Simultaneous parameters estimation of digital modulated signals
Full text
1
1
s
1
. ,
s
SIGNAL
PROCESSING
V:
Theo ies
and
Applica ions
L.
To"es,
E.
Masg au,
and
M.A.
Lagunas
(eds.)
© Else ie
Science
Publishe s
B.
V.,
1990
1835
SIMULTANEOUS PARAMETERS ESTIMATION OF DIGITAL MODULATED
SIGNALS
M.
Cab e a,
M.
A.
Lagunas
Dep .
o
Signal Theo y and Communica ions
ETSIT-UPC
P.O. Box 30.002-Ba celona, Spain
A maximum likelihood es ima e o equency ca ie , phase leakage and ampli ude
o
digi al PSK modula ed signals is p esen ed. The h ee op imum pa ame e s ob ained a e
analyzed and compa ed wi h subop imum ealiza ions in he synch oniza ion p ocess. In
o de o achie e an e icien sys em a inal s uc u e is closed whe e each me hod is
selec ed in o de o ge no only app op ia ed pe o mance as well as a ela i ely low
compu a ional load.
1.
INTRODUCTION
When
digi al
phase
and
equency
modula ed
signals
a e
ansmi ed
in
a
sa elli e
communica ions
en i onmen ,
he
p incipal
cha ac e is ics
o
he communica ions sys em ha e
o be eViewed. When he signal is ecei ed in Time
Di ision Mul iple Access (TDMA) mode, loop sys ems
as Phase locked loops and adap i e es ima ion
algo i hms a e no adequa e o be used because
o
he sho slo s
o
signal ecei ed pe iodically. The
con e gency
o
he
algo i hms
o
es ima e
pa ame e s
o
synch oniza ion could no be go
inside each slo
o
signal a i ing a he sys em.
He e, di e en kind
o
es ima ion me hods will be
analyzed, as in an op imum
sense
as in a
subop imum mode in he pa ame e s es ima ion
pa .
Fi s
o
all he en i onmen and kind
o
signal will
be
s udied. A p esen a ion
o
he
gene al sys em will be done wi h wo di e en
s ages. In he i s one, he signal is il e ed and
down con e ed o a low pass band signal.
In
he
second s age he synch oniza ion is implemen ed
by mean
o
es ima ing pa ame e s
o
he ca ie
signal as ampli ude, esidual equency and phase.
The comple e sys em can be summa ized by he
blocks diag am
o
igu e
1.
x(n)
y(nNs)
Figu e
1.
Basic scheme
o
he il e ing/es ima ion
p ocess.
The h ee pa ame e de ec o s can be implemen ed
join ly
o
sepa a ely inside he second s age.
Rega dless
ou
objec i e
is
he
simul aneous
es ima ion
o
he magni ude, phase and equency,
we may conclude
ha
equency ep esen s he
main di icul y and, o his eason, we will discuss
ini ially he design
o
he il e
o
igu e
1,
in
e ms
o
equency es ima ion only.
A e modula ion
o
he signal has been emo ed,
he es ima ion s age is done, selec ing a unc ion
objec i e o be minimized. The signal
is
modeled
as
a single one wi h equency, phase and magni ude
o be es ima ed. The pa ame e s a e ob ained in
e e y bu s
o
slo
o
signal, minimacing he Mean
Squa e E o (MSE) be ween he eal signal and i s
model, and hey esul op imum in a maximum
likelihood
sense,
which
allow
o deal
wi h
modula ed ca ie s and also p o ides na owband
in e e ence
ejec ion.
Phase and magni ude p ocess a e used in he
synch oniza ion
o
he sys em he e p esen ed,
bu
op imum
equency
ob ained
esul s
high
compu a ionally ine icien . Because
o
his o he
kind
o
equency es ima ion [2] wi h conside able
lowe compu a ional load, is used he e and
i
is
compa ed wi h op imum me hods o show ha wi h
Eb/No (bi ene gy o noise spec al densi y a io)
abo e 0 dB, bo h p ocess gi e
as
esul he same
pe o mance
.
The expe imen s ha e been done o e PSK-4 and
PSK-8 modula ed signals in colou ed noise, and
hey con i m he s a emen s
on
obus ness and
Eb/No h eshold e ec s done p e iously.
Taking
as
base he op imum es ima ion me hod,
o he kind
o
subop imal es ima ion o equency
and phase ha e been also p o ed. In a inal sys em,
each pa ame e de ec o , will ha e o be selec ed,
and he pa icula solu ion will usually depend on a
adeo
be ween
low
compu a ional
load
and
op imum pe o mance equi ed in he sys em.
