˆˆ

S o

r a n d r ’ a r e i n d e p e n d e n t

p h a s e i s u n i f o r m

F a s t p h a s e c h a n g e o n l y o c c u r w h e n a m p l i t u d e i s s m a l l

A v e r a g i n g o v e r a n d ’ g i v e s

r i s R a y l e i g h

r ^ i s z e r o m e a n G a u s s i a n w i t h v a r i a n c e b 2

53JPL

Level Crossings per Second

W e i n s e r t t h i s p d f i n

''' dr )R=R,f(r r = N 0

0

+

W e f i n d

+ D -1

N = 2 f

e

w h e r e

i s t h e f a d e m a r g i n = 2 p / ( R 02 )

L e v e l c r o s s i n g r a t e h a s a m a x i m u m f o r t h r e s h o l d s R 0 c l o s e t o t h e m e a n v a l u e o f

a m p l i t u d e . S i m i l a r l y f o r R i c e a n f a d i n g

+D R 0N = p f f ( R )

Calculate

54JPL

Average Fade / Nonfade Duration

F a d e d u r a t i o n s a r e r e l e v a n t t o c h o o s e p a c k e t d u r a t i o n

D u r a t i o n d e p e n d o n

F a d e m a r g i n = 2 p / R 02

D o p p l e r s p r e a d

A v e r a g e f a d e d u r a t i o n T F N T = ( R R )F 0P r

A v e r a g e n o n f a d e d u r a t i o n T N F+

N F 0N T = ( R R )P r

Calculate

55JPL

Average nonfade duration

N FD

T = = 2 f

1 P r ( o u t a g e )

L e v e l C r o s s R a t e

A v e r a g e n o n f a d e d u r a t i o n i s i n v e r s e l y p r o p o r t i o n a l t o D o p p l e r s p r e a d

A v e r a g e n o n f a d e d u r a t i o n i s p r o p o r t i o n a l t o f a d e m a r g i n

Calculate

56JPL

How to handle long fades when the user is stationary?

Analog Disconnect user

GSM Slow frequency hopping Handover, if appropriate Power control

DECT Diversity at base station Best channel selection by handset

IS95 Wide band transmission avoids most deep fades (at least in macro-cells) Power control

Wireless LANs Frequency Hopping, Antenna Diversity

57JPL

Optimal Packet length

We want to optimize

#User bitsEffective throughput = ---------------- Prob(success)

#Packet bits

This requires a trade off between

Short packets: much overhead (headers, sync. words etc).

Long packets:may experience fade before end of packet.

58JPL

Optimal Packet length

At 1200 bit/s, throughput is virtually zero:Almost all packets run into a fade. Packets are too long

At high bit rates, many packets can be exchanged during non-fadeperiods. Intuition:

Packet duration < < Av. nonfade duration

59JPL

Derivation of Optimal Packet length

A s s u m e e x p o n e n t i a l , m e m o r y l e s s n o n f a d e d u r a t i o n s( T h i s i s a n a p p r o x i m a t i o n : I n r e a l i t y m a n y n o n f a d e p e r i o d s h a v e

d u r a t i o n o f / 2 , d u e t o U - s h a p e d D o p p l e r s p e c t r u m )

S u c c e s s f u l r e c e p t i o n i f1 ) A b o v e t h r e s h o l d a t s t a r t o f p a c k e t2 ) N o f a d e s t a r t s b e f o r e p a c k e t e n d s

F o r m u l a :

P s u c c e x p e x p( ) = -1

- T

T = -

1-

2 f TL

N F

D L

w i t h T L p a c k e t d u r a t i o n

L a r g e f a d e m a r g i n : s e c o n d t e r m d o m i n a t e s : p e r f o r m a n c e i m p r o v e s o n l y s l o w l y w i t h i n c r e a s i n g O u t a g e p r o b a b i l i t y i s t o o o p t i m i s t i c Calculate

