The con olu ion heo em o he con inuous wa ele an o m
An onio F. P$e ez-Rend$on, Ra ael Robles
Dp o. Applied Ma hema ics I, Uni e si y o Se ille, Spain
Abs ac
We s udy he applica ion o he con inuous wa ele ans o m o pe o m signal 1l e ing p ocesses. We 1 s show ha he
con olu ion and co ela ion o wo wa ele u nc ions sa is y he equ i ed admissibili y and egu la i y condi ions. By using
hese new wa ele unc ions o analyze bo h con olu ions and co ela ions, espec i ely, we de i e con olu ion and co ela ion
heo ems o he con inuous wa ele ans o m and show hem o be simila o ha o o he join spa ial/spa ial– equency
o ime/ equency ep esen a ions. We hen in es iga e he e5ec o mul iplying he con inuous wa ele ans o m o a gi en
signal by a ela ed ans e unc ion and show how o pe o m spa ially a ian 1l e ing ope a ions in he wa ele domain.
Finally, we p esen nu me ical examples showing he u se u lness o applying he con olu ion heo em o he con inuous
wa ele ans o m o pe o m signal es o a ion in he p esence o addi i e noise.
Keywo ds: Con inu ou s wa ele ans o m; Con olu ion; Co ela ion; Space a ying 1l e ing; Signal es o a ion
1. In oduc ion
The ecei ed signal in a la ge numbe o signal p o-
cessing sys ems is modeled as a con olu ion be ween
he inpu signal and he impulse esponse o he sys em
wi h addi i e noise. Signal es o a ion (o decon o-
lu ion) p oblems ha e been mainly s udied unde he
assump ion ha he inpu signal and noise a e s a ion-
a y ( hou gh he e ms s a iona y and non-s a iona y
ha e some p ecise meanings in s ochas ic p ocesses,
in his wo k we use hese e ms in a quali a i e sense,
i.e., a s a iona y signal is a signal whose p ope ies
do no change much wi h ime o h ough space), and
con en ional es ima ion algo i hms such as Wiene o
Kalman 1l e s can be di ec ly applied. Howe e , he
p oblem o es o a ion o non-s a iona y signals such
as seismic da a, acous ic signals, mechanical ib a ion,
wi eless communica ion, ada and sona signals, e c.,
has ecen ly a ac ed he a en ion o signal p ocessing
esea che s. P ope ies o join ime– equency ep e-
sen a ions ha e allowed o 1nd op imal decon olu ion
1l e s o non-s a iona y signal es o a ion [4,14].
The wa ele ans o m is o in e es o he anal-
ysis o non-s a iona y signals because i p o ides an
al e na i e o classical linea ime– equency ep e-
sen a ions wi h be e ime and equency localiza ion
p ope ies [32]. On he o he hand, he ole o he
wa ele ans o ma ion is o emo e he non-s a iona y
p ope ies o he in ol ed signals; consequen ly, he
con en ional es ima ion algo i hms o s a iona y
signal p ocessing can be employed in each scale o
he wa ele domain [5]. Fo hese easons, se e al
Nomencla u e
˜
(w) Fou ie ans o m o unc ion (x)
E (x; w;) gene alized join dis ibu ion o unc ion wi h ke nel 
ˆ
(a; b) wa ele ans o m o unc ion wi h wa ele
L2(R) se o all measu able unc ions ha a e squa e in eg able
∗(x) complex conjuga e o unc ion (x)
·pno m de1ned in Lebesgue space Lp(R)
( ⊗g)(x) con olu ion o unc ions (x) and g(x)
( ∗g)(x) co ela ion o unc ions (x) and g(x)
( ⊗1g)(x; y) con olu ion in he 1 s a iable o unc ions and g
( ⊗2g)(x) con olu ion in he second a iable o unc ions and g
( ∗1g)(x; y) co ela ion in he 1 s a iable o unc ions and g
( ∗2g)(x) co ela ion in he second a iable o unc ions and g
( ⊗⊗g)(x) gene alized con olu ion o unc ions (x) and g(x)
M
-o de momen o wa ele (x)
F2{ˆ
}(a; w) Fou ie ans o m o ˆ
(a; b) in he second a iable (b↔w)
algo i hms ha e been de eloped in o de o pe -
o m signal es o a ion in he wa ele domain
[1,5,17,23,40,43], bu he lack o a con olu ion heo-
em o he con inuous o disc e e wa ele ans o m
p e en s us o 1nd op imal decon olu ion 1l e s in
he wa ele domain. The e o e, we hink ha he
es ablishing o such con olu ion heo ems, besides
hei heo e ical impo ance om a ma hema ical
poin o iew, ha e in e es ing applica ions in signal
p ocessing p oblems such as signal es o a ion.
