Signal P ocessing
19
(1990) 311-320
Else ie
311
A FLATNESS-BASED GENERALIZED OPTIMIZATION APPROACH TO
SPECTRAL ESTIMATION
Climen NADEU
Dep . de Teo ia del Senyal i Comunicacions, Uni e si a Po/i ecnica
de
Ca alunya, Ba celona, Spain
Miquel BERTRAN
Dep . de Ma emii ica Aplicada i Telemii ica, Uni e si a Poli ecnica de Ca alunya, Ba celona, Spain
Recei ed
12
Ap il 1988
Re ised 10 July 1989
Abs ac .
In
he
op imiza ion o mula ion, once he known au oco ela ions a e ixed as cons ain s,
he
objec i e unc ional
comple ely cha ac e izes he co esponding me hod
o
spec al es ima ion. A e obse ing
ha
his unc ional in ol es a so
o
spec al la ness maximiza ion,
he
ela ionship be ween he o m
o
i s in eg and and he salien ends
o
he a ious
me hods is poin ed ou . This se es as a basis o p opose a gene alized op imiza ion app oach
ha
encompasses some classical
es ima o s
and
o igina es new ones.
Zusammen assung.
In
de
Op imie ungs-Besch eibung cha ak e isie das Funk ional die jeweilige Me hode zu Spek alschal-
zung olls andig, wenn on es en Au oko ela ionswe en ausgegangen wi d.
Nach
dem Au zeigen
de
Ta sache, daB dieses
Funk ional eine
A
on Maximie ung de Flachhei des Spek ums beinhal e , wi d die Beziehung zwischen de Fo m des
In eg anden
und
den he aus agenden Rich ungen
de
e schiedenen Me hdden he ausges ell .
Au
diese G undlage wi d
ein e allgemeine e Op imie ungs-Ansa z o geschlagen, de einige klassische Scha ze einschlieB
und
neue he o b ing .
SchlieBlich wi d die B auchba kei
de
Vo s ellung
de
spek alen Flachhei ii die Spek alscha zung disku ie .
Resume.
Dans
un
con ex e
d'op imisa ion,la
onc ion objec i ca ac e ise comple emen la me hode co espondan e d'es ima-
ion spec ale des que les aleu s d'au oco ela ion connues
sen
ixees comme con ain es. Obse an que ce e onc ion
objec i implique
une
ce aine maximisa ion
du
ca ac e e pla
du
spec e (en anglais spec al la ness), la ela ion en e la
o me
de
!'in eg a ion e les ai s saillan s de di e en es me hodes es mise en e idence. Ceci se de base a une app oche
gene alisee de l'op imisa ion qui ecou e ce ains es ima eu s classiques e pe me
d'en
c ee
de
nou eaux.
Keywo ds. Spec al analysis, spec al modeling, maximum la ness.
1.
In oduc ion
The op imiza ion
o
a ia ional o mula i9n
is
an
in e es ing s a ing poin o spec al es ima ion
because a numbe
o
meaning ul es ima o s can
be
de i ed om i [1-3, 6, 8,
9,
12-14]. Succinc ly,
he app oach
is
as ollows. On he one hand, a
numbe
o
measu emen s ob ained om he signal
samples ca y in o ma ion abou he spec al
densi y unc ion
S(w)
o
he unde lying andom
This wo k was suppo ed
by
he
PRONTIC
g an
numbe
105/88.
0165-1684/90/$3.50 © 1990, Else ie Science Publishe s
B.V.
p ocess; in mos cases, hey a e he au oco ela ion
unc ion alues
n
iom lag 0 o
M.
In
he de e -
minis ic app oach [11], hese alues a e assumed
o be exac ly known, so hey ac ually ac as con-
s ain s
o
he op imiza ion p oblem.
On he o he hand, a cos measu e J ha , as i
will be shown in he ollowing,
is
conce ned wi h
he deg ee
o
emphasis gi en o each ype
o
spec-
al shape
is
de ined. The aim
is
o minimize he
unc ional
1
"'
J
=-
F[S(
w
)]
dw,
2'lT
_,
(1)
l
312
C.
Nadeu,
M.
Be an I A la ness-based op imiza ion
o
spec al es ima ion
subjec o he au oco ela ion cons ain s
1
"
. .
-
S(w)
eJWn
dw
=
n,
2'IT
_,
n
=0,
±1,
...
