TELKOM
NIKA Indonesia
n
Journal of
Electrical En
gineering
Vol. 12, No. 10, Octobe
r 20
14, pp. 7059
~ 706
9
DOI: 10.115
9
1
/telkomni
ka.
v
12i8.590
1
7059
Re
cei
v
ed Ma
rch 2, 2
014;
Re
vised June
24, 2014; Accepte
d
Jul
y
1
4
, 2014
Necessary Conditions of the Wave Packet Fr
ames w
i
th
Several Generators
Tao Zhao*, X
i
ao
y
u
Zhou
Dep
a
rtment of Mathematics a
nd Informatio
n
Sc
ienc
e, Hen
a
n
Univ
ersit
y
of Econom
ics an
d La
w
,
Z
hengz
ho
u 45
000
2, P. R. China
*Corres
p
o
ndi
n
g
author, e-ma
i
l
: taozhao
19
77
@16
3
.com
A
b
st
r
a
ct
The
m
a
in goal
of this paper is
to cons
ider
th
e nec
essary c
o
nditions
of wave pack
e
t system
s to
be
frames
in
hig
h
e
r di
mensi
ons.
T
he nec
essar
y
cond
itions
of
w
a
ve packet f
r
ames i
n
hi
gh
e
r
di
me
nsio
ns w
i
th
severa
l gen
era
t
ors are establ
i
s
hed,
w
h
ich in
clud
e the corre
spon
din
g
resu
lts of w
a
velet analysis a
nd Gab
o
r
theory as the
special cas
e
s. The
existing results are
generali
z
ed to
the
case of sev
e
ral generators
and
gen
eral l
a
ttices
.
Ke
y
w
ords
:
w
avel
et, the w
a
ve packet fra
m
e
,
necessary co
nditi
on
Copy
right
©
2014 In
stitu
t
e o
f
Ad
van
ced
En
g
i
n
eerin
g and
Scien
ce. All
rig
h
t
s reser
ve
d
.
1. Introduc
tion
Frame
s
we
re first introd
uce
d
by Du
ffin and Sch
aeffer (195
2
)
in the
co
ntext o
f
nonh
arm
onic Fouri
e
r
se
ri
es. Out
s
ide
of sign
al pr
o
c
e
ssi
ng, fram
es di
d not
seem to ge
ne
rate
much inte
re
st
until the grou
nd bre
a
ki
ng
work of Da
ub
echi
es
et al
. (1986
). Since
then, the theor
y
of frames be
gan to be m
o
re wi
dely st
udied. Tr
aditionally, frame
s
have bee
n
used in si
g
nal
pro
c
e
ssi
ng, image p
r
o
c
essing, data co
mpre
ssion,
a
nd sam
p
ling
theory. Re
ce
ntly, frames are
also
used to
mitigate the e
ffect of losse
s
in
p
a
cket
-b
ase
d
commu
nicatio
n
sy
ste
m
s a
nd he
nce to
improve the robu
stne
ss of
data tran
smission [Ca
s
a
zza and Kovae
v
i, 2003; Goyal
et al
., 2001],
and to
de
sig
n
high
-rate
co
nstellatio
n
with full dive
rsity in multipl
e
-antenn
a
cod
e
de
sign
[Ha
s
sibi
et al
., 2001]. We refer to the mono
grap
h of Da
ube
chie
s (1
9
92) or the rese
arch
-tutori
a
l
[Chri
s
ten
s
en,
2002] for ba
sic pro
p
e
r
ties
of frames.
An import
ant
example
ab
out frame i
s
wavele
t fram
e, whi
c
h i
s
o
b
tained
by transl
a
ting
and
dilating
a finite famil
y
of functio
n
s
. Wavelets
were int
r
odu
ced
rel
a
tively re
cently, in
the
begin
n
ing of
the 198
0. Th
ey attracted
con
s
id
era
b
le
intere
st from
the mathem
a
t
ical commu
n
i
ty
and from me
mbers of ma
ny diverse di
sci
pline
s
in which wavelets had promi
s
i
ng appli
c
atio
ns.
D
a
ub
ec
h
i
es
et al
. (1986) combin
ed th
e theory of the co
ntinuou
s wavel
e
t transform with the
theory of fra
m
es to d
e
fin
e
wavel
e
t fra
m
es fo
r
2
()
n
LR
. In
1990,
Dau
b
e
c
hie
s
(199
0)
obtaine
d
the first
re
sul
t
on the
ne
cessary
co
ndit
i
ons for
affine fram
es,
an
d then i
n
1
9
93, Chui a
n
d
Shi
(199
3) o
b
tain
ed an imp
r
ov
ed re
sult. After ab
out ten
years,
Ca
sa
zza an
d Chri
stense
n
(2
001
)
establi
s
h
ed a
stron
ger
con
d
ition whi
c
h
also
wo
rks fo
r wavelet fra
m
e. Re
cently
, Shi and his
co-
authors [Shi
and
Che
n
, 2005; Shi a
nd Shi, 200
5] obtaine
d the ne
ce
ssary conditio
n
s
an
d
s
u
ffic
i
ent c
o
nditions
of wavelet frames
.