Fil e ing o signals and images is a p esen ex-
ensi ely used in di5e en a eas o signal and im-
age p ocessing such as image enhancemen , image
es o a ion, signal coding and image analysis [3]. T a-
di ionally, mos op ical da a p ocessing sys ems only
pe o m space in a ian p ocessing ope a ions, ma h-
ema ically desc ibed in e ms o con olu ions, which
a e easily implemen ed h ough he Fou ie ans o m
(FT). The con olu ion heo em o his one s a es ha
he FT o he con olu ion o o signals is he p oduc
o hei espec i e FT, ha is
g(x)=+∞
−∞
d ( )h(x− )⇒
˜g(w)= ˜
(w)˜
h(w):(1)
The e o e, he 1l e ing ope a ion is gene ally educed
o mul iply he FT o he signal by he ans e unc ion
o he 1l e , and o make he in e se Fou ie ans o -
ma ion o he p oduc , which p o ides a 1l e ed e -
sion o he signal.
The FT pe o ms a global analysis o a gi en sig-
nal o image, and i is especially sui ed o s a iona y
signals. Ne e heless, in many cases o p ac ical in-
e es he signal unde in es iga ion is non-s a iona y
and has a di5e en equency con en a di5e en in-
s an s o localiza ions. One p oblem wi h he applica-
ion o he FT o such non-s a iona y signals is ha
high- equency e ms appea and hey a e no local-
ized, bu a e added e e ywhe e.
On he o he hand, signals can also be a5ec ed by
non-s a iona y p ocesses; o ins ance, op ical de ocus
may in oduce a spa ially a ian blu in he signal.
Space o ime- a ian 1l e ing [39] is ma hema ically
desc ibed as a gene alized con olu ion o a F edholm
in eg al equa ion o he 1 s kind:
g(x)=+∞
−∞
d ( )h(x; ):(2)
The con olu ion heo em no longe applies o hese
space o ime- a ying 1l e s and hus he Fou ie
domain is less use ul o hei applica ion [22].
The e o e, we come o he conclusion ha spa ially
a ian signals and p ocesses can be be e cha ac-
e ized by join ime– equency o spa ial/spa ial-
equency ep esen a ions [8,13,28].
Wigne [41] in oduced a bilinea dis ibu ion as a
join ep esen a ion o he phase space in Quan um
Mechanics. La e , Ville [37] de i ed he same dis ibu-
ion, nowadays called he Wigne –Ville dis ibu ion,
in he 1eld o signal p ocessing. Ano he way o ob ain
a join ep esen a ion is h ough he complex spec o-
g am, ha can be exp essed as a windowed FT, 1 s
in oduced by Gabo [10]. Bo h Wigne –Ville and Ga-
bo dis ibu ions belong o he Cohen class o bilinea
dis ibu ions [7], in which each membe is ob ained
by in oducing a pa icula ke nel in he gene alized
dis ibu ion de1ned as
E (x; w;)=
R3
ddd(; )
× (+=2) ∗(−=2)
×exp{2i(x −w −)}:(3)
The e exis con olu ion heo ems o he Wigne –
Ville dis ibu ion [6] as well as o o he join ep-
esen a ions belonging o Cohen class [26]. Such
heo ems s a e ha he join ep esen a ion o he
con olu ion o wo signals is he con olu ion in he
ime o space a iable, a e e y 1xed equency, o
he espec i e join ep esen a ions, ha is
Eg(x; w;)
=+∞
−∞
d E
( ; w;)Eh(x− ; w;):(4)
Join ep esen a ions also allow us o pe o m space
a ian 1l e ings on a gi en unc ion by mul iplying
i s join ep esen a ion by a unc ion T(x; w), which
can be ega ded as a space- a ian ans e unc ion,
and aking he in e se ans o ma ion. This p ocess
has been ca ied ou wi h he complex spec og am
[34] and he Wigne –Ville dis ibu ion [2,11,34].
