小波分析：ch2

Lecture Notes on Wavelets,Chapter 2,by D.Q,Dai,2003 1
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!f 2 L1(Rn),FouriersZ
Rn
f(t)e?i!tdt
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l ?,á
ì?
1f!)￥FourierMD,i:1^f(!).
??? ? ? ?1(Riemann-Lebesgue??? ? ? ?):?f 2 L1(Rn),5^f(!) ??,O?j!j! +1
H,^f(!) ! 0.
￡￡￡
ü
ü
ü,(?n=1) (1) ^f(!) ??b8" > 0,??f 2 L1,9N > 0
P
Z
jtj?N
jf(t)jdt <?4
V7
j ^f(! +h)? ^f(!)j
R
jtj?N
je?it(!+h)?e?it!jjf(t)jdt+ R
jtj?N
je?it(!+h)?e?it!jjf(t)jdt
< 2￠ "4 + R
jtj?N
je?it(!+h)?e?it!jjf(t)jdt:
??jtj? N
H,μ
je?it(!+h)?e?it!j = je?ith?1j? 2jt!j? 2Njhj:
yN,o1jhj sl,Aμ
j ^f(! +h)? ^f(!)j < "2 +2Njhj
Z
R1
jf(t)jdt < "
#^f(!) ??.
Lecture Notes on Wavelets,Chapter 2,by D.Q,Dai,2003 2
(2)Q￡^f(!) ! 0?j!j! +1(
6￡,n?.101),á
ìμ
j ^f(!)j
R
jtj?N
jf(t)jdt+
flfl
flfl
fl
R
jtj?N
e?it!f(t)dt
flfl
flfl
fl
< "4 +
flfl
flfl
fl
R
jtj?N
e?it!f(t)dt
flfl
flfl
fl:
|g 2 C1(jtj? N)
Pjjf?gjjL1(Rn) < ",5
flfl
flfl
flfl
fl
Z
jtj?N
e?it!f(t)dt
flfl
flfl
flfl
fl
+
flfl
flfl
flfl
fl
Z
jtj?N
e?it!g(t)dt
flfl
flfl
flfl
fl;
?
Z
jtj?N
e?it!g(t)dt = g(t)e
it!
i! j
N
N +
1
i!
Z N
N
g0(t)e?it!dt
μNw¤?f(!) ! 0.
á
ì I
n
IFourierMDbL1f
￥FourierMD?B?
?L1,??<￥Hq/μ
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I
I
IMMMDDD


:?f 2 L1(Rn)O?f 2 L1(Rn),5
f(t) = 1(2…)n
Z
Rn
f(!)ei!td!
￡￡￡


!n=1
á
ìμ
1
2…
Z
f(!)ei!td! = 1
2…
Z +1
1
Z +1
1
f(u)ei!(t?u)du
d!
á
ì?
°¤¨Fubini? ?,yf(u)e?iw(t?u)R2?? V.?? Vy0e2!2=4(? > 0),i
?l
I?(t) = 12…
Z Z
f(u)e2!2=4ei!(t?u)du
d!:
?Fubini? ?
51?us
μ
I?(t) = 12…
Z
f (!)e2!2=4ei!td!
y
flfl
fl?f (!)e2!2=4ei!t
flfl
fl?
flfl
fl?f (!)
flfl
flO?f V
? e?
l ?? ?
μ
lim?!0I?(t) = 12…
Z
f(!)ei!td!
Lecture Notes on Wavelets,Chapter 2,by D.Q,Dai,2003 3
6BZ

?Fubini? ?
51?!s
μ
I?(t) =
Z
g?(t?u)f(u)du
?
g?(t) = 12…
Z
eit!e2!2=4d!



g1(t) = 1p…e?t2.?g?(t) =1g1(1t)
OR g1(t)dt = 1
#I?(t) ! f(t)L1(R)

1
￡
).
??? ? ? ?3


:
!f;h 2 L1(Rn).5g = h?f 2 L1O
g(!) =?h(!)?f(!)
￡￡￡
??l
g(!) =
Z
e?it!
Z
f(t?u)h(u)du
dt:
yjf(t?u)h(u)jR2
 V
?Fubini? ?M
9D(t;u) ! (v = t?u;u)¤
g(!) = R R e?i(u+v)!f(u)h(u)dudv
= (R e?iv!f(v)dv)(R e?iu!h(u)du):
Table 1,FourierMDV:
f(t)?f(!)
I?f(t) 2…f(?!)
(f1?f2)(t)?f1(!)?f2(!)
ef1(t)f2(t) 12…(?f1f2)(!)
üMf(t?u) e?iuw?f(!)
??ei?tf(t)?f(!)
f(t=s) jsj?f(s!)
á
ì|?1L2(Rn)?)
lo
1N31)
? ?f
￥FourierMDb
!f 2 L1 \L2
5á
ìμ
??? ? ? ?4

1).

