Ch. 11 Fourier Series, Integrals, Ch. 11 Fourier Series, Integrals, Ch. 11 Fourier Series, Integrals, Ch. 11 Fourier Series, Integrals,

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EE ngineering Mathematics II ngineering Mathematics II

Prof. Dr. Yong-Su Na g (32-206, Tel. 880-7204)

Text book: Erwin Kreyszig, Advanced Engineering Mathematics, 9^th Edition, Wiley (2006)

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Ch. 11 Fourier Series, Integrals, Ch. 11 Fourier Series, Integrals, Ch. 11 Fourier Series, Integrals, Ch. 11 Fourier Series, Integrals,

and

and Transforms Transforms

11.1 Fourier Series

11.2 Functions of Any Period p=2Ly p

11.3 Even and Odd Functions. Half-Range Expansions 11 4 Complex Fourier Series

11.4 Complex Fourier Series 11.5 Forced Oscillations

11.6 Approximation by Trigonometric Polynomials 11.7 Fourier Integral

11.8 Fourier Cosine and Sine Transforms

11.9 Fourier Transform. Discrete and Fast Fourier Transforms 11.9 Fourier Transform. Discrete and Fast Fourier Transforms

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Ch. 11 Fourier Series, Integrals, Ch. 11 Fourier Series, Integrals, Ch. 11 Fourier Series, Integrals, Ch. 11 Fourier Series, Integrals,

and

and Transforms Transforms

11.1 Fourier Series

11.2 Functions of Any Period p=2Ly p

11.3 Even and Odd Functions. Half-Range Expansions 11 4 Complex Fourier Series

11.4 Complex Fourier Series 11.5 Forced Oscillations

11.6 Approximation by Trigonometric Polynomials 11.7 Fourier Integral

11.8 Fourier Cosine and Sine Transforms

11.9 Fourier Transform. Discrete and Fast Fourier Transforms 11.9 Fourier Transform. Discrete and Fast Fourier Transforms

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11.8 .8 Fourier Cosine and Sine TransformsFourier Cosine and Sine Transforms

((푸리에푸리에 코사인코사인 및및 사인사인 변환변환))

z Integral Transform (적분변환):

주어진 함수를 다른 변수에 종속하는 새로운 함수로 만드는 적분 형태의 변환 예) L l T f (Ch 6)

예) Laplace Transform (Ch. 6)

• Fourier Cosine Transform (푸리에 코사인 변환)(푸리에 사인 변환)

( )⁼^∞∫ ( ) ( )⁼ ^∞∫ ( )

0 0

2 cos

, cos

:

f x A w wxdw A w f v wvdv

적분 π 코사인

푸리에

( ) ( )

( ) ( ) ^∞∫ ( )

=

ˆ 2

ˆ . 2 f w w

A π _c

변환 사인

리에

하자 라

( ) ( ) ( )

( ) ∫ ( )

∫

= ∞

=

⇒

0

ˆ cos : 2

) (

, 2 cos

: ) (

wxdw w

f x

f

wxdx x

f w

f

f _c

c π

Transform Cosine

Fourier Inverse

Transform Cosine

Fourier

역변환 코사인

푸리에

변환 코사인

푸리에 F

( )⁼ ∫ ( )

0

cos :

) (

f x f_c w wxdw

Transform π Cosine

Fourier Inverse

역변환 코사인

푸리에

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• Fourier Sine Transform (푸리에 사인 변환)

( ) ^∞∫ ( ) ( ) ² ^∞∫ ( )

( ) ( ) ( ) ( )

( ) ( )

∫

=

0 0

2 ˆ

2 sin

, sin

:

w f w

B

wvdv v

f w

B wxdw w

B x

f π

하자 라

적분 사인

푸리에

( ) ( )

( )⁼ ( )⁼ ^∞∫ ( )

⇒

=

, 2 sin

ˆ : ) (

. 2

wxdx x

f w

f f

w f w

B

s s

π s

Transform Sine

Fourier 변환

사인

푸리에 F

하자 라

( ) ∫ ( )

∫

∞

=

0

ˆ sin 2

: ) (

f x f_s w wxdw

π π Transform

Sine Fourier Inverse

역변환 사인

푸리에 ∫

π 0

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Ex 1 Fourier Cosine and Fourier Sine Transforms

Ex. 1 Fourier Cosine and Fourier Sine Transforms

( ) ( )

( ) ^.