The con inuous wa ele ans o m (CWT) [12,15]
s ands o a scale-space ep esen a ion and is an e5ec-
i e way o analyze non-s a iona y signals and images.
Typical applica ions a e he de ec ion and cha ac e -
iza ion o singula i ies [25], ac al analysis [21,38],
pa e n ecogni ion [20], noise educ ion [18], image
p ocessing [27], geophysics [9], ae omagne ic p o-
cessing [31] and he analysis o biomedical [42] and
me eo ological [35] images. The exis ence o as al-
go i hms o he calculus o he CWT [24,36] gua an-
ees i s use o sol ing eal p oblems.
The CWT is a scale-space ep esen a ion o a gi en
unc ion which may be ega ded as a join ep esen-
a ion by iden i ying he scale wi h a equency a io
[15]. This ac in oduces an adap abili y in he anal-
ysis which join ep esen a ions lack. Ne e heless,
he e is no con olu ion heo em o he CWT, and
his limi s i s usabili y in image and signal p ocessing.
We ha e ound in he li e a u e some wo ks ela i e
o con olu ion ela ed 1l e ing and CWT, which a e
summa ized below.
In [19], gi en a unc ion g(x) as he con olu ion o
ano he unc ion (x) and a 1l e h(x), le ˆg(a; b) and
ˆ
(a; b) deno e he CWT wi h he same wa ele o g(x)
and (x), espec i ely. Then, i is shown ha
ˆg(a; b)=+∞
−∞
d ˆ
(a; )h(b− ):(5)
This means ha he CWT o he con olu ion be-
ween some unc ion and a 1l e is he con olu ion in
he ime o space a iable, a e e y 1xed scale, o he
CWT o he unc ion and he e y 1l e .
In [33], gi en he con olu ion g(x) as abo e, le
ˆ
(a; b) deno e he CWT o he unc ion (x) wi h
wa ele (x) and le ˆ
h(a; b) deno e he CWT o he
1l e h(x) wi h wa ele (x)= (−x). Then, i is shown
ha
g(x)= 1
C+∞
0
da
a2+∞
−∞
dˆ
(a; )ˆ
h(a; x −);
(6)
whe e he no maliza ion cons an C is de1ned in (21).
Tha is, he con olu ion (ins ead o i s CWT) be-
ween some unc ion and a 1l e is he sum o e all
scales o he con olu ion in he ime o space a iable,
a e e y 1xed scale, o hei espec i e CWT, bu wi h
di5e en wa ele s.
A di5e en app oach is gi en in [16], whe e
wa ele -based 1l e s, known as scale 1l e s, a e
gene a ed as ollows. Le ˆ
deno e he CWT o a
gi en unc ion wi h wa ele and le ˜
h(w) deno e
he FT o he 1l e h(x); hen, a scale 1l e (a)is
de1ned as
˜
h(w)=
a
(a)˜
∗(aw) (7)
and is calcula ed ia linea squa es me hods. Then, i
is shown ha
g(x)=
a
(a)ˆ
(a; x):(8)
A simila p ocess has been used o cons uc space
a ying scale 1l e s [29,30].
Le us ema k ha nei he esul can be ega ded
as a con olu ion heo em, because his mus o mu-
la e he con olu ion equa ion in he wa ele domain
en i ely, i.e., i mus show a ela ionship be ween he
CWT o wo unc ions and hei con olu ion. Ne e -
heless, hese wo ks poin ou he exis ence o such
a con olu ion heo em simila o ha o join ep e-
sen a ions as shown in (4) and he possibili y o use
di5e en wa ele s o e e y in ol ed unc ion. This
possibili y is deeply in es iga ed in Sec ion 3.
This pape is o ganized as ollows. In Sec ion 2we
de1ne some known opics abou Fou ie and con inu-
ous wa ele ans o ms and in oduce he no a ion we
will use in ollowing sec ions. In Sec ion 3we de i e
he con olu ion and co ela ion heo ems o he con-
inuous wa ele ans o m and show he e5ec o mul-
iplying his by some ans e unc ion. In Sec ion 4
we ini ially show he alidi y o he heo ems p o ed
in p e ious sec ion and hen apply hem o a compu e
simula ion o a signal es o a ion p oblem; compa -
isons wi h Fou ie based decon olu ion a e p o ided
showing ha he signal es o a ion me hod p oposed in
his pape yields be e es o a ions. Finally, we sum-
ma ize he main conclusions o his wo k in Sec ion 5.