Another mo
st
impo
rtant
co
ncrete
reali
z
a
t
i
on of frame
is Ga
bor fra
m
e. Ga
bor
systems
(Weyl-Heisen
berg
system
s) were
first intro
d
u
c
ed
by Gab
o
r (1
946).
They
a
r
e
gen
erate
d
by
modulatio
ns
and tra
n
sl
ations
of a fini
te family
of functio
n
s. In
2007, Shi a
n
d
Ch
en (200
7)
establi
s
h
ed some ne
w ne
ce
ssary con
d
itions fo
r G
abor frame
s
.
These co
nd
itions a
r
e al
so
sufficie
n
t for tight frame
s
. In pape
r [Li a
nd Wu, 2
001]
, Li and Wu
pre
s
ente
d
two new
suffici
ent
con
d
ition
s
for Gabo
r frame
via Fourie
r transfo
rm
. The
conditio
n
s th
ey prop
osed
were state
d
in
terms of the Fouri
e
r tran
sf
orm
s
of the Gabo
r sy
ste
m
's ge
neratin
g functi
on
s, and the con
d
itions
were
better than th
at of
Daube
chi
e
s.
F
u
rthe
rmo
r
e, i
n
pa
pe
r [Li
et al
., 2001], Li
et al
. e
s
tabli
s
hed
a ne
ce
ssary
con
d
ition an
d
two sufficien
t conditio
n
s
e
n
su
ring th
at the shift-invari
ant system i
s
a
frame for
2
()
n
LR
. As som
e
ap
plicatio
ns, th
e re
sults a
r
e use
d
to obtain some
kno
w
n
con
c
lu
sio
n
s a
bout wavel
e
t frame
s
and G
abor frame
s
.
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ISSN: 23
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046
TELKOM
NI
KA
Vol. 12, No. 10, Octobe
r 2014: 705
9
– 7069
7060
In pap
er [Co
r
doba
an
d F
e
fferman,
197
8
], aut
hors i
n
trodu
ced
wave
pa
cket
syste
m
s
by
applying
ce
rt
ain colle
ction
s
of dil
a
tion
s, modul
atio
n
s
and tran
slatio
ns to th
e Ga
u
ssi
an fun
c
tion
in
the
stu
d
y
of some cla
s
se
s of
sin
gula
r
i
n
tegral
op
erato
r
s. In
pa
per [
Labate
et al
.,
200
4], autho
rs
adopte
d
the
same
expression to
de
scribe any
colle
ction
s
of fun
c
tion
s which
are o
b
taine
d
by
applying the
same
ope
rati
ons to a finit
e
family
of functio
n
s. In fac
t, Gab
o
r
systems, wavelet
system
s a
n
d
the Fo
urie
r tran
sform
o
f
wavelet
sy
stem
s a
r
e
speci
a
l
cases of wave pa
cket
system
s.
Wa
ve pa
cket sy
stem
s
have
rece
ntly been
su
cce
ssfully
appli
ed to
some p
r
o
b
lem
s
in
harm
oni
c ana
lysis an
d ope
rator the
o
ry [Lacey and Th
iele, 1997; La
cey and Thi
e
l
e
, 1999].
The m
a
in g
o
a
l of this pap
er i
s
to
con
s
i
der th
e ne
ce
ssary
co
nditio
n
s of
multiwa
v
e packet
frame
s
in
hig
her dimen
s
io
ns. We esta
b
lish som
e
n
e
c
e
s
sary con
d
i
tions co
nditio
n
s
fo
r
the
wa
ve
packet
f
r
ame
s
of the different op
erat
or ord
e
r in
2
()
n
LR
with matrix dilations of
the form
()
(
)
(
)
Df
x
q
f
A
x
, where
A
is an a
r
bitra
r
y expandin
g
nn
matrix
with integer
coeffici
ents a
nd
||
qd
e
t
A
, which i
s
a ge
ne
raliza
t
ion of cl
assi
cal
wavelet f
r
ame
and
G
abor
frame. Of co
urse, our
wa
y combine
s
with so
me tech
niqu
es in
wavelet an
alysis a
nd time-
freque
ncy an
alysis. In parti
cula
r, we u
s
e
some thou
gh
ts of Chui an
d Shi (199
3) i
n
cla
ssifying t
he
necessa
ry co
ndition fo
r the
Gab
o
r frame
.
