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Chapter4

Synchronizatio

n

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Introductio

n

•Kindsof

synchron

izationfor

single-carrier

transm

ission

–Carrier

synchron

ization(carrier

recovery)

–Sym

bol

synchron

ization(tim

ingrecovery)

–Fram

eor

word

synchron

ization

•Thischapter

will

talkmore

abou

tcarrier

andsym

bol

synchron

ization

•Typ

esof

carrierrecovery

(CR)

–Pilot-aid

edandnon

-pilot-aid

ed

–Decision

-directed

(DD)andnon

-decision

-directed

(NDD)

•Typ

esof

timingrecovery

(TR)

–Extern

al-signal

timingandself

timing

–Decision

-directed

andnon

-decision

-directed

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•Join

tCRand/or

TRand/or

signal

detection

has

been

studied

.This

chapter

will

consid

erindivid

ual

CRandTRon

ly

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SeveralW

ordson

CarrierRecovery

•Practical

examples

of(an

aloganddigital)

single-carrier

systemsthat

transm

itpilots

(carriers)

–Color

subcarrier

inNTSCanalog

broadcast

TV

–Con

stellationbias

indigital

VSBfor

ATSCbroad

castHDTV

•Practical

examples

ofoth

ersystem

stran

smittin

gpilots

(carriers)

–Multicarrier

systems:

Pream

ble

“symbol”

andpilot

subcarriers

in

OFDM

transm

ission

–Spread

-spectru

msystem

s:Q-ch

annel

signal

structu

rein

3GPP

WCDMAair

interface

•Thischapter

will

not

address

pilot-aid

edCRspecifi

cally,but:

–Som

econ

cepts

underlyin

gnon

-pilot-aid

edCRmeth

odsalso

underlie

pilot-aid

edmeth

ods

–Hence,

understan

dingof

princip

lesbehindnon

pilot-aid

edCR

meth

odsshou

ldhelp

yourunderstan

dingof

pilot-aid

edmeth

ods

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Contents

•4.1

Decision

-directed

carrierrecovery

•4.2

Non

-decision

-directed

carrierrecovery

•4.3

Tim

ingrecovery

•4.4

Fram

esyn

chron

ization

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Introductio

n

•Typical

receiverstru

cture

with

decision

-directed

CR

Calculator

Dem

odulatorCarrier P

haseand F

requencyA

djustment

Carrier

Decision

Circuit

Phase E

rror

•Ithas

better

perform

ance

than

non

-decision

-directed

techniques

in

lowerror

rate(high

SNR)environ

ments.

(Thereason

forthiswill

be

relativelyclear

afterweintro

duce

non

-decision

-directed

meth

ods.

Butwou

ldyou

venture

agu

esson

why,at

thispoin

tin

time?)

•While

thisstru

cture

isintuitively

reasonable

anddiscu

ssioncan

proceed

with

it,let

usfirst

consid

eran

optim

ization-based

approach

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Principle

•Con

sider

thetran

smission

systemmodel:

fc’

Modulator

θ s(t)+n(t)

r(t)D

emod−

ulator

θ

Channel

ResponseH

(f)

f

|H(f)|

<H

(f)

f

σslope =

−2πτ

s (t)l

l r (t)

1π

cos/sin (2 fc’t + ’)

πcos/sin (2 fc t +

)

Signal B

and

•Usin

gcom

plex

(equivalen

tlow

pass)

representation

forsl (t)

and

rl (t),

wehave

s(t)=

ℜ[s

l (t)ej(2

πf′ct+

θ′)],

r(t)=

ℜ[r

l (t)ej(2

πfct+

θ)]

•If∠H(f)=

−2πfτ,then

r(t)=s(t−

τ)+n(t)

=ℜ[s

l (t−τ)e

j[2πf′c(t−τ)+

θ′]]+

n(t)

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•Now

with

∠H(f)=

−2πfτ+

2πf′c τ

+σin

(positive-freq

uency)

signal

band,

r(t)=

ℜ[s

l (t−τ)e

j(2

πf′ct+

θ′+

σ)]+

n(t)

•Hence,

ignorin

gtheeff

ectofn(t)

tentatively,

rl (t)

=sl (t−

τ)e

j[2π(f

′c−fc)t+

(θ′+

σ−θ)],sl (t−

τ)e

jφ(t),

where

φ(t)

iswhat

theCRcircu

itneed

sto

estimate

•In

thefollow

ingderivation

:

–Assu

methat

variationin

(f′c −

fc )t

isnegligib

leover

the

observation

interval

(eitherf′c ≈

fcor

theob

servationinterval

is

short)

andhenceφ(t)

may

bemodeled

asacon

stant

–For

conven

ience

with

outloss

ofgen

erality,let

τ=

0

•Q:Ifthetran

smitted

data

arekn

own,what

isaprop

erestim

ateofφ

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givenr(t)?