2. Backg ound
We de o e his sec ion o de1ne some known opics
abou FT and CWT and in oduce he no a ion we will
use in nex sec ion. Mos esul s in his sec ion come
om [12,15].
Gi en 1 6p¡∞, he Lebesgue space Lp(Rm)
is de1ned as he se o all measu able unc ions
:Rm→C o which
 p=Rm
d˜x| (˜x)|p1=p
¡∞:(9)
I is well known ha Lp(R) is a Banach space wi h
no m ·
pand L2(R) is a Hilbe space wi h inne
p oduc  ; g=Rdx (x)g∗(x), whe e (·)∗deno es
complex conjuga e.
The FT o a unc ion ∈L2(R) is de1ned as
˜
(w)=+∞
−∞
dx (x) exp{−2ixw}(10)
which also belongs o L2(R). The econs uc ion o -
mula is gi en by
(x)=+∞
−∞
dw˜
(w) exp{2ixw}:(11)
I ∈L1(R)∩L2(R), hen ˜
(w) is a bounded unc ion
in L2(R), i.e., he e exis s a posi i e eal numbe M
such ha |˜
(w)|6M o all w∈R.
I and ha e complex- alued unc ions de1ned on
R, hei con olu ion ⊗his de1ned as a new unc ion:
g(x)=( ⊗h)(x)=+∞
−∞
d ( )h(x− ) (12)
p o ided ha he in eg al exis s. I ∈L2(R) and
h∈L1(R), hen such an in eg al exis s almos e e y-
whe e and ⊗h∈L2(R) wi h  ⊗h2= 2·h1.
I and ha e complex- alued unc ions de1ned on
R2, i is possible o con ol e hem in jus one a iable
as ollows:
g1(x; y)=( ⊗1h)(x; y)
=+∞
−∞
d ( ; y)h(x− ; y);(13)
g2(x; y)=( ⊗2h)(x; y)
=+∞
−∞
d (x; )h(x; y − );(14)
p o ided ha bo h in eg als exis . I ∈L2(R2) and
h∈L1(R2), hen such in eg als exis almos e e y-
whe e and g1;g
2∈L2(R2).
An ope a o qui e ela ed o he con olu ion is
he so-called co ela ion ope a o . I and ha e
complex- alued unc ions de1ned on R, hei co e-
la ion ∗his de1ned as a new unc ion
g(x)=( ∗h)(x)=+∞
−∞
d ( )h( +x);(15)
p o ided ha he in eg al exis s. I ∈L2(R) and
h∈L1(R), hen such an in eg al exis s almos e e y-
whe e and ∗h∈L2(R) wi h  ∗h2= 2·h1.
When ≡h, ∗ is known as he au oco ela-
ion o , while o he wise ∗his known as he
c oss-co ela ion be ween and h.
I and ha e complex- alued unc ions de1ned on
R2, i is possible o co ela e hem in jus one a iable
as ollows:
g1(x; y)=( ∗1h)(x; y)
=+∞
−∞
d ( ; y)h( +x; y);(16)
g2(x; y)=( ∗2h)(x; y)
=+∞
−∞
d (x; )h(x; +y);(17)
p o ided ha bo h in eg als exis . I ∈L2(R2) and
h∈L1(R2), hen such in eg als exis almos e e y-
whe e and g1;g
2∈L2(R2).
Gi en wo unc ions ∈L2(R) and h∈L2(R2),
hei gene alized con olu ion is de1ned as he F ed-
holm in eg al equa ion o he 1 s kind:
g(x)=( ⊗⊗h)(x)=+∞
−∞
d ( )h(x; ) (18)
which is also known as a space- a ian 1l e ing in
op ics and enginee ing li e a u e [2,11], as opposed o
he space in a ian 1l e ing gi en by he con olu ion
de1ned in (12).