0

0 의 푸리에 코사인및 푸리에 사인변환을 구하라

함수 ⎩⎨⎧ < <

= k x a

x

f ( ) ⎩⎨0 (x > a)

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Ex 1 Fourier Cosine and Fourier Sine Transforms

Ex. 1 Fourier Cosine and Fourier Sine Transforms

( ) ( )

( ) ^.

0

0 의 푸리에 코사인및 푸리에 사인변환을 구하라

함수 ⎩⎨⎧ < <

= k x a

x

f ( ) ⎩⎨0 (x > a)

( )

⎟⎞

⎜⎛

=

= ^k

∫

^wxdx ^k ^aw

w f

a 2 sin

2 cos ˆ

( )

⎞

⎛

⎟⎠

⎜⎝

∫

^wxdx ^k _w

k w

f

a c

1 2

2

cos

0 π

π

( )

⎟

⎠

⎜ ⎞

⎝

= ⎛ −

= ^k

∫

^wxdx ^k _w ^aw

w f

a s

cos 1

sin 2 ˆ 2

0 π

π

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z Linearity

Linearity

( ) ( ) ( )

(^af ^bg) ^a ( )^f ^b ( )^g

g b

f a bg

af _c _c

c

F F

F

F F

F

+

= +

+

= +

z Cosine and Sine Transforms of Derivatives

(^af ^bg) ^a _s( )^f ^b _s( )^g

s F F

F + = +

( )

'

,

x f

x x

f

∗

연속 구분

유한구간에서 모든

가

적분가능 절대

상에서 축

연속이고 가

( ) ( )

{ } { ( )} ² ( ) { ( )} { ( )}

0

,

f f

f

x f

x → ∞ →

∗

F F

때 일

P !

( )

{ } { ( )} ( ) { ( )} { ( )}

( )

{ ^'' } { ( )} ² ^'( )⁰ { ^''( )} { ( )} ² ( )⁰

'

, 2 0

'

2

2 f x f f x w f x wf

w x

f

x f w x

f f

x f w x

f _s _s _c

c π

+

−

=

−

=

⇒

F F

F

F Prove!

( )

{ ^'' } { ( )} ^'( )⁰ ^, { ^''( )} { ( )} ( )⁰

_c f x w _c f x f _s f x w _s f x wf π

π ⁼ ⁻ ⁺

−

= F F F

F

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Ex 3 An Application of the Operational Formula (9) Ex. 3 An Application of the Operational Formula (9)

( ) ( ⁰) ^.

의 푸리에 코사인 변환을 구하라

함수 f x = e⁻^ax a > in two ways

( )

^'' ( ) ^''( )

,

의하여 e⁻^ax = a²e⁻^ax이므로a² f x = f x 미분에

( ) ( )^'' ( ) ² ^'( )⁰ ( ) ²

² = = − ² − = − ² +

⇒ a _c f _c f w _c f f w _c f a

π

π ^F

F F

F

( )

( ) ²

² ²

⎞

⎛

= +

⇒ a w _c f a

F π

( )

² ⁽ ⁰⁾

₂ ₂ ⎟ >

⎠

⎜ ⎞

⎝

⎛

= +

∴ ⁻ a

w a

e ^ax a

c π

F

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Ex 3 An Application of the Operational Formula (9) Ex. 3 An Application of the Operational Formula (9)

( ) ( ⁰) ^.

의 푸리에 코사인 변환을 구하라

함수 f x = e⁻^ax a > in two ways

( )

^'' ( ) ^''( )

,

의하여 e⁻^ax = a²e⁻^ax이므로a² f x = f x 미분에

( ) ( )^'' ( ) ² ^'( )⁰ ( ) ²

² = = − ² − = − ² +

⇒ a _c f _c f w _c f f w _c f a

π

π ^F

F F

F

( )

( ) ²

² ²

⎞

⎛

= +

⇒ a w _c f a

F π

( )

² ⁽ ⁰⁾

₂ ₂ ⎟ >

⎠

⎜ ⎞

⎝

⎛

= +

∴ ⁻ a

w a

e ^ax a

c π

F

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PROBLEM SET 11.8

HW 4 18 HW: 4, 18

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11.9 .9 Fourier Transform. Fourier Transform. Discrete Discrete and Fast Fourier Transforms and Fast Fourier Transforms

((푸리에푸리에 변환변환. . 이산이산 및및 고속고속 푸리에푸리에 변환변환))

z Complex Form of the Fourier Integral (푸리에 적분의 복소형식)

