与分析基础课件第三章

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FourierC???5

FourierC?

F(f)(λ)=

1

Z

√2π

Z∞f(x)e

−∞

−iλxdx

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f(x)=

1 ∞

√2π −∞

F(f)(λ)e

iλxdλ

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FourierC?"y&???z??U?,????

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FourierC???J?&?3?????&E,=???

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???A??6u&?3???5?;

?m&???UC?K????;????UC??K

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??:?????w∈L2(R)??I??,XJ?vtw∈L2(R).

1 Z

?%??????I??w,??%t∗???4w?O???

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R

t∗=

kwk22

t|w(t)|2dt,

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w

4 = 1 (Z

(t−t∗)2|w(t)|2dt)1/2.

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kwkL2 R

~:pd.??g

(t)=1 e−t2

?I??,??%????O?

α

t∗=0,4gα=√α.

2√πα 4α

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w(at)?%????O?t∗/a?4w/|a|;

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aw(t)?%????O?t∗?4w;

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w(t−t0)?%????O?t∗+t0?4w;

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w(t)eiωt?%????O?t∗?4w;

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????z

?Y?C?

??C?

??e

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??FourierC?(STFT)

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Z

??:g,gˆ??I??,Kf∈L2(R)??FourierC????

Sf(ω,b)=

R

f(t)g(t−b)e−iωtdt.

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AO/,I???pd.??

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t

1 −2

gα(t)=

?,?A??FourierC?

2√παe

4α.

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Gαf(ω,b)=

??GaborC?.

f(t)gα(t b)e−iωtdt

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R

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bg,gˆ?%????O?t∗,4g?ω∗,4gˆ.

Z

-Wω,b(t)=g(t−b)eiωt,KdSTFT???

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Sf(ω,b)=

f(t)Wω,b(t)dt.

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R

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duWω,b???I??,??%????O?t∗+b?4g,u?

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Sf(ω,b)??f3?mI

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[t∗+b−4g,t∗+b+4g]

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-Vω,b(η)=Wcω,b(η)=eiωbe−iηbgˆ(η−ω),KVω,b????k?%

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ω∗+ω???4gˆI??.du

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JJ

Sf(ω,b)=hf,Wω,bi=hfˆ,Vω,bi,

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u?Sf(ω,b)???f3??I

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[ω∗+ω−4gˆ,ω∗+ω+4gˆ]

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n???,??FourierC?Sf(ω,b)??&?3??I

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[t∗+b−4g,t∗+b+4g]×

[ω∗+ω−4gˆ,ω∗+ω+4gˆ],

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???z&E.?X????m???(b,ω)??,

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?X??I.???I?k?C??24g????

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44g4gˆ.d,dSTFT????

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f(t)=1Z Z eiωtg(t−b)Sf(ω,b)dωdb

2π R R

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?STFT(???f?&E.

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STFT??5

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STFTI?/G????C,????'.

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STFT??Nl?z,???U???|?.

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Z

??:XJψ∈L2(R)?vNN5^?

K?ψ???.

∞

Cψ:=2π

−∞

ψ(λ)2

Page11of62

|b |

|λ| dλ<+∞,

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~^??

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1, 0≤t<1,

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1

ψ(t)=

−1,2

2

≤t<1,

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0, ??.

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b

4i λsin2λ

ψ(λ)=

Marr?($?xlf?)

√2π

e−i2 4

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λ

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2

1 2 −t/2

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ψ(t)=

√2π(1−t)e

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λ

√

ψ(λ)=

λ

2π

e

2

b

1 2−2

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ψ(t)=(h∗χ[0,1])(t),

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??h?Haar?.

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t, 0≤t<1,

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ψ(t)=

1−t,1

t−2,3

2

2

3

≤t<2,

≤t<2,

2

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0, ??.

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NN5^?

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b

?*??bψ∈L1(R),Kψ??Yk.??.u?dNN5^

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|b |

Z

?

∞ψ(λ)2

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b R

−∞

|λ| dλ<+∞,

Page14of62

?ψ(0)=0,= ∞

−∞

ψ(t)dt=0.dd??ψ?k??5.q??

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ψ∈L2(R),??`5?kP~5.?d,??ψ???.

,??3k??mu"???/?u".??“?”??

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??d5.

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bψ∈L2(R)TL1(R),??vXe^?

Z∞

−∞

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ψ(t)dt=0;

C

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|ψ(t)|≤

(1+|t|)1+,C>0,>0;

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Kψ?vNN5^?,?ψ???.

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??:bψ????,????????S

ψa,b(t)=|a|−1/2ψ(

t−b),a

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??a,b∈R,a6=0.?uf∈L2(R),??Y?C????