Gi en a 1xed unc ion ∈L2(R), called wa ele ,
conside i s ansla ions and dila ions de1ned as
ab(x)= 1
|a|1=2x−b
a(19)
wi h a; b ∈Rand a= 0. These unc ions a e scaled
so ha hei L2(R) no ms a e independen o a. The
CWT o a unc ion ∈L2(R) wi h wa ele is now
de1ned as
ˆ
(a; b)=+∞
−∞
dx (x) ∗
ab(x)
=1
|a|1=2+∞
−∞
dx (x) ∗x−b
a:(20)
A wa ele unc ion (x) is said o be admissible i
0¡C
=+∞
−∞
dw|˜
(w)|2
|w|¡∞:(21)
I his is he case, he CWT ˆ
(a; b) is in e ible on i s
ange, and an in e se ans o m is gi en by he ela ion
(x)= 1
C+∞
−∞
dadb
a2ˆ
(a; b) ab(x);(22)
whe e he in eg al o e ais unde s ood o exclude a=0
(in eg a e o e 0 ¡6|a|¡∞, hen ake he limi
→0).
The mos impo an p ope ies o wa ele s a e
admissibili y (21) and egula i y condi ions. As
can be seen in (20), he wa ele ans o m o a
one-dimensional unc ion is wo-dimensional and o
mos p ac ical applica ions his is no a desi able p op-
e y. The e o e, one imposes some addi ional condi-
ions on he wa ele unc ions in o de o make he
wa ele ans o m dec ease quickly wi h scale. These
a e he egula i y condi ions and hey s a e ha he
wa ele unc ion should ha e some smoo hness and
concen a ion in bo h ime and equency domains.
Regula i y is a qui e complex opic which is ela ed
o he concep o anishing momen s; he numbe o
anishing momen s equi ed depends hea ily on he
applica ion. The momen s o a wa ele ∈L2(R) a e
de1ned as
M
=+∞
−∞
d
 ( ):(23)
Le Lbe he Hilbe space o all measu able unc-
ions H(a; b) ha a e squa e in eg able wi h espec o
he weigh unc ion |a|−2, and le Fbe he space o
all wa ele ans o ms wi h espec o a 1xed wa ele
unc ion (x). I has been shown [15] ha Fis a
p ope subspace o L; ha is, no e e y unc ion
H(a; b)∈Lis he CWT o some unc ion ∈L2(R).
I we apply he econs uc ion o mula (22) osucha
unc ion H(a; b):
h(x)= 1
C+∞
−∞
dadb
a2H(a; b) ab(x);(24)
hen he unc ion h(x) belongs o L2(R) and i
is he unique unc ion such ha o any o he
unc ion ∈L2(R)i isH(a; b)−ˆ
(a; b)L¿
H(a; b)−ˆ
h (a; b)L. The unc ion h(x) is known
as he leas -squa es app oxima ion in L2(R) o he
in e se ans o m o H(a; b).
I is impo an o no e ha in (20) and (22) he
wa ele unc ions a e no speci1ed. This is a di5e ence
be ween he CWT and he FT, o o he ans o ms. The

heo y o wa ele ans o ms deals wi h he gene al
p ope ies o he wa ele s and he wa ele ans o ms
only. The choice o he wa ele unc ion depends on
he pa icula applica ion. Fo space– equency anal-
ysis, we need a wa ele ha is op imally localized in
e ms o bo h spa ial wid h and equency bandwid h.
Fo smoo h signals, we gene ally wan a wa ele ha
is i sel smoo h and he e o e has good equency lo-
caliza ion. In con as , signals ha con ain discon i-
nui ies a e be e analyzed using wa ele s wi h good
spa ial localiza ion o accu a ely map apid changes
in he signal. The e a e se e al o he use ul p ope ies
ha he wa ele unc ion can possess, as i s o hogo-
nali y, complexi y, wid h o shape.
3. The con olu ion heo em
In [33] i is poin ed ou he possibili y o use di -
e en wa ele unc ions o he signal and he 1l e .
Assuming ha we ha e selec ed he bes wa ele s o
bo h, wha is hen he bes choice o he con olu-
ion o co ela ion? I is somewha na u al o con-
side he con olu ion o co ela ion o he espec i e
wa ele s p o ided o sa is y he admissibili y condi-
ion and ha e a suPcien numbe o anishing mo-
men s (a leas , he same numbe as he wa ele used
o he o iginal signal).
Theo em 1. Le 1∈L2(R)and 2∈L1(R)∩L2(R)be
wo admissible wa ele s wi h N1and N2 anishing mo-
men s, espec i ely,and conside wo new wa ele s
3=( 1⊗ 2)and 4=( 1∗ 2). Then,bo h 3and
4a e admissible wa ele s oo and ha e N1+N2 an-
ishing momen s.