Also, we
di
scu
s
s ne
ce
ssa
r
y co
ndition
s
for othe
r
wav
e
packet f
r
ame
s
with the
dif
f
erent
ope
rat
o
r
ord
e
r. Al
so, we
fu
se
some
way
s
in
wavel
e
t an
al
ysis
and Ga
bor th
eory and
we
mainly borro
w som
e
thou
ghts
in cl
assif
y
ing the sufficient con
d
ition
s
of
the wavelet frame in
pa
pers [Shi a
nd
Ch
en, 20
07; Li
a
nd
Wu, 2
001;
Li
et al
., 200
1; Shi an
d Sh
i,
2005].
2. Prelimilari
es
In this section, some notati
ons and
som
e
results
whi
c
h w
ill be used later are introduced.
Throughout this pape
r, the followi
ng notations
will
be used.
n
R
and
n
Z
den
ote the
set of n
-
dimen
s
ion
a
l
real
numb
e
rs and the
set of intege
rs,
respe
c
tively.
2
()
n
LR
is the
sp
ace of all
squ
a
re
-inte
g
rable fun
c
tion
s, and
,
and
·
‖‖
denote th
e in
ner p
r
o
d
u
c
t and no
rm in
2
()
n
LR
,
respe
c
tively, and
()
n
lZ
denotes
the spa
c
e of
all squ
a
re
su
mmable
seq
u
ences.
For
12
(,
,
,
)
n
n
x
xx
x
R
, define:
22
2
12
||
n
x
xx
x
We d
enote
b
y
n
T
the n-dim
e
nsio
nal torus.
By
()
pn
LT
we d
eno
te the spa
c
e
of all
n
Z
-p
erio
di
c
function
s
(.
.
,
f
ie
f
is 1-pe
riodi
c in e
a
ch vari
able
)
su
ch that
|(
)
|
n
p
T
fx
d
x
.
We u
s
e the F
ourie
r tran
sfo
r
m in the form:
2·
ˆ
()
(
)
,
n
ix
R
f
fx
e
d
x
W
h
er
e
·
den
otes th
e
sta
ndard in
ne
r
prod
uct
in
n
R
, and
we
ofte
n omit it
wh
en
we
ca
n
unde
rsta
nd th
is from the ba
ckgro
und. So
metimes,
ˆ
()
f
is defined by
f
F
.
The exp
andi
n
g
matri
c
e
s
m
ean th
at all ei
genvalu
e
s
ha
ve magnitu
de
gre
a
ter th
an
1. We
denote
the
set of the
exp
andin
g
mat
r
ices
a
s
n
E
. Let
()
n
GL
R
denote
the
set of all
nn
non-
sing
ular
(o
r i
n
v
e
rtible
) ma
trice
s
with
re
al entrie
s
. F
o
r
()
n
A
GL
R
, we den
ote by
*
A
the
transpo
se of
.
A
It is obvious t
hat
*
n
A
E
. For
()
n
B
GL
R
we
denote by
1
B
the invertible
matrix of
B
. For the sa
ke of
simpli
city, we denote
*1
()
A
by
A
♯
.
Let us recall t
he definition
of frame.
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TELKOM
NIKA
ISSN:
2302-4
046
Ne
ce
ssary Condition
s of the Wa
ve Pa
cket Fram
e
s
with Several G
enerators (T
a
o
Zhao
)
7061
Defini
tion 1.
Let
H
be a sep
a
rabl
e Hilb
ert
spa
c
e. A se
quen
ce
{}
ii
N
f
of elements of
H
is a frame for
H
if
there exist
con
s
tants
0
CD
s
u
c
h
that for all
f
H
, we have:
2
22
,.
i
iN
Cf
f
f
D
f
(1)
The num
be
rs
,
CD
are called l
o
we
r an
d up
per fra
m
e bo
und
s, re
spe
c
t
i
vely (the larg
est
C
and
the sm
alle
st
D
for which (1) hold
s
a
r
e
th
e optimal
fra
m
e bo
und
s).
Tho
s
e
se
qu
ences which
satisfy only the uppe
r ineq
u
a
lity in (1) are
call
ed Be
sse
l
sequ
en
ce
s. A frame is tig
h
t if
CD
.
If
1
CD
, it is called a Parseval fra
m
e.
Let
f
T
denote t
he synth
e
si
s operator of
{}
ii
N
ff
, i.e.,
()
f
ii
i
Tc
c
f
for eac
h
seq
uen
ce of
scal
a
r
s
()
ii
N
cc
. T
hen the fra
m
e ope
rato
r
*
()
ff
Sh
T
T
h
asso
ciat
e
d
wit
h
{}
ii
N
f
is a bo
und
e
d
, invertible,
and po
sitive
operator m
a
p
p
ing of
H
on itself. This
pro
v
ides
the recon
s
tru
c
tion form
ula:
11
,,
.
