engineering mathematics iiocw.snu.ac.kr/sites/default/files/note/4341.pdf · 2018-01-30 · ch. 11...
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EEngineering Mathematics IIngineering Mathematics II
Prof. Dr. Yong-Su Na g(32-206, Tel. 880-7204)
Text book: Erwin Kreyszig, Advanced Engineering Mathematics,
9th Edition, Wiley (2006)
Ch. 11 Fourier Series, Integrals, Ch. 11 Fourier Series, Integrals, Ch. 11 Fourier Series, Integrals, Ch. 11 Fourier Series, Integrals, andand TransformsTransforms
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 Series11.4 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 Transforms11.9 Fourier Transform. Discrete and Fast Fourier Transforms
2
Ch. 11 Fourier Series, Integrals, Ch. 11 Fourier Series, Integrals, Ch. 11 Fourier Series, Integrals, Ch. 11 Fourier Series, Integrals, andand TransformsTransforms
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 Series11.4 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 Transforms11.9 Fourier Transform. Discrete and Fast Fourier Transforms
3
1111.8 .8 Fourier Cosine and Sine TransformsFourier Cosine and Sine Transforms((푸리에푸리에 코사인코사인 및및 사인사인 변환변환))
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)
( ) ( ) ( ) ( )∫∫∞∞
==00
cos2 ,cos : wvdvvfwAwxdwwAxfπ
적분코사인푸리에
( ) ( )
( ) ( ) ( )∫∞
=
2ˆ
. ˆ 2 wfwA cπ
변환사인리에
하자라
( ) ( ) ( )
( ) ( )∫
∫
∞
=
==⇒0
cosˆ2:)(
,cos2 : )(
wxdwwfxf
wxdxxfwff cc π
TransformCosineFourierInverse
TransformCosineFourier
역변환코사인푸리에
변환 코사인 푸리에 F
( ) ( )∫=0
cos : )( wxdwwfxf cπTransformCosineFourier Inverse역변환 코사인 푸리에
1111.8 .8 Fourier Cosine and Sine TransformsFourier Cosine and Sine Transforms((푸리에푸리에 코사인코사인 및및 사인사인 변환변환))
• Fourier Sine Transform (푸리에 사인 변환)
( ) ( ) ( ) ( )∫∫∞∞ 2( ) ( ) ( ) ( )
( ) ( )
∫∫
=
==00
ˆ2
sin2 ,sin :
wfwB
wvdvvfwBwxdwwBxfπ
하자라
적분사인푸리에
( ) ( )
( ) ( ) ( )∫∞
==⇒
=
,sin2ˆ : )(
. 2
wxdxxfwff
wfwB
ss
sπ
Transform SineFourier 변환 사인 푸리에 F
하자라
( ) ( )∫
∫
∞
=
0
sinˆ2 : )( wxdwwfxf sπ
π
Transform SineFourier Inverse역변환 사인 푸리에 ∫0π
1111.8 .8 Fourier Cosine and Sine TransformsFourier Cosine and Sine Transforms
Ex 1 Fourier Cosine and Fourier Sine Transforms
((푸리에푸리에 코사인코사인 및및 사인사인 변환변환))
Ex. 1 Fourier Cosine and Fourier Sine Transforms
( ) ( )( ) .
00
구하라변환을사인푸리에및코사인푸리에의함수⎩⎨⎧ <<
=axk
xf ( ) ( )0⎩⎨ > ax
1111.8 .8 Fourier Cosine and Sine TransformsFourier Cosine and Sine Transforms
Ex 1 Fourier Cosine and Fourier Sine Transforms
((푸리에푸리에 코사인코사인 및및 사인사인 변환변환))
Ex. 1 Fourier Cosine and Fourier Sine Transforms
( ) ( )( ) .