Parseval?
T

!f;g 2 L2(Rn)\L1(Rn)
5
hf;gi = (2…)?n
D?
f;?g
E
:
Lecture Notes on Wavelets,Chapter 2,by D.Q,Dai,2003 4
2).

Plancherel
T

kfk = (2…)?n2
f
,
￡
?h = f?g
,?g
(t) = g(?t)b? ?é¤
h(!) =?f(!)￠?g
(!)
|×
T?¨?h(0)¤
Z
f(t)g(t)dt = h(0) = (2…)?n
Z
h(!)d! = (2…)?n
f;?g
(!)
:
!f 2 L2(Rn)
/
á
ì?l
￥FourierMD
iùs?f(!)￥Vr
TbTofjtj? Ng￥
+?f
′N(t) =
(
1; jtj? N;
0; jtj > N:
5A?μfN = ′Nf 2 L1 \L27OkfN?fkL2 ! 0(N !1)
yN
?Plancherel
T
¤

cfN? cfN0
L2
! 0;(N;N0 !1)
?L2 (Rn)￥?!?
ig 2 L2

P¤
^f
N ! g
?lg1?f
5yfN 2 L1

[
^f
N (!) =
Z
jtj?N
e?it!f (t)dt
#L2?
μ
f (!) = lim
N!1
Z
jtj?N
e?it!f (t)dt:
è:
!f (t) = e?t2
p?f(!).
A?f^(!) = Rjtj?N e?t2e?it!dt
@±sZ?2?f0(!) + !?f (!) = 0.
[?f (!) = Ke?!2=4,
f (0) = p…
#?f (!) = p…e?!2=4
1?f
￥?5?D
￥FourierMDvl￥1",-
Xü
a
üf 2 L1
H

￥FourierM
D1 ??f

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á
ìμ
??? ? ? ?5:?R+1?1
flfl
fl?f(!)
flfl
fl(1+j!jp)d! < +1,5f#°?pQ?
??μ?b
￡￡￡
yF(f(k)(t)) = (i!)k?f(!),?Nw¤
Lecture Notes on Wavelets,Chapter 2,by D.Q,Dai,2003 5
flflf(k)(t)flfl? 1
2…
Z flfl
fl?f(!)
flfl
flj!jk d! < +18k? p
yNf(k)(t) ??
μ?.
www


1
?j?f(!)j? K=(1+j!jp+1+?)(? > 0)
5f 2 Cp,
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5f 2 C1.
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2??????eee ? ? ?
0 ?D?
of
f 2 L2(R)
í
B? ì0￥
bê?t￥à
q
á11kfk2 jf(t)j2bM
?￥!)￥
á
^12…kfk2
flfl
flfl^f(!)
flfl
flfl
2
ì0￥
ü (ê?
^

u = 1kfk2
+1Z
1
tjf(t)j2dt

ù?′

ü ( :
= 12…kfk2
Z +1
1
!
flfl
fl?f(!)
flfl
fl
2d!
M?￥Zμ′1
( t)2 = 1kfk2
+1Z
1
(t?u)2jf(t)j2dt
( !)2 = 12…kfk2
1Z
1
(!)2
flfl
fl?f(!)
flfl
fl
2 d!
v￥ t
1? ì0ê?￥????,v￥ !
1? ì0
￥????.
??? ? ? ?6





Heisenberg??????eee ? ? ?


:
2t 2!? 1=4
O?|? ?
?O??
f(t) = aei?t?b(t?u)2;(u;?;a;b) 2R2 ￡C2
Lecture Notes on Wavelets,Chapter 2,by D.Q,Dai,2003 6
￡￡￡


o￡ptf(t) ! 0￥f?
?f￥
ü (
Ha
ê?
^u?
5e?i?tf(t + u)M?￥
ü
(
Ha
ê??
^0byNá
ì
!u =?=0.?F(f0(t)) = i!?f(!)DPlancherel
T,[#
?Schwarz??
T
2t 2! = 12…kfk4 R jtf(t)j2 dtR
flfl
fl!?f(!)
flfl
fl
2 d!
= 1kfk4 R jtf(t)j2 dtR jf0(t)j2 dt
1kfk4(R
flfl
fltf0(t)f(t)
flfl
fldt)2
1kfk4[R t2[f0(t)f(t)+f0(t)f(t)dt]2
= 1kfk4[R t2(jf(t)j2 )0]dt]2
= 1kfk4[R t2(jf(t)j2 )0dt]2
= 14kfk4[R (jf(t)j2)dt]2
= 1=4:
1¤??
T
????¨Schwarz??
T
H
A??|? ?by79b 2C
P¤
f0(t) =?2btf(t):
?N¤
f(t) = ae?bt2; a 2C:
N
Ha
￥??
T?9¤?
T.
/
￥? ?V
üá
ì?
/
HB
=×
=]
Hμ?|"￥f
b÷B?1μ
??? ? ? ?7:
!f(t) 6= 0μ?|"
5?f(!)?
?BuW
10b ?
1
f(!) 6= 0μ?|
"
5f(t)?
?BuW
10b
￡￡￡
?￡-B2
b
!?fμc?[?b;b]￥?|"
5
f(t) = 12…
Z b
b
f(!)ei!td!:
?f(t) = 0;8t 2 [c;d]
t0 = c+d2)pmQ?