( ) ¹ ^∞∫ ( )

( )^x [^A( )^w ^wx ^B( )^w ^wx]^dw

f ⁼ ∫^∞ ⁺

0

sin cos

( ) ( )

( ) ∫ ( )

∫

∞

−

= f v wvdv w

A

1 1 cos π

0 ( ) ∫ ( )

∞

−

= f v wvdv w

B 1 sin

π

( )^x ^f ^v ^wx ^wv ^dv ^dw

f ⁼ ¹ ∫ ∫^∞ _⎢⎣^⎡ ^∞ ⁽ ⁾^cos( ⁻ ⁾ _⎥⎦^⎤ ^Prove!

x i

x

e^ix = cos + sin

( )^x ^f ^v ^wx ^wv ^dv ^dw

f ∫ ∫

∞

− ⎢⎣ −∞ ( )cos( ) ⎥⎦

2π

0 )

sin(

)

1 ( ⎥⎦⎤ =

⎢⎣⎡ −

∫ ∫∞ ∞

dw dv wv wx

v f

Prove!

x i

x

e cos + sin 0

) sin(

)

2 ∫ ∫⎢⎣ ( ⎥⎦

∞

− −∞ f v wx wv dv dw

π

( )^x ^f ( )^v ^e ⁽ ⁾^dv^dw

f ^{( )}^x ¹

∫ ∫

^{∞ ∞} ^f ^{( )}^v ^e⁻^iw⁽^x⁻^v⁾^dv^dw ^Prove!

f

∫ ∫

∞

− −∞

= 2π ^Prove!

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( ) ¹ ^∞∫ ( )

( )^x [^A( )^w ^wx ^B( )^w ^wx]^dw

f ⁼ ∫^∞ ⁺

0

sin cos

( ) ( )

( ) ∫ ( )

∫

∞

−

= f v wvdv w

A

1 1 cos π

0 ( ) ∫ ( )

∞

−

= f v wvdv w

B 1 sin

π

( )^x ^f ^v ^wx ^wv ^dv ^dw

f ⁼ ¹ ∫ ∫^∞ _⎢⎣^⎡ ^∞ ⁽ ⁾^cos( ⁻ ⁾ _⎥⎦^⎤ ^Prove!

x i

x

e^ix = cos + sin

( )^x ^f ^v ^wx ^wv ^dv ^dw

f ∫ ∫

∞

− ⎢⎣ −∞ ( )cos( ) ⎥⎦

2π

0 )

sin(

)

1 ( ⎥⎦⎤ =

⎢⎣⎡ −

∫ ∫∞ ∞

dw dv wv wx

v f

Prove!

x i

x

e cos + sin 0

) sin(

)

2 ∫ ∫⎢⎣ ( ⎥⎦

∞

− −∞ f v wx wv dv dw

π

( )^x ^f ( )^v ^e ⁽ ⁾^dv^dw

f ^{( )}^x ¹

∫ ∫

^{∞ ∞} ^f ^{( )}^v ^e⁻^iw⁽^x⁻^v⁾^dv^dw ^Prove!

f

∫ ∫

∞

− −∞

= 2π ^Prove!

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( )^x ^f ( )^v ^e ^dv ^e ^dw

f ^∞

∫ ∫

^iwv ^iwx

∞

−

∞

−

= − ]

2 [ 1 2

1

π

π _∞ _∞

( )

^w ^f

( )

^x ^e ^dx

f^ˆ

^{( )}

^w ⁼ ¹ ^∞

∫

^f

^{( )}

^x ^e⁻^iwx^dx

f

∫

∞

−

= 2π

( )

¹

∫

^∞ ^ˆ

( )

ⁱ

( )

x f

( )

w e dw

f

∫

^iwx

∞

−

= −

2 1

π

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( ) ^f ( ) ⁽ ⁾^d ^d

f ¹ ∫ ∫^{∞ ∞} ⁻^iw ^x⁻^v

) (C l F i I t l 적분

푸리에

• 복소

•

( )^x ^f ( )^v ^e ⁽ ⁾^dv^dw

f ∫ ∫ ^iw ^x ^v

∞

− −∞

= 2π :