(Wψf)(a,b)=hf,ψa,bi

=|a|−1/2Z∞

−∞

f(t)ψ(

t−ba

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bψ,φ???,f,g∈L2(R),K

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(Wψ(αf+βg))(a,b)=α(Wψf)(a,b)+β(Wψg)(a,b);

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(Wψ(Tcf))(a,b)=(Wψf)(a,b−c),??Tcf(t)=f(t−c);

(Wψ(Dcf))(a,b)=

√1(Wψf)(a,b),??Dcf(t)=1f(t),c>0;

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c

c

c

c

c

(Wαψ+βφf)(a,b)=α(Wψf)(a,b)+β(Wφf)(a,b);

(WTcψ(f))(a,b)=(Wψf)(a,b+ca);

√

c

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(WDcψ(f))(a,b)=

1(Wψf)(ac,b).

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1Z+∞Z+∞ 1

Parseval?

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bf,g∈L2(R),ψ????,K

Cψ −∞

(Wψf)(a,b)(Wψg)(a,b)a2dadb=hf,gi.

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−∞

AO/,

1Z+∞Z+∞

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21 2

Cψ −∞

|(Wψf)(a,b)|

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−∞

a2dadb=kfkL2.

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y?dFourierC?

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1

b Z −1/2

t−b

−itλ

1/2

−ibλb

ψa,b(λ)=

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?

√2π

|a|

R

ψ( )e

a

dt=|a| e

ψ(aλ)

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(Wψf)(a,b)=hf,ψa,bi

bZ

=hfˆ,ψa,bi

=|a|1/2

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R

II

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fˆ(λ)ψb(aλ)eibλdλ.

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Fa(λ)=fˆ(λ)ψb(aλ),Ga(λ)=gˆ(λ)ψb(aλ),u?

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(Wψf)(a,b)=|a|1/2Fca(b),

(Wψg)(a,b)=|a|1/2Gca(b).

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d???

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Z Z 1

( (Wψf)(a,b)(Wψg)(a,b)db)a2da

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ZRZR 1

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ZR

R

=Z( Fca(b)Gca(b)db)|a|da

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= (

ZRZR

Fa(λ)Ga(λ)dλ)

1

da

|a|

Page20of62

= ( fˆ(λ)gˆ(λ)|ψb(aλ)|2dλ)1da

R

R

Z ˆ Z |ψb(aλ)|2

|a|

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= f(λ)gˆ(λ)(

R R

|a|

da)dλ

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=Cψhf,gi.

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d(Wψf)(a,b),a,b∈R,?f

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bψ???,f∈L2(R),K3f?Y:k????

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Z

f(t)=1 ∞

Cψ −∞

Z∞|a|−1/2ψ(t−b)(W

a

ψ

−∞

dbdaf)(a,b)a2 .

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d(Wψf)(a,b),a>0,b∈R,?f

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bψ???,???v

2π

dλ=2π

λ

Z+∞|ψb(λ)|2

0

Z+∞|ψb(−λ)|2 1

0

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TitlePage

Z Z

dλ=

λ

2Cψ<∞,

K???f,g∈L2(R),k

2 +∞ +∞ 1

(Wψf)(a,b)(Wψg)(a,b)

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Page22of62

dadb=hf,gi.

Cψ 0 −∞ a2

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?????f∈L2(R),3f?Y:k????

f(t)=

2Z+∞Z+∞

|a|−1/2ψ(

t−b)(Wf)(a,b)

ψ

a

dbda

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ψ 0 −∞

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b

bψ?????,??ψ9?FourierC?ψ??I??,?

?%????O?

t∗ =

1 Z

TitlePage

t|ψ(t)|2dt,

4ψ=

2

kψk2

Z

1L R

kk

(

II

JJ

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(t−t∗)2|ψ(t)|2dt)1/2,

Z

ψL2

ω∗ = 1

L

kψˆk22

R

Page23of62

R

ω|ψˆ(ω)|2dω,

Z

4ψˆ

1

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R

= ˆ (

kψkL2

(ω−ω∗)2|ψˆ(ω)|2dω)1/2.

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b??S?J??U?ω∗??,?3?Sψa,b???

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a

dψ???I???,ψa,b(t)=|a|−1/2ψ(t−b)????I??,

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??%????O?at∗+b,a4ψ.d?Y?C???

(Wψ

f)(a,b)=Z∞

−∞

f(x)ψa,b

II

JJ

(t)dt

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?,(Wψf)(a,b)??&?3?mI

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[b+at∗−a4ψ,b+at∗+a4ψ]

S??z&E.