¤
f(m)(t) = 12…
Z b
b
f(!)(i!)mei!t0d! = 0
??t 2R
Z 7ei!(t?t0)
¤
f(t) = 12… Rb?b?f(!)ei!(t?t0)ei!t0d!
= 12… Pm [i(t?t0)]mm! Rb?b?f(!)!mei!t0d! = 0
?Df 6= 0
±b
Lecture Notes on Wavelets,Chapter 2,by D.Q,Dai,2003 7
LLL???
H
H
H???MMM"""dddDDD
r
r
rooo
I
n
bWXDYbL?"d??B?L?
0L
L,X ! Y
f(t) ! g(t) = (Lf)(t):
N"d
^
H?M￥
?T?it0
μ
f(t?t0) ! g(t?t0):
V[￡
ü
?L?
H?M"d
if
h

P¤
g(t) = (h?f)(t) =
Z +1
1
h(t?u)f(u)du:
V7
á
ìμ
g(!) =?h(!)?f(!):
?"
L?
H?M"d
^
q×

{?￥B?V
rb
? f
h(!)
q bW!￥
s?
á
ìó
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?f
h(!)￥
sY"??

ú
?W


á
ìsY?
1?Y
úY
{Y
rob
è è è


?h(!) = ′[;?](!)1?Y
robiO
^ ?X?Y
ro
?
^y1
??1
=?
?zO"?ú
?zb
C I
n? ?X?Y
ro
á
3￥GibbsC`b
!f? = f?h?1? ?X?Y
robh? = ′[;?]
roa￥?|b
?f 2 L2(R)
H
μf? ! fL2(R).?
^y1?f? =?f ￠′[;?],O?Plancherel
Tμ
jf?f?j2 = 12…
Z +1
1
j?f(!)? bf?(!)j2 = 12…
Z
j!j>?
j?f(!)j2d! ! 0
 !1b
?ft0? ??
H
f?t0)á
3Gibbs??bá
ì? I
nX/￥
?

f(t) = fc(t)+Ju(t?t0)
?fc(t)t0 ??
u(t) = 1
?t? 0
0,
b9'ft0)￥K
1Jb
M?￥f?(t)1
f?(t) = fc?h?(t)+Ju?h?
?B[t0￥
#×
=
l ??fc(t)b?=[
á
ìμ





555





Gibbs


:8? > 0,μ
Lecture Notes on Wavelets,Chapter 2,by D.Q,Dai,2003 8
u?h?(t) =
Z t?
1
sinx
…x dx
￡￡￡
?h?(t) = sin(?t)=(…t),¤
u?h?(t) =
+1Z
1
u(t)sin?(t?x)…(t?x) dx =
+1Z
0
sin?(t?x)
…(t?x) dx
M
9Da
'¤￡b
f
S(?t) =
tR
1
sinx
…x dx￥m^?m.
KvGibbs???
3t=§…=?
??G ?b
A = S(…)?1 =
…Z
1
sint
…t dt?1 … 0:045:
Ka
á
ìμ
f(t)?f?(t) = JS(?(t?t0))+?(?;t)
(?;t) ! 0?j?j!1
t0￥

#×
=b
á
ì[ è03
dsO
q￥à
Q
roh|"vlW￥1"bL?h
^?|"￥

'supph = [a;b](?1 < a < b < 1)bá
ì[?

?BV?1 è ?
a
üb
!f1?
L

7g
^YV
V??
V?7¤?￥4?′b
üA
g
H Yt = T￥
′???
L1
Q?ft = T
H￥?
L
b?h￥|"￥?é
g(T)G ??f(t)uWt 2 [T?b;T?a]￥
′by7?4?V?
uWéb?a?l
g(T)??
L￥Q?fb#4?V?￥sO
q
Dh￥|"￥vlμ1búsO
q"d??l|"￥
ro
Q-g?b
4|||"""??? ? ? ?
I
nDiracs?–(t).?
T
μ

–(!) =
Z
–(t)e?it!dt = 1
y7
–t0(t) = –(t?t0)
μ
b–t
0(!) = e
it0!:
Dirac
x0

c(t) =
+1X
n=?1
–(t?nT)
Lecture Notes on Wavelets,Chapter 2,by D.Q,Dai,2003 9
FourierMD1
c(!) =
X
n
e?inT!
??? ? ? ?8

Poissonp
T

s?il/
μ
X
n2Z
e?inT! = 2…T
X
n2Z
–(!? 2k…T ):
￡￡￡
y?c(!)[2…=T1?
ù
1￡

T
o3￡?cK??[-…=T
…=T]
H??2…=T–.1N
o
3￡?i_f
`(!)

