MELECON
'85/
Volume
11:
Digi al Signal P ocessing
A. Luque, A.R. Figuei as Vidal,
V.
Cappellini (eds.l
Else ie Science Publishe s
B.
V.
(No h·Holland)/
C) IEEE
1985
..
;·
:-;.-
.
:.
::
.!
-{
~
:·=·
·
.:
·:
.·--
.
~
..
-.
.
..
,.______
..
-· : -;: __
~
:
~-~
--
~-
·
. . .·
:-
. _
__
, .. .
.-
·-.-.
..
-
- ~
-
~
-
;:-
·· ';";" .
..
...:
.•
_
:
_._
;
~
.
~~:
.
. - . :
.·:..-
·· -
107
PARAMETRIC
ENVELOPE
IN
LPC
SPEECH
CODERS
A.
lob eno,
M.A.
Iagunas,
F.
Vall e d 1.
P ocesado
de
sei'lal
en
ca unicaciones
E. T.
S.
I.
Telecoounicaci6n
Apdo 30002
08071
Ba celona
Spain
Abs ac :
Du ing
he
las
decade,
many
e o s
ha e been
de o ed
o
he
ela i e
impo ance
o
associa ed
lmc ions
like
magni ude
a l
);:hase
o
Fou ie
T ans o ms
in
image
and
signal
ba lwid h
educ ion.
'Ihe
epo ed
' IO k
deals
wi h
he
impo ance
o
he
eal
en elope
and
ins an aneoJS
equency
in
signal
analysis/sin esis
p oblems.
In
his
pape
au ho s
show a me hod
o
pa ame ize
he
en elope
and
ins an aneaJS
~ equency
o
a
eal
signal.
'Ibis
me hod
is
e y
closed
o
spec al
analysis
me hods
m
he
sense
ha
wi h
an
app opia e
s udy,
il e
do ain
and
equency
da ain
can
be
analyced
in
a
simila
way.
I.
INTRODUCTION
Signal
p ocessing
echniques
a e
cha ac-
e ised
in
he
equency
domain
by
means
o
he
magni ude
and
he
phase
o
he
asso-
cia ed
Fou ie
ans o m
o
he
signals
unde
analysis.
I
is
clea
ha
he e
a e
no
simila i ies
be ween
ime
domain
and
equency
domain
signal
ep esen a ions,
in
he
sense
ha
one
domain
uses
a
eal
signal
and
he
o he
wo
signals.
Many
au ho s
ha e
epo ed
some
in e es ing
esul s
which
deals
wi h
his
opic
and
acing
he
ques ion
o
edundancy
in
he
ep esen a ion
o
a
eal
signal
in
he
equency
domain
-1-
,-2-
•
In
some_ way
we
can
ecognize
ha
ega dless
o
he
p e iosly
men ioned
wo k
in
signal
p ocessing,
we
a e
e y
amilia;
wi h
magni ude/phase
ep esen a ion
in
he
equency
domain
ins ead
o
eal
and
imagina y
pa s
o
he
complex
Fou ie
ans o m.
In
o he
wo ds,
no
ma e
edundancy,
a
designe
can
always
ecognize
be e ,
a
low
pass
il e
in
a
magni ude-
-phase
plo
han
om
he
cu es
co es-
ponding
o
hei
eal
and
imagina y
pa .
The
poin
is,
up
o
wha
deg ee
he
eal
pa
o
he
anali
yc
signal,
he
gi en
signal
x(
),
p o ides
a
be e
ep esen-
a ion.
o
he
phenomena
unde
s udy.
Follow1ng
he
equency
domain
expe ience,
in
a
e y
heu is ical
way,
he
conclussion
will
be
ha
magni ude
and
ins an aneous
phase
a e
a
good
in o ma ion
suppo
o
~he
designe .
I'n
o he
wo ds,
en elope
and
1ns an aneous
equency,
looks
like,
hey
dese e
he
same
impo ance
in
he
ime
domain
ha
magni ude
and
phase
in
he
equency
domain.
In
his
pape ,
he
au ho s
will
explo e
he
po en ial
o
such
ep esen a ion
and
he
di icul ies
a ound
handling
i
in
CH
2185-7/85/0000-0107(8)/$01.00
© 1985 IEEE
signal
p ocessing
p oblems.
To
be
mo e
concise,
en elope
and
ins an aneous
e-
quency
a c
compu ed
o
simula ed
signals
using
wo
app oaches
and
he
p oblems
which
a ise
a e
epo ed.
