wsp 19 mm1 - compatibility modekom.aau.dk/~pe/education/menu/8sem/wsp_19_mm1 - compatibility...

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(c) Patrick Eggers 201904/02/2019

MM1: Short Term Fading (narrowband)

Wireless System PerformancePatrick Eggers,

5/2-2019, C3-204, 12.30

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(c) Patrick Eggers 2019

Formals• Exam format

– Mini projects in groups• Part I : signal domain• Part II : link domain• Part III : network domain• Each part to be handed in shortly after each part of

course is over. Exact dates yet to be setteled

– Oral exam• Based on course and mini projects• Pass/no pass

:04/02/2019 2

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Contents (changes can occur, check moodle)

1. Narrowband multipath2. Wideband multipath 3. DiversityCa 2 Lectures:Cellular concept, Channel allocationResource allocation, Link adaptationCa 2 LecturesWireless ad hoc networkingMAC protocols for ad hoc networks

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Part 1

Part 2

Part 3

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Litterature part I• Book = Parsons 2nd Ed + articles

– Not ordered in book store as very few students usually buy hard copy

• Parson several sources.. 2sections in moodle

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Expected course effort

• 5ECTS·avg. 27-28h = 140h = ‘visible’ at exam– IF you have all prerequistes in ‘active memory’.

ELSE you need ADDITIONALLY to invest time catching up

• Stochastic processes• fundamental physics and geometric relations (phase vs amplitydeu

etc)• Some network and radio systems fundamentals

– Experience:min 150h/ca 10 topics = 15h= 2 day / topic!, fx

• 2-4½h preparation = USE UP FRONT TIME = ‘tail wind’• 4½ confrontation and exercises• 4-6h Mini project+ 2h exam preparation and exam

04/02/2019

Ét ECTS point svarer til en arbejdsindsats for en "normal-studerende" på cirka 27-28 timer

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I: Contineous to discrete world transition• Contineous signal –> discrete info.

• Decision/quantisation

• ’Noise’ (thermal, interference etc)

HW

Antenna

Signal (de)mapping

(de)ModulationRadio waves

Detection

0110111011

Bits

= 00 = 10 = 01 = 11

= 00 ~ 00 = ??

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Signal -> link transition : channel impact SNR• AWGN -> Fading

• Channel changes BER statistics drastically•http://www.raymaps.com/index.php/bit-error-rate-of-qpsk-in-rayleigh-fading/ber2/

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Channel impact : Doppler/Speed• Channel phase dynamics : Irreducible

BER floor

•http://ratnuu.wordpress.com/2011/03/03/the-theoretical-ber-under-fading/

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BLER - PER• Block ER -> Packet ER/delay etc

– Progression to higher and higher abstraction level:

• Signal imperfection: Ch(s)->signal SNR, CIR, ISI …• Link imperfection: signal(s)->data BER, FER/BLER,..• Network imperfection: link(s)->connection PER, Delay, Data rate..

•http://wireless.agilent.com/rfcomms/refdocs/wcdma/wcdma_meas_wblerror_desc.html

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AMC : Adaptation to Phy layer

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Quality aware routing metrics/ cost func• Hops, delays. Tranmissions, loss, etc

• Packet error rate• Depend on FER, BLER• Depend BER• Depend dynamics of SNR, SIR etc04/02/2019 11

Wireless Mesh Networks (Architectures and Protocols)Hossain, Ekram; Leung, Kin K. (Eds.)XXIV, 333 p. 2007, Springerpp 227-243 Quality-Aware Routing Metrics in Wireless Mesh NetworksC. E. Koksal

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(c) Patrick Eggers 201904/02/2019 12

Part I Layout : Signal domain

Basicpropagation

effects

Shorttermnarrowband

Shorttermwideband

MeasurementsPathloss

Shadow fading

DiversityMicro

NB & WB

HandsetAntennas

Measuremenst

BLOK 1 : LINK BUDGET/MEASUREMENTS,voluntary background

BLOK 2 : SHORT TERM EFFECTS, MM1-MM2

9sem WCSBLOK 3 : ANTENNAS & Propagation

BACKGROUND

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(c) Patrick Eggers 201904/02/2019 13

General +Specific Part I• General layout schedule, time and place• Moodle

• PE menus : http://kom.aau.dk/~pe/education/menu/8sem

• Block 2 : Short term variations • MM1 : Narrowband multi path• MM2 : Wideband multi path• MM3 : Diversity

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Outline• Rayleigh distribution: physical interpretation• Properties of the Rayleigh distribution

– Mean, Median, Variance– Distribution of power

• Other commonly used distributions• Doppler spectrum• Stochastic description of short term fading

– Phase and power gradients– Correlation function

• Practical considerations

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Notations : be CONTEXT AWARE• Space

– x,y,x Cartesian coordinates– θ,ϕ,r Polar coordinates– L, l length– D, d distance– R, r run (length) or range– H, h height– α,β,θ,ϕ,ψ etc angle

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Notation: be CONTEXT AWARE• Signal

