Introduction Chapter 4dwlin/courses/21digcom/hnotes... · 2021. 6. 21. · 4.1 Decision-directed...
Transcript of Introduction Chapter 4dwlin/courses/21digcom/hnotes... · 2021. 6. 21. · 4.1 Decision-directed...
Lin
:Digita
lCommunicatio
n169
✬✫
✩✪
Chapter4
Synchronizatio
n
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:Digita
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n170
✬✫
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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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✬✫
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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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✬✫
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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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✬✫
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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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✬✫
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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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✬✫
✩✪
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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✬✫
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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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✬✫
✩✪
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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✬✫
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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
Fall2021
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CommLabEE
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:Digita
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n196
✬✫
✩✪
•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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CommLabEE
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:Digita
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n197
✬✫
✩✪
–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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CommLabEE
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Lin
:Digita
lCommunicatio
n198
✬✫
✩✪
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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:Digita
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n199
✬✫
✩✪
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
Fall2021
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CommLabEE
NYCU
Lin
:Digita
lCommunicatio
n200
✬✫
✩✪
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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CommLabEE
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Lin
:Digita
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n201
✬✫
✩✪
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)?
Fall2021
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CommLabEE
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Lin
:Digita
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n202
✬✫
✩✪
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
Fall2021
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CommLabEE
NYCU
Lin
:Digita
lCommunicatio
n203
✬✫
✩✪
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
Fall2021
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CommLabEE
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Lin
:Digita
lCommunicatio
n204
✬✫
✩✪
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
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
Fall2021
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CommLabEE
NYCU
Lin
:Digita
lCommunicatio
n205
✬✫
✩✪
=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)
}
Fall2021
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CommLabEE
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Lin
:Digita
lCommunicatio
n206
✬✫
✩✪
•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
Fall2021
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CommLabEE
NYCU
Lin
:Digita
lCommunicatio
n207
✬✫
✩✪
,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
Fall2021
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1896
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CommLabEE
NYCU
Lin
:Digita
lCommunicatio
n208
✬✫
✩✪
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
Fall2021
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Lin
:Digita
lCommunicatio
n209
✬✫
✩✪
Contents
•4.1
Decision
-directed
carrierrecovery
•4.2
Non
-decision
-directed
carrierrecovery
•4.3
Tim
ingrecovery
•4.4
Fram
esyn
chron
ization
Fall2021
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CommLabEE
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Lin
:Digita
lCommunicatio
n210
✬✫
✩✪
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
Fall2021
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CommLabEE
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Lin
:Digita
lCommunicatio
n211
✬✫
✩✪
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:
Fall2021
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CommLabEE
NYCU
Lin
:Digita
lCommunicatio
n212
✬✫
✩✪
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
Fall2021
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CommLabEE
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Lin
:Digita
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n213
✬✫
✩✪
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
Fall2021
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CommLabEE
NYCU
Lin
:Digita
lCommunicatio
n214
✬✫
✩✪
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:
Fall2021
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1896
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CommLabEE
NYCU
Lin
:Digita
lCommunicatio
n215
✬✫
✩✪
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
?
Fall2021
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CommLabEE
NYCU
Lin
:Digita
lCommunicatio
n216
✬✫
✩✪
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
Fall2021
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CommLabEE
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Lin
:Digita
lCommunicatio
n217
✬✫
✩✪
•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).
Fall2021
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CommLabEE
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Lin
:Digita
lCommunicatio
n218
✬✫
✩✪
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
Fall2021
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CommLabEE
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Lin
:Digita
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n219
✬✫
✩✪
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
Fall2021
A
1896
ES
CommLabEE
NYCU
Lin
:Digita
lCommunicatio
n220
✬✫
✩✪
Contents
•4.1
Decision
-directed
carrierrecovery
•4.2
Non
-decision
-directed
carrierrecovery
•4.3
Tim
ingrecovery
•4.4
Fram
esyn
chron
ization
Fall2021
A
1896
ES
CommLabEE
NYCU
Lin
:Digita
lCommunicatio
n221
✬✫
✩✪
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