Ingrid Daubechies and Fabrice Planchon

Description

Niveau: Supérieur, Doctorat, Bac+8
Adaptive Gabor transforms Ingrid Daubechies and Fabrice Planchon 1 Program in Applied and Computational Mathematics, Princeton University, Princeton NJ 08544-1000, USA Abstract We aim to provide time-frequency representations of a 1D signal where the window is locally adapted to the signal, thus providing a better readability of the representation. Introduction All the linear transforms which allow to depict a signal in phase space ( = time-frequency plane) have a blurring eect, because they typically introduce an auxiliary function which can be chosen arbitrarily and which serves as a \template: the window for a Gabor transform, the wavelet for a wavelet transform. Various attempts have been made to correct such blurring arti- facts while retaining the interesting properties of such transforms. Let us cite reallocation methods ([11]), or squeezing ([6]). On the other hand, bi- linear transform such as the Wigner transform don't introduce extraneous \templates, and have less blurring for some classes of signals. In partic- ular, the Wigner transform is well localized for linear chirps. However, for more complicated signals, interferences appear due to the quadratic nature of the transform; often these are diÆcult to separate from the interesting components in the representation.

• wigner transform don'


• gabor transform


• shows blurring


• time-frequency plane


• various adaptive


• such


• linear chirps

Sujets

• Planchon
• Princeton University
• Gabor transform

Informations

• Publié par: profil-zyak-2012
• Nombre de lectures: 53
• Langue: English

Extrait

Adaptiv
e
Gab
or
transform
onen
t
transforms
hirps.
Ingrid
cost
Daub
([11]),
ec
of
hies
quadratic
and
correct
F
Gab
abrice
et
Planc
bi-
hon
ha
1
is
Program
signals,
in
dicult
Applied
metho
and
they
Computational
aim
Mathematics,
e
Princeton
URA
Univ
75
ersit
On
y
h
,
duce
Princeton
blurring
NJ
ular,
08544-1000,
calized
USA
er,
Abstract
ear
W
transform;
e
the
aim
represen
to
this
pro
([4,
vide
a
time-frequency
linear
represen
v
tations
that
of
simple
a
d'Analyse
1D
ersit
signal
place
where
Cedex
the
squeezing
windo
other
w
transform
is
the
lo
in
cally
\templates",
adapted
e
to
some
the
In
signal,
Wigner
th
ell
us
linear
pro
w
viding
more
a
terferences
b
to
etter
of
readabilit
these
y
separate
of
teresting
the
in
represen
Adaptiv
tation.
exist
In
text
tro
h
duction
]),
All
suer
the
com-
linear
compared
transforms
W
whic
pro
h
adaptiv
allo
represen
w
eha
to
ell
depict
1
a
Lab
signal

in
189,
phase
e
space
Curie,
(
BP
=
P
time-frequency
ds
plane)
or
ha
([6]).
v
the
e
hand,
a
linear
blurring
suc
eect,
as
b
Wigner
ecause
don't
they
tro
t
extraneous
ypically
and
in
v
tro
less
duce
for
an
classes
auxiliary
signals.
function
partic-
whic
the
h
transform
can
w
b
lo
e
for
c
c
hosen
Ho
arbitrarily
ev
and
for
whic
complicated
h
in
serv
app
es
due
as
the
a
nature
\template":
the
the
often
windo
are
w
to
for
from
a
in
Gab
comp
or
ts
transform,
the
the
tation.
w
e
a
ds
v
in
elet
con
for
to
a
suc
w
problems
a
10
v
but
elet
usually
transform.
from
V
high
arious
putational
attempts
when
ha
to
v
transforms.
e
e
b
at
een
viding
made
arious
to
e
correct
or
suc
tations
h
b
blurring
v
arti-
w
facts
on
while
signals,
retaining
curren
the
address:
in
oratoire
teresting
Num
prop
erique,
erties
CNRS
of
Univ
suc

h
Pierre
transforms.
Marie
Let
4
us
Jussieu
cite
187,
reallo
252
cation
aris
metho
1and
can
b
This
1)-
It
e

computed
(
at
(
reasonable
hirps,
cost.
suc
T
simply
ec
2
hniques
or
of
j
this
when
t
frequency
yp
x
e

ha
the
v
of
e
=
also

b
e
een
Wigner-Ville
dev

elop
)
ed
(
in
v
[1],
calized
where
=
they
x
in
W
tro
the
duce
for
the
w
so-called
;
w
transform.
arp
windo
ed
)(
Gab

or
2
rep-

resen
kno
tation.
energy
Other
con
adaptiv
with
e
)
metho
)(
ds
(
are
f
presen
j
ted
2
in
(
[13
lik
],
is
whic
along
h
\sharpness"
are

based
p;q
on
f
an
dx:
adaptiv
will
e
a
Wigner-Ville

transform
or
(using
(0
dieren
or
t
often
windo
el
ws).
calling
This
Gab
later
Wigner-Ville
approac
h
h
reads
ho

w

ev
e
er

do
)
es
2
not
+
solv
:
e
w
the
that
problem
obtain
of
the
cross-term
b
eects.
olving
1
of
Wigner-Ville
Gaussian
transform

and
j
Gab
(
or
q
trans-
=
form

W
W
e
:
recall
wh
that,

giv
)
en
ws
a
to
1D
).
signal
simple
f
linear
(
Wigner-Ville
x
erfectly
),
a
its
instan
Wigner-Ville
while
transform
)
is
Z
giv
g
en
;
b
(
y
)
W
(
(
)
f
(4)
)(
e
x;
call

h
)
transform
=
(
Z
;
e
)-Gab
i
transform;
u
the
f
;
(
Gab
x
transform
+
e
u
drop
2
lab
)
(0
f
1),
(
it
x
the
u
or
2
The
)
transform
du:
suc
(1)
a
Let's
w
de