60JPL

Average fade duration

1

η

1exp

π

η-

f 2 = T

D

F

A F D i s i n v e r s e l y p r o p o r t i o n a l t o D o p p l e r s p r e a d

F a d e d u r a t i o n s r a p i d l y r e d u c e w i t h i n c r e a s i n g m a r g i n , b u t t i m e b e t w e e n f a d e s

i n c r e a s e s m u c h s l o w e r

E x p e r i m e n t s : F o r l a r g e f a d e m a r g i n s : e x p o n e n t i a l l y d i s t r i b u t e d f a d e d u r a t i o n s

R e l e v a n t t o f i n d l e n g t h o f e r r o r b u r s t s a n d d e s i g n o f i n t e r l e a v i n g

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Conclusion

• The multipath channel is characterized by two effects: Time and Frequency Dispersion

• Time Dispersion effects are proportional to speed and carrier frequency

• System designer need to anticipate for channel anomalies

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Statistical properties of a Rayleigh signal

Parameter Probability Distribution

I(t1) and Q(t1) i.i.d. Gaussian, Zero mean

I(t1) and Q(t2) Correlated jointly Gaussian

I(t2) given I(t1) Gaussian

Amplitude r Rayleigh

r(t2)givenr(t1) Ricean

r’(t1)derivative r Gaussian, Independent of amplitude

Phase Uniform

Derivative of Phase Gaussian, Dependent on amplitude

Power Exponential

The nice thing about jointly Gaussian r.v.s is that the covariance matrix fully describes the behavior

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Taylor Expansion of Amplitude

• Rewrite the Channel Model as follows

• Tayler expansion of the amplitude

• gn(t) = gn(0)+ gn

(1) (t-t) + gn(2) (t-t)2/2 + .. .

• gn(q) : the q-th derivative of amplitude wrt time, at instant t = t.

• gn(p) is a complex Gaussian random variable

• To address frequency separation ns (later):

)(}exp{)()( tntjtgatr cnn

1

0

)( })(exp{wI

itiisci

qi

qn jTnjDjg

)(}))((exp{)(1

0tntjTtjDatr

wI

iiicin

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Doppler Multipath Channel

Simplified baseband model for narrowband linear modulation

Received signal r(t) = an g(t) + n(t), with time-varying channel amplitude g(t)

Channel model:

Iw reflected waves, each has the following properties:

• Di is the amplitude

I is the Doppler shift

• Ti is the path delay

Note g(t) is not an impulse response;

It is the channel amplification and phase

Signal parameters:• linear modulation • an is the user data• c is the carrier frequency

• n(t) is the noise

1

0

)( })(exp{)(wI

iiisci

qi

qn tjTnjDjtg

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Doppler Multipath Channel

p-th Derivative gp(t) of the channel w.r.t. to time t

All derivatives are gaussian rv’s if Iw and Di i.i.d.

The covariance matrix fully describes the statistical properties

1

0}exp{

)( wI

iiic

pii

pp

p

tjTjDjdt

tgd

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Doppler Multipath Channel

Correlation between p-th and q-th derivative

1

0

1

0

)*()(

}exp{1E

)()(E

w wI

i

I

kkikicki

qk

pi

qpq

q

p

p

tjTTjDDj

dt

tdg

td

tdg

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Doppler Multipath Channel

ddTT

DDI rmsrmsAi

iiw

exp2

1E

lim *

)dθθcos(2ωθcos2::τ ffidtTiA ii

0

2

0

)(*)(

exp

cos2

)1(E

dT

dT

j

c

vgg

rms

qp

rms

qpq

qp

qps

qpcq

mp

n

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Random Complex-Gaussian Amplitude

It can be shown that for p + q is even

and 0 for p + q is odd.

• This defines the covariance matrix of subcarrier amplitudes and derivatives,

• OFDM: allows system modeling and simulation between the input of the transmit I-FFT and output of the receive FFT.

srms

qpqqp

Dq

mp

n Tmnj

j

qp

qpfvv

)(1

)1(

!)!(

!)!1(2E )*()(