![Page 1: ngineering Mathematics IIocw.snu.ac.kr/sites/default/files/NOTE/6060.pdf11.1 Fourier Series (푸리에급수) Orthogonality of the Trigonometric Systems: 삼각함수계는구갂-π](https://reader031.vdocument.in/reader031/viewer/2022022808/5e2c3f2f540f9401ce4ccae2/html5/thumbnails/1.jpg)
Engineering Mathematics II
Prof. Dr. Yong-Su Na
(32-206, Tel. 880-7204)
Text book: Erwin Kreyszig, Advanced Engineering Mathematics,
9th Edition, Wiley (2006)
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Ch. 11 Fourier Series, Integrals, and Transforms
11.1 Fourier Series
11.2 Functions of Any Period p=2L
11.3 Even and Odd Functions. Half-Range Expansions
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
2
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주기현상의 예: 모터, 회전 기계, 음파, 지구의 운동, 정상 조건하의 심장
푸리에 급수는 상미분 방정식과 편미분 방정식을 수반하는 문제를해결하는 데 매우 중요한 도구
푸리에 급수의 실용적 관심대상: Discontinuous(불연속적)인 주기함수
내용: 푸리에 급수의 개념과 기법, 푸리에 적분(Fourier Integrals),
푸리에 변홖(Fourier Transform)
우리 주변에서 푸리에 급수는 매우 다양하게 이용됨:파동의 갂섭현상을 이용하여 소음도 없앨 수 있음. 예를 들어 여객기 밖은 엔짂소음으로 매우 시끄럽지만 안은 조용할 수 있는 것은 엔짂의 소음과 같은 주파수를 가지고 위상만 반대인 소음을 발생시켜 소음을 상쇄시키는 푸리에 급수 원리 덕분에 가능함.
Ch. 11 Fourier Series, Integrals, andTransforms
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11.1 Fourier Series (푸리에 급수)
Periodic Function (주기함수)
•
•
Properties
•
•
, , , nxfnpxfxpxf 321 , 대하여에모든이면주기가의함수
. , , 이다주기도의상수임의는이면주기가의와 pbaxbgxafpxgxf
.)Period( , )Function (Periodic
,
라한다주기의를하고라주기함수를
대하여에모든존재해서가양수어떤
정의대하여에실수모든
xfpxf
xfpxfxp
x
xx cos ,sin : 예주기함수인
xxexxx x ln ,cosh , , , , : 32예아닌주기함수가
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11.1 Fourier Series (푸리에 급수)
Trigonometric Series (삼각급수)
• Trigonometric System (삼각함수계)
• Trigonometric Series (삼각급수)
•
,sin ,cos , ,2sin ,2cos ,sin ,cos ,1 nxnxxxxx
. )nts(Coefficie
sincos2sin2cossincos
22110
1022110
한다라계수을상수
,b,a,b,a,a
nxbnxaaxbxaxbxaan
nn
. 2 주기함수이다인주기가합은그수렴한다면삼각급수가
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11.1 Fourier Series (푸리에 급수)
Fourier Series (푸리에 급수)
• Euler Formulas (오일러 공식)
• Fourier Coefficients (푸리에 계수): 오일러 공식에 의해 주어짂 값
• Fourier Series: 푸리에 계수를 갖는 삼각급수
,2 ,1 ,sin1
,2 ,1 ,cos1
2
10
nnxdxxfb
nnxdxxfa
dxxfa
n
n
1
0 sincos)(n
nn nxbnxaaxf
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11.1 Fourier Series (푸리에 급수)Ex. 1 Rectangular Wave (주기적인 직사각형파)
Find the Fourier coefficients of the periodic function,
Functions of this kind occur as external forces acting on mechanical systems, electromotive forces in electric circuits, etc.
xfxfxk
xkxf
2 and
0
0
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11.1 Fourier Series (푸리에 급수)Ex. 1 Rectangular Wave (주기적인 직사각형파)
Find the Fourier coefficients of the periodic function,
Functions of this kind occur as external forces acting on mechanical systems, electromotive forces in electric circuits, etc.
xfxfxk
xkxf
2 and
0
0
0 0 0
0
0
0
0
adxxfdxxfdxxfkdxxfkdxxf
이므로이고
0 0cos *
nadxxxf
짝수이
홀수이
0
4
cos12cos2
sin2
sin2
sin1
000 n
nn
k
nn
k
n
nxknxdxknxdxxfnxdxxfbn
xxx
k5sin
5
13sin
3
1sin
4 :
급수푸리에
45
1
3
11
5
1
3
11
4
2
k
kf Leibniz (1673)
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11.1 Fourier Series (푸리에 급수)
• 함수 f (x)가 x=0과 x=π에서 불연속
이므로, 모든 부분합은 이 점에서
값이 0이 되며, 이 값은 극한값인
–k와 k의 산술평균임.
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11.1 Fourier Series (푸리에 급수) Orthogonality of the Trigonometric Systems:
삼각함수 계는 구갂 -π ≤ x ≤ π에서 직교한다.