Also,
and
dealing
wi h
a
p oblem
o
bandwidh
educ ion
in
a
signal
communica ion
sys em,
i
is
shown
he
ad an ages
o
well-known
spec al
es ima-
ion
p ocedu es
in
pa ame ic
models
o
en elope
and
ins an aneous
phase.
The
pape
is
o ganized
as
ollows:
Sec ion
II
p esen s
an
in oduc ion
o
he
gene al
concep s
o
in e es
abou
anali yc
signals.
Sec ion
III
epo s
he
ela ion-
ship
be ween
spec al
es ima ~on
p ocedu es
and
en elope
smoo hing.
A e
Sec ion
III,
he
nex
one
will
show
undamen al
ques ions
abou
en elope
and
ins an aneous
equency.
Finally,
some
p elimina y
esul s
a e
in oduced.
II
ANALITYC
SIGNAL
Any
signal
x( )
has
an
associa ed
complex
signal
a (
)
being
i s
Fou ic
ans o m
wice
hl
ig h
hand
side
o
he
o iginal
X(w)
• [ 2 X(w) ;
w>O
A (w)=
X 0 ;
w<O
(1)
Clea ly,
a (
)
is
a
complex
signal,
and
i s
ime
o mula ion
includes
as
eal
componen
he
o iginal
da a
signal
x( )
and
as
imagina y
componen
he
so-called
Hilbe
T ans o m
o
i .
ax( )
=
x( )
+ j
hx( )
whe e
hx( )=(l/
{
00
(X( '
)/( - '
))d '
(
2)
(3)
This
las
o mula
s eems
om
he
de ini-
ion
o
he
analy ic
signal
in
he
equen-
: ! .
...
. . ·. .
.......
·
~
..
--~-:..
..
· ..
,._
....
___
..
.
...
..
........
·-.
..
. .
108
A.
Mo eno
e
si.
cy
domain.
Mo e
conc e ly,
de i ~d
om
he
causal
condi ion
o
A (w).
I
1s
easy
~o
conclude
h!"~'
a
)
ol~::·s
he
same
ole
1n
he
ime
domainx
ha
X(w)
doe:;; .'.!!
he
e-
quency domain.
Howe e ,
he
main
conce n
o
::his wo k
a e
no
x( )
and
h
( ),
bu
he
al e na i e
ep esen a ion
s~own
in
(4),
a (
)
,. x(
)+jh
( )
= e
( )
exp
~
x( )
(4)
X X X
being
e (
)
and
41
(
)
he
al eady
en elop/
and
ins a~ aneous
phase.
e e ed
e 2 (
)
= x 2 (
)
+h
2 (
)
= I ax (
)
I 2
X X
(S.a)
~
( )
=
an-
1
(h
( )/x( ))
= Phase
X X
I
is
wo hwhile
o
men ion
ha
gi en
x(
)
he e
is
no
an
uniquiness
·
in
inding
unc ions
e
( )
and
~
( ).
To
check
his,
jus
add
s~me
esidu~l
only
o
he
ima-
gina y
oa
o
(2).
The
uniquiness
?
ex~ 2
cos(
(
))
in
ep esen 1ng
x(
)
l1e::;
wl
u
he
'*inimum
phase
condi ion
o
a
( ).
Exac ly,
he
en elope/phase
ep esen ~ ion
is
conec ed
wi h
he
anali yci y
o
a ( ;)
in
he
uppe
hand
side
o
he
(
plane.
¥ ~us
no
poles
can
be
inside
he
uppe
hand
s de
o
plane
in
o de
o
gua an ee
he
causa-
li y
cons ain
o
A
(w)
holds.
Wi h
espec
he
i~o ance
o
bo h
pa a-
me e s,
no e
ha
he
en elope
p o ides
in onna ion
conce ning
o
he
ime
ene gy
dis ibu ion
and
he
ins an aneous
equency
wi h
ze o-c ossing
in o ma ion.
III
SPA
METUODS
IN
ENVELOPE
REPRESENTATION
F om
he
de ini ion
o
en elope
o
a
gi en
analy ical
signal
i
can
be
in e ed
ha
he
squa e
o
he
en elope
can
be
iewed
as
a
ime
domain
pe iodog am.
In
o he
wo ds,
i
we
know
he
Fou ie
T ans o m
o
he
analy ic
signal
Ax(w),
we
can
ob ain
he
en elope
in
he
same
way
as
we
ob ain
he
Pe iodog am
o
a
gi en
da a
signal.
Fou ie
A
(w)
-------~
ax( )
-~1·1
2
-•en elope
~
.