￥|"c?[-…T
…T ]
μ
<?c;?` >= lim
N?>1
Z +1
1
NX
n=?N
e?inT!?`(!)d! = 2…T?`(0):
s
=)
￥1
NX
n=?N
e?inT! = sin[(N +1=2)T!]sin(T!=2):
yN
<?c;?` >= lim
N?>1
2…
T
Z …=T
…=T
sin[(N +1=2)T!
…!
T!=2
sin(T!=2)
`(!)d!
:(!) =
(
’(!) T!=2sin(T!=2); k!k < …=T;
0;
b7?(t)1
(!)￥
IFourierMD
y2!?1 sin(a!)
^′[?a;a](t)￥FourierMD
?Parseval?
Tμ
<?c;?’ > = lim
N!1
2…
T
R+1
1
sin[(N+12)T!]
…!
(!)d!
= lim
N!1
2…
T
R(N+1
2)T
(N+12)T?(t)dt
= 2…T R+1?1?(t)dt = 2…T(0) = 2…T?`(0):
?
Poissonp
TóFourier"
D|"′-W￥1"b
1?f￥Bá|"′f (nT)
 VV1Diracf
f (nT)–(t?nT)by7
fd (t) =
X
n
f (nt)–(t?nT)
??\–(t?nT) = e?inT!

[μ
fd(!) = X
n
f(nt)einT!:
á
ìμ
Lecture Notes on Wavelets,Chapter 2,by D.Q,Dai,2003 10
??? ? ? ?
^fd (!) = 1T Pk ^f?!? 2k…T ￠b
￡￡￡:s?￥il/
μ
hf (nt)–(t?nT);’(t)i = f (nT)’(nT)

hf (t)–(t?nT);’(t)i = f (nt)’(nt)
#f (nt)–(t?nT) = f(t)–(t?nT).y7fd (t) V?1
fd (t) = f(t)
X
n
–(t?nT) = f(t)c(t)
?FourierMD￥e?é
^f
d (!) =
1
2…
^f?^c
?Poisson
T
μ
^c(!) = 2…
T
X
k
–
!? 2…kT
?^f?–(!) = ^f (!)
? ?¤￡.
?
VBf
g(t)/
￥?
ù
B?ZE
^
gc(t) =
X
k
g
t? 2…kT
:
??? ? ? ?9

Shanon(1949)
Whittaker(1935)
:
!?f￥|"c?[?…=T;…=T].5
f(t) =
X
n
f(nT)hT(t?nT)
?
hT(t) = sin…t=T…t=T,
￡￡￡
y?hT = T′[?…=T;…=T]
?-B? ?¤
hT ￠?fd = 1
T
hT ￠X
k
f(!? 2k…
T ) (?)
f￥|"?é


T·
?f(!).|
IFourierMD¤
Lecture Notes on Wavelets,Chapter 2,by D.Q,Dai,2003 11
f(t) = hT?fd
= hT?P
n
f(nT)–(t?nT)
= P
n
f(nT)hT(t?nT):
Nyquist|||"""
q
q
q





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aliasing


:
!supp?f ‰ [?B;B]b|"?éTDB￥1"
1T = …=B.?-1Nyquist|"
qb
?T > …B
5?f(!?2k…T )￥|"D[?B;B]
tk 6= 0d bby7



T·
￥FourierM
D
hT?fd = ′[?…=T;…=T]
+1X
k=?1
f(!? 2h…
T )
[-B,B]

?]f(!).??á
?r?b
è è è

!f(t) = cos(!0t) = (ei!0t + e?i!0t)=2,…=T < !0 < 2…=T.?f(!) = …[–(!? !0) +
–(! +!0)]
#μB = !0b?
H
fd (!)?hT (!) = …′[?…=T;…=T] (!) +1P
k=?1
￡–?!?!
0? 2k…T
￠+–?! +!
0? 2k…T
￠?
= …￡–?!? 2…T +!0￠+–?! + 2…T?!0￠?
[
fd?hT (t) = cos
2…
T?!0
t
:

?r?
|ú
!0M?
?
2…T?!0 2 [?…=T;…=T].
???
?¨?
?
.{
^??^h"￥.
/
á
ìóB?/í2Tbó?f
f
¨FourierMD|?[?…=T;…=T]￥f
f/
íf
?f =??Plancherel
T
á
ìμ
ff
2 = 1
2…
R+1
1
flfl
fl?f (!)f (!)
flfl
fl
2
d!
= 12… Rj!j>…=T
flfl
fl?f(!)
flfl
fl
2 d! + 1
2…
R
j!j<…=T
flfl
fl?f (!)f
flfl
fl
2
d!
??=[10
H
N  ?Kl
yN
f(!) =?f(!)′
[?…=T;…=T] =
1
T
hT(!)?f(!):
'?f = 1=Tf?hT.
1?|"? ?μò?￥w<
 è?

1
f￥?]K?

2
|"￥?