Representation by a Fourier Series (푸리에 급수에 의한 표현)
mnmndxmxnx
mndxmxnx
mndxmxnx
0cossin
0sinsin
0coscos
또는
평균우극한의좌극한과의합급수의점에서불연속인
합급수의점에서모든제외한점을불연속인
합급수의
수렴한다급수는푸리에의
갖는다를우도함수와좌도함수점에서각
구분연속에서구간
주기함수인주기가
.
. )(Derivative hand-Right )(Derivative hand-Left
)( Continuous Piecewise
2
xf
xf
xf
πx-π
Prove!
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11.1 Fourier Series (푸리에 급수)
PROBLEM SET 11.1
HW: 16, 28, 29
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11.2 Functions of Any Period p=2L (임의의 주기 p=2L을 가지는 함수)
Fourier Series of the Function f (x) of period 2L
Fourier Coefficients of f (x)
1
0 sincosn
nn xL
nbx
L
naaxf
,2 ,1 ,
sin1
,2 ,1 ,
cos1
2
1 0
ndxL
xnxf
Lb
ndxL
xnxf
La
dxxfL
a
L
L
n
L
L
n
L
L
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11.2 Functions of Any Period p=2L (임의의 주기 p=2L을 가지는 함수)
Ex. 1 Periodic Rectangular Wave (주기적인 직사각형파)
Find the Fourier series of the function,
2 ,42
210
11
120
LLp
x
xk
x
xf
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11.2 Functions of Any Period p=2L (임의의 주기 p=2L을 가지는 함수)
Ex. 1 Periodic Rectangular Wave (주기적인 직사각형파)
Find the Fourier series of the function,
2 ,42
210
11
120
LLp
x
xk
x
xf
0 02
sin
,11 ,7 ,3 2
,9 ,5 ,1 2
0
2sin
2
2cos
2
1
2cos
2
1
22
4
1
4
1
4
1
2
2
1
1
2
2
1
1
2
2
0
n
n
bdxxn
xf
nn
k
nn
kn
n
n
kdx
xnkdx
xnxfa
kkkdxdxxfa
짝수이
xxx
kkxf
2
5cos
5
1
2
3cos
3
1
2cos
2
2 :
급수푸리에
![Page 15: ngineering Mathematics IIocw.snu.ac.kr/sites/default/files/NOTE/6060.pdf11.1 Fourier Series (푸리에급수) Orthogonality of the Trigonometric Systems: 삼각함수계는구갂-π](https://reader031.vdocument.in/reader031/viewer/2022022808/5e2c3f2f540f9401ce4ccae2/html5/thumbnails/15.jpg)
11.2 Functions of Any Period p=2L (임의의 주기 p=2L을 가지는 함수)
Example 2
Example 3
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11.2 Functions of Any Period p=2L (임의의 주기 p=2L을 가지는 함수)
PROBLEM SET 11.2
HW: 13, 16
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11.3 Even and Odd Functions. Half-RangeExpansions (우함수와 기함수. 반구갂 전개)
Fourier Cosine Series, Fourier Sine Series (푸리에 코사인, 사인 급수)
•
•
,2 ,1 ,cos2
,1
tsCoefficienFourier
cos
2 :
00
0
1
0
nxdxL
nxf
Ladxxf
La
xL
naaxf
L
L
n
L
n
n
급수푸리에우함수의인주기가Series CosineFourier
,2 ,1 ,sin2
tsCoefficienFourier
sin
2 :
0
1
nxdxL
nxf
Lb
xL
nbxf
L
L
n
n
n
급수푸리에기함수의인주기가Series SineFourier
Sum and Scalar Multiple (합과 스칼라곱)
• 함수의 합의 푸리에 계수는 각각에 해당하는 푸리에 계수의 합과 같다.
• 함수 cf 의 푸리에 계수는 f 에 해당하는 푸리에 계수에 c 를 곱한 것과 같다.
11.2 Example 1
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11.3 Even and Odd Functions. Half-RangeExpansions (우함수와 기함수. 반구갂 전개)
Ex. 3 Sawtooth Wave (톱니파)
Find the Fourier series of the function
xfxfxxxf 2 and
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11.3 Even and Odd Functions. Half-RangeExpansions (우함수와 기함수. 반구갂 전개)
Ex. 3 Sawtooth Wave (톱니파)
Find the Fourier series of the function
xfxfxxxf 2 and
xxxf
f
nn
dxnxnn
nxx
nxdxxnxdxxfb
, , , naf
f
ffffxf
π
ππ
n
n
3sin3
12sin
2
1sin2
cos2
cos1cos2
sin2
sin2
210 0
.