•A:Areason

able

estimation

criterionisto

maxim

izetheaposterio

ri

(MAP)prob

ability

f(φ

|rl (t),s

l (t))

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Sim

plifi

catio

n

•ByBayes

rule,

f(φ

|rl (t),s

l (t))=f(r

l (t)|φ,s

l (t))f(φ

|sl (t))

f(r

l (t)|sl (t))

•With

akn

ownsl (t)

andagiven

receivedsign

alwaveform

rl (t),

the

denom

inator

f(r

l (t)|sl (t))

iscom

mon

toall

values

ofφandmay

be

disregard

ed

•Usually,

itisreason

able

toassu

mestatistical

independence

betw

een

φandsl (t),

andhencef(φ

|sl (t))

=f(φ)

•Twocom

mon

assumption

son

f(φ)

–Least

favorable

situation

:f(φ)isuniform

over[0,2π

)

–φisdeterm

inistic

butunkn

own—

equivalen

tto

assuming

uniform

f(φ)

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•Thustheop

timal

estimate

isgiven

bytheML(m

aximum-likelih

ood)

criterion:

maxφ

f(r

l (t)|φ,s

l (t))

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Furth

erSim

plifi

catio

n

•Recall

vector-space

representation

ofsign

al+

relevantnoise

compon

ents.

Thusweseek

amath

ematical

expressionfor

thePDF

f(r|φ

,s),where

weassu

methat

theob

servedsign

alhasN

complex

dim

ension

s,e.g.,

Nsym

bol

perio

dsof

PAM,PSK,or

QAM

•Since

theab

oveform

ulation

isin

termsof

equivalen

tlow

pass

quantities,

(generalized

)com

plex

expon

ential

Fou

rierseries

expansion

ismore

natu

ralthan

(generalized

)trian

gular

Fou

rierseries

expansion

.Thusrandsab

oveare

complex

vectorsof

series

coeffi

cients

•Ifnoise

iscolored

(non

-white),

then

expressionof

thePDFreq

uires

intro

duction

ofaddition

alnotation

s,which

weavoid

forsim

plicity

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•For

zero-mean

white

Gaussian

noise

n(t),

wehave

f(r|φ

,s)=

1

(2πN

0 )N

exp

{

−1

2N0 ‖r−

sejφ‖

2

}

=1

(2πN

0 )N

exp

{

−1

2N0[‖r‖

2−2ℜ

(rHs e

jφ)+‖s‖

2] }

where

superscrip

tH

denotes

Herm

itiantran

spose

(complex

conjugate

transpose)

•Observation

s

–Themultip

licativefactor

1(2

πN

0)N

iscom

mon

toallφandhence

has

noeff

ecton

MLestim

ation

–Thequantities‖r‖

2and‖s‖

2in

theexp

onentare

alsocom

mon

toallφandhence

donot

affect

MLestim

ation,eith

er

–Exp

onential

function

ismon

otoneincreasin

g.Hence

max{ex

p(a)}

isequivalen

tto

max{a}

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•Thustheop

timal

decision

rule

isequivalen

tto

maxφ

ℜ(r

Hs e

jφ),

which

formulates

theop

timization

problem

indiscrete-tim

e,

baseb

andquantities

•Altern

atively,wehave:

ℜ(r

Hs e

jφ)=

ℜ(∫

T0

r∗l (t)s

l (t)ejφdt)

,T0bein

gob

servationinterval,

=ℜ(∫

T0 [r

l (t)ej(2

πfct+

θ)

︸︷︷

︸

,aI(t)+

jaQ(t)

]∗[s

l (t)ej(2

πfct+

θ+φ)

︸︷︷

︸

,bI(t)+

jbQ(t)

]dt)

=

∫

T0 [a

I (t)bI (t)

+aQ(t)b

Q(t)]d

t

=

∫

T0 {[r

lI (t)cos(2π

fc t+θ)−

rlQ(t)

sin(2π

fc t+θ)]