T ans o m
Fcu ie 2
(6)
x( )
------•
X(w)
--•l'l
-•Pe iodog am
T ans o m
F om
he
p e ious
simila i y
we
can
conclude
ha
he
mos
amilia
p ocedu es
applied
in
pa ame ic
spec al
es ima ic:m
could
also
be
appl~d
o e
A
(w)
o
ob a n
an
es ima ion
o
e
( ).
x
Conc e ly,
he
m~
popula
maximum
en o-
py
echnique
can
be
used
o e
a
sampled
e sion
o
he
·causal
signal
A
(w).
I
A
(w)
is
gi en
as
a
da a
egis e
engh
o
W2
samples
wi h
index
1
(i.e.
Ax(l);
1 =
0,
N/2-1),
hen
a
linea
p edic o
o
::oe icien s
a
(q)
(q
=
l,Q)
can
be
designed
by
minimizing
he
squa e
e o
:
(1).
Thus
he
linea
p edic o
is:
"' Q
A ( 1 ) =
I:
(
q)
Ax
(
1-q)
( 7 )
X
q=l
The
p~dic o
esidual
:(l)
is
de ined
as
A
(1)-A
(1)
and
he
quan i y
o
be
m nimiz~d
is:
N2
2
I:
l :(lll
(8)
l=N1
.
The e
a e
many
well-known
p ocedu es
in
he
li e a u e
o
sol e
(8)
depending
on
he
choice
o
N1
and
N2•
We
selec
he
p ocedu e
o
co ~la ion,
ega dless
i
is
no
ecommended when
he
signal
unde
analysis
is
de e minis ic
in
na u e,
as
i
is
he
case
due
o
he
causal
cha ac e
o
A
(1).
Anyway,
he
consequences
de i ed
~om
he
use
o
Le inson
algo i hms
in
he
minimiza ion
o
(8)
will
be
g ea e
deg ee
o
smoo hing
o
low
esolu ion,
in
e ms
o
spec al
es ima ion,
u -
he
esul ing
pa ame ic
en elope
ep esen a ion.
O he
p ocedu es
can
be
ca ied
o e
he
p oblem
p c iosly
s a ed
in
(7)
and (8)
(sec
o
example-s~.
As
conce ns
wi h
his
pape ,
i
is
no
e y
ele an
he
spec al
es ima ion
p ocedu e
selec ed
and
he
eade
could
change
i
acco dingly
wi h
he
desi ed
ea u es
in
he
esul ing
en elope
es ima e.
The
en elope
es ima e,
one~
coe icien s
a(.)
ha e
been
ob ained,
can
be
de i ed
assuming
he
whi e
cha ac e
o
he
esidual
sequence
(1),
Wi h
his
assump ion
he
squa e
magni ude
o
he
in e se
disc e e
Fou ie
ans o m
e(n)
is
cons an ,
so
ha ;
£
(l)=a(,)
*A(,)
X
e
(1)=
8
(,)
•
ax(n)
(
9)
and
because
he
whi e
cha ac e
o
d
1)
is
assumed,
(10)
being
k
he
a e age
po~.Je
o
dl)
(i.e.
he
minPmum
o
he
objac
i e
in
he
design
p ocess
(8))
In
summa y,
2
K0 = E I 1 d 1
>I
1
;~(n)
= K0
/l8
Cnll
2
whe e Q
(ll.a)
(ll.b)
8(n)
=
1+!
a(q)
exp
(j2nqn
/N) (12)
q=l
The
es ima e
shown
in
(ll.b)
ha e
been
applied
succes ully
o
oice
speech
eco ds.
In
Fig
1
he
eade
can
see
such
!..
·.·
·.
..
-
~
. .
...
. .
_.
.·
::
·.:.
·.
·:
.:.
..
.
:=
·
····
•
:.
~
:
..
~
-
.
<
:-~-
<
: .
...
:-
.
:·
.
..
.
,.
.: .. ·.
Pa ame ic En elope
in
LPC Speech Code s 109
en elope
es ima e
compa ed
wi h
he
ac ual
~n clope
and
he
da a
signal
eco d
•.
The
da a
leng h
was
32
ms
and
he.
l1ne~
p edic ion
o de
20 •
I
can
be Vlewed 1n
his
plo
ha
he
LP
es ima e
p o ides
a
smoo hed
e sion
o
he
ac ual
en elope.
The
eplica
ob ained
wi h
he
es ima e
esul s
accu a e
when
he
o de
0
is
almos
double
han
he
numbe
o
pe iods
included
in
he
o iginal
signal.
(a)
(b)
(c)
Fig.l.
(a)
Voiced
speech
signal,
(b)
en clope,(c)
pa ame ic
en elope.