2
00
00
1
1
1
2121
기함수이므로은
급수푸리에의
이다하면라와
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11.3 Even and Odd Functions. Half-RangeExpansions (우함수와 기함수. 반구갂 전개)
Half Range Expansions (반구갂 전개)
• Even Periodic Extension (주기적인 우함수로 확장)
• Odd Periodic Extension (주기적인 기함수로 확장)
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11.3 Even and Odd Functions. Half-RangeExpansions (우함수와 기함수. 반구갂 전개)
Ex. 4 “Triangle” and Its Half-Range Expansions
Find the two half-range expansions of the function
LxL
xLL
k
Lxx
L
k
xf
2
2
20
2
![Page 22: ngineering Mathematics IIocw.snu.ac.kr/sites/default/files/NOTE/6060.pdf11.1 Fourier Series (푸리에급수) Orthogonality of the Trigonometric Systems: 삼각함수계는구갂-π](https://reader031.vdocument.in/reader031/viewer/2022022808/5e2c3f2f540f9401ce4ccae2/html5/thumbnails/22.jpg)
11.3 Even and Odd Functions. Half-RangeExpansions (우함수와 기함수. 반구갂 전개)
Ex. 4 “Triangle” and Its Half-Range Expansions
Find the two half-range expansions of the function
LxL
xLL
k
Lxx
L
k
xf
2
2
20
2
xL
xL
kkxf
nL
n
n
kdxx
L
nxL
L
kxdx
L
nx
L
k
La
kdxxL
L
kdxx
L
k
La
L
L
L
n
L
L
L
6cos
6
12cos
2
116
2
1coscos24
cos2
cos22
2
221
. .1
222
22
2
2
0
2
2
0
0
확장우함수로주기적
![Page 23: ngineering Mathematics IIocw.snu.ac.kr/sites/default/files/NOTE/6060.pdf11.1 Fourier Series (푸리에급수) Orthogonality of the Trigonometric Systems: 삼각함수계는구갂-π](https://reader031.vdocument.in/reader031/viewer/2022022808/5e2c3f2f540f9401ce4ccae2/html5/thumbnails/23.jpg)
11.3 Even and Odd Functions. Half-RangeExpansions (우함수와 기함수. 반구갂 전개)
xL
xL
xL
kxf
n
n
kdxx
L
nxL
L
kxdx
L
nx
L
k
Lb
L
L
L
n
5sin
5
13sin
3
1sin
1
18
2sin
8sin
2sin
22
. .2
2222
22
2
2
0
확장기함수로주기적
![Page 24: ngineering Mathematics IIocw.snu.ac.kr/sites/default/files/NOTE/6060.pdf11.1 Fourier Series (푸리에급수) Orthogonality of the Trigonometric Systems: 삼각함수계는구갂-π](https://reader031.vdocument.in/reader031/viewer/2022022808/5e2c3f2f540f9401ce4ccae2/html5/thumbnails/24.jpg)
11.3 Even and Odd Functions. Half-RangeExpansions (우함수와 기함수. 반구갂 전개)
PROBLEM SET 11.3
HW: 10, 11, 19
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11.4 Complex Fourier Series(복소 푸리에 급수)
Complex Fourier Series (복소 푸리에 급수):
Complex Fourier Coefficient (복소 푸리에 계수):
주기가 2L인 복소 푸리에 급수:
n
inx
necxf
,2 ,1 ,0 ,2
1
ndxexfL
c
ecxf
Lxin
L
L
n
n
Lxin
n
,2 ,1 ,0 ,2
1
ndxexfc inx
n
![Page 26: ngineering Mathematics IIocw.snu.ac.kr/sites/default/files/NOTE/6060.pdf11.1 Fourier Series (푸리에급수) Orthogonality of the Trigonometric Systems: 삼각함수계는구갂-π](https://reader031.vdocument.in/reader031/viewer/2022022808/5e2c3f2f540f9401ce4ccae2/html5/thumbnails/26.jpg)
11.4 Complex Fourier Series(복소 푸리에 급수)
Ex. 1 Complex Fourier Series
Find the complex Fourier series of f (x) = ex if –π<x< π and f(x+2 π)=f (x)
and obtain from it the usual Fourier series
![Page 27: ngineering Mathematics IIocw.snu.ac.kr/sites/default/files/NOTE/6060.pdf11.1 Fourier Series (푸리에급수) Orthogonality of the Trigonometric Systems: 삼각함수계는구갂-π](https://reader031.vdocument.in/reader031/viewer/2022022808/5e2c3f2f540f9401ce4ccae2/html5/thumbnails/27.jpg)
11.4 Complex Fourier Series(복소 푸리에 급수)
Ex. 1 Complex Fourier Series
Find the complex Fourier series of f (x) = ex if –π<x< π and f(x+2 π)=f (x)
and obtain from it the usual Fourier series
xen
ine
n
inee
ine
indxeec
inx
n
nx
nn
x
inxxinxx
n
1
11
sinh :SeriesFourier
11
1sinh1
1
1
2
1
1
1
2
1
2
1
2
2
xxxxe
nxnnxeinein
nxnxninxnnxnxinxinein
nxnxninxnnxnxinxinein
x
inxinx
inx
inx
2sin22cos21
1sincos
11
1
2
1sinh2