P oo . Since 1∈L2(R) and 2∈L1(R), we know
ha bo h 3;
4∈L2(R) oo. De1ne
Ci=+∞
−∞
dw|˜
i(w)|2
|w|(25)
o i=1;:::;4. Since 1(x) and 2(x) a e admissi-
ble wa ele s, hen acco ding o (21) we know ha
0¡C
1;C
2¡∞. Taking in o accoun some p ope ies
o he FT, we ha e
˜3(w)= ˜
1(w)˜
2(w);(26)
˜4(w)= ˜
1(w)˜
∗
2(w);(27)
om which we can 1nd
|˜
3(w)|=|˜
4(w)|=|˜
1(w)||˜
2(w)|:(28)
The e o e, we ge
0¡C
3=C4=+∞
−∞
dw|˜
1(w)|2|˜
2(w)|2
|w|:(29)
Since 2∈L1(R), i s FT ˜
2(w) is a bounded unc ion,
so ha he e exis s a posi i e eal numbe Msuch ha
|˜
2(w)|6M o all w∈R. Then, we a i e a :
0¡C
3=C46M2+∞
−∞
dw|˜1(w)|2
|w|
=M2C1¡∞:(30)
The e o e, he new de1ned wa ele s 3and 4sa is y
he admissibili y condi ion (21). Calcula e now he
momen o wa ele 3, de1ned as
M3
=+∞
−∞
d
 3( ):(31)
By inse ing (12) o 3( ) in o abo e equa ion and
changing he o de o in eg a ion, we 1nd
M3
=+∞
−∞
dx
1(x)+∞
−∞
d
 2( −x):(32)
I we pe o m he change o a iables y= −xin o
he inne in eg al and expand (x+y)acco ding o
New on binomial expansion, we a i e a
M3
=


k=0 
k+∞
−∞
dxx
k 1(x)
×+∞
−∞
dyy
−k 2(y);(33)
ha is
M3
=


k=0 
kM1
kM2
−k:(34)
An analogous easoning p o ides
M4
=


k=0
(−1)k
kM1
kM2
−k:(35)
Le be 6N1+N2.I k6N1, hen M 1
k=0 because 1
has N1 anishing momen s; o he wise, −k6N1+
N2−k6N1+N2−N1=N2, so ha M 2
−k=0 because
2has N2 anishing momen s. The e o e, all addends
in (34) and (35) a e ze o and M 3
=M 4
= 0 o all
6N1+N2.
Since bo h he con olu ion and co ela ion o wo
admissible wa ele s sa is y he equi ed admissibili y
and egula i y condi ions, we may use hem o analyze
ou con ol ed and co ela ed signals. Once he signal
and he 1l e ha e been analyzed wi h wo di5e en
wa ele unc ions and hei con olu ion has been an-
alyzed wi h he con olu ion o he chosen wa ele s,
wha is he ela ion be ween he h ee ob ained CWT?
As i was poin ed ou in Sec ion 1, such a ela ionship
is qui e simila o he con olu ion heo em o join
ep esen a ions shown in (4).
Theo em 2 (Wa ele con olu ion heo em). Le
∈L2(R)and h∈L1(R)∩L2(R)be wo admissi-
ble wa ele s,and le ˆ
and ˆ
h hdeno e he CWT o
wo unc ions ∈L2(R)and h∈L1(R)∩L2(R)wi h
wa ele s and h, espec i ely.I g=( ⊗h)and
g=( ⊗ h), hen
ˆg g(a; b)= 1
|a|1=2(ˆ
⊗2ˆ
h h)(a; b):(36)
P oo . The CWT o g(x), gi en by (20), may be
ew i en as
ˆg g(a; b)= 1
|a|1=2
R3
dxd dy ( )h(x− )
× ∗
(y) ∗
hx−b
a−y:(37)
Pe o ming he change o a iables =x− and $=
b+ay − ,(37) may be ew i en as
ˆg g(a; b)= 1
|a|3=2+∞
−∞
d$+∞
−∞
d ( )
× ∗
 −(b−$)
a
×+∞
−∞
dh() ∗
h−$
a;(38)
om which i is easy o 1nd
ˆg g(a; b)= 1
|a|1=2+∞
−∞
d$ˆ
(a; b −$)ˆ
h h(a; $)
=1
|a|1=2(ˆ
⊗2ˆ
h h)(a; b):(39)
The co ela ion o wo signals is an ope a ion qui e
ela ed o he con olu ion. In ac , he co ela ion he-
o em o he FT is qui e simila o he con olu ion
heo em; he only di5e ence is he complex conjuga e
ha appea s in he FT o he 1l e , ha is, ˜g(w)=
˜
(w)˜
h∗(w). The e o e, i is hoped ha he CWT sa is-
1es a co ela ion heo em simila o he p e ious con-
olu ion heo em.