·[slI (t)

cos(2πfc t+θ+φ)−

slQ(t)

sin(2π

fc t+θ+φ)]

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+[r

lI (t)sin

(2πfc t+θ)

+rlQ(t)

cos(2πfc t+θ)]

·[slI (t)

sin(2π

fc t+θ+φ)+slQ(t)

cos(2πfc t+θ+φ)]}d

t

=12

∫

T0 {[r

lI (t)slI (t)

cosφ−rlI (t)s

lQ(t)

sinφ

+rlQ(t)s

lI (t)sin

φ+rlQ(t)s

lQ(t)

cosφ]+

[same]}d

t,

drop

pingdou

ble-freq

uency

terms,

=2

∫

T0

aI (t)b

I (t)dt

=2

∫

T0 ℜ

[rl (t)e

j(2

πfct+

θ)]·ℜ

[sl (t)e

j(2

πfct+

θ+φ)]dt

=2

∫

T0

r(t)·[sign

almodulated

with

cos/sin(2π

fc t+θ+φ)]dt

,2

∫

T0

r(t)sφ(t)d

t,

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which

givesaform

ulation

incon

tinuou

s-time,

passb

andquantities

•Wenow

givesom

eexam

ples

ofadaptive

CRtech

niques

based

on

thetwoab

oveform

ulation

s

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Disc

rete-T

imeAdaptiv

eIm

plementatio

n

•Prin

ciples

–Doon

e-shot

phase

estimate

based

onob

servationover

thelast

symbol

interval

–Update

theaverage

oftheon

e-shot

estimates

forafinal

phase

estimate

–Use

thefinal

phase

estimate

fordem

odulation

inthecurren

t

symbol

interval

•Below

only

consid

eron

e-andtwo-d

imension

alsign

aling:

PAM,

PSK,QAM

•Geom

etricinterpretation

ofmax

φ ℜ(r

∗sejφ)

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Re

Im

srφopt^

–se

jφiscou

nterclo

ckwise

rotationofson

thecom

plex

plan

eby

anangle

φ

–Since

ℜ(a

∗b)=aI b

I+aQbQ,ℜ

(r∗se

jφ)isinner

product

ofthe

two-d

imension

alvectors

representin

grandse

jφ

–With

length

sof

twovectors

keptunchanged

,inner

product

is

maxim

um

when

they

arecolin

ear

–Thusop

timumφisgiven

bytheangle

betw

eenvectors

representin

grands

•Therefore,

optim

ization-based

approach

leadsto

ourearlier

intuitively

reasonable

CRstru

cture

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Calculator

Dem

odulatorCarrier P

haseand F

requencyA

djustment

Carrier

Decision

Circuit

Phase E

rror

•Exam

ple

—QAM

with

purely

baseb

andDD

CR:

Filter

Decision

Circuit

Phase E

rror

X+

j

PS

F

PS

F

r(t)

XX

cos

-sinr rIQ

s

φexp(-j )

^

Calculator

Loop

•Exam

ple

—QAM

with

baseb

and-D

Dpassb

and-correction

CR:

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✬✫

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Loop

j

PS

F

PS

F

r(t)

XX

cos

-sin

rI

+V

COrQ

Decision

Circuit

Phase Error

s

Calculator

Filter

•Perform

ance

analysis

uses

PLL(phase-lo

ckedloop

)con

cepts,

which

wedonot

have

timeto

gointo

inthiscou

rse.Ihop

ethat

youhave

learned

somebasic

PLLcon

cepts

intheprereq

uisite

course

Prin

ciples

ofCommunica

tionSystem

s

•Note

that

CRfor

PAM

alsoneed

sQ-bran

ch(to

obtain

rQ)

•2π/M

phase

ambigu

itywhere

Mdependson

rotational

symmetry

of

signal

constellation

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Contin

uous-T

imeAdaptiv

eIm

plementatio

n

•Recall

optim

izationprob

lem

maxφ

ΛL(φ),

maxφ

∫

T0

r(t)sφ(t)d

t

•Differen

tiating∫

T0

wrt

(with

respect

to)φandsettin

gtheresu

ltto

zeroyield

sddφΛL(φ)=

−∫

T0

r(t)[slI (t)

sin(2π

fc t+θ+φ)

+slQ(t)

cos(2πfc t+θ+φ)]dt=

0

•Exam

ple

—PAM:

sφ(t)

=slI (t)

cos(2πfc t+θ+φ)where

slI (t)

=k∑

n=−∞

Ing(t−

nT)

forkT≤t≤

(k+1)T

.Assu

meg(t)

=Π(

t−T/2

T

)

forsim

plicity.