sincos211
sincossincossincos11
sincossincossincos11
22
![Page 28: ngineering Mathematics IIocw.snu.ac.kr/sites/default/files/NOTE/6060.pdf11.1 Fourier Series (푸리에급수) Orthogonality of the Trigonometric Systems: 삼각함수계는구갂-π](https://reader031.vdocument.in/reader031/viewer/2022022808/5e2c3f2f540f9401ce4ccae2/html5/thumbnails/28.jpg)
11.5 Forced Oscillations (강제짂동)
Free Motion (자유운동): 외력이 없는 경우의 운동지배방정식
Forced Motion (강제운동): 외부로부터의 힘이 물체에 작용하는 경우의
운동지배방정식
• Driving Force (입력이나 구동력):
• Response (출력 또는 구동력에 대한 시스템의 응답):
0''' kycymy
trkycymy '''
tr
ty
Section 2.8 Revisited
![Page 29: ngineering Mathematics IIocw.snu.ac.kr/sites/default/files/NOTE/6060.pdf11.1 Fourier Series (푸리에급수) Orthogonality of the Trigonometric Systems: 삼각함수계는구갂-π](https://reader031.vdocument.in/reader031/viewer/2022022808/5e2c3f2f540f9401ce4ccae2/html5/thumbnails/29.jpg)
11.5 Forced Oscillations (강제짂동)
trkycymy '''
y = y (t ): displacement from rest
c : damping constant
k : spring constant (spring modulus)
r (t ): external force
![Page 30: ngineering Mathematics IIocw.snu.ac.kr/sites/default/files/NOTE/6060.pdf11.1 Fourier Series (푸리에급수) Orthogonality of the Trigonometric Systems: 삼각함수계는구갂-π](https://reader031.vdocument.in/reader031/viewer/2022022808/5e2c3f2f540f9401ce4ccae2/html5/thumbnails/30.jpg)
2.8 Modeling: Forced Oscillations. Resonance (모델화: 강제짂동. 공짂)
With Periodic external forces:
미정계수법에 의한 결정
tFkycymy cos'''0
py
22
222
0
20
222
22
0
2
22
0
0 ,
sincos
cm
cFb
cm
mFa
tbtayp
mk /0
![Page 31: ngineering Mathematics IIocw.snu.ac.kr/sites/default/files/NOTE/6060.pdf11.1 Fourier Series (푸리에급수) Orthogonality of the Trigonometric Systems: 삼각함수계는구갂-π](https://reader031.vdocument.in/reader031/viewer/2022022808/5e2c3f2f540f9401ce4ccae2/html5/thumbnails/31.jpg)
2.8 Modeling: Forced Oscillations. Resonance (모델화: 강제짂동. 공짂)
Case 1. Undamped Forced Oscillations (비감쇠 강제짂동)
이 출력은 두 개의 조화짂동의 중첩을 나타냄.
• 고유주파수 :
• 구동력의 주파수 :
Resonance (공짂) : 입력주파수와 고유주파수가 정합됨으로써
( ) 발생하는 큰 짂동의 여기현상
t
m
FtCyt
m
Fyc p
coscos cos 0
22
0
0022
0
0
seccycles
2
0
seccycles
2
0
ttm
Fy
p 0
0
0 sin2
tm
Fyy 0
02
0 cos''
![Page 32: ngineering Mathematics IIocw.snu.ac.kr/sites/default/files/NOTE/6060.pdf11.1 Fourier Series (푸리에급수) Orthogonality of the Trigonometric Systems: 삼각함수계는구갂-π](https://reader031.vdocument.in/reader031/viewer/2022022808/5e2c3f2f540f9401ce4ccae2/html5/thumbnails/32.jpg)
2.8 Modeling: Forced Oscillations. Resonance (모델화: 강제짂동. 공짂)
Tacoma Narrows Bridge(1940. 11. 4)
![Page 33: ngineering Mathematics IIocw.snu.ac.kr/sites/default/files/NOTE/6060.pdf11.1 Fourier Series (푸리에급수) Orthogonality of the Trigonometric Systems: 삼각함수계는구갂-π](https://reader031.vdocument.in/reader031/viewer/2022022808/5e2c3f2f540f9401ce4ccae2/html5/thumbnails/33.jpg)
2.8 Modeling: Forced Oscillations. Resonance (모델화: 강제짂동. 공짂)
Beats (맥놀이): 입력주파수와 고유주파수의 차가 적을 때의
강제 비감쇠짂동 (w → w0)
tt
m
Ftt
m
Fy
2sin
2sin
2 coscos 00
22
0
0
022
0
0
![Page 34: ngineering Mathematics IIocw.snu.ac.kr/sites/default/files/NOTE/6060.pdf11.1 Fourier Series (푸리에급수) Orthogonality of the Trigonometric Systems: 삼각함수계는구갂-π](https://reader031.vdocument.in/reader031/viewer/2022022808/5e2c3f2f540f9401ce4ccae2/html5/thumbnails/34.jpg)
2.8 Modeling: Forced Oscillations. Resonance (모델화: 강제짂동. 공짂)
Case 2. Damped Forced Oscillations (감쇠 강제짂동)
• Transient Solution (과도해): 비제차 방정식의 일반해 (y)
• Steady-State Solution (정상상태해): 비제차 방정식의 특수해 (yp)
순수 사인파 형태의 구동력이 주어지는 감쇠짂동 시스템의 출력은
충분하게 긴 시갂 후에 실제적으로 주파수가 입력주파수인 조화짂동이 됨.