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dψba,b(ω)=|a|1/2e−ibωψb(aω)??ψba,b???I??,??%

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????O?ω∗/a?4ψˆ/a.u?d?C???L?

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(Wψf)(a,b)=hfˆ,ψba,bi

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?,?C??kL&??????5?U?,???&

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?3??I

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[ω∗−4ψb,ω∗+4ψb]

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a a a a

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(??C??I

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?Y?C??k????zA,(Wψf)(a,b)??&?3

?m-????????/??I

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[b+at∗−a4ψ,b+at∗+a4ψ]×

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[ω∗−4ψb,ω∗+4ψb]

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a a a a

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2a4ψ, 44ψ4b. (

??I??? ??? up?&E??

ψ

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ua>0),?mIg?C?;u$?&E?(?u?

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3?Y?C?(Wψf)(a,b)??I

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a

a

a

a

[ω∗−4ψb,ω∗+4ψb]

?,a

=1,??bω∗=34

,K??I?

j 2j

ψb

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ψ

ψ

[2j+14b,2j+24b].

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u?,?C?

1

(Wψf)(2j,b)

??&?3??

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[2j+14b,2j+24b]

ψ ψ

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???&E.??,

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(0,∞)=[[2j+14b,2j+24b]

ψ ψ

j∈Z

Page28of62

?????(0,∞)???y?.?d,d?C??

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1

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(Wψf)(2j,b),j∈Z,

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5^?ω∗=34ψb??N??v.

bψ?????.-ψe(t)=eiαtψ(t),Kψbe(ω)=

?,ψbe?%????O?

ωe∗=ω∗+α,

4ψbe=4ψb.

ψe

?α=34ψb−ω∗,=?ωe∗=34b.

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ψb(ω−α).u

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??&?f∈L2(R)??Ud?Y?C?(Wψf)(a,b)3l?z

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????aj=1/2j,j∈Ze??

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??????ψ∈L2(R)????,XJ3??~?A

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X+∞

?B,?v0<A≤B<+∞,???5^?

A≤

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j=−∞

|ψb(2−jω)|2≤B

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??.?f∈L2(R),???C????

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j j/2 1

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(Wψf)(b)=2 (Wψf)(2j,b),b∈R,j∈Z.

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+∞

X

A≤

j=−∞

II

JJ

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|ψb(2−jω)|2≤B

du?u??f∈L2(R)

1X jAkfk2≤ kWfk2

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2

≤Bkfk.

2 2π

ψ 2 2

j∈Z

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?nbψ???,K???v

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Z |b− |

Aln2≤

ω dω≤Bln2,

Z+∞|ψb(ω)|2

0

Aln2≤

+∞ψ( ω)2

Page32of62

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0

ω dω≤Bln2,

Z |b |

??.AO/,XJA=B,Kk

+∞ψ(ω)2

Cψ=2π

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dω=4πAln2.

−∞ |ω|

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y?dC?O??

Z2|ψb(2−jω)|2

dω=

Z2−j+1

|ψb(ω)|2

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dω.

1

u?,d??5^?

1

Z2A

ω

XZ2−j+1

2−j

|ψb(ω)|2

ω

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JJ

Z2B

lk

ωdω≤

j∈Z

2−j

ω dω≤

dω.

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Page33of62

1 ω

Aln2≤

+∞ψ(ω)2

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0

Z |b |

ω dω≤Bln2.

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aq/,|^

Z−1|ψb(2−jω)|2

dω=

Z2−j+1

|ψb(−ω)|2

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dω.

−2

9??5^??

Z−1A

−ω

XZ2−j+1

2−j

|ψb(−ω)|2

ω

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JJ

Z−1 B

−2

lk

−ωdω≤

j∈Z

2−j

ω dω≤

dω.

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−2 −ω

Aln2≤

+∞ψ( ω)2

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0

Z |b− |

ω dω≤Bln2.

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???

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??bψ?????.?

ψc∗(ω)= X+∞

k=−∞

II

JJ

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ψb(ω)

Page35of62

|ψb(2−kω)|2

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Fourier_C?ψ∗∈L2(R)?ψ???.

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?nbψ?????.K????ψ∗?????

?,???v

X

1 +∞

B≤

j=−∞

|ψc∗(2−jω)|2≤ .

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JJ

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1

A

d,???f∈L2(R)k????

X

1 +∞

f(t)=

2π

j=−∞

Z+∞

−∞

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ψ

(Wjf)(b)2jψ∗(2j(t−b))db.

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y?dψ∗FourierC??

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ψc∗

(2−j

ω)=

ψ(2−jω)

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b

.