,
ii
i
i
ii
hh
f
f
h
f
f
h
H
(2)
Whe
r
e
1
ii
f
Sf
. The family
{}
ii
N
f
is also
a frame fo
r
H
,
calle
d the ca
noni
cal du
al frame
of
{}
ii
N
f
. If
{}
ii
N
g
is any se
quen
ce in
H
which
sat
i
sf
ie
s:
11
,,
,
,
ii
i
i
ii
hh
g
f
h
f
g
h
H
(3)
It is called an
alternate d
ual
frame of
{}
ii
N
f
.
In
this pape
r,
we will work with
th
ree
families
of u
n
itary op
erat
ors on
2
()
n
LR
. Let
n
A
E
and
,(
)
n
B
CG
L
R
. The first
one con
s
ist
s
of the dilation operator
22
:(
)
(
)
nn
A
D
LR
LR
defined
by
1
2
()
(
)
(
)
A
Dx
q
f
A
x
with
||
qd
e
t
A
. The se
con
d
on
e
con
s
i
s
ts of all
translatio
n
o
perato
r
s
22
:(
)
(
)
,
,
nn
n
Bk
TL
R
L
R
k
Z
Define
d by
()
(
)
(
)
Bk
Tf
x
f
x
B
k
.
The third o
n
e
con
s
ist
s
of the modulatio
n operator.
22
:(
)
(
)
,
,
nn
n
Cm
E
LR
LR
m
Z
Define
d by
2·
()
(
)
(
)
.
iC
m
x
Cm
E
fx
e
f
x
Let
PZ
and
n
QR
. Let
SP
Q
. Then, we h
a
ve
n
SZ
R
. Again, l
e
t:
{:
}
pp
n
A
AP
E
and
()
n
BG
L
R
.
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ISSN: 23
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046
TELKOM
NI
KA
Vol. 12, No. 10, Octobe
r 2014: 705
9
– 7069
7062
For the func
tions
2
()
,
1
,
2
,
,
ln
LR
l
L
,
we
will consider
the wave packet system
defined by the followin
g
:
1,
2
,
,
,
,
(
,
)
{(
)
}
n
p
l
AB
m
lL
m
Z
p
S
DE
T
x
(5)
Let
()
,
{
0
}
j
p
AA
j
Z
S
Z
. Then, we obtain th
e wavelet
syste
m
s. On th
e o
t
her
side, we ca
n get the G
abor
system
s wh
en the
set
{:
}
pp
A
AP
only con
s
i
s
ts of the
elementa
r
y m
a
trix
.
E
This si
mple
ob
serva
t
ion al
ready
s
ugge
sts
that the
wave pa
cket system
s
provide greater flexibility th
an the wave
l
e
t system
s or t
he Gabor sy
stem
s.
By chan
ging
the order of
the ope
rato
rs, we
can
also define
the
followin
g
on
e
-
to-o
ne
function syste
m
s
from
n
SZ
into
2
()
n
LR
:
1
,,
()
|
(
)
,
,
(
,
)
,
p
ln
pm
A
B
m
x
DT
E
x
m
Z
p
S
2
,,
()
|
(
)
,
,
(
,
)
,
p
ln
pm
A
B
m
x
ED
T
x
m
Z
p
S
3
,,
()
|
(
)
,
,
(
,
)
,
p
ln
pm
B
m
A
x
ET
D
x
m
Z
p
S
4
,,
()
|
(
)
,
,
(
,
)
,
p
ln
pm
B
m
A
x
TD
E
x
m
Z
p
S
5
,,
()
|
(
)
,
,
(
,
)
.
p
ln
pm
B
m
A
x
TE
D
x
m
Z
p
S
(
6
)
Then, we
wil
l
give the definitions of wa
ve packet
multiwavelet
frame an
d the frame
wave pa
cket multiwavelet.
Defini
tion 2.
We say that the wave p
a
cket syst
em d
e
fined
by (5) is a wave p
a
cket
multiwavelet
frame if it i
s
a fram
e for
2
()
n
LR
. Then, th
e f
unctio
n
s
12
(,
,
,
)
M
is
calle
d a fram
e wave pa
cke
t
multiwavelet.
For
other wave pa
cket
system
s
(1
5
)
i
i
de
fined by (6),
we
ca
n def
ine the
corre
s
p
ondin
g
wave pa
cke
t
frames an
d t
he frame
wav
e
packet
s
like
definition 2.2
.
In orde
r to prove the main
results to be
pre
s
ente
d
in
next sectio
n, we ne
ed the f
o
llowin
g
lemmas.
Lemma 2.1.
Suppo
se
that
1
{}
kk
f
is a family
of elements i
n
a Hilbert space
H
suc
h
that there exi
s
t con
s
tants
0
CD
sat
i
sf
y
i
ng
(
2
)
f
o
r all
f
belon
ging to
a
den
se
su
bset
D
of
H
. Then, the sam
e
inequ
alities (2
) a
r
e
true for all
f
H
; t
hat is
,
1
{}
kk
f
is a frame for
.