1836
2.
SIGNAL
ENVIRON:MENT
The ype
o
signal used in his wo k
is
desc ibed.
The kind
o
modula ion is PSK. The in o ma ion
o
he symbols is con ained in he phase, added o he
phase
o
he ca ie signal.
Fo
N
Symbols
by
bu s , he ecei ed o m will be
as
is
shown
in
(1),
whe e p(.) is he esul ing pulse in he ecei e
wi hou in e symbol in e e ence.
N
x ( )
=
A.
L,cos[( od
+ oN)
+9+9s]p( -sT
8)
s=1
(1)
T s is he symbol pe iod, 9 s is he modula ion phase
and 9 he leakage phase. I is assumed ha a
p e ious
ansla ion
om
he
eal
ca ie
equency
o
c o he nominal equency
o
N
has
been done abou he signal. The e is a esidual
unknown equency
Clld.
This unce ain y has o be
es ima ed o ge app op ia ed accu acy.
The signal is sampled wi h a sampling pe iod T
no malized o be one. Taking
in
conside a ion he
modula ion p ocess, he sampling
is
done
o
se
N
+
c
a ound 0.25 no malized alue a he sampling
equency. Nss will be he numbe
o
samples by
symbol. In e e y bu s he e a e N symbols o
p ocess he signal (2).
Q
x(i)=
L,A.cos[( oc + oN)n+9+9s]p(n-s.Nss)
+
n(i)
s=1
(2)
Wi h his kind
o
signal,
we
ha e N symbols and
N
.N
ss samples in e e y bu s .
3.
FILTERING
STAGE
In he i s s age
o
he ecei e he signal is
il e ed o ob ain a sample by symbol o each
g oup
o
Nss samples. I is assumed han accu a e
iming synch oniza ion is go in his s age. Wi h a
ec o no a ion a he ou pu il e i will
be
y(n)=AH.X(n)
(3)
whe e
A.
H is he il e esponse and X (n) is he
signal ec o co esponding o a single symbol.
Taking
as
he il e o de Q, he same
as
he
numbe
o
samples by symbol Nss , he e will be N
samples
o
signal y(n) a he il e ou pu .
Inside each symbol, he signal can be conside ed
as
a
single
sinusoid
wi hou
any phase change.
Recei ing he signal wi h he s ee ing ec o a
he nominal equency
o
N , a he ou pu
o
he
il e , he e will be he signal y(n), whe e he
equency will be
jus
he unknown dopple
equency.
Q
1~
.
y(n)
= Q
£..J
x(n-i)expU oN~
(4)
i=O
In hese condi ions, he il e esponse A (n)
is
he
s ee ing ec o
.S..
A=
.S.
=[1,exp(-j oN),
.......
(-j oN(Q-1)]
(5)
Analysing y(n), analy ically will be
as
(6)
y(n)=
A.expUcp(n)] +
n(n)
=
A(n).expU'I (n)]
(6)
The ins an aneous magni ude A(n) is he signal
ampli ude wi h noise and 'I' (n) is he ins an aneous
phase. e(n) is he ou pu noise con ained in
'l'(n).
Q:l
'l'(n)= od.Q(n-1)
+
9n
+ 9 +Old 2 +
e(n)
(7)
An
app oxima ion is assumed in (7). The phase o
each sample is he cen al phase
o
he measu e
snapsho . Really in he case
o
no in e e ences
and only addi i e Gaussian noise (7) will be he
exac phase a he il e ou pu .
Being ou objec i e o es ima e A and
cp(n)
om a
single
snapsho
X..(n)
he e is an al e na i e
o
il e he signal x(n). I is ound, de i ing he
maximum likelihood es ima e o pa ame e s A and
'I'. The il e esponse is hen ob ained
as
a esul o
minimacing
he
di e ence
be ween
he signal
ec o and a sinusoid model o i , and i esul s
as
a
da a dependen ec o . Bu in ideal condi ions,
we
mean, wi hou o he sinusoid in e e ences, whi e
noise,
s a iona i y
and
signal
au oco ela ion
ma ix wi hou
es ima ion
e o s, he maximum
likelihood il e esul s jus
as
he il e p esen ed
in (5), he FFT p ocessing o he cu en snapsho
[1]. In
o he
wo ds, o he abo e men ioned
assump ion he op imum il e ing is no longe
da a dependen and can
be
implemen ed
as
a
DFT
p ocesso
o
leng h Q.