?￥f?μ÷￥ê4


Lecture Notes on Wavelets,Chapter 2,by D.Q,Dai,2003 12
3
f
￥ê|b
5 ? ? ????Fourier)))
L2[?…;…]?,??
=< f;g >= 12… R…?… f(t)g(t)dtb
??? ? ? ?10
fe?ik!gk2Z
^L2[?…;…]￥S??b
￡￡￡
????^b1￡
^
á
ì￡
ü
L?FL2[?…;…]?b
!’ 2 C10(?…;…)5￡
’(!) =
X
k
< ’(?);e?ik? >e?ik!;8! 2 [?…;…]:
1N
9
?s
SN(!) =
NX
N
< ’(?);e?ik? > e?ik!
=
NX
N
1
2…
Z …
…
’(?)eik?d?e?ik!
= 12…
Z …
…
’(?)
NX
N
eik(!)d?:
Possionp
Tó
lim
N!+1
NX
N
eik(!) = 2…
X
k
–(!?2…k)
ysupp’ ‰ [?…;…];#
lim
N!+1
SN(!) = ’(!):
1￡fe?ik!gk2Z￥L?FL2 [?…;…]?
á
á
ì￡
ü8a 2 L2 [?…;…]
i??
[
TSN

P¤ka? SNk <?,8? > 0.???| ?? V±f
L2 [?…;…]?

[i
`
P¤ka?` k <?=2
??B?￥Bá
l ??
μsup
!2[?…;…]
jSN(!)?`(!)j=2;?ó
kSN?`k2 = 12…
Z …
…
jSN(!)?`(!)j2d!
2
4
y7
ka?SNk?ka?`k+k`?SNk:
Lecture Notes on Wavelets,Chapter 2,by D.Q,Dai,2003 13
?N
8ffng2 l2
μFourier)
f (!)L2=
X
n
fne?i!n 8f 2 L2 (?…;…]
OFourier"
fn =< f;e?i!n >= 12…
Z …
…
f (!)ei!nd!
á
ìgμ ??Plancherel?
T
X
n
jfnj2 = kfk2 = 12…
Z …
…
jf (!)j2d!
111???
l
l
l ? ? ????:

o)
L2il/￥
l ??b???
l ??
μ

1) Dubois-Reymond(1873)/
 ??f

Fourier)

t???

2) Carleson(1966)￡
ü
Luzin?X

1913

?f 2 L2
H

￥Fourier)
+))
l
?b
6 ? ? ????FourierMMMDDDDDD y y y
EEE
???ììì   :
!?fD?h1N?"'￥?

 I
n ?? 
(?fh)n =
+1X
p=?1
fp?hn?p:
19
0? n? N
H￥′
á
ì?fnD?hn0? n? N[?￥′bB?ZE
^[?
ùNü
fD?h
'
7
fn =?f(n mod N); hn =?h(n mod N):
ì￥?ì 1

(f?h)n =
N?1X
p=0
fphn?p =
N?1X
p=0
fn?phn?p
f?h9
^?
ù1Nb





555
?ì 
0(Lf))n = (f?h)n￥+?′1h￥ ??FourierMD,+?f
1ˉ·
(ek)n = ei2…kn=Nb
Lecture Notes on Wavelets,Chapter 2,by D.Q,Dai,2003 14
￡￡￡
y(Lek)n = PN?1p=0 ei2…k(n?p)=Nhp = ei2…kn=N ￠
N?1P
p=0
e?2…kp=Nhp =?(ek)n,?è
=
N?1P
p=0
hpe?i2…kp=N1hp￥ ??FourierMD.
?
ù1N
?￥?
???
=

< f;g >=
N?1X
n=0
fngn;f = (f0;f1;￠￠￠ ;fN?1);g = (g0;g1;￠￠￠ ;gN?1):
??? ? ? ?11,(ek)n = ei2…kn=N;0 6 k < N
^??b
￡￡￡
?-

5
o3￡???
7?
^A?￥.
?i￥?
ù?
f = (f0;f1;￠￠￠ ;fN?1),????
μ
f =
N?1X
k=0
< f;ek >
kekk2 ek:
?lf￥ ??FourierMD(DFT)1
bfk =< f;ek >=
N?1X
n=0
fne?i2…knN,
??kekk2 = N
#á
ìμo
I ??FourierMD(IDFT)p:
fn = 1N
N?1X
k=0
bfkei2…knN,
?￥???
¤Plancherel
T
kfk2 =
N?1X
n=0
jfnj2 = 1N
N?1X
k=0
jbfkj2:
??l! = e?i2…=N
5 V?l′DFT ?1:
F =
1
pN!kn;0? k;n? N?1:
A?μF?1 = Fb
??? ? ? ?12

?ì 
:
!fDh￥?
ù1N
5g = f?h￥DFT1
gk =?fk ￠?hk:
?N? ? V A
?ì  V[3
d1 ??
q
rob
Lecture Notes on Wavelets,Chapter 2,by D.Q,Dai,2003 15
y y y
FourierMMMDDD(FFT)