H
For p
r
oof of L
e
mma 2.1, pe
ople can refe
r to the book [Dau
be
chie
s, 1990].
Therefore, we will consi
d
er t
he following set of functi
ons:
2
ˆ
ˆ
()
:
(
)
a
n
d
.
ha
s com
p
a
c
t support in
{
0
}
nn
n
fL
R
f
L
R
f
D
R
‚
The followi
ng
result i
s
well
kno
w
n, we ca
n find it in [D
aube
chi
e
s, 1
992].
Lemma 2.2
D
is a den
se
su
bset of
2
()
.
n
LR
The followi
n
g
useful fa
cts ca
n be fo
und in pap
er [Christe
nse
n
and Rahi
mi, 2008,
Lemma 2.2].
Lemma 2.3
Let
()
n
A
GL
R
,
,
n
yz
R
and
2
()
n
f
LR
. Then the followi
ng hold
s
:
(1)
ˆ
ˆ
()
,
(
)
,
yz
z
Tf
E
y
f
E
f
T
f
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TELKOM
NIKA
ISSN:
2302-4
046
Ne
ce
ssary Condition
s of the Wa
ve Pa
cket Fram
e
s
with Several G
enerators (T
a
o
Zhao
)
7063
ˆ
()
;
A
A
Df
D
f
♯
(2)
2·
,
iz
y
yz
z
y
TE
f
e
E
T
f
*1
,;
Ay
A
A
y
A
Ay
A
y
D
E
f
E
Df
D
T
f
T
Df
(3)
2·
ˆ
()
;
iz
y
yz
z
y
TE
f
e
T
E
f
(4)
ˆ
()
(
)
(
)
Ay
Ay
A
DT
f
E
D
f
♯♯
1
1
2·
2
ˆ
||
(
)
.
iA
y
detA
f
A
e
♯
3. Nece
ssary
Condition
s of Wav
e
Packet Fr
ames
Motivating by the fundament wo
rks [C
hriste
nse
n
, 2002; Chui and Shi
,
1993;
Dau
b
e
c
hie
s
,
1990], we
will
give a n
e
cessary
co
nditio
n
of wave pa
cket fram
e
defined by
(5
)
for highe
r dim
ensi
on with a
n
arbitrary expan
sive matri
x
dilation in the followin
g
.
Theorem 3.1
.
Suppose th
at wave pa
cket system:
1
,
2
,
,,
,
(
,)
{(
)
}
n
p
l
AB
m
lL
m
Z
p
S
DE
T
x
(7)
Define
d by (5
) is a fram
e with frame bou
nds
1
A
and
2
A
, then we have:
2
12
1(
,
)
|(
)
|
,
.
.
,
L
l
p
lp
S
bA
A
b
A
a
e
♯
(
8
)
Whe
r
e
b
det
B
.
Proof. Becau
s
e wave pa
cket system:
1
,
2
,
,,
,
(
,)
{(
)
}
n
p
l
AB
m
lL
m
Z
p
S
DE
T
x
Is a frame wit
h
frame bo
un
ds
1
A
and
2
A
, for all
2
()
n
f
LR
, we have:
22
2
1
2
1(
,
)
|,
|
.
n
p
L
l
AB
m
mZ
lp
S
Af
f
D
E
T
A
f
‖‖
‖‖
(
9
)
Let
ˆ
()
c
f
CR
and
ˆ
f
have comp
act
su
pport.
Let
||
pp
qd
e
t
A
. Accordin
g to Lemma 2
.
3 and Planch
e
ral theo
rem,
we have:
2
(,
)
|,
|
n
p
l
AB
m
mZ
pS
fD
E
T
2
(,
)
|,
|
n
p
l
AB
m
mZ
pS
fD
E
T
FF
2
(,
)
ˆ
|,
|
n
p
l
Bm
mZ
A
pS
fD
T
E
♯
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NI
KA
Vol. 12, No. 10, Octobe
r 2014: 705
9
– 7069
7064
1
ˆ
|(
)
(
)
nn
l
p
mZ
R
pP
Q
p
qf
A
♯
2(
)
2
|
p
iBm
A
ed
♯
*
ˆ
|(
(
)
)
nn
pp
mZ
R
pP
Q
qf
A
22
()
|
li
B
m
ed
(
1
0
)
Whe
r
e we ch
ange vari
able
s
by
p
A
♯
in the last equality.
We a
s
s
e
rt:
*
22
ˆ
|(
(
)
)
()
|
nn
j
p
mZ
R
pP
Q
li
B
m
qf
A
ed
([
0
,
1
]
)
(,
)
2
*
|
()
|
.