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Then

Ideally, = 0

Integrationover

(t-T,t)

Decision

Circuit

Ik-1

^

PSFg(t)

XL

oopFilter

VC

O

X

kT

X

-sin(2 fc t + )

r(t)

DelayT

ππθ + φ

θ + φcos(2 fc t +

)

–180

◦phase

ambigu

ity

•Exam

ple

—M

-PSK:

sφ(t)

=slI (t)

cos(2πfc t+θ+φ)−

slQ(t)

sin(2π

fc t+θ+φ)

where

slI (t)

=A

k∑

n=−∞

g(t−

nT)cos

θn,slQ(t)

=A

k∑

n=−∞

g(t−

nT)sin

θn

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✬✫

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with

θn∈{2πm

M+ψ;m

=0,1,···

,M−

1}for

someψ

for

kT≤t≤

(k+1)T

.Assu

meg(t)

=Π(

t−T/2

T

)

forsim

plicity.

Then

= 0

Decision

Circuit

^θk-1

XX +

πθ + φ

cos(2 fc t + )

X

-sin(2 fc t + )

r(t)

πθ + φ

XIntegration

over(t-T

,t)

Integrationover

(t-T,t)

kT kT

DelayT

DelayT

sin

cos

PSF

PSFg(t)

g(t)

_

FilterL

oopV

CO

ideally

–2π/M

phase

ambigu

ity

•Exercise:

Try

toderive

contin

uou

s-timeDD

loop

sfor

QAM

and

OQPSK.What

kindof

phase

ambigu

itydoes

eachhave?

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Contents

•4.1

Decision

-directed

carrierrecovery

•4.2

Non

-decision

-directed

carrierrecovery

•4.3

Tim

ingrecovery

•4.4

Fram

esyn

chron

ization

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✬✫

✩✪

Nth

-PowerCircuits

•Basic

structu

re

BP

F at

( ) N

N fc

Narrow

bandP

hase/N

X

Carrier

Recovered

r(t)

•Exam

ple

—M

-PAM:

s(t)=A∑

n

Ing(t−

nT)cos(2π

fc t+

φ),In∈{±

1,±3,···

,±(M

−1)}

–Wehave

E[s

2(t)]=A

2(M2−

1)

3

∑

n

g2(t−

nT)cos

2(2πfc t+φ)

=A

2(M2−

1)

6

∑

n

g2(t−

nT)[1

+cos(4π

fc t+2φ

)]

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✬✫

✩✪

–Hence

obtain

Kcos(4π

fc t+2φ

),for

someK,at

narrow

band

BPFou

tput

–180

◦phase

ambigu

ity

•Exam

ple

—M

-PSK:

s(t)=A∑

n

g(t−

nT)cos(2π

fc t+φ+θn)

where

θn∈{2πm

M+ψ;m

=0,1,···

,M−1}

–Weget

E[s

M(t)]

=A

M∑

n

gM(t−

nT)E

[cosM(2π

fc t+φ+θn)]

where

E[cos

M(···)]

=E

{[ej(2

πfct+

φ+θn)+e−j(2

πfct+

φ+θn)

2

]M}

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✬✫

✩✪

=1

2M

−1cos(2π

Mfc t+Mφ+Mψ)+low

erfreq

uency

terms

–Hence

obtain

Kcos(2π

Mfc t+Mφ+Mψ),for

someK,at

narrow

bandBPFou

tput

–Geom

etricalinterpretation

:

φM

-th Power of

Original C

onstellationM

-th Power of

(collapsed to one point)R

eceived Constellation

(rotated by M )φ

Original C

onstellationR

eceived Constellation

(rotated by )

–Observation

:Raise

toNth

pow

erifcon

stellationshow

s“an

gular

perio

dicity”

(rotational

symmetry)

ofperio

d2π/N

–2π/M

phase

ambigu

ity

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•Exam

ple

—Square

M-Q

AM:

s(t)=A∑

n

[Inc g(t−

nT)cos(2π

fc t+

φ)−Ins g(t−

nT)sin

(2πfc t+

φ)]

where

Inc ,I

ns ∈

{±1,±

3,···,±

( √M

−1)}

–Con

stellationshow

sangu

larperio

dicity

ofperio

dπ/2,

indicatin

g

4th-pow

ercircu

itshou

ldwork

-A

dm

in

f1(t)

f2(t)