00
구하라변환을사인푸리에및코사인푸리에의함수⎩⎨⎧ <<
=axk
xf ( ) ( )0⎩⎨ > ax
( ) ⎟⎞
⎜⎛== ∫
awkwxdxkwfa sin2cos2ˆ ( )
⎞⎛
⎟⎠
⎜⎝∫ w
kwxdxkwf
a
c
122
cos0 ππ
( ) ⎟⎠⎞
⎜⎝⎛ −
== ∫ wawkwxdxkwf
a
scos12 sin2ˆ
0 ππ
1111.8 .8 Fourier Cosine and Sine TransformsFourier Cosine and Sine Transforms
Linearity
((푸리에푸리에 코사인코사인 및및 사인사인 변환변환))
Linearity
( ) ( ) ( )
( ) ( ) ( )gbfabgaf
gbfabgaf ccc
FFF
FFF
+=+
+=+
Cosine and Sine Transforms of Derivatives
( ) ( ) ( )gbfabgaf sss FFF +=+
( )
( ) '
,
xf
xxf
∗
∗
연속구분유한구간에서모든가
적분가능절대상에서축연속이고가
( )
( ){ } ( ){ } ( ) ( ){ } ( ){ }2
0 ,
fffff
xfx →∞→∗
FFFF
때일
P !( ){ } ( ){ } ( ) ( ){ } ( ){ }
( ){ } ( ){ } ( ) ( ){ } ( ){ } ( )02''0'2''
' ,02'
22 wfxfwxffxfwxf
xfwxffxfwxf cssc π
+
−=−=⇒
FFFF
FFFF Prove!
( ){ } ( ){ } ( ) ( ){ } ( ){ } ( )0'' ,0''' wfxfwxffxfwxf sscc ππ+−=−−= FFFF
1111.8 .8 Fourier Cosine and Sine TransformsFourier Cosine and Sine Transforms((푸리에푸리에 코사인코사인 및및 사인사인 변환변환))
Ex 3 An Application of the Operational Formula (9)Ex. 3 An Application of the Operational Formula (9)
( ) ( ) . 0 구하라변환을코사인푸리에의함수 >= − aexf ax in two ways
( ) ( ) ( )'' '' , 22 == −− xfxfaeae axax 이므로의하여미분에
( ) ( ) ( ) ( ) ( ) 20'2'' 222 +−=−−==⇒ afwffwffa cccc ππFFFF
( ) ( ) 2 22
⎞⎛
=+⇒ afwa c πF
( ) ( )0 2 22 >⎟⎠⎞
⎜⎝⎛
+=∴ − a
waae ax
c πF
1111.8 .8 Fourier Cosine and Sine TransformsFourier Cosine and Sine Transforms((푸리에푸리에 코사인코사인 및및 사인사인 변환변환))
Ex 3 An Application of the Operational Formula (9)Ex. 3 An Application of the Operational Formula (9)
( ) ( ) . 0 구하라변환을코사인푸리에의함수 >= − aexf ax in two ways
( ) ( ) ( )'' '' , 22 == −− xfxfaeae axax 이므로의하여미분에
( ) ( ) ( ) ( ) ( ) 20'2'' 222 +−=−−==⇒ afwffwffa cccc ππFFFF
( ) ( ) 2 22
⎞⎛
=+⇒ afwa c πF
( ) ( )0 2 22 >⎟⎠⎞
⎜⎝⎛
+=∴ − a
waae ax
c πF
1111.8 .8 Fourier Cosine and Sine TransformsFourier Cosine and Sine Transforms((푸리에푸리에 코사인코사인 및및 사인사인 변환변환))
PROBLEM SET 11.8
HW 4 18HW: 4, 18
1111.9 .9 Fourier Transform. Fourier Transform. Discrete Discrete and Fast Fourier Transformsand Fast Fourier Transforms((푸리에푸리에 변환변환. . 이산이산 및및 고속고속 푸리에푸리에 변환변환))
Complex Form of the Fourier Integral (푸리에 적분의 복소형식)
( ) ( )∫∞1
( ) ( ) ( )[ ]dwwxwBwxwAxf ∫∞
+=0
sincos( ) ( )
( ) ( )∫
∫∞
∞−
= wvdvvfwA
1
cos1π
0 ( ) ( )∫∞−
= wvdvvfwB sin1π
( ) dwdvwvwxvfxf ∫ ∫∞
∞
⎥⎦⎤
⎢⎣⎡ −= )cos()(1
Prove!
xixeix sincos +=
( ) dwdvwvwxvfxf ∫ ∫∞−
∞− ⎥⎦⎢⎣)cos()(
2π
0)sin()(1=⎥⎦
⎤⎢⎣⎡ −∫ ∫
∞∞
dwdvwvwxvf
Prove!
xixe sincos +0)sin()(2 ⎥⎦⎢⎣∫ ∫
∞−∞−
dwdvwvwxvfπ
( ) ( ) ( ) dwdvevfxf vxiw∫ ∫∞ ∞
−−1Prove!( ) ( ) ( ) dwdvevfxf ∫ ∫
∞− ∞−
=π2
Prove!