ˆ
((
)
)
n
p
B
pS
l
q
p
b
n
sZ
Bs
d
fA
B
s
♯
♯
♯
(
1
1
)
For fixed
(,
)
pS
, we
have:
*
([
0
,
1]
)
ˆ
|(
(
)
)
(
)|
nn
l
p
Bs
Z
fA
B
s
Bs
d
♯
♯
♯
*
([
0
,
1
]
)
ˆ
|(
(
)
)
(
nn
l
p
sZ
B
fA
B
s
♯
♯
)|
Bs
d
♯
*
([
0
,
1]
)
ˆ
|(
(
)
)
(
)
|
nn
l
p
sZ
B
s
B
f
Ad
♯♯
*
ˆ
|(
(
)
)
(
)
|
n
l
p
R
f
Ad
1
*2
2
ˆ
(|
(
(
)
)
|
)
n
p
R
fA
d
1
2
2
(|
(
)
|
)
n
l
R
d
.
(
1
2
)
Whe
r
e the fo
urth ineq
ualit
y is obtained
by using
Cau
c
hy-S
ch
warz'
s
ineq
uality.
Thus we can define
a
fun
c
tion
:
p
FR
C
by:
*
ˆ
()
(
(
)
)
n
pp
sZ
Ff
A
B
s
♯
()
,
.
.
.
l
B
sa
e
♯
(
1
3
)
()
p
F
is
n
B
T
♯
-pe
r
iodi
c, and the ab
ove
argu
ment gi
ves that
1
()
(
[
0
,
1
]
)
n
p
FL
B
♯
. In
fact, we even
have
2
()
(
[
0
,
1
]
)
n
p
FL
B
♯
. To s
e
e this
, we firs
t s
ee that:
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
Ne
ce
ssary Condition
s of the Wa
ve Pa
cket Fram
e
s
with Several G
enerators (T
a
o
Zhao
)
7065
2*
2
2
ˆ
|
(
)|
|
(
(
)
)|
|(
)
|
.
n
n
pp
sZ
l
sZ
Ff
A
B
s
Bs
♯
♯
(
1
4
)
Since
ˆ
()
c
f
CR
, the fu
nction:
*2
ˆ
|(
(
)
)
|
n
p
sZ
fA
B
s
♯
Is boun
ded.
Acco
rdi
ng to above argum
ent, we ea
sily get
2
()
(
[
0
,
1
]
)
.
n
p
Fx
L
B
♯
Then, a
cco
rdi
ng to the definition of
()
p
F
, we have:
*2
ˆ
((
)
)
(
)
n
li
B
m
p
R
f
Ae
d
*
([
0
,
1]
)
ˆ
((
)
)
(
)
nn
l
p
sZ
B
s
B
fA
♯
♯
2
iB
m
ed
*
([
0
,
1]
)
ˆ
((
)
)
(
nn
l
p
sZ
B
fA
B
s
♯
♯
2
)
iBm
Bs
e
d
♯
*
([
0
,
1
]
)
ˆ
((
)
)
()
n
n
p
sZ
B
l
fA
B
s
Bs
♯
♯
♯
2
iB
m
ed
2
([
0
,
1
]
)
()
.
n
iBm
p
B
Fe
d
♯
(
1
5
)
Parseval's e
q
uality sho
w
s t
hat:
22
([
0
,
1
]
)
|(
)
|
nn
iB
m
p
mZ
B
Fe
d
♯
2
([
0
,
1]
)
1
|(
)
|
;
n
p
B
F
d
b
♯
(
1
6
)
Combi
n
ing
(1
5), (16
)
and t
he definition
of
()
p
F
, we obtain
that:
*2
2
ˆ
|(
(
)
)
(
)
|
nn
li
B
m
p
mZ
R
fA
e
d
*
([
0
,
1
]
)
1
ˆ
|(
(
)
)
nn
p
Bs
Z
fA
B
s
b
♯
♯
2
()
|
.
l
B
sd
♯
(
1
7
)
So, we obtain
(11). Th
us, we compl
e
te the assertio
n.
Cho
o
s
e
0
R
to be Lebe
sg
ue p
o
int of the function
2
(,
)
|(
)
|
l
p
pS
A
♯
.
Letting
()
B
ò
den
ote the ball of radiu
s
0
ò
about the origi
n
and
ò
be s
u
ffic
i
ently s
m
all,
define
f
ò
by:
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NI
KA
Vol. 12, No. 10, Octobe
r 2014: 705
9
– 7069
7066
()
0
1
ˆ
()
(
)
.
|(
)
|
B
f
B
òò
ò
Therefore, we obtain:
22
ˆ
1.
ff
‖‖
‖‖
òò
Thus, we
hav
e:
2
0
(,
)
|(
)
|
l
p
pS
A
♯
0
2
||
0
(,
)
1
lim
|
(
)
|
.
|(
)
|
l
p
pS
Ad
B
♯
ò
ò
ò
(
1
8
)
From the d
e
finition of
f
, (9), (10
)
and (11
)
, we have:
0
2
||
1(
,
)
1
|(
)
|
|(
)
|
L
l
p
lp
S
Ad
B
♯
ò
ò
22
([
0
,
1
]
)
1(
,
)
ˆ
|
(
)|
|
(
)|
n
L
l
B
lp
S
f
Ad
♯
♯
ò
([
0
,
1]
)
1(
,
)
*2
|
ˆ
((
)
)
(
)
|
nn
L
p
Bs
Z
lp
S
l
p
q
f
AB
s
B
s
d
♯
♯♯
ò
2
1(
,
)
|,
|
n
p
L
l
AB
m
mZ
lp
S
bf
D
E
T
ò
2
,
bA
(
1
9
)
Whe
r
e the thi
r
d equ
ality is obtaine
d by changi
ng varia
b
les
*
()
p
A
.