?DFT￥?l
°¤9
N?FourierMD31N2QˉeEDF
E
? ?¨ y
FourierMD

o31O(N log2 N)Q
b/
á
ìw?B? y
E#
9
￥ˉ?b
?N1
}

??f2k
?f2k+11
f2k = N=2?1P
n=0
(fn +fn+N=2)e?i2…kn=(N=2) (?)
f2k+1 = N=2?1P
n=0
e?i2…n=N(fn?fn+N=2)e?i2…kn=(N=2) ()



V
ü

}
q?9
N=2??
fen = fn + fn+N=2￥DFT7¤b7


V
ü

q?9
N/2??￥?
ù?
fon = e?i2…n=N(fn?fn+N=2)¤?b
[é1N￥DFT￥
?
?N=2?￥DFT￥
F
O(N)Q
b?B
E y
1
LC
DFTb
7?IDFT
?1"
T
ˉfn = 1
N
N?1X
k=0
ˉ?f
ke?i2…kn=N
VYVDFT¤?b
999
ˉˉˉ???

!C(N)19
N?DFT
31￥
Q
byf
^ˉ￥
9
feDfo3NQˉ
FEDN2QˉeEb
!K ￠N
^M?￥?
Q

5μ
C(N) = 2C(N2 )+KN:
?l = log2 N;T(l) = C(N)N
¤
T(l) = T(l?1)+K
????￥FourierMD11
&

[C(1) = 0bV7μ
T(l) = Kl'C(N) = KN log2 N:
aaa??????
m^) ???¨?
B???￥?
^
L￥?|b
L?
fn
μ
fN
2?k
=?fN
2 +k;k = 0;1;￠￠￠ ; N2?1
? ? ??????????MMMDDD(DCT):
?l
C(k;n) =
8
<
:
1p
N; k = 0;0? n? N?1q
2
N cos
…(2n+1)k
2N ; 1? k? N?1;0? n? N?1;
Lecture Notes on Wavelets,Chapter 2,by D.Q,Dai,2003 16
?
fun;0? n? N?1g￥DCT?l1

vn = fin
N?1X
n=0
un cos
…(2n+1)k
2N; 0? k? N?1
?fi0 =
q
1
N
fik =
q
2
N
1? k? N?1.
IMD1

un =
N?1X
k=0
fikvk cos
…(2n+1)k
2N; 0? n? N?1:
á
ìμC?1 = CT.

￡
üC￥?1?? ?
Qc =
2
66
66
66
4
1?fi?fi ￠￠￠
fi 1?fi
￠￠￠ ￠￠￠ ￠￠￠
fi 1?fi
￠￠￠?fi 1?fi
3
77
77
77
5
￥+?_

7333 g g gFourierMMMDDD
1946
MGabor??

L￥3 gFouriere0b
!g(t)1
L?f

?l
gu;?(t) = ei?tg(t?u):
L?kgk = 15?i(u;?) 2R2á
ìμ
kgu;?k = 1.
f
f 2 L2(R)￥3 gFourierMD1
(Sf)(u;?) =< f;gu;? >=
+1Z
1
f(t)g(t?u)e?i?tdt
NMDμ
H9?1
HFourierMDb
/
a
ü
?(u;?),3 gFourierMD
H—
ü
μM]￥sO
qb
gu;?￥
ü (ê?11=kgu;?k2R tjgu;?(t)j2dt = 1=kgu;?k2(R(t? u)jg(t? u)j2dt + uR jg(t?
u)j2dt) = u,yg?
R tjg(t)j2dt = 0.
] ?
??g
^
L￥
 V
¤gu;?¤
ü ( 1?.
1? ut
gu;?(t) = ei?tg(t?u)
[u1??
( ut )2 = R+1?1 (t?u)2jgu;?(t)j2 dt
= R+1?1 t2jg(t)j2 dt
Lecture Notes on Wavelets,Chapter 2,by D.Q,Dai,2003 17
[ t?G ??u.
1!,dgu;?(!) =?g(!)e?iu(!)
[?1??b
(?!)2 = 12… R+1?1 (!)2
^
jgu;?(!)j2d!
= 12… R+1?1 !2j^g(!)j2d!
?G ?.V7

u;?
￥Heisenberg0￥vlD??G ??

u;?
b
×××
T
T
T:?

u;?
R2?M
H
gu;?￥Heisenberg0-???
H—
ü


[ V[
?
V3 gFourierMDˉe^fb
??? ? ? ?13:?f 2 L2(R)
kgk = 1
O1
}f

5
f(t) = 12…
Z +1
1
Z +1
1
Sf(u;?)g(t?u)ei?td?du
Oμ]
TZ
+1
1
jf(t)j2 dt = 12…
Z +1
1
Z +1
1
jSf(u;?)j2 d?du
￡￡￡