,
±M,
(2)
whe e he
a ea
o
S(w)
will
be
no malized he e o
uni y, i.e., 0 =
1,
wi hou loss
o
gene ali y.
Ob iously, once
he
cons ain s (2) a e speci ied,
he pe o mance
o
he spec al es ima ion
me hod
a ising om he abo e
app oach
is comple ely
cha ac e ized
by
F(S),
so we ha e o ace wi h he
p oblem
o
designing a
p ope
cos unc ion
F(S).
We may ega d his p oblem as
ha
o
measu ing
he dissimila i y be ween
he
shape
o
he spec um
es ima e
S(w)
and
he la spec um S0
(w)
=
1.
In
ac ,
among
all he spec a
ha
ma ch
he
gi en
au oco ela ions (2), he maximally la
o
smoo h
spec um
could
be
a sensible choice because i
would
be
maximally close o he whi e noise
powe
densi y unc ion [3].
On
he
o he
hand,
i
he
spec al
a ea
0 is he only cons ain , i.e.
M=
0,
S(w)
= 1 wha e e
F(S)
is [9]. The e o e,
maximum la ness seems o
be
a common endency
o
he me hods a ising om he op imiza ion
app oach
which
is
only
bounded
by
he con-
s ain s.
Fo
his eason, i was exploi ed
in
[1, 2]
as a uni ying p inciple o spec al es ima ion.
Fi s
o
all, conside
he
well-known maximum
en opy
me hod
[3] (he ea e we will call i
MEM1)
ha
maximizes he en opy measu e gi en
by
1
"
-log
S(w)
dw,
2T
_,
(3)
which belongs o he gene al class
o
unc ionals
J
o
(1) when
F(S)
=-logS.
(4)
I
has o en
been
ega ded as a maximum la ness
me hod
[3] mainly
due
o
he
ac
ha
i maximizes
he en opy
o
he associa ed Gaussian p ocess
wi h he
idea
o
app oaching
he whi e noise p o-
cess and, consequen ly,
he
whi e noise ( la ) spec-
um.
Signal P ocessing
The e exis s
an
al e na i e e sion
o
he maxi-
mum
en opy p inciple (we will e e o
i
as
MEM2) which s a s om a di e en la ness
measu e [9]. Since a
powe
spec al densi y
ac ually is a p obabili y densi y unc ion
o
a an-
dom a iable
2T ,
whe e makes sense in e ms
o
he ins an aneous equency
o
he associa ed
andom
(no
necessa ily Gaussian) p ocess [14],
we can also use he ela i e en opy measu e
be ween
S(w)
and
S0
(w)
[7], namely
1
"
S(w)
E(S,
S0)
=-
S(w)
log-S
( ) dw.
2'IT
_,
0 w (5)
Since (5) is a measu e
o
closeness be ween he
wo densi y unc ions
S(w)
and
S0(w), a la ness
measu e is ob ained
by
se ing S0( w) =
1.
Thus, he
MEM2
a ises om he op imiza ion
app oach
o
F(S)
=SlogS.
(6)
Con e sely o he maximum en opy app oach,
we can a oid any e e ence o he en opy
o
he
p ocess
and
jus
no ice he spec um i sel , so
ha
we me ely
hink
o
la ness om a geome ical
poin
o
iew. Then, he Euclidean measu e
o
sepa a ion om he cons an spec um
(7)
appea s as a sensible unc ional
J,
leading o he
classical Blackman-Tukey
me hod
wi h ec-
angula window which ex apola es wi h ze oes
he au oco ela ion unc ion beyond
M.
Ob iously,
i s co esponding cos unc ion
F(S)
is
F(S)
=
(S
-1?.
(8)
The o egoing spec al es ima ion me hods a e
h ee di e en ways
o
aiming
a
maximum la ness
es ima es. Howe e , he e a e many
o he
possibili ies. The
pu pose
o
his
pape
is
no
only
he explo a ion
o
new me hods,
bu
mainly o
poin
ou , in a
a he
quali a i e manne ,
he
in luence
o
he o m
o
he unc ion
F(S)
on
he
pe o mance
o
he me hods om he iewpoin
o
la ness in
o de
o a i e a a gene alized
app oach.
C.
Nadeu, M. Be an I A la ness-based op imiza ion
o
spec al es ima ion 313
The
pape
is o ganized as ollows.
In
Sec ion 2,
he
op imiza ion
p oblem
is sol ed
o
ind
he
spec um es ima e associa ed
o
each unc ion
F(S)
o a gi en se
o
cons ain s
n,
n =
0,
±1,
...