Theo em 3 (Wa ele co ela ion heo em). Le
∈L2(R)and h∈L1(R)∩L2(R)be wo admissi-
ble wa ele s,and le ˆ
and ˆ
h hdeno e he CWT o
wo unc ions ∈L2(R)and h∈L1(R)∩L2(R)wi h
wa ele s and h, espec i ely.I g=( ∗h)and
g=( ∗ h), hen
ˆg g(a; b)= 1
|a|1=2(ˆ
∗2ˆ
h h)(a; −b):(40)
P oo . The CWT o g(x), gi en by (20), may be
ew i en as
ˆg g(a; b)= 1
|a|1=2
R3
dxd dy ( )h( +x)
× ∗
(y) ∗
hy+x−b
a:(41)
Pe o ming he change o a iables = +xand $=
b+ −ay, Eq. (41) may be ew i en as
ˆg g(a; b)= 1
|a|3=2+∞
−∞
d$+∞
−∞
d ( )
× ∗
 −($−b)
a
×+∞
−∞
dh() ∗
h−$
a;(42)
om which i is easy o 1nd
ˆg g(a; b)= 1
|a|1=2+∞
−∞
d$ˆ
(a; $ −b)ˆ
h h(a; $)
=1
|a|1=2(ˆ
∗2ˆ
h h)(a; −b):(43)
Acco ding o abo e heo ems, he CWT allows us o
pe o m any linea 1l e ing. Al hough his ans o m
is a oo edundan o yield as algo i hms, he main
ea u e o Theo ems 2and 3is ha he con olu ion
and co ela ion ope a o s apply a e e y scale indepen-
den ly o each o he , and his ac allows us o deeply
unde s and he e5ec o he 1l e on e e y scale o he
gi en signal. This p ope y may be e y use ul in de-
con olu ion ela ed p oblems, whe e i will be possi-
ble o design a di5e en algo i hm o e e y scale, de-
pending on he p ope ies o he 1l e a such a scale
and he condi ion numbe o he co esponding in e se
p oblem.
Despi e o p e ious conside a ions, he main d aw-
back o Theo ems 2and 3is ha he con olu ion and
co ela ion ope a o s do no disappea in he wa ele
domain, and his implies no ime educ ion in he ap-
plica ion o such ope a o s wi h espec o he spa ial
domain. This is no he case wi h he FT. The con olu-
ion and co ela ion heo ems o he FT a e powe ul
ools o pe o m space in a ian 1l e ing p ocesses be-
cause hey educe hese p ocesses o simply mul iply
he FT o he signal by he ans e unc ion associa ed
wi h he 1l e .
I would be desi able o pe o m some kind o 1l-
e ing by simply mul iplying he CWT by a ans e
unc ion; o join ep esen a ions his p oduc pe -
o ms a spa ially a ian 1l e ing on he gi en signal.
The simila i y be ween he con olu ion heo ems o
join ep esen a ions and CWT leads o hink abou
he possibili y o pe o m such space a ying 1l e ing
ope a ions in he wa ele domain.
Le ˆ
deno e he CWT o a unc ion ∈L2(R)
wi h an admissible wa ele ∈L2(R) and le
H∈L2(R2). De1ne G(a; b)= ˆ
(a; b)H(a; b), which
may no be he CWT o any unc ion in L2(R); he
leas squa e app oxima ion o such a unc ion is ob-
ained by applying he econs uc ion o mula (22) o
G(a; b). The e o e,
g(x)= 1
C+∞
−∞
dadb
a2ˆ
(a; b)
×H(a; b) x−b
a(44)
wi h C de1ned in (21). By inse ing he de1ni ion
o ˆ
(a; b) acco ding o (20) in o abo e equa ion and
changing he o de o in eg a ion, we 1nd
g(x)= 1
C+∞
−∞
d ( )
×+∞
−∞
dadb
|a|5=2H(a; b)
× ∗ −b
a x−b
a:(45)
I we de1ne
h(x; )= 1
C+∞
−∞
dadb
|a|5=2H(a; b)
× ∗ −b
a x−b
a(46)
p o ided he abo e in eg al o con e ge, hen g(x)=
( ⊗⊗h)(x) is he unc ion in L2(R) whose CWT
wi h wa ele is as close as possible o he p oduc
ˆ
(a; b)H(a; b).