Let
0
ò
, using th
e definition of
Lebe
sgu
e
po
int, we get:
2
02
1(
,
)
|(
)
|
.
L
l
p
lp
S
Ab
A
♯
(20)
Acco
rdi
ng to the definition
of Lebe
sgu
e
poin
t, by the simila
r tech
ni
que of Chui
and Shi
[1993], we ob
tain:
2
01
1(
,
)
|(
)
|
.
L
l
p
lp
S
Ab
A
♯
(21)
We leave the
assertio
n to reade
rs.
Comp
ari
ng wi
th (20) a
nd (2
1), by chan
gi
ng variabl
es
by
0
, we have (9).
Therefore, we have com
p
l
e
ted
the pro
o
f
of Theorem
3.1.
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
Ne
ce
ssary Condition
s of the Wa
ve Pa
cket Fram
e
s
with Several G
enerators (T
a
o
Zhao
)
7067
Remar
k
3.1.
In partic
u
lar, let
A
the el
eme
n
tary matrix
E
i
n
the T
heo
re
m 3.1, then,
we
obtain th
e n
e
c
e
s
sary
co
ndi
tion of th
e G
a
bor fram
es
a
s
the follo
win
g
, whi
c
h
is a
ge
nerali
z
atio
n o
f
the kno
w
n re
sult [Ch
r
isten
s
en, 20
02] in highe
r dimen
s
ion
s
.
Corollar
y
3.1
Let
,(
)
n
BC
G
L
R
. Suppose that the Gabor sy
stem.
,
{(
)
}
n
l
Ck
B
m
km
Z
ET
x
Is a frame wit
h
frame bo
un
ds
1
A
and
2
A
, then:
2
12
|(
)
|
,
.
.
,
n
l
kZ
bA
Ck
bA
a
e
Whe
r
e
b
det
B
.
On the othe
r side, let :
{:
,
(
)
}
j
n
PA
j
Z
A
G
L
R
And
{0
}
Q
in the Theorem 3.1, then, we obtai
n the
nece
ssary con
d
ition
of the wavele
t frames
as the follo
wi
ng, whi
c
h is a
generalizatio
n of
Chui an
d
Shi [1993] in highe
r dimen
s
ion
s
.
Corollar
y
3.2
Let
,(
)
.
nn
A
EB
G
L
R
Suppose that wavele
t system.
,
{(
)
}
n
j
AB
m
jZ
m
Z
DT
x
Is a frame wit
h
frame bo
un
ds $A_1
$ an
d $A_2$, the
n
:
*2
12
|(
)
|
,
.
.
,
lj
jZ
bA
A
b
A
a
e
Whe
r
e
||
bd
e
t
B
.
Remar
k
3.2.
In the following, we will
discuss ne
cessary
con
d
itions for oth
e
r wave
packet fram
e
s
(1
5
)
i
i
defined by
(2.6)
with the different ope
rator order.
For w
a
v
e
pa
c
k
et
sy
st
e
m
s
1
,
from Lemm
a
2.3, we have:
2·
()
()
.
pp
li
B
m
l
AB
m
A
B
m
DT
E
x
e
D
E
T
x
If wave pack
e
t s
y
s
t
em.
1
,
2
,
,,
,
(
,)
{(
)
}
n
p
l
AB
m
lL
m
Z
p
S
DT
E
x
Define
d by (2
.6) is a fram
e for with fram
e
bound
s
1
A
and
2
A
, then, from
Theo
rem 3.1
and (3.2
2), th
e inequ
ality (3.4) hold
s
.
For w
a
v
e
pa
c
k
et
sy
st
e
m
s
2
,
from (2
) of Le
mma 2.3, we
have:
()
()
.
pp
ll
AB
m
A
B
m
A
E
DT
x
D
E
T
x
♯
If wave pack
e
t s
y
s
t
em
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NI
KA
Vol. 12, No. 10, Octobe
r 2014: 705
9
– 7069
7068
1
,
2
,
,,
,
(
,)
{(
)
}
n
p
l
AB
m
lL
m
Z
p
S
DT
E
x
Define
d by (2
.6) is a fram
e for with fram
e
bound
s
1
A
and
2
A
, then, in the
same
way,
the inequ
ality (3.4) hol
ds.