,
±M.
In
Sec ion 3
he
second de i a i e
o
F(S)
is p esen ed as a sui able ool o explain-
ing
he
salien de e minis ic ea u es
o
he
co e-
sponding
es ima o . This ac is
used
in Sec ion 4
o
p opose
an
unbounded
amily
o
me hods
ha
includes
he
h ee abo e
men ioned
BTM, MEM1
and
MEM2.
Some illus a i e examples a e gi en
in Sec ion 5
and,
inally, a gene alized
app oach
is
p esen ed
in
Sec ion 6, along wi h
an
algo i hm
o
de e mine
he
disc e e spec al es ima e.
2. Sol ing he op imiza ion p oblem
The
minimiza ion
o
he
unc ional (1) wi h con-
s ain s (2)
can
be ca ied
ou
by means
o
Lag ange mul iplie s An, n = 0,
±1,
...
,
±M,
by
minimizing
he
in eg al
2
~
J:"'
{
F[S(w)]
+
n=~M
An[ n-
S(w)
ej"'"J} dw.
(9)
I
he
de i a i e
o
he
in eg and
wi h espec
o
S(w)
is
made
equal
o
ze o,
an
ex emal
o
(9) is
ob ained.
The
co esponding spec um e i ies
he
equali y
M
F'[S(w)]=
L
Anej"'"=P(w),
(10)
n=-M
whe e
F'
is
he
de i a i e
o
F wi h espec
o
S
and
P(
w) is a eal
and
e en igonome ic poly-
nomial
o
o de
M.
I
can
be
shown [3] ha ,
i
he e is a solu ion
o
he
cons ained minimiza ion
p oblem, i is
unique
and
gi en
by
S(w)
=
G[P(w
)],
(11)
whe e G is
he
in e se unc ion
o F'.
No e
ha ,
in
gene al, G is
no
linea
and
he
model
is
no
a ional.
In
o de
o
ind
he
spec um es ima e
S(w)
we
should
subs i u e (11)
in
(2), ob aining,
in
gene al,
a sys em
o
M+
1
non-linea
equa ions.
The
M+
1
a iables
An
can
be
de e mined
by means
o
an
i e a i e algo i hm. Then,
S(w)
is
compu ed
om
P(w)
wi h (11). Thus,
he
co esponding
au oco ela ion unc ion R (
n)
ag ees wi h i s
known
alues
n
up
o
M
and
ex apola es
hem
up
o
in ini y.
3. Analy ical compa ison
o
me hods
Gi en a pa icula se
o
cons ain s
n,
n = 0,
±1,
...
,
±M,
he
spec al es ima ion
me hods
esul ing om
he
abo e
app oach
only di e
by
hei
cos unc ion
F(S).
Howe e , cons ain s
could
be
di e en om au oco ela ions
( o
example, ceps al coe icien s [8]), so, wi h
he
objec
o
compa ing me hods,
he
ype
o
con-
s ain s
should
be
included. This
can
be accom-
plished
by
using as subjec
o
compa ison
he
spec al
model
since i is a consequence
o
bo h
F(S)
and
he
ela ionship be ween
he
unc ion o
which
co espond
he
cons ain s, i.e.
he
au oco ela ion unc ion
in
ou
case,
and
he
spec-
um.
Fo
example,
i
ceps al coe icien s we e
used,
F'(S)
in (10)
should
be
subs i u ed
by
SF'(S)
o
ob ain
he
spec al
model
whe eas
F(S)
would
be
le unchanged.
spec al
model
is
jus
de e mined
by
he
i s
de i a i e
o
he
cos unc ion acco ding
o
(10).
Consequen ly, each
me hod
o
spec al es ima ion
a ising om
he
op imiza ion
app oach
is
cha ac-
e ized
by
F'(S).
Howe e
F'(S)
+ K1 leads
o
he
same spec al
model
as
F'(S)
because he cons an
K1 may
be
included as
pa
o
he
coe icien A0
in
(10). Thus,
due
o
he
ac
ha
bo h
unc ions
F'(S)
and
F'(S)
+ K1 ha e
he
same de i a i e
wi h espec
o
S,
we
can
choose
(S),
he
second
de i a i e
o
F(S)
wi h espec o
S,
as
he
unc ion
ha
bes
cha ac e izes
he
spec al model.
Equa ion
(10)
can
hen
be
ew i en in
he
ollow-
ing way:
J
S(w)
(x)
dx=P(w),
XJ
(12)
whe e
he
cons an x1 is a bi a y.