1111.9 .9 Fourier Transform. Fourier Transform. Discrete Discrete and Fast Fourier Transformsand Fast Fourier Transforms((푸리에푸리에 변환변환. . 이산이산 및및 고속고속 푸리에푸리에 변환변환))
Complex Form of the Fourier Integral (푸리에 적분의 복소형식)
( ) ( )∫∞1
( ) ( ) ( )[ ]dwwxwBwxwAxf ∫∞
+=0
sincos( ) ( )
( ) ( )∫
∫∞
∞−
= wvdvvfwA
1
cos1π
0 ( ) ( )∫∞−
= wvdvvfwB sin1π
( ) dwdvwvwxvfxf ∫ ∫∞
∞
⎥⎦⎤
⎢⎣⎡ −= )cos()(1
Prove!
xixeix sincos +=
( ) dwdvwvwxvfxf ∫ ∫∞−
∞− ⎥⎦⎢⎣)cos()(
2π
0)sin()(1=⎥⎦
⎤⎢⎣⎡ −∫ ∫
∞∞
dwdvwvwxvf
Prove!
xixe sincos +0)sin()(2 ⎥⎦⎢⎣∫ ∫
∞−∞−
dwdvwvwxvfπ
( ) ( ) ( ) dwdvevfxf vxiw∫ ∫∞ ∞
−−1Prove!( ) ( ) ( ) dwdvevfxf ∫ ∫
∞− ∞−
=π2
Prove!
1111.9 .9 Fourier Transform. Fourier Transform. Discrete Discrete and Fast Fourier Transformsand Fast Fourier Transforms((푸리에푸리에 변환변환. . 이산이산 및및 고속고속 푸리에푸리에 변환변환))
Complex Form of the Fourier Integral (푸리에 적분의 복소형식)
( ) ( ) dwedvevfxf iwxiwv∫ ∫∞
∞−
∞
∞−
−= ]21[
21
ππ ∞ ∞
( ) ( ) dxexfwf iwx∫∞
−=1ˆ ( ) ( ) dxexfwf ∫
∞−
=π2
( ) ( ) i∫∞
ˆ1( ) ( ) dwewfxf iwx∫∞−
−=21π
1111.9 .9 Fourier Transform. Fourier Transform. Discrete Discrete and Fast Fourier Transformsand Fast Fourier Transforms((푸리에푸리에 변환변환. . 이산이산 및및 고속고속 푸리에푸리에 변환변환))
Complex Form of the Fourier Integral (푸리에 적분의 복소형식)
( ) ( ) ( ) ddff vxiw∫ ∫∞ ∞
−−1)( I t lF iC l적분푸리에복소•
•
( ) ( ) ( ) dwdvevfxf vxiw∫ ∫∞− ∞−
=π2
: )( IntegralFourier Complex 적분 푸리에복소
( ) ( ) ( ) dxexfwff iwx∫∞
−==1ˆ:)( FTransformFourier변환푸리에
•
( ) ( ) ( ) dxexfwff ∫∞−π2
: )( FTransformFourier 변환 푸리에
( ) ( ) dwewfx f iwx∫∞
−= ˆ1 : )( TransformFourier Inverse역변환 푸리에
Existence of the Fourier Transform (푸리에 변환의 존재)
( ) ( )ff ∫∞−2
)(π
( )
( ) ( ) ( ) ( )
xxf
i∫∞1ˆˆ
존재하며는변환푸리에의
구분연속구간에서유한모든가능이고적분절대상에서축가
( ) ( ) ( ) ( ) dxexfwfwfxf iwx∫∞−
−=⇒π2
1 , 존재하며는변환푸리에의
1111.9 .9 Fourier Transform. Fourier Transform. Discrete Discrete and Fast Fourier Transformsand Fast Fourier Transforms((푸리에푸리에 변환변환. . 이산이산 및및 고속고속 푸리에푸리에 변환변환))
Ex. 1 Fourier Transform
( ) ( ) . 0 1 1 구하라변환을푸리에함수의인구간에서는이외의그이고에서 ==< xfxfx