No e
ha
he
co esponding
analy ic
signal
o
he
so-called
pa ame ic
en elope
is
always
a minimum
phase
signal
in
he
sense
ha
a
(~)
has
no
ze os
in
he
uppe
hand
side
o
he
complex
plane
~
•
This
p ope y
gua an ees
ha
he
loga i m
o
he
es ima e
en elope
and
he
phase
o
a
(n)
a e
a
Hilbe
T ans o m
pai .
X
IV
INSTA."lTANEOUS
FREQUENCY
FROM
PARAMETRIC
ENVELOPE.
In
he
p eceding
sec ion
i
is
in oduced
how
en elope
in o ma ion
could
be
smoo hed
by
using
well-known
L.PC
echniques.
T~is
poin
is
e y
in e 'cs lng.
as
conce n~ w~ h
da a
a e
educ ion
o
s1gnal
ansm1ss1on
pu poses:
The
ques ion
which
emains
is
ha ,
a
non
minimum
phase
signal
~eods
also,
o
be
eco e ed
a
he
ece1 e ,
ins an aneous
phase
in o ma ion.
To
ealice
how
ins an aneous
phase
o
equency
can
be
ep esen ed
by
a
ini e
se
o
pa ame 2 s,
i
wi
11
be
in e es ing
o
explo e
how
classical
equency
disc i-
mina o s
wo k
o
de ec
ins an aneous
equency.
.
Conside ing
he
linea
sys em
depl ed
in
Fig.'2,
i
he
inpu
signal
is
gi en
wi h
e
( )
almos
cons an ,
he
ou pu
signal
wh.J.
be:
.
/
h( ')
e
( - ')
cos
<I>
( - '
)d ';
X ~ X
- e J
~
X( )
h( ')cos<l>
x( - ')
d '
-~
_
..
(13)
:
..
·
.·
·.· . . :· -.
-.
. . ·
..
::
·. . .
~'
..
. . .
· :
·:·
:· ·:·
·-~·
.
-.
-
..
..
'1',.
- >
Fig
2.
Ins an aneous
equency
. hough
linea
sys ems.
Thus,assuming
ha
wo
e ms
o
Taylo '
s
se ies
o
he
phase
e m
is
adequa e
o
ep esen
i
in
he
con olu ion
in eg al,
< >x( - ')
= < >x( ) -
'
~
x( )
he
ou pu
signal
will
be:
(14)
(15)
~
being
H(.)
he
ans e
esponse
o
he
linea
sys em
(i.e.
he
Fou ie
T ans o m
o
h( )),
F om
(15)
i
is
easy,
o
conclude
ha ,
being
e (
)
almos
cons an ,
he
en elope
o
y(
)
xis
gi en
om
ans e .
esponse
and
he
ins an aneous
equency
ljlx
o
he
inpu
signal.
This
easonin'J
d9
no
p ;eclu-
de
he
anali yci y
o
e
( ).H(
~
x)exp(J
~
xl'
hus
o
a oid
his
as~ump ion
he
aoo e
compu a ions
will
be
ca ied
o e
he
complex
signal
exp(j<l>x>·
The
men ioned
complex
signal
is
ob ained
om
he
quo ien
o
he
gi en
signal
a.x.( )
and
i s
ac ual
en elope
ob ained
wi n
a
Hilbe
ans o m.
x
( )
= a
( )/ex( )
m x (16)
No e
ha
x
( )
is
no
analy ic
in
gene al.
Only
n
he
case
o
ex( )
band
limi ed
o
w
and
cos
(cp
(
ll
no
spec
aly
o e lapped
iR
his
band,
~
( )
will
ha e
an
analy ic
cha ac e .
This
cWse
is
no
longe
ue
o
many
p ac ical
signals
x( ).
Rega dless
o
he
analy ic
condi ion
o
x
( ),
i
i
is
a~plied
o
a
linea
sys em
hT l,
he
ou ~u
signal
will
be:
•
ym( )
..
H(<l>xl
exp(j<Px)
(17)
As
a
consequenc~
we
could
use
p ocedu es
o
smoo h
E
{I
y
I}
in
he
same
ashion
we
use
hem
in
h~
p e ious
s:lc ion.
Fu he -
mo e
i
H(.)
is
a
de 'i a i c
in
he
ime
domain
he
magni ude
o
1
Ym( )
1
wil~
be
ap oxima ely
l$
( ll
•
In
summa y,
as.ln
a
classic
disc ilbina o ,
using
a
llnea
sys em
wi h
cons an
slope
equency
'es-
ponse
we
can
ob ain
he
ins an aneous
e-
quency
as
he
magni ude
o
a
complex
signal.