Vol.
19,
No. 4, Ap il
1990
314
C.
Nadeu,
M.
Be an I A la ness-based op imiza ion
o
spec al es ima ion
The ollowing example is illus a i e.
Conside
wo di e en unc ions
Fa(S) =
Kz(S
-1)
2,
Fb(S) =
Kz(S
2
-1).
(13)
(14)
Thei i s de i a i es di e by
an
addi i e
cons an
o
alue -
2K
2• Howe e ,
hei
second de i a i es
a e iden ical, as well as
hei
spec al models
(which
co espond
o
he
BTM).
The e o e, assuming
he
cons ain s ", n =
0,
±1,
...
,
±M,
he
way in which
he
cos unc ion
F(S)
a ou s a spec al
shape
wi h espec
o
ano he
one
is de e mined
by
i s cu e con exi y.
In
ac , a ze o alue
o
i s second de i a i e
(S)
o all S implies a linea unc ion
F(S)
so
ha
he
unc ional J
in
(1) only
depends
on
0,
he
a ea
o
S(w
).
Hence,
when (S)
= 0 o all
S,
S(w) may
show
any
shape consis en wi h he cons ain s.
Mo eo e ,
when
(S)
has a cons an non-ze o
alue K3, we
can
choose x1 such
ha
(15)
esul ing
in
F(1
+AS)=
F(l-
AS), o AS
:s;;
1.
(16)
Since
he
mean
alue
o
S(w) is 0 =
1,
(16) means
ha
spec al peaks
(AS>
0)
and
alleys
(AS<
0)
a e iden ically ea ed
by
he
spec al
model
as
long as 0
:s;;
S
:s;;
2.
A his
poin ,
once
he
signi icance
o
he
second
de i a i e
(
S)
has
been
shown,
i
is wo h compa -
ing di e en spec al es ima o s om
he
poin
o
iew
o
hei
associa ed unc ions. Since
he
spec-
al models emain una ec ed when a
cons an
is
added
o
F(S),
F'(S)
o
bo h,
o
when
(S)
is
mul iplied
by
a cons an , a no maliza ion is
needed
o
emo e
he
a bi a i y
o
he
compa ison.
Fo
his eason, we will
ix
he
alues
o
he
unc ions
F,
F'
and
a
he
mos cha ac e is ic poin , namely
S =
1.
The
imposed
condi ions a e
F(1)
=0,
(17)
F'(1)
=0
(18)
(l)
=
1.
(19)
Signal P ocessing
Condi ion
(17) o ces
he
cos alue J in (1)
o
be
ze o
when
S(
w)
= 1 o all w ( la spec um).
Con-
di ion (18) ensu es
ha
F(S)
has
an
ex emal
a
S =
1,
which has
o
be
a minimum
in
o de
o
weigh nega i ely
any
spec al de ia ion om
uni y. Finally, condi ion (19) makes a con exi y
no maliza ion
ha
will allow us
o
compa e
he
ea men
o
peaks wi h espec
o
alleys
and
ice
e sa.
Le us show
he
e ec
o
using
he
abo e condi-
ions
on
he
h ee unc ions
F(S)
in (4), (6)
and
(8). The esul ing unc ions a e shown
in
Table
1,
along wi h
hei
i s
and
second
de i a i es. The
unc ional J co esponding
o
he
MEM1 is exac ly
he
I aku a-Sai o
measu e
o
sepa a ion
be ween
wo spec a [5]
when
one
o
hem
is
S(
w)
and
he
o he
is
he
cons an
spec um
S(w)
=
1.
Fu he -
mo e, i
can
be
shown
ha
all
F(S)
a e nonnega i e
unc ions. Also no ice
ha
he
second de i a i es
show a egula o m o all me hods, namely,
Sk(w ), k = 0,
-1
and
-2.
The
unc ions
F(S)
and
(S)
co esponding o
he
h ee spec al es ima ion me hods a e
plo ed
in Fig.
1.
Obse e
ha
he
MEM1
weigh s alleys
(S
< 1)
in
he
cos
measu e
mo e
han
he
BTM
and
he
opposi e
occu s
a
peaks.
The
MEM2
lies
be ween
he
o he
wo, consis en ly wi h esul s
epo ed
in [9].
The
second de i a i es a e
mono onic
unc ions
and
hey a e such ha , gi en
a alue
o
S,
a g ea e con exi y implies a g ea e
weigh ing in
he
cos unc ion.