A3A

-A-3A

A

3A

-3A

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–Wehave

E[s

4(t)]=

A4

[

∑

n

E(I

4nc )g

4(t−

nT)cos4(2

πfc t

+φ)

+6∑

n

E(I

2nc )E

(I2ns )g

4(t−

nT)cos2(2

πfc t

+φ)sin

2(2πfc t

+φ)

+∑

n

E(I

4ns )g

4(t−

nT)sin

4(2πfc t

+φ)

]

–Hence

obtain

Kcos(8π

fc t+4φ

),for

someK,at

narrow

band

BPFou

tput

–90

◦phase

ambigu

ity

•May

use

PLLto

implem

entnarrow

bandBPF

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✬✫

✩✪

at M fc

BP

FX

LoopF

ilter

VC

Oat M

fc

–BPFatMfcmay

have

wider

bandthan

narrow

bandBPFin

circuitnot

usin

gPLL

–Better

perform

ance

whenfcisnot

accurately

know

n

•Other

non

linearity

than

Nth

pow

ermay

becon

sidered

,e.g.,

absolu

te

value,

aslon

gas

itspow

erseries

expansion

contain

sNth-pow

erterm

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Nth

-PowerComputatio

nin

Baseband

•Com

putation

ofNth

pow

erin

baseb

andandphase

control

in

passb

and:

|r| e^

j(φ−φ)

^

j

PS

F

PS

F

r(t)

XX

cos

-sin

rI

+V

COrQ

Decision

Circuit

s

FilterL

oop

Phase=φ

( ) N+

PhaseE

stimator

φ

–Key

work

incom

putin

g()N

isvector

rotationin

complex

plan

e.

Treatin

gthevector

as2-D

realvector,

wehave

[

ℜ{ejN

(φ−φ̂)}

ℑ{ejN

(φ−φ̂)}

]

=

[

cos(φ

−φ̂)

−sin

(φ−

φ̂)

sin(φ

−φ̂)

cos(φ

−φ̂)

]N

−1[

cos(φ

−φ̂)

sin(φ

−φ̂)

]

–Ifphase

errorφ−φ̂issm

all,then

phase

estimator

may

simply

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✬✫

✩✪

takeim

aginary

part

ofejN

(φ−φ̂),givin

g

ℑ[e

jN

(φ−φ̂)]=

[sin(φ

−φ̂)

cos(φ−φ̂)]

·

cos(φ−φ̂)−

sin(φ

−φ̂)

sin(φ

−φ̂)

cos(φ−φ̂)

N−2cos(φ

−φ̂)

sin(φ

−φ̂)

•Exam

ple

—M

-PSK:LetN

=M

inab

ovefigu

re

•Exam

ple

—M

-PAM

–ℑ[e

j2(φ

−φ̂)]=

2cos(φ

−φ̂)sin

(φ−φ̂)

–Com

pare

Costas

loop

,which

youmay

have

learned

inPrin

ciples

ofCommunica

tionSystem

s:

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φ−φ

X

VC

OLP

F

X

r(t)

Phase=φ

2 cos

−2 sin

φ̂X

2

LPF

LPF

a (t)/2 x sin2( )+n’(t)

^φ−φ

^a(t)cos( )+

n1(t)φ−φ^

a(t)sin( )+n2(t)

•Can

youdesign

agen

eralpurely-b

asebandarch

itecture

andgive

somespecialization

sfor

common

modulation

meth

ods?

•Con

sider

theeff

ectof

thenoise

terminr(t)

when

takingtheNth

pow

er.From

this,

canyou

seewhyDD

CRshou

ldperform

better

than

NDD

CRin

high

SNR(or

inthesitu

ationwith

lowdecision

errorrates)?

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How

AboutOptim

alEstim

atio

n?