Then, from T
heorem 3.1 a
nd (3.23
)
, we
have:
Corollar
y
3.3
Suppose tha
t
wave packet
system
1
,
2
,
,,
,
(
,)
{(
)
}
n
p
l
AB
m
lL
m
Z
p
S
ED
T
x
Define
d by (2
.6) is a fram
e with frame b
o
und
s
1
A
and
2
A
, th
en we have:
2
12
1(
,
)
|(
(
)
)
|
,
.
.
,
L
l
p
lp
S
bA
A
b
A
a
e
♯
where
||
bd
e
t
B
.
4. Conclusio
n
Frame
s
pl
ay an impo
rta
n
t role in
sign
al proce
ssi
ng, imag
e pro
c
e
s
sin
g
, data
comp
re
ssion,
and sam
p
lin
g theory.
The mai
n
go
al of this p
a
per i
s
to
con
s
ide
r
the
ne
cessary
con
d
itions
of wave
packet
system
s to be frame
s
in highe
r dimen
s
ion
s
. The n
e
ce
ssary con
d
itions for all
kinds of wa
ve
packet f
r
ame
s
of th
e diffe
rent
ope
rator order in
hig
her
dime
nsi
o
ns with arbit
r
ary
expa
ndin
g
matrix dilatio
n
s a
r
e
esta
bli
s
he
d, whi
c
h i
n
clu
de the
co
rre
sp
ondi
ng result
s of
wav
e
let analy
s
is
and
Gabo
r the
o
ry
as th
e spe
c
i
a
l ca
se
s. So
me techniq
u
e
s
an
d
ways i
n
wavel
e
t an
alysis
and ti
me-
freque
ncy an
alysis a
r
e co
mbined.
Ackn
o
w
l
e
dg
ment
The a
u
tho
r
s
woul
d like to
expre
s
s their
gratitude
to t
he referee fo
r his
(o
r h
e
r) v
a
luabl
e
comm
ents a
n
d
sug
g
e
s
tion
s that lead to a signifi
cant i
m
provem
ent of our man
u
script.
Referen
ces
[1]
Casazz
a PG, O Christens
en.
W
e
y
l
-H
eis
enb
erg frames for
subsp
a
ces of
2
()
.
LR
).
Am
er. Math.
Soc.,
200
1; 129: 14
5
-
154.
[2]
Casazz
a PG, J Kovaev
ic. Equ
a
l-n
o
rm tight frames
w
i
th
eras
ures.
Adv. Com
p
ut. Math.,
2003; 1
8
: 38
7
-
430.
[3] Christensen
O.
An introducti
o
n
to frames a
n
d
Ries
z
b
a
ses
.
Birkhaus
er, Boston. 20
02.
[4]
Christe
n
sen
O, A Ra
himi. F
r
a
m
e pro
perti
es
of
w
a
ve
p
a
cket
s
y
stems
in
2
()
.
d
L
R
Adv. Com
put. Mat
h
.,
200
8; 29: 101-
111.
[5]
Chui
CK,
XL
S
h
i. Ine
qua
lities
of Little
w
o
o
d
-P
ale
y
t
y
p
e
for fr
ames a
nd
w
a
v
e
lets.
SIAM J. Math. Anal.,
199
3; 24: 263-
277.
[6]
Cord
oba A, C
F
e
fferman.
W
a
ve pack
e
ts an
d F
ouri
e
r inte
g
r
al o
perators.
Comm. Partial Differrenti
a
l
Equati
ons. 19
7
8
; 3: 979-1
005.
[7]
Czaja
W
G, Kutyn
i
ok,
D Sp
ee
gle. T
he Ge
om
et
r
y
of sets
of
prameters
of
w
a
ve
pack
e
ts.
Appl. Co
mp
ut
.
Har
m
on. An
al.,
2006; 2
0
:108-
125.
[8] Daubechies
I.
T
en
Lecture
s on W
a
v
e
l
e
ts.
CBMS-NSF
Reg
i
o
nal
Co
nferenc
e S
e
ri
es in
Ap
pli
e
d
Mathematics.
1
992; 61, SIAM, Phila
del
ph
ia.
[9]
Dau
bech
i
es I. T
he
w
a
vel
e
t transform, time-freque
nc
y
loca
lizatio
n an
d si
gna
l ana
l
y
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IEEE Trans.
Inform., T
heory
.
1990; 36: 96
1
-
100
5.
[10]
Dau
bech
i
es
I, A Grassman
n,
Y Ma
yer. P
a
i
n
l
e
ss n
o
n
o
rthog
ona
l e
x
pa
nsio
n
s
. J. Math. Phys.,
198
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1-12
83.
[11]
Duffin
RJ, AC
Schaeffer. A
cl
ass of
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nhar
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F
ouri
e
r
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366.
[12]
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hoer
y
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