No
only
he
ela i e ea men
o
peaks
and
alleys cha ac e is ic
o
e e y spec al es ima o
can
be
an icipa ed om
he
unc ions
used
in
he
Table 1
No malized cos unc ions
o
he op imiza ion app oach co e-
sponding o he h ee p e iously known spec al es ima ion
me hods, and hei i s " wo de i a i es.
MEM1
MEM2
BTM
F(S)
-logS+(S-1)
Slog
S-(S-1)
!(S
-1)
2
F'(S)
-(1/S)+1
JogS
S-1
(S)=
F"(S)
1/Sz
1/S
1
C.
Nadeu,
M.
Be an I A la ness-based op imiza ion o spec al es ima ion 315
2
~----------------~
--
BlM
-----
MEM2
.......
MEM1
§:1
LL
.
·._
.
'' _
..
-····
.."''./;::.<>/
0
~--~~~----~--~
0 2 3
(a)
2
,-~--------------,
,.
·.
·
,_·.
--
BlM
----·
MEM2
.......
MEM1
(J)j
;.
-
-
'·
·-::.-·.,
·
..
:::::···-··-
...............
_____
,_
····-~
···-
.........
.
0
1------ ----------~
0 2 3
(b)
Fig.
1.
(a) No malized cos unc ions
F(S)
co esponding !'
he BTM
(-),
MEM2
(----)and
MEMl(·
···);(b)
hei
second de i a i es
(S).
op imiza ion app oach, bu also o he ea u es
ela ed wi h he geome ical shape
o
spec a. Le
us show wo examples. O he examples will appea
in Sec ion
5.
Fi s
o
all conside he well-known possibili y
o
he BTM o p oduce spec al es ima es wi h
nega i e alues. This e ec has
an
explana ion in
e ms
o
i s associa ed unc ion since F'(O)
is
ini e.
In
ac , when he slope
o
F(S)
a S = 0
is
in ini e
he spec um
is
d i en away om ze o by he
minimiza ion p ocess [3].
As
i was poin ed ou in [6, 9] MEM2 spec a
can exhibi e y deep alleys when he e exis
p ominen peaks. This e ec can also be explained
by obse ing he MEM2 unc ions. On he one
hand, F'(O)
is
in ini e so, unlike in he BTM, nega-
i e
o
ze o alues
o
he spec um a e no allowed.
On he o he , unlike in he MEM1, F(O)
is
ini e
and i
is
e y close o uni y inside he in e al
[
0,
s],
whe e s «
1,
so S ( w) can app oach he ze o
alue a some equency bands wi hou inc easing
no iceably he unc ional J ha has o be minim-
ized. This
is
specially ue when he au oco ela ion
cons ain s o ce he spec um es ima e o ha e
p ominen peaks since in his case J has a high
alue ha
is
only sligh ly a ec ed by he deg ee
o
dep h
o
spec al alleys.
4. A amily
o
spec al es ima ion me hods
The egula o m
o
he second de i a i es
(S)
shown in Table 1 sugges s he possibili y
o
de ining a amily
o
spec al es ima o s cha ac e -
ized by a eal cons an g and he ollowing simple
exp ession
o
he second de i a i e
(20)
which e i ies (19) and encompasses he abo e
conside ed BTM, MEM2
and
MEM1, espec i ely,
o g=O,
-1
and
-2.
Pe o ming a double in eg a ion and imposing
(17), (18), he amily
o
cos unc ions
(s)
1
(Sg+2
)
Fg
(g+1)(g+2)
-1
1
--(S-1)
(21)
g+1
ollows. They a e alid o all eal alues
o
g
excep o g =
-1
and g =
-2
which a e singula
poin s in he amily ( hei co esponding unc ions
a e shown in Table 1).
In
a s ic sense, his amily should comp ise
only hose es ima o s
ha
gua an ee non-nega i i y
o
spec a. Since a su icien condi ion
is
o show
an
in ini e alue
o
F~(O),
we
should es ic he
amily o alues
g,;:;;
-1.
Howe e , i will also be
wo h s udying he beha iou
o
he me hods co e-
sponding o g >
-1.
Vol.
19,
No.
4,
Ap il
1990

316
C.
Nadeu, M. Be an I A la ness-based op imiza ion
o
spec al es ima ion
Figu e 2 shows Fg(S)
and
g(S) o se e al
in ege alues
o
g.