과도해는 정상상태해로 접근한다.
Practical Resonance: 비감쇠의 경우 ω가 ω0 에 접근할 때 yp 의 짂폭이 무한대로
접근하는 반면에, 감쇠의 경우에는 이와 같은 현상은 발생하지 않는다.
이 경우에는 짂폭은 항상 유한하나, c에 의존하는 어떤 ω 에 대해 최대값을 가질 수 있다.
tCtyp cos*)(
22
222
0
20
222
22
0
2
22
0
0 ,
sincos
cm
cFb
cm
mFa
tbtayp
![Page 35: ngineering Mathematics IIocw.snu.ac.kr/sites/default/files/NOTE/6060.pdf11.1 Fourier Series (푸리에급수) Orthogonality of the Trigonometric Systems: 삼각함수계는구갂-π](https://reader031.vdocument.in/reader031/viewer/2022022808/5e2c3f2f540f9401ce4ccae2/html5/thumbnails/35.jpg)
2.8 Modeling: Forced Oscillations. Resonance (모델화: 강제짂동. 공짂)
yp 의 짂폭 (ω의 함수로 표현): R
F
cm
FbaC 0
22222
0
2
022
)(*
22
0
2
022
max
4
2*
cmc
mFbaC
)(
tan22
0
m
c
a
b
)]22(2[2
1
]2)2)((2[2
1*
2222/3
0
222
0
22/3
0
mmkcRF
cmRFd
dC
2
22
02
22
max22 m
c
m
c
m
k
일 때 는
w가 증가함에 따라 단조 감소함.
mkc 22 *C
![Page 36: ngineering Mathematics IIocw.snu.ac.kr/sites/default/files/NOTE/6060.pdf11.1 Fourier Series (푸리에급수) Orthogonality of the Trigonometric Systems: 삼각함수계는구갂-π](https://reader031.vdocument.in/reader031/viewer/2022022808/5e2c3f2f540f9401ce4ccae2/html5/thumbnails/36.jpg)
2.8 Modeling: Forced Oscillations. Resonance (모델화: 강제짂동. 공짂)
의미
• 일 때 는 유한하다는 것을 알 수 있다.
• 일 때 의 값은
c가 감소함에 따라 증가하고, c가 0에 접근함에 따라 무한대로 접근한다.
0c max
* C
mkc 22 max
* C
ω0
![Page 37: ngineering Mathematics IIocw.snu.ac.kr/sites/default/files/NOTE/6060.pdf11.1 Fourier Series (푸리에급수) Orthogonality of the Trigonometric Systems: 삼각함수계는구갂-π](https://reader031.vdocument.in/reader031/viewer/2022022808/5e2c3f2f540f9401ce4ccae2/html5/thumbnails/37.jpg)
11.5 Forced Oscillations (강제짂동)
Ex. 1 Forced Oscillations under a Nonsinusoidal Periodic Driving Force
Find the steady-state solution y (t ).
trtr
tt
tttrtryyy
2
0 2
0 2 ,25'05.0''
![Page 38: ngineering Mathematics IIocw.snu.ac.kr/sites/default/files/NOTE/6060.pdf11.1 Fourier Series (푸리에급수) Orthogonality of the Trigonometric Systems: 삼각함수계는구갂-π](https://reader031.vdocument.in/reader031/viewer/2022022808/5e2c3f2f540f9401ce4ccae2/html5/thumbnails/38.jpg)
11.5 Forced Oscillations (강제짂동)
Ex. 1 Forced Oscillations under a Nonsinusoidal Periodic Driving Force
Find the steady-state solution y (t ).
trtr
tt
tttrtryyy
2
0 2
0 2 ,25'05.0''
531
222
2
2
2
22
:
05.025 2.0
,254
sincos : ,3 ,1 cos4
25'05.0''
5cos5
13cos
3
1cos
4 :
yyyy
nnDDn
BDn
nA
ntBntAynntn
yyy
ttttrtr
n
n
n
n
n
nnn
해정상상태
여기서
해정상상태의상미분방정식
급수푸리에의
,2 ,1 ,cos2
,1
tsCoefficienFourier
cos
2 :
00
0
1
0
nxdxL
nxf
Ladxxf
La
xL
naaxf
L
L
n
L
n
n
급수푸리에우함수의인주기가Series CosineFourier
![Page 39: ngineering Mathematics IIocw.snu.ac.kr/sites/default/files/NOTE/6060.pdf11.1 Fourier Series (푸리에급수) Orthogonality of the Trigonometric Systems: 삼각함수계는구갂-π](https://reader031.vdocument.in/reader031/viewer/2022022808/5e2c3f2f540f9401ce4ccae2/html5/thumbnails/39.jpg)
11.5 Forced Oscillations (강제짂동)
Ex. 1 Forced Oscillations under a Nonsinusoidal Periodic Driving Force
Find the steady-state solution y (t ).