– h impulse response– H frequency transfer function– f frequency (Doppler)– t time– τ time delay– λ wave length– k = 2π/λ , wave number– x + jy etc = Re + j·Im, complex signal– r exp(jϕ) etc = amplitude ∠phase– P, p power04/02/2019

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Notation: be CONTEXT AWARE• Physics

– c = 3·108 m/s, speed of light (Vacuum)– E electrical field strength V/m– H magnetic field strength A/m

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Notations: be CONTEXT AWARE• Stochastics & Statistics

– P, Pr, F, cdf, cumulative distribution function– p, f, pdf, density distributiion function– E expectation– S power spectral density– R correlation (function)– ρ correlation coefficient– μ mean– σ2 , s2 , S2 variance– Δ difference– Signal parameters as random variables (x,y r,ϕ

etc)04/02/2019

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• Subscripts– 0,o origen, reference– b, bs base station– m, ms mobile station (terminal)– tx transmitter– rx receiver– c carrier, center frequency, coherent part– coh coherent– d, D, ν Doppler– r random, diffuse part– x cross (across)– m mean (or median)04/02/2019

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Notations: be CONTEXT AWARE• Abbreviations

– LOS line-of-sight– NLOS non-line-of-sight– XPD cross polarisation discrimination– PDP power delay profile– FCF frequency coherence function– DS delay spread (also στ etc)– BW bandwidth (signal) or beam width (antenna) !!– WSS wide sense stationary– US uncorrelated scattering– NB vs WB narrowband vs wideband

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Propagation channel elements

II. Short term fading

(vector interference)I. Path loss

(global decay d-n)

I: Shadow fading

(blocking, local mean)

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The two-source model (unidirectional)

2sin2

22

00

000 2

k

exrekLkxja

jeeaejaeaexE

xjkLj

kLkxjkLkxjkLjkLkxjkxj

x

L

’Frozen time’ – only look at space dependance

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What does it look like?

-2 -1.5 -1 -0.5 0 0.5 1 1.5 20

0.5

1

1.5

2

x in

|E|

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2-100

-50

0

50

100

x in

Pha

se in

deg

rees

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The 2 source model: random directions

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The two source model: 2 directionsAPMS


Many incoming waves

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Rayleigh fading

Central limit theorem:•x, y are sums of a large number of random variables.

•x,y can be assumed to be

•Gaussian distributed,

•independent,

•zero mean

•of the same variance

•Joint probability density function (pdf)

2

22

2

2

2

2

22

2

22

22 2

12

12

1,

yxyx

eeeypxpyxp

x y

8

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Rayleigh fading: from cartesian to polar

rrJ

rryxry

rx

ryxJ

ryrx

xy

ryx

erjyxE j

22

222

sincoscossinsincos

,,

sincos

arctan

2

2

2

22

22

2

22

2

2,,

2

1,

r

yx

eryxpJrp

eyxp

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Rayleigh fading: from joint to the marginals

21,

,

2,,

0

22

2

0

22

2

2

2

2

2

drrpp

erdrprp

eryxpJrp

r

r

Uniform distribution of the angle

Rayleigh distribution of the envelope

Phase and envelopeare related as?

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Pdf of the Rayleigh distribution

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Amplitude

PD

F

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Cdf of the Rayleigh distribution

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

AmplitudeC

DF

-40 -30 -20 -10 0 10 2010

-4

10-3

10-2

10-1

100

Amplitude in dB

CD

F

•Log-log representation ->RULE OF THUMB! … what!!??

Is the curve fully correct? .. Compare withlater slides

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Distribution of the power

2

20

2

2

2

2

22

2

0

2200

20

22

21Pr

1PrPrPr

P

P

PP r

r

edx

xPdxp

edrerPrPrPP

errp

If the envelope is Rayleigh distributed, then the power is exponentially distributed.

CDF

PDF

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Other distributions: Ricean

QIQI cyjcxjccjyxcjyxs

In phase and quadrature components are still Gaussian, but NOT zero mean!

Following the same process as before, we find that the envelope is Ricean distributed:

2

22

0

Rayleigh'offset '

22

2

22

crIerrp

cr

I0 is modified Bessel function of the zero-th order.

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Parameters of the Ricean distribution

• Ricean K factor (ratio of constant vs. scattered energy)

2

2

2c

PPK

r

c

10

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Doppler shiftDoppler shift -

R

d

far fieldR>>d

•Envelope r, envelope or power gradient•Phase , phase gradient d/dd

•Max. Doppler shift : fd,max = 1/ [c/m]•Actual Doppler shift : fd= fd,max cos() [c/m]•Temporal : fd [Hz] = fd,[c/m] v[m/s]

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Power azimuth distribution:

Antenna Gain:

How much power from each direction:

Multiple sources: Doppler spectrum

G

Gpx

y

What is the Doppler frequency:

α

p

ffff d cosmax,Power spectrum due to a small range of angles da:

22

max,

2

max,max,max,

max, 1sincos

ff

GpGpbfS

df

ffdffd

fdddf

dGpGpbdffS

d

ddd

d

What ‘trick ‘ to get the |.| form?