All he cos unc ions a e
necessa ily non-nega i e since hey a e con ex and
i s minimum alue
is
ze o. F om he sequence
o
cu es i
is
appa en
ha
spec al peaks a e mo e
a ou ed o lowe alues
o
g and he same occu s
wi h alleys o g ea e alues
o
g.
I
F(S)
in (1)
is
subs i u ed by Fg(S)
o
(21), a
la ness measu e
o
S(w)
is
ob ained. To make
mo e appa en ha i eally measu es sepa a ion
om he la spec um, we obse e ha Fg(S)
is
equi alen o
1
[(
s)g+
2 J
Fg(S, S0) = (g + 1
)(g
+ 2)
So
-1
__
1
(~-1)
g+1
S0 (22)
0 2 3
(a)
0 2 3
(b)
Fig.
2.
(a) Plo
o
Fg(S) o se e al alues
o
g.
(b) Plo
o
/g(S)
o he same alues
o
g.
Signal P ocessing
when S0(w) = 1 o all
w.
This exp ession also
shows
ha
peaks (S > S0) and alleys (S
<So)
a e
di e en ly weigh ed by he measu e, he ype
o
weigh ing depending
on
he alue
o
g.
Mo eo e ,
no ice ha ,
i
a p io spec al es ima e S0( w)
di e en om he cons an spec um exis s [12], i
is
eadily inco po a ed in o he op imiza ion
app oach using Fg(S, S0) ins ead
o
Fg(S). No e
ha
he unc ion J co esponding o g =
-2
is he
I aku a-Sai o dis ance be ween S and S0 since he
limi
o
he i s e m when g
~
-2
is
-log(S/
S0
).
Acco ding o (11), he spec al models a ising
om he op imiza ion app oach and Fg(S) a e
S(w) =
[(g
+
1)P(w)
+
1F
1(g+l
g;C-1.
(23)
The spec al model o he MEM2 (g =
-1)
can
be ob ained om
j_
1(S) = s-1
o
aking he limi
when g
~
-1
in (23).
As
i can be obse ed om
Table 1 and conside ing (10), i co esponds o a
polynomial modeling
o
he log spec um. Table
1 also shows he models associa ed o he BTM
and he MEM1, which a e a ional models. Fo
he BTM, he ob ained au oco ela ion unc ion
R(n)
is
ze o beyond M and ma ches he gi en
alues
n
om 0 o M.
In
he MEM1,
an
i e a i e
algo i hm
is
no equi ed o ind he ex apola ed
au oco ela ions o , equi alen ly, he spec um,
because he e exis s he e icien Le inson-Du bin
algo i hm [3]. Any o he eal g di e en om 0
and
-2
gi es ise o a non a ional model and needs
an
i e a i e algo i hm o ind S (
w).
No e om (10) ha he polynomial coe icien s
An
a e he Fou ie 's se ies coe icien s
o
F'[S(w
)],
which a e ze o o
In
I>
M.
Hence, e e y me hod
eme ging om he op imiza ion app oach
is
equi alen o mul iplying he Fou ie se ies
coe icien s
o
he i s de i a i e
o
he exac spec-
um by a window
o
leng h
2M
+
1.
The window
depends
on
he gi en au oco ela ions
n,
since he
esul ing coe icien s mus p ese e hese da a. The
BTM
is
an
excep ion since, in his case, he ( ec-
angula ) window
is
di ec ly applied o he
au oco ela ion unc ion.
C.
Nadeu,
M.
Be an I A la ness-based op imiza ion
o
spec al es ima ion 317
5.
Examples
Now we will illus a e he s a emen s
o
he
p e ious sec ions wi h some examples. Pa -
icula ly, we desi e
o
show he ela ionship
be ween he pa ame e g
o
he abo e-men ioned
amily
o
me hods
and
he pe o mance
o
he
co esponding me hods.
All he esul s
o
his sec ion we e ob ained by
means
o
an
i e a i e algo i hm based
on
he
New on-Raphson
app oach [9], excep o he
cases g = 0
and
-2
which ha e simple special
algo i hms.
Fo
small alues
o
lgl,
h ee i e a ions
usually su ice o sol e he non-linea sys em
o
equa ions (2) accu a ely, when he ini ial solu ion
is compu ed om he
da a
n
wi h he p ocedu e
desc ibed in [9]. Howe e , he con e gence
becomes mo e di icul when
lgl
g ows, specially
i
he spec um shows a la ge ampli ude ange.