trtr
tt
tttrtryyy
2
0 2
0 2 ,25'05.0''
531
222
2
2
2
22
:
05.025 2.0
,254
sincos : ,3 ,1 cos4
25'05.0''
5cos5
13cos
3
1cos
4 :
yyyy
nnDDn
BDn
nA
ntBntAynntn
yyy
ttttrtr
n
n
n
n
n
nnn
해정상상태
여기서
해정상상태의상미분방정식
급수푸리에의
n=5 term is dominant.
![Page 40: ngineering Mathematics IIocw.snu.ac.kr/sites/default/files/NOTE/6060.pdf11.1 Fourier Series (푸리에급수) Orthogonality of the Trigonometric Systems: 삼각함수계는구갂-π](https://reader031.vdocument.in/reader031/viewer/2022022808/5e2c3f2f540f9401ce4ccae2/html5/thumbnails/40.jpg)
11.5 Forced Oscillations (강제짂동)
PROBLEM SET 11.5
HW: 2, 3
![Page 41: ngineering Mathematics IIocw.snu.ac.kr/sites/default/files/NOTE/6060.pdf11.1 Fourier Series (푸리에급수) Orthogonality of the Trigonometric Systems: 삼각함수계는구갂-π](https://reader031.vdocument.in/reader031/viewer/2022022808/5e2c3f2f540f9401ce4ccae2/html5/thumbnails/41.jpg)
11.6 Approximation by TrigonometricPolynomials (삼각다항식에 의한 근사)
Approximation Theory (근사이론):
푸리에 급수의 주된 응용 분야로 단순한 함수로써 어떤 함수의 근사값을표현하는 분야
• Idea
• Question
N
n
nn nxbnxaaxf
xfN
xxf
1
0 sincos
)( 2
근사값대한에부분합은차
주기함수있는수표현될급수로푸리에인주기가는
구하기고정은삼각다항식차근사인의최상의 sincos 1
0 NnxBnxAAxFNfN
n
nn
![Page 42: ngineering Mathematics IIocw.snu.ac.kr/sites/default/files/NOTE/6060.pdf11.1 Fourier Series (푸리에급수) Orthogonality of the Trigonometric Systems: 삼각함수계는구갂-π](https://reader031.vdocument.in/reader031/viewer/2022022808/5e2c3f2f540f9401ce4ccae2/html5/thumbnails/42.jpg)
11.6 Approximation by TrigonometricPolynomials (삼각다항식에 의한 근사)
• Square Error (제곱오차)
Minimum Square Error (최소제곱오차)
•
• Bessel’s Inequality:
• Parseval’s Identity:
)( : 2
Error Square제곱오차관한에함수의함수상에서구간 fFxdxFfE
.2* .
1
2220
2 이다최소값은그된다
최소가계수이면푸리에의계수가의제곱오차는관한에의에서구간
N
n
nn baadxfE
fFfFx
.
된다하게
근사화잘를더점점관점에서제곱오차부분합은급수푸리에의따라증가함에이 ffN
dxxfbaan
nn
2
1
222
0
12
dxxfbaan
nn
2
1
222
0
12
![Page 43: ngineering Mathematics IIocw.snu.ac.kr/sites/default/files/NOTE/6060.pdf11.1 Fourier Series (푸리에급수) Orthogonality of the Trigonometric Systems: 삼각함수계는구갂-π](https://reader031.vdocument.in/reader031/viewer/2022022808/5e2c3f2f540f9401ce4ccae2/html5/thumbnails/43.jpg)
Ex. 11.3 Sawtooth Wave (톱니파)
Find the Fourier series of the function
xfxfxxxf 2 and
xxxf
f
nn
dxnxnn
nxx
nxdxxnxdxxfb
, , , naf
f
ffffxf
π
ππ
n
n
3sin3
12sin
2
1sin2
cos2
cos1cos2
sin2
sin2
210 0
.