•Recall

MAPcriterion

forDD

CR:max

φf(φ

|rl (t),s

l (t))

•Corresp

ondingcriterion

forNDD

(non

-decision

-directed

)CR:

maxφ

f(φ

|rl (t)),

i.e.,maxim

izingtheaverage

(ormargin

al)aposterio

riPDFover

the

signal

set:f(φ

|rl (t))

=∑

i

f(φ

|rl (t),s

li (t))P(s

li (t)|rl (t)),

where

sli (t)

denotes

ithelem

entin

theset

ofsign

alwaveform

s

(which

isfinite

orcou

ntab

lyinfinite)

•Sim

ilarassu

mption

sandderivation

asfor

DD

CRagain

leadto

ML

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estimation

:

maxφ

f(r

l (t)|φ)=

maxφ

∑

i

f(r

l (t)|φ,s

li (t))P(s

li (t)|φ)

•Itisintuitively

reasonable

toassu

meindependence

betw

een

baseb

andsign

alwaveform

sandtheunkn

owncarrier

phase

φ,i.e.,

P(s

li (t)|φ)=P(s

li (t))∀i

•Unfortu

nately,

there

isusually

nosim

ple

closed-form

solution

tothe

above

MLestim

ationprob

lem,becau

seP(s

li (t))isnon

-Gaussian

•Tosim

plify,

consid

erapproxim

atingsl (t)

asGaussian

•Tocon

tinue,

itiscon

venien

tto

use

vector-space

formulation

.

Tentatively,

assumeob

servationover

onesym

bol

interval

only.

Since

most

common

modulation

meth

ods(PAM,square

QAM,PSK)have

one“com

plex

dim

ension

”in

equivalen

tlow

pass

representation

,we

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consid

ersuch

signals

inAWGN:

f(r|φ

,s)=

1

2πN

0e−

|r−

sejφ|2

2N

0

where

randsare

thegen

eralizedFou

rierseries

coeffi

cients

forrl (t)

andsl (t),

respectively

•Exam

ple

—M

-PAM:sisreal.

Let

ithave

PDF

f(s)

=1

√2πVe−

s2

2V.

Then

f(r|φ

)=

∫

f(r|φ

,s)f(s)d

s

=1

√

8π3N

20V

∫

exp

(−|r−

sejφ| 2

2N0

−s2

2V

)

ds

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✬✫

✩✪

=1

√

8π3N

20V

∫

exp

[−12

(1N0+

1V

)

s2+

ℜ(re

−jφ)

N0

s−|r| 22N

0

]

ds

=1

√

8π3N

20V

∫

exp

{

−N

0+V

2N0 V

[

s−Vℜ(re

−jφ)

N0+V

]2}

ds

·exp

{

N0+V

2N0 V

[Vℜ(re

−jφ)

N0+V

]2−

1

2N0 |r| 2

}

•Since√

N0+V

2πN

0 V

∫

exp

{

−N

0+V

2N0 V

[

s−Vℜ(re

−jφ)

N0+V

]2}

ds=

1,

wegetf

(r|φ)=

1

2π√

N0 (N

0+V)e−

|r|2

2N

0exp

{V[ℜ(re

−jφ)] 2

2N0 (N

0+V)

}

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✬✫

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•Thus

max

φf(r

|φ)∼

max

φ[ℜ

(re−jφ)] 2

∼max

φ

[∫

T

0

r(t)g(t)

cos(2

πfc t

+φ)dt

]

2

where

thesecon

dsim

ilaritymay

beundersto

odgeom

etrically:

φR

e[r exp(-j )]φ

Re[r exp(-j )]

φ

Re,

cosR

e,cos

-sin-sin

Im,

Im,

φ

φ

rr

r exp(-j )

•Exten

dingob

servationinterval

toN

symbol

perio

dslead

sto

maxφ

f(r|φ

)∼

maxφ

N−1

∑k=0

[∫

(k+1)T

kT

r(t)g(t−

kT)cos(2π

fc t+φ)dt

]2

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✬✫

✩✪

,ΛL(φ)

•Differen

tiatingΛL(φ)wrtφandsettin

gtheresu

ltto

zeroyield

s

−N

−1

∑k=0

[∫

(k+1)T

kT

r(t)g(t−

kT)cos(2π

fc t+φ)dt

·∫

(k+1)T

kT

r(t)g(t−

kT)sin

(2πfc t+φ)dt

]

=0

•Thusabaseb

and/p

assbandadaptive

implem

entation

isas

follows:

kT

VC

OS

um over

[k,k-N+

1]X

X Xr(t)

cos

-sin

Decision

Circuit

φ̂

PSFg(T

-t)

PSFg(T

-t)

kT

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✬✫

✩✪

Com

pare

Costas

loop

:

φ−φ

X

VC

OLP

F

X

r(t)

Phase=φ

2 cos

−2 sin

φ̂X

2

LPF

LPF

a (t)/2 x sin2( )+n’(t)

^φ−φ

^a(t)cos( )+

n1(t)φ−φ^

a(t)sin( )+n2(t)

•Can

youderive

purely

baseb

andadaptive

implem

entation

based

on

max

φ[ℜ(re

−jφ)] 2?