. . ; .•
.·
.
• -
',
•- ' I
~
• ... · .
...
..
. , . ..
'•· . . ·
'·
..
110 A. Mo eno
e
al.
No e
ha
o
eco e
exac ly
he
ins an-
aneous
~equency
i
mus be
posi i e.
This
p~oblem
could
be
a oided
by
adequa e
scaling
and
ca~ i~e
modula ion.
Anyway
he
designe
mus
gua an ee
he
basic
assump-
ion
o
no
ab up
phase
changes
in
he
ins an aneous
phase
(low
ins an aneous
equency)
o
de i e
he
p e ious
esul s.
To
check
ou
he
esul ing
pe o mance
o
he
p ocedu e,
a 128
da a
s
ample
eco d
wi h
ins
an an
eous
equency
e olu ion
as
is
shown
bellow,
was
used.
[
0.125
: n =
1,
32
i(n)
linea :
n = 33,
95
0.25
: n = 96,128
The
linea
sys em
was
implemen ed
by a
ans e
unc ion
ha
a ies
linea y
whi h
he
equency.
The
ins an aneous
equency
wich
esul s
can
be
iewed
in
Fig.
3
oge he
wi h
he
o iginal
signal
x(n).
Fig
2. Top,
o iginal
signal.
Bo om,
ins an aneous
~equency
de ec ed.
I
can
be iewed some
side
e ec s
due
o
he
1"indow
e ec s
in
he
de ec ed
ins an aneous
equency.
I
is
cspec ed
ha
his
dis o sion
wich
also
appea s
in
he
en elope
will
no
appea
in
a
con inous
p ocessing
o
he
signal
unde
analysis.
Anyway, om
he
esul s
ob ained
by
he
au ho s
he
app oxima ion
in
he
cen a
zone,
in
a
block
p ocessing
ashion,
is
accou a c
enough
o
ep oduce
he
o iginal
signal.
CONCLUSIONS
The 'main
con ibu ion
o
his
wo k
esides
in
e ealing
he
in e es
o
hese
unc-
ions
associa ed
o
any
signal
o
be
p o-
cessed.
No
essen ial
e o s
in
he
pas
ha e been
de o ~d
o
such
ep esen a ion
o€
signals,
jus
in
modul
a
ion
p oblems
some
au ho s
epo ed
in e es ing
esul s
in
such
ield
Cu en ly,
ime- equency
ep esen a ions
exhibi
a
new
look
o
he
p oblem.
We
eally
belie e
ha
his
unc ions
dese e
mo e
a en ion,
because
i
ia
well
ecognized
ha
essen ial
in o ma ion
is
in ol ed
inside
he
en elope
o
he
ins an aneous
phase
e olu ion.
In
his
pape
i
is
epo ed
how
en elope
and
ins
an
:<'
n
~'=''..!S
!::-equenc
a c
sui
abl!!
o
pa ame ic
me hods which
a e
ami 1 i
i'J
in
spec al
es ima ion
p oblems.
Ne e heless
he
main
guidelines
o
wo k
o e
hese
associa ed
uc ions
ha e
been
shown,
u he
wo k
mus
be
de o ed
o
signal
p ocessing
ools
in
o de
o
alle ia e
unce ainly
in
en elope/phase
ep esen a ion
o
signals.Also
side
e ec s
o
di e
e
n s
ways
o
ob ain
en elope
and
phase
will
be
explo ed
in
he
u u e.
REFERENCES
-1-
Cha les
Se homie .
'Ins an aneous
equency
and
Ene gy
Dis ibu ion
o
a
signal'.
Signal
P ocessing
5.(1983).
-2-
Pe
iclis
Y.
K onas
and
Nicola
Papp.
'Ins an aneous
En elope and
Phase
ex ac ion
om
eal
signals:
Theo y,
implemen a ion,
and
an
applica
ion
o
EEG
analysis'.
Signal
P ocessing
2
{1980).
-3-Alan
V.
Oppenhcim,
Ronald
w.
Sha e .
'Digi al
signal
P ocessing'
P en ice
Hall.l975.
-4-He be
s.
Voelcke .
'Towa d a
uni ied
Theo y
o
Modula ion.
Pa !.
Phase-En elo-
pe
Rela ionships'.
P oceedings
o
he
IEEE
ol.54:
n.3.
Ma ch 1966.
-5-S.M.Kay
and
S.L.Ma plc,J .
'Spec um
Analysis-
A Mode n
Pe spec i e'
P oceedings
o
he
IEEE
ol.
69:
n.ll.
No embe
19
81.
·'
.
"-
-
I
i
L
I
.
a;