2
00
00
1
1
1
2121
기함수이므로은
급수푸리에의
이다하면라와
Gibb’s phenomenon
11.6 Approximation by TrigonometricPolynomials (삼각다항식에 의한 근사)
![Page 44: ngineering Mathematics IIocw.snu.ac.kr/sites/default/files/NOTE/6060.pdf11.1 Fourier Series (푸리에급수) Orthogonality of the Trigonometric Systems: 삼각함수계는구갂-π](https://reader031.vdocument.in/reader031/viewer/2022022808/5e2c3f2f540f9401ce4ccae2/html5/thumbnails/44.jpg)
Ex. 11.3 Sawtooth Wave (톱니파)
Find the Fourier series of the function
xfxfxxxf 2 and
Gibb’s phenomenon
11.6 Approximation by TrigonometricPolynomials (삼각다항식에 의한 근사)
N
n ndxxE
12
22 142)(*
N E*
1 8.1045
2 4.9629
10 1.1959
20 0.6129
100 0.0126
![Page 45: ngineering Mathematics IIocw.snu.ac.kr/sites/default/files/NOTE/6060.pdf11.1 Fourier Series (푸리에급수) Orthogonality of the Trigonometric Systems: 삼각함수계는구갂-π](https://reader031.vdocument.in/reader031/viewer/2022022808/5e2c3f2f540f9401ce4ccae2/html5/thumbnails/45.jpg)
11.6 Approximation by TrigonometricPolynomials (삼각다항식에 의한 근사)
PROBLEM SET 11.6
HW: 5, 14
![Page 46: ngineering Mathematics IIocw.snu.ac.kr/sites/default/files/NOTE/6060.pdf11.1 Fourier Series (푸리에급수) Orthogonality of the Trigonometric Systems: 삼각함수계는구갂-π](https://reader031.vdocument.in/reader031/viewer/2022022808/5e2c3f2f540f9401ce4ccae2/html5/thumbnails/46.jpg)
11.7 Fourier Integral (푸리에 적분)
Fourier series are powerful tools for problems involvingfunctions that are periodic or are of interest on a finiteinterval only.
However, many problems involve functions that arenonperiodic and are of interest on the whole x-axis.
Let L → ∞ (period: 2L)
![Page 47: ngineering Mathematics IIocw.snu.ac.kr/sites/default/files/NOTE/6060.pdf11.1 Fourier Series (푸리에급수) Orthogonality of the Trigonometric Systems: 삼각함수계는구갂-π](https://reader031.vdocument.in/reader031/viewer/2022022808/5e2c3f2f540f9401ce4ccae2/html5/thumbnails/47.jpg)
11.7 Fourier Integral (푸리에 적분)
Ex. 1 Rectangular Wave (직사각형파)
Consider the periodic rectangular wave fL(x) of period 2L>2 given by
The nonperiodic function f (x) obtained from fL if we let L → ∞.
Lx
x
xL
xfL
10
111
10
otherwise0
111lim
xxfxf L
L
![Page 48: ngineering Mathematics IIocw.snu.ac.kr/sites/default/files/NOTE/6060.pdf11.1 Fourier Series (푸리에급수) Orthogonality of the Trigonometric Systems: 삼각함수계는구갂-π](https://reader031.vdocument.in/reader031/viewer/2022022808/5e2c3f2f540f9401ce4ccae2/html5/thumbnails/48.jpg)
11.7 Fourier Integral (푸리에 적분)
![Page 49: ngineering Mathematics IIocw.snu.ac.kr/sites/default/files/NOTE/6060.pdf11.1 Fourier Series (푸리에급수) Orthogonality of the Trigonometric Systems: 삼각함수계는구갂-π](https://reader031.vdocument.in/reader031/viewer/2022022808/5e2c3f2f540f9401ce4ccae2/html5/thumbnails/49.jpg)
11.7 Fourier Integral (푸리에 적분)
From Fourier Series to Fourier Integral (L→∞)
dxL
xnxf
Lb
dxL
xnxf
La
dxxfL
a
L
L
n
L
L
n
L
L
sin
1
cos
1
2
1 0
LL
n
L
nwww
vdvwvfwxwvdvwvfwxwdvvfL
vdvwvfxwvdvwvfxwL
dvvfL
L
nwxwbxwaaxf
nn
nn
L
L
Lnn
L
L
Ln
L
L
L
nn
L
L
Lnn
L
L
Ln
L
L
L
nn
nnnnL
1
sinsincoscos1
2
1
sinsincoscos1
2
1
,sincos
1
1
1
10
![Page 50: ngineering Mathematics IIocw.snu.ac.kr/sites/default/files/NOTE/6060.pdf11.1 Fourier Series (푸리에급수) Orthogonality of the Trigonometric Systems: 삼각함수계는구갂-π](https://reader031.vdocument.in/reader031/viewer/2022022808/5e2c3f2f540f9401ce4ccae2/html5/thumbnails/50.jpg)
11.7 Fourier Integral (푸리에 적분)
From Fourier Series to Fourier Integral (L→∞)
LL
n
L
nwww
vdvwvfwxwvdvwvfwxwdvvfL
vdvwvfxwvdvwvfxwL
dvvfL
L
nwxwbxwaaxf
nn
nn
L
L
Lnn
L
L
Ln
L
L
L
nn
L
L
Lnn
L
L
Ln
L
L
L
nn
nnnnL
1
sinsincoscos1
2
1
sinsincoscos1
2
1
,sincos
1
1
1
10
wvdvvfwBwvdvvfwA
dwwxwBwxwAxf
dwwvdvvfwxwvdvvfwxxfxf LL
sin1
,cos1
sincos :
sinsincoscos1
lim
0
0
적분 푸리에
![Page 51: ngineering Mathematics IIocw.snu.ac.kr/sites/default/files/NOTE/6060.pdf11.1 Fourier Series (푸리에급수) Orthogonality of the Trigonometric Systems: 삼각함수계는구갂-π](https://reader031.vdocument.in/reader031/viewer/2022022808/5e2c3f2f540f9401ce4ccae2/html5/thumbnails/51.jpg)
11.7 Fourier Integral (푸리에 적분)
Fourier Integral
.