4
ESTIMATION
STAGE
The a ailable signal samples in his s age, a e
shown in (6) and (7). The e a e Nss samples
o
y(n), a sample by symbol, and he objec i e is
o
es ima e he esidual equency
ca ie
od, he
phase leakage a and also he magni ude
A.
The
modula ion phase
On
is emo ed om he signal.
In o de o emo e he modula ion, we need
o
mul iply he measu ed phase by he numbe
o
modula ion le els
M.
This will emo e om cp(n)
he
con ibu ion
o
modula ion
s eps
be ween
symb
co e
exis i
algo1
Cons
new
as
i
y(n)
An·~
Being
objecl
~=
n
whe e
ph as(
and
objec
noise
case
o
l
ph
as
aS
SO(
wheD
dB
eql
Taki
e m:
like!
Mag
A=
I
he
he
• en m
one
A:
Thi:
ph a
Co
he
lea!
ph!
Th~
~
p e:
o
I he
F
[
sig1
,,
~
"
~
~
I.
e
symbols bu i will in oduce
27
s eps in he
co~ ~spon~ing
phase. To emo e he
27
s eps
ex s l!lg m he new phase, a phase unw apping
algo .l hD?-
can be used when
i
is necessa y.
Cons de ~ng
he abo e explained ope a ions, he
new a a lable signal o p ocess pa ame e s will be
as
i is shown
in
(8)
y(n)=AnH.expj(M odn
+Me
+~ od+
e'(n))
=
An'·expj<p(n)
(8)
Being he Mean Squa e E o (MSE) he selec ed
objec i e, he p oblem is o mula ed in (9)
N
~
=
Ll
y(n)
- A '(n).exp(j'l
'(n)l2)
(9)
n=l
whe e 'l '(n) is equal
o
~
+ a(n-1). Iden i ica ion
o
phase e and equency
(l)
d wi h he pa ame e s a
and
~
is i ial. I should be no ed ha he abo e
objec i e is no an op imum c i e ia
i
he phase
noise is no gaussian; bu , e en in he non gaussian
case he associa ed pe o mance is ecognized. Due
o he
way
y(n) is o med, i he e a e no w ong
phase s ep co ec ion in he unw apping, he
associa ed noise o he phase emains gaussian,
when he Eb/No he e is high enough, abo e 2 o 3
dB
and he MSE es ima ion o magni ude
A,
equency
IDd
and phase e emain op imum.
Taking de i a i es
o
(9)
wi h espec A'and
q>
'in
e ms
o
~
and
a,
and se ing o ze o, he maximum
likelihood es ima ions ob ained a e he ollowing.
Magni ude es ima ion:
N
A=~
LA(i).cose'(n)
i=l
(10)
he classical es ima e o magni ude (11) esul s
he op imum es ima e MSE whene e e '(n) is small
enough o assume he second e m
o
he sum as
one o all he measu emen s done
1
A:N
N
L I
y(n)l
i=l
(11)
This es ima e can be used
o
alida e equency and
phase es ima es, depending on he A le el.
Conce ning he phase de i a i e, and aking i s
he case
o
de i a i e wi h espec he phase
leakage
~
(12) is ob ained o compu e he op imum
phase.
The ob ained op imum phase is
jus
he same
p esen ed
by
Vi e bi in
[3]
also
as
he op imum, bu
o he case o
no
dopple equency,
a=O.
I
is jus
he phase ob ained, in he Fou ie ans o m
o
he
signal e alua ed a a equency.
1837
N-1
eal(
,y(mTs)exp(-ja(m-1)))
Phase: 9 =
~
g-1
~~~
(12)
imag(
L
y(mTs)exp(
-ja(m-1
)))
m=O
The equency es ima e is ob ained om he
de i a i e
o
(9) wi h espec
o
pa ame e
a.
A e
some algeb a i can be shown ha he op imum a
jus se s ou he condi ion
o
ind an ex ema
o
he
pe iodog am
o
y(.) o he squa e magni ude
o
i s
Fou ie T ans o m. Thus, he op imum way o
de e mine he equency leakage
a,
will be a DFI' o
he
il e
ou pu
samples
o
measu emen s
a ailable.
Th ee op imum solu ions o magni ude, equency
and phase es ima es ha e been gi en. They
ep esen he maximum likelihood es ima es wi h
he only d awback o he equency compu es
o
i s compu a ional load. Wi h he DFT me hod i
would equi e a lo
o
samples o ge a BER o he
o de o lQ-4 •
The p eceden me hod gi e he op imum od
es ima ion bu wi h a conside able compu a ion
cos .