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Example of Doppler spectrum

21

p

41

GOmni directional source distribution

Omni directional antenna

22

max,22

max,

141

21

2ffff

fSdd

This is also known as the bathtub spectrum.

Put the limits

on the axes!

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Outage probability for Rayleigh fading

2

20

0

2

20

2

20

2

2

2

0

2

0

22

00

22

1Pr

r

rrr rr

r

eedrerdrrprr

errp

Approximation

2

20

0

2

202

0

2Pr

sorder termhigher 2

111Pr 2

20

rrr

rerrr

The expressions are more complicated for other distributions, but still calculable, at least numerically!!!!

For r0 small

PDF

CDF

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Phase gradient• Gradients: Derivatives with respect to time/ space

22222

1

1

1

1

arctan

yxyxxy

yyxxy

xydt

xyd

xydt

tdt

txtyt

icidiid

icidiid

iidicidi

iiii

Tftfafy

Tftfafx

vTftf

ayax

22cos2

,22sin2

cos2,22

sin,cos

,,

,,

,,

Also Gaussian !!!

What does this mean?

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Joint distributions

yyxxpJrrp ,,,,,,

|J|: laborious to calculate but doable

p(x,x’,y,y’): multi-dimensional Gaussian

From that we can calculate p(r,r’,φ,φ’)

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Reminder: multi-variable Gaussian distr.

iii NX ,

**,,cov jijiXXji XXEXX

ji

XXX

X

μXμXμX

μX

12/12/

2121

21exp

21,

det,

,

21

22212

12111

Tn

XXXXXX

XXXXXX

XXXXXX

Tn

Tn

Np

XXX

nnnn

n

n

Correlated Gaussian random variables

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Multivariable Gaussian distribution• The variables are x,y,x’, y’.•

d

d

ddi

EkvxyEyyExxE

yE

fEEaEkvxE

yExE

cos'0''

cos

2

22

2222222

222

22

2

2

2

2

det

,

000000

00

d

dd

dd

d

d

If the spectrum is symmetric

Units??

What then??

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From cartesian to polar• For a symmetric spectrum

,,,,

21exp

2,,,,,,

cossinsinsincoscos

21exp

21,,,

2

222

2

2

222

2

2

2

22

2

22

222

rprpprrp

rrrryxyxpJrrp

rJrryryrrxrx

yxyxyxyxp

dd

dd

Note what?, wrt the variables

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Random FM = φ’ (integrate over r, r’, φ)• Stochastic Random FM: Student’s t distribution• PDF For general Doppler spectrum

• PDF For classical bathtub Doppler spectrum

• CDF For general Doppler spectrum

• The Doppler spread : defined as standard deviation

2/32

212

1

fd

d

fd Sf

Sp

2/32

maxmax 221

221

dd ffp

221

21Pr

dfd

d

fxS

fxx

Doppler spread

Doppler mean

What has been done here?

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Power gradient• For common receivers, we care more about the power

gradient than the envelope gradient.• Power gradient:• Log-Student’s t distributed, i.e.

P’dB=10log(P’) is Student’s t distributed with PDF

By appropriate transformation :P’ is double-sided exponential (Laplace) distributed as

rrPrP 22

ddsss

dBmPs

sPp

l

dBl

ldB

6859.810ln/102

/2

2/32'2

2'

WradmesP

Pp d

d

sPP

/2

1''

•Symmetric

•Bounded at P’=0

?

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Rayleigh fading distribution functions

x

yQuadrature components

Phase Envelope

UncorrelatedGaussian (Normal)

Uniform Rayleigh

=arctan(Y/X) r=sqrt(X +Y )2 2

r

0 d

Jacobi transform

J=d(x,y)/d(r, )

p(r, )=p(x,y)|J|

Random-FMStudent's t

d /dd

d

'

Power gradientLog-Student's t

d|r| /dd2

d

20log(r)'

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Reminder: definition of correlation

2222

**

,vEvEuEuE

vEuEvuEvu

For Gaussian random variables:

envelopecomplexpower 2

Often needed

Easier to treat analyticallyWhy??

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Spatial correlation• Correlation theorem:

Power density spectrum Auto-correlation

• How to find the Doppler spectrum from meas:

• For uniform angle of arrival• When does the correlation =0?

2π∆d/λ=2.512 or equivalently ∆d=0.4λ

fSFddrdrEdR

smRHzscmcS

enveloper1

,/,/

dJdRr 20

2fHfS 2

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Warning• If you get this wrong, you fail:

– Rayleigh: because of many paths– Auto correlation= Bessel if/because of

uniform angle of arrival (i.e. omni directional)

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Fast (short) & slow (shadow) fading• 40 uncorrelated samples of a Rayleigh

process are enough to estimate the mean within 1dB.

• Assuming decorrelation every 0.5λ, this would mean sampling over a window of 20λ (notice 1D vs 2D sampling)

• Correlation length of the log-normal (shadow) fading: 5-10m

• Trade-off: window length vs sufficient samples

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