•ForM

-PSK

andM

-QAM,unfortu

nately,

assumingsto

becom

plex

Gaussian

(with

independentreal

andim

aginary

parts)

will

not

work,

becau

sesuch

PDFiscircu

larlysym

metric

oncom

plex

plan

e,andis

hence

indiscrim

inate

inphase

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✬✫

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Contents

•4.1

Decision

-directed

carrierrecovery

•4.2

Non

-decision

-directed

carrierrecovery

•4.3

Tim

ingrecovery

•4.4

Fram

esyn

chron

ization

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✩✪

Archite

ctu

ralAlte

rnativ

es

•Extern

al-signal

timingandself

timing.

Exam

ples

ofform

er:

–Tran

smitter

andreceiver

both

derive

timinginform

ationfrom

a

master

clock,

such

asnetw

orkclo

ckor

GPS—

butstill

need

sto

deal

with

transm

issiondelay

–Tran

smitter

sendsasyn

csign

alvia

asep

aratechannel

–Tran

smitter

superim

poses

apilot

toneon

data

stream

–Derive

symbol

synchron

izationfrom

framemarkers

•Decision

-directed

andnon

-decision

-directed

timingrecovery

•Con

tinuou

s-timeor

discrete-tim

epro

cessing.

Exam

ple

sampling

ratesof

latter:

–Low

integral

multip

leof

baudrate,

e.g.,2

–Baudrate

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✬✫

✩✪

Notio

nofEyeDiagrams

•Atsom

epoin

talon

gthetran

smission

path

,overlay

together

all

symbol

perio

dsof

thesign

alwaveform

,where

thetran

smitted

signal

shou

ldcon

tainall

possib

lesym

bol

sequences.

Then

weob

tainan

eyediagram

forthat

poin

t

of an eyeresem

bling the shape

•How

toob

tainan

eyediagram

?

–Com

putation

al:softw

aresim

ulation

oftran

smission

system

–Exp

erimental:

probingof

actual

systemwith

anoscilloscop

e

•Typical

position

ingof

TRcircu

itfor

single-carrier

transm

ission:

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✬✫

✩✪

path feasible

RecoveryC

ircuit

Tim

ing

Processing

Decision

Circuit

Further

Dem

od−ulator

path only if DD

for DD

or ND

D

where

“furth

erpro

cessing”

isadded

proleptically

inanticip

ationof

thediscu

ssionin

ch.6andto

capture

thetyp

icalstru

cture

of

practicalreceivers.

Weneed

not

becon

cerned

with

itsdetails

for

now

•Ifthere

isno“fu

rther

processin

g”before

decision

,then

intuitively

oneshou

ldsam

ple

atmaxim

um

verticaleye

openingfor

maxim

um

noise

resistance,

where

verticaleye

openingmay

bedefined

several

ways,

dependingon

which

ismore

appropriate:

–Minim

um

opening(w

orst-casecon

dition

)

–Som

ekin

dof

averageop

ening

•Som

erelated

notion

s(assu

mingno“fu

rther

processin

g”before

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✬✫

✩✪

decision

)

–Sensitivity

totim

ingerror:

change

invertical

eyeop

eningdueto

subop

timal

timing(i.e.,

samplinginstan

t)

–Maxim

um

tolerance

totim

ingerror:

dependson

width

ofeye

–Maxim

um

ISI(in

tersymbol

interferen

ce)at

samplinginstan

t:

amou

ntof

verticaleye

closure

there

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✬✫

✩✪

Practic

alSelf

Tim

ing

•Typical

operatin

gstru

cture

ofTRcircu

it:

Instant Waveform

InstantS

ampling

Adjust

Check Sam

pling-

Characteristics

Against D

esired

where

desired

waveform

characteristics

atsam

plinginstan

tsmay

be

determ

ined

based

onheuristics

orbased

onmath

ematical

optim

ization.Exam

ples:

–Sym

metry

abou

tsam

plinginstan

t

–Particu

larasym

metric

shap

esab

outsam

plinginstan

t

•Afreq

uently

referenced

schem

e—

early-lategate

synchron

izer:

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✬✫

✩✪

early

..