) (Complex Fourier Integral 적분

푸리에 복소

( )^f ⁼ ^f^ˆ( )^w ⁼ ¹ ^∞∫ ^f ( )^x ^e⁻^iwx^dx

: ) (Fourier Transform F 변환

푸리에

•

( )^f ^f ( )^w ∫ ^f ( )^x ^e ^dx

∞

π −

: 2 ) (Fourier Transform F 변환

푸리에

( )^x ^f ( )^w ^e ^dw

f ⁼ ¹ ^∞∫ ^ˆ ⁻^iwx

: ) (InverseFourier Transform 역변환

푸리에

z Existence of the Fourier Transform (푸리에 변환의 존재)

( ) ^f ( )

f ∫

∞

2 −

)

( π

( )

( ) ( ) ( ) ( )

x x

f

∫ i

1 ∞

ˆ ˆ

존재하며 는

변환 푸리에

의

구분연속 구간에서

유한 모든

가능이고 적분

절대 상에서

축 가

( )^x ^f ( )^w ^f ( )^w ^f ( )^x ^e ^dx

f ∫ ^iwx

∞

−

= −

⇒ 2π

1 ,

의 푸리에변환 는 존재하며

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Ex. 1 Fourier Transform

( ) ¹ ( ) ⁰ ^.

1에서 = 이고 그 이외의 구간에서는 = 인 함수의 푸리에변환을 구하라

< f x f x

x

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( ) ¹ ( ) ⁰ ^.

1에서 = 이고 그 이외의 구간에서는 = 인 함수의 푸리에변환을 구하라

< f x f x

x

Ex. 1 Fourier Transform

iwx 1

1

1 ¹ ¹

( )

(

^e ^e

)

iw iw dx e

e w

f ^iw ^iw

iwx iwx

2 1 2

1 2

ˆ 1

1 1

1 π π

π = − ⁻

⋅ −

=

= ⁻

−

∫

−

w w iw

w

i 2 sin

2 sin 2

π ⁼ π

−

= −

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z Linearity. Fourier Transform of Derivatives

• 푸리에 변환의 선형성푸리에 변환의 선형성

(^af ^bg) ^aF( )^f ^bF( )^g

F

+ = +

.

변환은 선형연산이다 푸리에

• 도함수의 푸리에 변환

( )^x ^x

f

∗ 함수 가 축상에서연속

( )

( )^x

f x

x x

f

0 ,

'

→

∞

→

∗

때 일

절대적분가능 상에서

축 가

( ) ( )

{^f ^x } ^iw {^f ( )^x } {^f ( )^x } ^w {^f ( )^x }

f

F F

F

F ' , '' ²

= = −

⇒ Prove!

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Ex. 3 Application of the Operational Formular (9)

2의 푸리에 변환을 구하라

−x

. 푸리에 변환을 구하라

x 의

xe

( )

⁽ ⁰⁾

2

1 ²₄

2 = ⁻ >

− e a

e ^ax ^w ^a

F

( )

2a

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2의 푸리에 변환을 구하라

−x

Ex. 3 Application of the Operational Formular (9) .

푸리에 변환을 구하라

x 의

xe

( )

⁽ ⁰⁾

2

1 ²₄

2 = ⁻ >

− e a

e ^ax ^w ^a

F

( )

^⎧ ¹

( )

^⎫ ¹

{ } ( )

¹

( )

2a

( ) ( ) { } ( ) ( )

2 2

1 1

2 ' 1

2

1 _x _x _x

x

iw

e iw e

e

xe⁻ ⁻ = − ⁻ = − ⁻

⎭⎬

⎫

⎩⎨

⎧−

= F F F

F

4 4

2 2

2 2 2

1 2

1

^w iw e ^w

e

iw ⁻ = − ⁻

−

=

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z Convolution (합성곱)

( ^f )( ) ^∞∫ ^f ( ) ( )^d ^∞∫ ^f ( ) ( )^d

) (C l i

• 합성곱

• Convolution Theorem (합성곱 정리)

( ^f ^g)( )^x ∫ ^f ( ) (^p ^g ^x ^p)^dp ∫ ^f (^x ^p) ( )^g ^p ^dp

∞

−

∞

−

=

−

=

∗ :

) (Convolution 합성곱

Convolution Theorem (합성곱 정리)

( ) ( )

( ) ( ) ( )

x x

g x

f , ,

와 가 구분연속이고 유한하며 축상에서 절대적분가능

함수

( f g) F( ) ( )f F g

F 2π

∗ =

⇒ Prove using x-p=q

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PROBLEM SET 11.9

HW 14 HW: 14