7g?(t) = g(t)ei?t,5?g?(!) =?g(!).?Sf(u;?)1

Sf(u;?) = R+1?1 f(t)g(t?u)e?i?tdt
= R+1?1 f(t)g(t?u)e?i?(t?u)dt￠e?iu?
= R+1?1 f(t)g(u?t)ei?(u?t)dt￠e?iu?(g1
})
= e?iu?(f?g?):
[
1?u￥FourierMD1
(cSf)(!;?) =?f(! +?)?g?(! +?) =?f(! +?)?g(!):
V7
?f 2 L1(R)
H

1
2…
R+1
1
R+1
1 Sf(u;?)g(t?u)￠e
i?tdud?(g
^
L￥)
= 12… R+1?1 ei?t 12… R+1?1 cSf(! +?)?g(!)￠eit!d!d?(g(t?u) =?g(!)e?it!)
= 12… R+1?1 ei?t 12… R+1?1?f(! +?)j?g(!)j2 ￠eit!d!d?
= 12… R+1?1 12… R+1?1?f(! +?)j?g(!)j2 ￠eit(!+?)d!d?
= 12… R+1?1 12… R+1?1?f(! +?)j?g(!)j2 ￠eit(!+?)d?d!
= 12… R+1?1 12… R+1?1?f(?)eit?d?j?g(!)j2 d!
= f(t):
f =2 L1(R)
H
¨
á?ZE V￡¤b
?=
T
?Plancherel
T
μ
Lecture Notes on Wavelets,Chapter 2,by D.Q,Dai,2003 18
1
2…
R+1
1
R+1
1 jSf(u;?)j
2dud?
= 12… R+1?1 12… R+1?1
flfl
fl?f(! +?)?g(!)
flfl
fl
2d!d?
= 12… R+1?1
flfl
fl?f(! +?)
flfl
fl
2d? = kfk2,
×
T V1
f(t) = 12…
Z +1
1
Z +1
1
< f;gu;? >gu;?(t)d?du:
DFourierMD?B"
Sf(u;?)
^o?￥
y7?Φ 2 L2(R2)

?B?
^
?f 2
L2(R)￥3FourierMD￥^.





555:
!Φ 2 L2(R2)b5if 2 L2(R)
P¤Φ(u;?) = Sf(u;?)?O??
Φ(u0;?0) = 12…
Z +1
1
Z +1
1
Φ(u;?)K(u0;u;?0;?)dud?
?K(u0;u;?0;?) =< gu;?;gu0;?0 >b
￡￡￡
)
!9f

P¤Φ(u;?) = Sf(u;?)b|×
T}?f
i?ig
^
L￥
¤

Sf(u0;?0) =
Z +1
1
1
2…
Z +1
1
Z +1
1
Sf(u;?)gu;?(t)d?du
gu0;?0(t)dt
1?t￥s
'1Kb
(?lf

?×
T

μ

f(t) = 12…
Z +1
1
Z +1
1
Φ(u;?)g(t?u)ei?td?du
?Φ￥?é
 Vw¤Φ(u;?) = Sf(u;?).
8 ? ? ????333 g g gFourierMMMDDD
á
ì)
 ??3 gFourierMD??
(?;u)￥ ??ù5b
!?0;t0? 0b?i(m;n) 2
Z2,
7
gm;n(t) = eim?0tg(t?nt0):
?? ??￥ O
9?1Gabor O
Weyl-Heisenberg Ob
??3 gFourierMD?l1
(Swinf)(m;n) =< f;gm;n >=
Z 1
1
f(t)g(t?nt0)e?im?0tdt:
á
ì1?￥ù5
^
8"￥Hq/
fgm;ng?L2(R)￥ Ob
n5
á
ìóB?A
1Hqb
Lecture Notes on Wavelets,Chapter 2,by D.Q,Dai,2003 19
??? ? ? ?14



!g1ó?￥3 gf

?0;t0? 0b?fgm;ng?
a/?1B;A￥ O
5
A? 2…?
0
X
n
jg(t?nt0)j2? B;a:e:
￡￡￡


?i￥T > 0,??1=p2Te?im…t=T
^L2[?T;T]￥S??
#μ
X
m
flfl
flfl
Z T
T
f(t)e?im…t=Tdt
flfl
flfl
2
= 2T
Z T
T
jf(t)j2dt;8f 2 L2:
!t?
^Pnjg(t?nt0)j2￥Lebesgue?(x
^G￥Lebesgue?
?Rx+hx?h jG(t)?G(x)jdt = o(h)(h !
0)),Ot?=(…=?0)?
^?
b|? sl
5?it 2 [?…=?0;…=?0]
i·B￥?
fi0