Pk∈Z|ψb(2−kω)|2

X

j∈Z

|ψc∗(2−jω)|2 =

=

Pj∈Z|ψ(2−jω)|2

II

JJ

I

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PkZ|ψb(2−kω)|2)2

b

( ∈

1

Page37of62

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.

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1

A

?d,ψ∗???,??v

Pk∈Z|ψb(2−kω)|2

X

1 +∞

B≤

j=−∞

|ψc∗(2−jω)|2≤ .

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-gj(t)=2jψ(−2jt),Kk

?

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j j/2 1

????z

?Y?C?

??C?

??e

(Wψf)(b)=2

=2jR

(Wψf)(2j,b)

Rf(t)ψ(2j(t−b))dt

TitlePage

dgˆj(ω)=ψb(2−jω)?

=(f∗gj)(b).

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JJ

X

1+∞

2π

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ψ

b c

j=−∞

Z+∞

−∞

(Wjf)(b)2jψ∗(2j(t−b))db

Page38of62

1X+∞

√ Z+∞

=

2π

1

j=Z−∞

2π fˆ(ω)ψ(2−jω)ψ∗(2−jω)eitωdω

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+∞

ˆ

−∞

=√2π

f(ω)e dω

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itω

−∞

=f(t).

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k

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bj,k=

2jb0,j,k∈Z,

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??b0>0????~?,?????.????\

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0

ψb;j,k(t)=2j/2ψ(2jt−kb0),j,k∈Z,

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????f∈L2(R),|^?Y?C?l?&E

1

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(Wψf)(2j,bj,k)=hf,ψb0;j,ki

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?en?(Duffin,Schaeffer,1952)

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??:{φj,j∈Z}?Hilbert?mH?S,XJ3~?

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A,B,?v0<A≤B<+∞,????f∈H,k

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Akfk2≤ |hf,φji|2≤Bkfk2,

j∈Z

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K?{φj,j∈Z}?H????e,A,B???e?!e..

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XJA=B,K?{φj,j∈Z}?H?;?e.

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?Y?C?

??C?

??e

?

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~H=R2,e1=(0,1),e2=(−√3/2,−1/2),e3=(√3/2,−1/2).

II

JJ

???x=(x1,x2)∈R2,k

3

I

J

2

X|hx,ei| 3 2 2 3kxk2.

j=1

j =2(x1+x2)=2

Page41of62

u?,{e1,e2,e3}?R2?;?e,??A=B=3/2. w,,

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{e1,e2,e3}??5?',l??R2?.

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?e?IO?

?nHilbert?mH?S{φj,j∈Z}?IO??

?7?^??{φj,j∈Z}?H;?e,?A=B=1,

kφjk=1,j∈Z.

y?(??5)d{φj,j∈Z}?HA=B=1;?e?,

X

???f∈H,k

????z

?Y?C?

??C?

??e

?

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TitlePage

II

JJ

AOf=φk,Kk

|hf,φji|2=kfk2.

I

J

Page42of62

j∈Z

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kφkk2= |hφk,φji|2.

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X

j∈Z

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??kφkk=1,??Pj6=k|hφk,φji|2=0.lhφk,φji=δjk,=

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{φj,j∈Z}?H?IO8.

????z

?Y?C?

??C?

?e?f????e

?

??e

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??{φj,j∈Z}?Hilbert?mH???e.??5?f

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F:H→l2,Ff={hf,φji}j∈Z,∀f∈H,

II

JJ

X

??e{φj,j∈Z}???f.?F???f

I

J

F∗:l2→H,F∗c= cjφj,∀c={cj}j∈Z∈l2,

Page43of62

j∈Z

X

??e{φj,j∈Z}n??f.-T=F∗F,K

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T:H→H,Tf= hf,φjiφj,∀f∈H.

j∈Z

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?T??e{φj,j∈Z}?e?f.

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??e

????z

?Y?C?

??C?

?e?f5?

?

???e?fT??,?

hTf,fi=

X

2

j∈Z

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|hf,φji|.

II

JJ

I

J

?d,?e^?du,?u??f∈H

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Ahf,fi≤hTf,fi≤Bhf,fi,

=

??I???f.

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AI≤T≤BI,

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?

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?

T∗=T,??

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Akfk≤kTfk≤Bkfk,f∈H.

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JJ

T?_,??T−1?v

I

J

Page45of62

(T−1)∗=T−1,

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1kfk≤kT−1f 1 fk,f∈H,

B k≤Ak

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≤

1 −1 1

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BI≤T AI.

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??C?

??e

?

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?nb{φj,j∈Z}?Hilbert?mH???A,B??e.