Al hough we ha e no ound an in e se o mula o
ob aining he ans e unc ion H(a; b) which co e-
sponds o a gi en ke nel h(x; ), his esul aises an
in e es ing ques ion. Is i possible o 1nd use ul 1l-
e s such ha hei Fou ie ans e unc ions canno
be well app oxima ed by a ional unc ions bu hei
ac ion is diagonal in he wa ele domain? We hink
ha his ques ion migh be o in e es o he signal
p ocessing communi y, so ha u u e wo k may elay
on i .
4. Nume ical examples
In his sec ion, wo nume ical examples a e gi en
o illus a e (i) he alidi y o he heo ems p o ed in
p e ious sec ion and (ii) he usabili y o such heo-
ems in signal es o a ion p oblems in he p esence o
addi i e noise. In bo h examples, he es signals ha e
been analyzed wi h he 1 s and second de i a i es o
he well–known no mal dis ibu ion wi h ze o mean
and uni a iance, gi en by
1(x)=2
√xexp −x2
2(47)
and
2(x)= 2
3√(1 −x2) exp −x2
2;(48)
espec i ely. I can be shown ha bo h wa ele s sa is y
he admissibili y condi ion (21) wi h coePcien s C1=
44:5466 and C2=29:6977, he wa ele 1(x) has a
unique anishing momen and he wa ele 2(x) has
wo anishing momen s.
In he ollowing examples we ha e also employed
he con olu ion 3= 1⊗ 2and he co ela ion 4=
1∗ 2o he p e ious wa ele s, whose analy ical ex-
p essions a e gi en by
3(x)= 1
2√6x(x2−6) exp −x2
4(49)
and 4(x)=− 3(x) because o he symme y p ope -
ies o 1(x) and 2(x). Bo h new wa ele unc ions
also sa is y he admissibili y condi ion (21) wi h co-
ePcien s C3=C4=52:6379 and ha e h ee anish-
ing momen s, in acco dance wi h Theo em 1. All ou
wa ele unc ions a e depic ed in Fig. 1.
-4 -2 0 2 4
x
-0.6
-0.3
0
0.3
0.6
ψ1(x)
Wa ele unc ion
-4 -2 0 2 4
x
-0.3
0
0.3
0.6
ψ2(x)
Wa ele unc ion
-6 -3 0 3 6
x
-0.8
-0.4
0
0.4
0.8
ψ3(x)
Wa ele unc ion
-6 -3 0 3 6
x
-0.8
-0.4
0
0.4
0.8
ψ4(x)
Wa ele unc ion
(a) (b)
(c) (d)
Fig. 1. Wa ele unc ions used o analyze he es signals: (a) 1 s de i a i e o a gaussian dis ibu ion o ze o mean and uni a iance,
1, (b) second de i a i e o a gaussian dis ibu ion o ze o mean and uni a iance, 2, (c) con olu ion o p e ious wa ele s, 3= 1⊗ 2,
and (d) co ela ion o he 1 s wo wa ele s, 4= 1∗ 2.
Example 1. Conside a es signal de1ned as
(x) = exp −(x−1)2
6cos(3(x−1)) (50)
whose plo is shown in Fig. 2(a). The alidi y o he
wa ele co ela ion heo em is es ablished by calcula -
ing he au oco ela ion o he es signal in he wa ele
domain using Theo em 3and compa ing he esul
wi h he au oco ela ion ob ained using he co ela ion
heo em o he FT.
The gene a ed es signal is analyzed wi h wa ele s
1(x) and 2(x) (all in eg als ha e been nume ically
e alua ed using he midpoin quad a u e ule). Ac-
co ding o Theo em 3, bo h CWT a e co ela ed a
e e y 1xed scale in o de o ob ain he CWT o he
au oco ela ion o he es signal wi h wa ele 4(x).
Finally, such an au oco ela ion is ob ained by ap-
plying he econs uc ion o mula (22) o i s CWT.