P¤
[t;t?+?] ‰ [?…=?0;…=?0]+2…fi=?0b|f(t) = (2?)?1=2′[t;t?+?](t).A?μkfk = 1.
?
H
á
ìμ
P
m;nj < f;gm;n > j
2
= Pm;njRR f(t)g(t?nt0)e?im?0tdtj2
= Pm;njR…=?0…=?0 Pfi f(t+2…fi=?0)g(t+2…fi=?0?nt0)e?im?0(t+2…fi=?0)dtj2
= Pm;njR…=?0?…=?0 Pfi f(t+2…fi=?0)g(t+2…fi=?0?nt0)e?im?0tdtj2
= 2…=?0PnR…=?0?…=?0 jPfi f(t+2…fi=?0)g(t+2…fi=?0?nt0)j2dt
= 2…=?0(2?)?1PnR…=?0?…=?0 jPfi ′[t;t?+?](t+2…fi=?0)g(t+2…fi=?0?nt0)j2dt
= 2…=?0(2?)?1PnR…=?0?…=?0 ′[t;t?+?](t+2…fi0=?0)g(t+2…fi0=?0?nt0)j2dt
= 2…=?0(2?)?1PnR(1+2fi0)…=?0(?1+2fi0)…=?0 ′[t;t?+?](t)jg(t?nt0)j2dt
= 2…=?0(2?)?1PnRt?+?t jg(t?nt0)j2dt:
?fgm;ng
^μ
a/?B;A￥ O
i
7? ! 0
'¤￡b
?[?t0=2;t0=2]
s
5 V¤fgm;ng?1 O￥÷e?￥A1Hq
A? (?0t0)?12…kgk2? B:
f
gm;nD
￥FourierMDdgm;n μ??

dgm;n(!) =?g(!?m?0)e?int0!eimn?0t0:
?? ?
?? ?14￥￡
ü
á
ìw¤
Pfgm;ng?1 O

××￥A1Hqb
A? 2…=t0
X
n
j?g(!?m?0)j2? B;a:e:
á
ì9μ?/￥2Tb
??? ? ? ?15
?fgm;n(t)g
^ O
5fdgm;n(!)g9
^ O
OμM]￥ O?b
￡￡￡
ü
ü
ü
?Plancherel
T
¤
2…j < f;gm;n > j2 = j <?f; dgm;n > j2:
Lecture Notes on Wavelets,Chapter 2,by D.Q,Dai,2003 20
?2…kfk2 = k?fk2,'¤￡b
0t0 > 2…
H
?3 gf
g 2 L2?? V
Pfgm;ng?1 Obá
ì?0t0 = 4…
Hó
B?Q èbB?f?
 V? IDaubechies,pp.107-108.
á
ì|?0 = 2…;t0 = 2.
7
f(t+fi) = (?1)fig(t?fi?1);0? t < 1;fi 2 Z:
5μf 2 L2.?f? gm;n;8m;n.
Y
L

< f;gm;n > = R1?1f(t)g(t?2n)e?i2…mtdt
= R10 Pfi f(t+fi)g(t+fi?2n)e?2…imtdt;
7Oá
ìμ
P
fi f(t+fi)g(t+fi?2n)
= Pfi(?1)fig(t?fi?1)g(t+fi?2n)
= Pfl(?1)2n?fl?1g(t+fl?2n)g(t?fl?1)(
7fi = 2n?fl?1)
=?Pfl(?1)flg(t?fl?1)g(t+fl?2n):
?=
TD?
1
T?|MQ
y7??
,b
[Pfi f(t + fi)g(t+fi?2n) = 0,V7<
f;gm;n >= 0.#fgm;ng? V
?1 Ob
0t0 = 2…
H
o
μ
H

??μ￥ Ob??Balian
Low? ?
Yb￡
ü
VnDaubechies,pp,109-112.
??? ? ? ?16(Balian-Low)

!g
^B?3 gf

?0t0 = 2…,
Pfgm;ng?1L2(R)￥ O
5A
μR(tg(t))2dt = 1R(!?g(!))2d! = 1.
è
g(t) =
(
1; 0? t? 1;
0;;g(t) = sin(…t)=(…t)
Pgm;n(t) = e
2…imtg(t?n)?S??
b?B?f?μR(!j?g(!)j)2d! = 1
7?=?f?μR(tjg(t)j)2dt = 1.
[
1¤?
H

?z￥ O
á
ìo
?0t0 < 2…￥K?/ ?ùpb'
P?0t0 < 2…
9?
￡?3 gf
g
Pfgm;ng?1 Ob I
nf
g(t) = ′[0;1](t),|
?t0 > 1,?|"c?[1;t0]￥L2f
f(t)
??g(t?nt0)￥|"[nt0;1 + nt0]Df(t)￥|
"[1;t0]￥?"1 b"

[fD
μfgm;ng??
?5?0?lbD5?N
á
ì
￡
ü


V? IDaubechies,pp,83-84

?0t0 < 2…
H
 V[/ μz
H

?￥3f
g
P¤fgm;ng?1 Ob
92B/
'??á
ì￡
ü

0t0 > 2…
H

àμ Oi

0t0 = 2…
H
 Oi
?
H

?μ

0t0 < 2…
H
 V[/ μ
z
H

?￥ Ob