II

JJ

?e,T??A?e?f.-

I

J

φej=T−1φj,j∈Z.

Page46of62

e e

K{φj,j∈Z}?H?B−1,A−1??e.?e.?{φj,j∈

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Z}?{φj,j∈Z}???e.

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?

??C?

??e

y?:???f∈H,k

hf,φeji=hf,T−1φji=hT−1f,φji.

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l

II

JJ

e

Pj∈Z|hf,φji|2 =Pj∈Z|hT−1f,φji|2

=kF(T−1f)k2

I

J

=hT−1f,fi.

Page47of62

2d

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1I≤T−1≤1I

B A

?

B

ji|

≤

1kfk2≤X|hf,φe 2

j∈Z

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1

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Ak

fk2.

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??C?

??e

?

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?e?m

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e

?nb{φj,j∈Z}?Hilbert?mH???e,??{φj,j∈

II

JJ

X e

Z}?????e.K???f∈H,k

I

J

Page48of62

f= hf,φjiφjj∈Z

?

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j

j

f=Xhf,φeiφ.

j∈Z

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y????f∈H,k

f=T−1Tf=T−1(

X

j∈Z

hf,φjiφj)

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?Y?C?

??C?

??e

?

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?u1????,??k

X

= hf,φji(T−1φj)

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X e

X

j∈Z

II

JJ

I

J

= hf,φjiφj.j∈Z

Page49of62

X

j∈Z

hf,φejiφj=

X

j∈Z

hf,T−1φjiφj

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= hT−1f,φjiφj

j∈Z

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=T(T−1f)=f.

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??e

?

?e?mA?5

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XJf=Pj∈Zcjφj,??{cj}j∈Z∈l2?vcj?uhf,φeji.

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JJ

K

I

J

X|cj|2>X|hf,φeji|2.

?`?,

j∈Z

j∈Z

Page50of62

f= hf,φjiφj.

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X e

j∈Z

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?|^?e{φj,j∈Z}?m?,?“!?”?m??.

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XJ???f∈H,f=Pj∈Zhf,φjiuj.K???f∈H,

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JJ

X|huj,fi|2≥X|hφej,fi|2.

?`?,

j∈Z

j∈Z

I

J

f= hf,φjiφjj∈Z

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X e

???X????m??“!?”.

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??C?

??e

?

??e

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???u??ψ∈L2(R),XJS

ψb0

;j,k(t)=2j/2ψ(2jt−kb0),j,k∈Z,

II

JJ

I

J

?L2(R)???e,K?ψ)?L2(R)????e

Page52of62

{ψb0;j,k,j,k∈Z}k???b0.

0

?nbψ∈L2(R))?L2(R)????e{ψb;j,k,j,k∈Z}

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k???b0,??e.?A,B.Kψ?????,?

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Xb

b0A≤2π |ψ(2−jξ)|2≤b0B.

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j∈Z

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??e

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??e?

?e?f

???e

Tf= hf,ψb0

X

j,k

;j,kiψb0

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?

TitlePage

;j,k,f∈L2(R)

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JJ

I

J

0

???

ψeb0;j,k

=T−1(ψb;j,k

),j,k∈Z.

Page53of62

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X e

f= hf,ψb0;j,kiψb0;j,k

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X

j,k

f=

j,k

hf,ψeb0;j,kiψb0;j,k.

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??e

????z

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??C?

?O?

?

;?e,=A=B.

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JJ

?e?f

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T=AI

???e

ψe

=T−1(ψ

1

)= ψ

I

J

Page54of62

,j,k∈Z.

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???

b0;j,k

b0;j,k

Ab0;j,k

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X

1

f=

A

j,k

hf,ψb0;j,kiψb0;j,k=

2

X

A+B

j,k

hf,ψb0;j,kiψb0;j,k.

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??C?

??e

?

?;?e,A≈B.

?e?f

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≈

A+B

T I

II

JJ

2

≈

???e

T−1 2 IA+B

???

ψeb0;j,k≈

f=

X e

j,k

2

I

J

Page55of62

A+Bψb0;j,k,j,k∈Z.

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hf,ψb0;j,kiψb0;j,k≈

hf,ψb0;j,kiψb0;j,k.

2 X

A+B

j,k

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?N/,

?

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A+B

-R=I−2 T,Kk

TitlePage

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JJ

A+B

A+B

A−BI≤R≤B−AI.

A+B

u??kRk≤B−A

<1,l

T−1=

2

(I−R)−1=

2 X+∞

I

J

Page56of62

Rn.

A+B

???,???e?L??

2 X+∞

A+B

n=0

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ψeb0

;j,k=T−1(ψb

;j,k)=

A