Fi s
o
all, le us conside he spec um plo ed
in Fig. 3(a), which is o med adding wo Gaussian
shapes
o
he same a ea
and
wi h opposi e sign o
uni y. We assume exac ly known he ou i s
alues
o
i s au oco ela ion unc ion and ex apo-
la e hem o se e al in ege alues
o
g.
The esul -
ing es ima es a e shown in Figs. 3(b)
and
3(c).
These esul s clea ly show he abo e asse ions
abou he signi icance gi en by each me hod o
high
and
low alues
o
S(w
).
I
g1 is lowe
han
g2, he me hod co esponding o g1 has a sha pe
peak
and
a la e alley
han
he me hod co e-
sponding o g2•
Th ee pa icula obse a ions can be no iced
om he esul s. Fi s ly, S (
w)
becomes nega i e
o some w when g >
0,
a ac ha is qui e possible
o g >
-1,
in gene al, as shown in he p e ious
sec ion. Secondly, he ampli ude ange is minimal
o g = 0
and
g ows when
lgl
inc eases. Thi dly,
he lowe g is he highe he esolu ion capabili y
o
he me hod is, due o i s g ea e pe missi eness
a
peaks. This claim is illus a ed in Fig. 4, using
a spec um wi h wo e y close Gaussian peaks.
While he
MEMl
(g =
-2)
can no esol e he
peaks, a smalle alue
o
g (g =
-4)
p oduces a
spec um
ha
can sepa a e hem.
2,---------~~--~
Exac spec um
0'------"'=""---------------l
0 1
(a)
0 1
(b)
Fig.
3.
(a) Exac spec um; (b), (c) spec um es ima es, o
M=3,
ob ained, wi h
g=3,
2,
1,
0,
and
g=O,
-1, -2, -3,
espec i ely.
To comple e his se
o
examples,
we
shall con-
side a new spec um which is plo ed in Fig. 5(a).
As
he spec um in Fig. 3(a), i consis s
o
wo
Gaussian unc ions
o
opposi e sign; howe e , in
his case, he ampli udes
o
he peak
and
he alley
a e loga i hmically equi alen , so he
peak
is
no iceably highe . Figu e 5(b) shows he spec a
co esponding o g = 0
and
g =
-5
ob ained using
he i s wen y au oco ela ions as cons ain s.
In
his example,
we
can obse e a clea e ec
o
leakage ha is mo e accen ua ed o g = 0 o such
an
ex en
ha
he alley is
no
be e cha ac e ized
Vol.
19,
No. 4, Ap il !990
318
C.
Nadeu,
M.
Be an I A la ness-based op imiza ion
o
spec al es ima ion
5 Exac
4
spec um
3
2
V
0
0 1
(a)
3,----------,
-g=-4
----
g=-2 (MEM1)
2
0~-------~
0 1
(b)
Fig.
4.
(a) Exac spec um; (b) spec um es ima es, o
M=
10,
ob ained wi h g =
-4
(-)and
g =
-2
(--
-).
han
o g =
-5.
These di e ences
o
leakage can
also be explained om
F(S)
due o he ac
ha
he luc ua ions essen ially co espond o spec al
alues lowe
han
1, so hey a e mo e a ou ed by
high alues
o
g (i.e., g = 0)
han
by low alues
(i.e. g =
-5).
6. Gene alized op imiza ion app oach
Un il now we es ic ed ou sel o he amily
o
unc ions Fg(S). This amily shows e y in e es ing
p ope ies. Fi s
o
all, i de ines a ow
o
in ini e
me hods including h ee basic app oaches o spec-
um
es ima ion (BTM,
MEMl
and
MEM2)
among hem. Mo eo e , all he unc ions a e
Signal P ocessing
dB
10
Exac spec um
5
0
-5
·1
0
0 1
(a)
dB
10
-g=O
(BTM)
--
--9=-5
5
0
-~.
-·
..
").
-5
0 1
(b)
Fig.
5.
(a) Exac spec um; (b) spec um es ima es, o
M=
19,
ob ained wi h g = 0
(-)
and
g =
-5
(-
- -
).
mono onic o
S(w)
;;;.1
o
S(w):;;:;
1
and
he e a e
no c ossings be ween hem excep o
S(w)
=
1.
These egula cha ac e is ics lead o well speci ied
changes
in
he
beha iou
o
he
es ima es when g
is a ied.