1111.9 .9 Fourier Transform. Fourier Transform. Discrete Discrete and Fast Fourier Transformsand Fast Fourier Transforms((푸리에푸리에 변환변환. . 이산이산 및및 고속고속 푸리에푸리에 변환변환))
( ) ( ) . 0 1 1 구하라변환을푸리에함수의인구간에서는이외의그이고에서 ==< xfxfx
Ex. 1 Fourier Transform
iwx 11111
( ) ( )eeiwiw
edxewf iwiwiwx
iwx
21
21
21ˆ
1
1
1 πππ−
−=
−⋅== −
−
−
−
−∫
ww
iwwi sin2
2sin2
ππ=
−−
=
1111.9 .9 Fourier Transform. Fourier Transform. Discrete Discrete and Fast Fourier Transformsand Fast Fourier Transforms((푸리에푸리에 변환변환. . 이산이산 및및 고속고속 푸리에푸리에 변환변환))
Linearity. Fourier Transform of Derivatives
• 푸리에 변환의 선형성푸리에 변환의 선형성
( ) ( ) ( )gbfabgaf FFF +=+. 선형연산이다변환은푸리에
• 도함수의 푸리에 변환
( ) xxf ∗ 연속상에서축가함수
( )( )xfx
xxf
0,
'
→∞→∗
∗
때일
절대적분가능상에서축가
( )
( ){ } ( ){ } ( ){ } ( ){ }xfwxfxfiwxf
f
FFFF 2'' ,' −==⇒ Prove!
1111.9 .9 Fourier Transform. Fourier Transform. Discrete Discrete and Fast Fourier Transformsand Fast Fourier Transforms((푸리에푸리에 변환변환. . 이산이산 및및 고속고속 푸리에푸리에 변환변환))
Ex. 3 Application of the Operational Formular (9)2
구하라변환을푸리에의x− . 구하라변환을푸리에의xxe
( ) )0( 21 4
22
>=−− aee a
waxF( )2a
1111.9 .9 Fourier Transform. Fourier Transform. Discrete Discrete and Fast Fourier Transformsand Fast Fourier Transforms((푸리에푸리에 변환변환. . 이산이산 및및 고속고속 푸리에푸리에 변환변환))
2
구하라변환을푸리에의x−
Ex. 3 Application of the Operational Formular (9)
. 구하라변환을푸리에의xxe
( ) )0( 21 4
22
>=−− aee a
waxF
( ) ( ) ( ){ } ( )111 ⎫⎧
( )2a
( ) ( ) ( ){ } ( )22
2222
1121'
21'
21 xxxx
iw
eiweexe −−−− −=−=⎭⎬⎫
⎩⎨⎧−= FFFF
4422
2221
21
wweiweiw−−
−=−=
1111.9 .9 Fourier Transform. Fourier Transform. Discrete Discrete and Fast Fourier Transformsand Fast Fourier Transforms((푸리에푸리에 변환변환. . 이산이산 및및 고속고속 푸리에푸리에 변환변환))
Convolution (합성곱)
( )( ) ( ) ( ) ( ) ( )dfdff ∫∫∞∞
)(C l i합성곱•
• Convolution Theorem (합성곱 정리)
( )( ) ( ) ( ) ( ) ( )dppgpxfdppxgpfxgf ∫∫∞−∞−
−=−=∗ : )( nConvolutio합성곱
Convolution Theorem (합성곱 정리)
( ) ( )( ) ( ) ( )
xxgxf , , 가능절대적분상에서축유한하며연속이고구분가와함수
( ) ( ) ( )gfgf FFF π2 =∗⇒ Prove using x-p=q
1111.9 .9 Fourier Transform. Fourier Transform. Discrete Discrete and Fast Fourier Transformsand Fast Fourier Transforms((푸리에푸리에 변환변환. . 이산이산 및및 고속고속 푸리에푸리에 변환변환))
PROBLEM SET 11.9
HW 14HW: 14