As
an al e na i e o equency es ima ion,
Kay's app oach [2] minimizes
an
objec i e unc ion
be ween
a ini e
consecu i e
samples phase
di e ences ec o
(a)
and
aL.
The compu a ional
load
o
he me hod is e y low. I is synch onous
wi h he symbol pe iod, and ge s he C ame -Rao
bound when Eb/No is abo e 6
dB,
possible h eshold
o i s use.
The esul ing algo i hm, ye p ese ing he
men ioned p ope ies, becomes op imum e en o
signal con amina ed wi h colou ed noise. This is o
capi al impo ance aking in mind ha many
~~
cu en ly TDMA sys ems in communica ions will
"- equi e o ack ampli ude, phase and equency
om symbol
o
symbol, wi h a maximum
o
16
symbols in he bu s and wi h no mo e han 4
samples pe symbol.
The de e mina ion
o
ampli ude es ima es
p o ides an use ul quali y index o he phase and
equency es ima es. This is due
o
he ole played
by
ca ie , o sinusoid, magni ude in he h eshold
e ec . No e ha , in o de o ob ain gaussian noise
in he phase es ima es, he inpu signal
o
noise
a io is abo e 0
dB
is equi ed. Once his cons ain
holds, he equency and phase es ima es achie e
he C ame - Rao bound.
A
his poin , i is clea ha
na ow band in e e ences, colou ed noise o
sinusoid modula ion des oy his p ope y and,
as
a
consequence, he o e -all sys em ails.
As
a subop imum al e na i e o he op imum phase
and equency es ima ions, linea eg ession can
be used wi h he phase
o
he emo ed modula ion
signal, o ob ain a and
~
es ima es. In his case he
algo i hms wo k di ec ly wi h he signal phase
samples.
1838
Summa izing he inal s uc u e o he ecei e ,
he gene al p ocess o synch onize he TDMA
signal is shown in igu e
2.
..
R.M.
~
Op imum
~
Op imum
~
Op imum
equency
phase
magni ude
y(n)
4
Unw appin~
•
L.R.
algo i hm
Figu e 2. Gene al Recei e
5.
RESULTS
AND
COMMENTS
Simula ions esul s we e ob ained o e PSK-4 and
PSK-8 modula ed signals in colou ed noise.
In he igu e 3, o a 4-PSK signal wi h
N=l8
symbols, N
88
=5
samples/symbol,
Wd=.005,
6=
10
2, he
no malized e o s a e compu e o Eb/No om -10
dB
o
10
dB, wi h 50 andomized ials o each
Eb/No alue.
Fo
each signal,
he
maximum,
a e age and minimum es ima e e o s a e compu e
o pa ame e s
COd,
a and
A.
Th esholds
o
Eb/No o wo k, could be es ima ed
om igu e
3.
In
hese condi ions 1
dB
o Eb/No
is
app op ia ed o ge low alues
o
e o s. Fo o he
alues
o
pa ame e s o es ima e, he app op ia ed
h esholds can also be p ocessed and hey always
esul
be e
han
he equi ed Eb/No in PSK
ansmission
i
d
is g ea e han 0,01
(5%
o
he
ca ie
equency).
6.
REFERENCES
[1]
M.
A. Lagunas,
M.
Cab e a. "Open Loop Join
Pa ame e
Es ima ion
o
Bu s
Communica ion Mode". ESA (Eu opean Spa ial
Agency)
Repo ,
ESA-ESTEC
Noo wijk
(Ne he lands), Feb ua y 1990.
[2]
S.
Kay, "S a is ically/compu a ionally E icien
F equency Es ima ion". IEEE-ICASSP 88, pape
E3.5,
New
Yo k 1987.
[3]
A.
J. Vi e bi, A. M. Vi e bi, "Nonlinea
Es ima ion
o
PSK Modula ed Ca ie Phase
wi h
Applica ions
o
Bu s
Digi al
T ansmission". IEEE T .
on
IT, Vol IT_29 n2 4,
July 1983.
This wo k has been suppo ed
by
ESA and PRONTIC
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3.
4PSK signal,
COd=.OOS,
6=:10
2,
18
symbols
and
5 samples/symbol. Values
o
Eb/No om -10
dB
o
+
10
dB.
1
S/GNI
L.
To
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