Howe e , we may de ine o he unc ions
F(S)
di e en om Fg(S)
and
use hem o ob ain new
me hods wi h gi en cha ac e is ics.
In
any case,
we should impose wo su icien condi ions
on
hei
de i a i es, namely
(1)
F'(S)
~
oo,
S->0
(2) (S);;;. 0 o all
S.
Condi ion (1) gua an ees he posi i i y
o
S(w)
and
condi ion (2) ensu es ha ,
i
an
ex emal
o
he cons ained minimiza ion p oblem exis s, i is
unique (as shown in [3]).
C.
Nadeu,
M. Be an / A la ness-based op imiza ion o spec al es ima ion 319
Fo
ins ance,
we
could use he unc ion
Fs(S)
=![F_
1(S) +
F_iS)]
=!(S
-1)
logS.
(24)
No e ha (24) can also be ob ained a e aging he
en opy unc ions (4)
and
(6). Mo eo e , i s unc-
ional J esul s om he symme iza ion
o
he
ela i e en opy measu e (5), i.e.,
!(E(S,
S0
)+E(S
0, S))
by equa ing S0 o
1.
Suppose also he ollowing unc ion
4
"'T
S-1
F(S)=-- g--+S-1
"'T
2
S+1
' (25)
which e i ies he abo e wo condi ions in addi ion
o (17)-(19). Figu e 6 shows he es ima e ob ained
wi h his unc ion o he spec um depic ed in Fig.
3(a) and, as be o e,
M=
3.
We can obse e ha
he esul ing spec um es ima e lies be ween hose
co esponding o g =
-2
(MEMl)
and
g =
-3;
his ac can be explained conside ing
ha ,(S)
is
a mono onic unc ion like
/g(S)
and he slope
o
,(S)
in he cen al poin S = 1 lies be ween hose
co esponding o
j_
2
(S)
and
_
3(S), since
j;(l)
=
-2.74
and ~(l)
=g.
In
ac , he spec um ob ained
by
Fg(S)
o g =
-2.74
isually coincides wi h he
spec um
o
F1(S) in Fig.
6.
Now
we
desi e o go u he in o he a emp
on
gene alizing he me hodology
o
spec al es ima-
ion wi hin he amewo k gi en by he op imiz-
0 1
Fig.
6.
Spec um es ima e ob ained om
F,(S)
using he same
cons ain s as hose
o
es ima es shown in Fig.
3.
a ion app oach.
In
ac , i is possible o selec any
kind
o
unc ion
F(S)
e i ying he o egoing con-
di ions
o
o selec
(S)
and
o in eg a e i wice
in o de o ob ain
F'(S)
and
F(S).
Mo eo e , he
unc ions may be de ined bo h analy ically o
nume ically. This app oach is qui e di ec ; i does
no
need any in e p e a ion
o
he unc ional (
1)
in e ms
o
en opy
o
o he concep s,
i
only
equi es he design
o
a unc ion acco ding o he
ype
o
ea men desi ed o he di e en spec al
shapes.
Un o una ely, he algo i hm
o
New on-
Raphson may no be use ul in his gene al
app oach because unc ion
Gin
(11) may no be
known. Ne e heless, we can always use a nume i-
cal p ocedu e by disc e izing he a iable w so
ha
he alues
S(k)
o
he spec um in he N + 1
equencies
wk
equally dis ibu ed be ween 0
and
"'T
a e he a iables o be de e mined by means
o
a cons ained op imiza ion algo i hm. The aim
is
hen o minimize
N
I
F[S(k)],
(26)
k=-N
subjec o he au oco ela ion cons ain s
__
1_
£
S(k)
ej[2'1Tkn/(2N+l)]
=
2N
+1
k=-N
n•
n=0,±1,
...
,±M
(27)
and
he posi i i y cons ain (i equi ed)
S(k);;;.O,
k=O,
±1,
...
,
±N.
(28)
In
a p e ious in es iga ion [10], he same p ob-
lem was sol ed wi h a sequen ial quad a ic p o-
g amming algo i hm [ 4] which uses he g adien
o
(26).
Fo
example, he spec um in Fig. 6 was
compu ed using his algo i hm wi h N =
128.
Un o una ely, he algo i hm does no exhibi a
con e gence as good as he New on-Raphson's
one. Howe e , o he algo i hms which a e mo e
speci ic
and
show a be e con e gence pe o m-
ance ha e ecen ly been de eloped [15].
Vol. 19, No.
4,
Ap il 1990
l'l
I
I