Delay

VC

Clock

LPF

+_

late

Delay +∆

Delay -∆

–Based

onsym

metric

waveform

abou

tsam

plinginstan

t—

appropriate

choice

fortran

smission

overAWGN

channel

after

match

edfilterin

g

–How

manysam

ples

per

symbol

perio

dare

need

ed?

–Isitdecision

-directed

ornon

-decision

-directed

?

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✬✫

✩✪

Decisio

n-D

irected

Tim

ingRecovery

•Recall

transm

issionsystem

model:

fc’

Modulator

θ s(t)+n(t)

r(t)D

emod−

ulator

θ

Channel

ResponseH

(f)

f

|H(f)|

<H

(f)

f

σslope =

−2πτ

s (t)l

l r (t)

1π

cos/sin (2 fc’t + ’)

πcos/sin (2 fc t +

)

Signal B

and

•Relation

betw

eenequivalen

tlow

pass

representation

ofsystem

input

andou

tput:

rl (t)

=sl (t−

τ)e

jφ(t)

where

τ(or

more

exactly,som

ethingequivalen

t)iswhat

theTR

circuitneed

sto

estimate

•For

notation

alcon

venien

ce,let

φ(t)

=0in

thefollow

ingderivation

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✬✫

✩✪

•MAPestim

ateofτismax

τf(τ|r

l (t),sl (t))

whensl (t)

iskn

own

•Sim

ilarassu

mption

sandderivation

asfor

CRlead

sto

MLestim

ate

maxτ

f(r

l (t)|τ,sl (t))

andfinally

maxτ

ℜ(∫

T0

r∗l (t)s

l (t−τ)dt

)

,maxτ

ΛL(τ),

where

T0istheob

servationinterval

•Exam

ple

—M

-PAM:sl (t−

τ)isreal

andgiven

by

sl (t−

τ)=

∑

k

Ik g(t−

kT−τ).

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✬✫

✩✪

Then

ΛL(τ)=

∫

T0

rlI (t)

[∑

k

Ik g(t−

kT−τ)

]

dt

=∑

k

Ik

∫

T0

rlI (t)g

(t−kT−τ)dt,

∑

k

Ik y

(τ+[k

+1]T

)

where

y(t)

,rlI (t)∗

g(T

−t)

•Settin

gdΛL(τ)/d

τ=

0yield

s

∑

k

Ik ·

ddτy(τ

+[k

+1]T

)=

0

•Discrete-tim

eadaptive

implem

entation

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✬✫

✩✪

τ̂kT

+M

atchedF

ilterg(T

−t)

y(t)

XD

ifferenti−ation

Circuit

Decision

VC

CD

iscrete−

Loop Filter

Tim

e

k−1

^Ir (t)lI

•Heuristic

interpretation

–Theunity

differen

cein

timeindexes

forIk−1andy(τ

+kT)

accounts

forthelen

gth-T

delay

inmatch

edfilterin

g

–Multip

licationofy(·)

byIk−1squares

outthesign

ofIk−1

–Differen

tiationofy(·)

andsettin

gthetim

eaverage

ofits

product

with

Ik−1to

zerolan

dthesam

plinginstan

tat

maxim

um

eye

openingdefined

byy(·)

•Differen

tiationmay

beapproxim

atedby

differen

ceof

earlyandlate

samples

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n220

✬✫

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Contents

•4.1

Decision

-directed

carrierrecovery

•4.2

Non

-decision

-directed

carrierrecovery

•4.3

Tim

ingrecovery

•4.4

Fram

esyn

chron

ization

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TwoW

aysofFrameSynchronizatio

n

•Use

markers,

where

amarker

word

may

appear

inits

fullin

onedata

frameor

bespread

overmultip

lefram

es

A codew

ord may spread over m

ultiple frames

Codew

ords with good autocorrelation properties so

Data

their positions can be distinguished.

–May

becalled

“external-sign

alfram

ing”

•Use

regular

error-control

codes

(ECC)or

special

self-synchron

ization

codes

self-syncingregular ecc or

regular ecc orregular ecc or

codeword

self-syncingcodew

ordself-syncing

codeword

Fall2021

A

1896

ES

CommLabEE

NYCU

Lin

:Digita

lCommunicatio

n222

✬✫

✩✪Fall2021

A

1896

ES

CommLabEE

NYCU