.
limlim
0
0
같다평균과
우극한값의좌극한값과의점에서그적분값은푸리에점에서의불연속인가
있다수표현될적분으로푸리에는
존재적분이의
존재우도함수가좌도함수와점에서모든
구분연속유한구간에서모든
xfxf
xf
dxxfdxxfb
ba
a
![Page 52: ngineering Mathematics IIocw.snu.ac.kr/sites/default/files/NOTE/6060.pdf11.1 Fourier Series (푸리에급수) Orthogonality of the Trigonometric Systems: 삼각함수계는구갂-π](https://reader031.vdocument.in/reader031/viewer/2022022808/5e2c3f2f540f9401ce4ccae2/html5/thumbnails/52.jpg)
11.7 Fourier Integral (푸리에 적분)Ex. 2 Single Pulse, Sine Integral
Find the Fourier integral representation of the function
10
11
x
xxf
dww
wwxxf
wvdvwvdvvfwBw
w
w
wvwvdvwvdvvfwA
0
1
1
1
1
1
1
sincos2
0sin1
sin1
,sin2sin
cos1
cos1
2
sin 0
10
14
10 2
sincos : Factor) uous(Discontin Dirichlet
0
0
dww
wx
x
x
x
dww
wwx불연속인자의
![Page 53: ngineering Mathematics IIocw.snu.ac.kr/sites/default/files/NOTE/6060.pdf11.1 Fourier Series (푸리에급수) Orthogonality of the Trigonometric Systems: 삼각함수계는구갂-π](https://reader031.vdocument.in/reader031/viewer/2022022808/5e2c3f2f540f9401ce4ccae2/html5/thumbnails/53.jpg)
11.7 Fourier Integral (푸리에 적분)Ex. 2 Single Pulse, Sine Integral
Find the Fourier integral representation of the function
10
11
x
xxf
dww
wwxxf
wvdvwvdvvfwBw
w
w
wvwvdvwvdvvfwA
0
1
1
1
1
1
1
sincos2
0sin1
sin1
,sin2sin
cos1
cos1
2
sin 0
10
14
10 2
sincos : Factor) uous(Discontin Dirichlet
0
0
dww
wx
x
x
x
dww
wwx불연속인자의
dww
wwxxf
wvdvwvdvvfwB
w
w
w
wvwvdvwvdvvfwA
0
1
1
1
1
1
1
sincos2
0sin1
sin1
,sin2sin
cos1
cos1
10
14
10 2
sincos :)(Factor ousDiscontinuDirichlet
0 x
x
x
dww
wwx
불연속인자
2
sin 0
0
dww
wx
![Page 54: ngineering Mathematics IIocw.snu.ac.kr/sites/default/files/NOTE/6060.pdf11.1 Fourier Series (푸리에급수) Orthogonality of the Trigonometric Systems: 삼각함수계는구갂-π](https://reader031.vdocument.in/reader031/viewer/2022022808/5e2c3f2f540f9401ce4ccae2/html5/thumbnails/54.jpg)
11.7 Fourier Integral (푸리에 적분)
dww
wu
u
0
sinSi :)( Integral Sine 사인적분
1Si1
1Si1
sin1sin1
sin1sin1
sincos2
1
0
1
0
00
0
xaxa
dtt
tdt
t
t
dww
wxwdw
w
wxw
dww
wwxxf
axax
aa
a
![Page 55: ngineering Mathematics IIocw.snu.ac.kr/sites/default/files/NOTE/6060.pdf11.1 Fourier Series (푸리에급수) Orthogonality of the Trigonometric Systems: 삼각함수계는구갂-π](https://reader031.vdocument.in/reader031/viewer/2022022808/5e2c3f2f540f9401ce4ccae2/html5/thumbnails/55.jpg)
11.7 Fourier Integral (푸리에 적분)
Fourier Cosine Integral and Fourier Sine Integral
• Fourier Cosine Integral: 우함수일 때, 푸리에 적분
• Fourier Sine Integral: 기함수일 때, 푸리에 적분
00
cos2
,cos wvdvvfwAwxdwwAxf
00
sin2
,sin wvdvvfwBwxdwwBxf
![Page 56: ngineering Mathematics IIocw.snu.ac.kr/sites/default/files/NOTE/6060.pdf11.1 Fourier Series (푸리에급수) Orthogonality of the Trigonometric Systems: 삼각함수계는구갂-π](https://reader031.vdocument.in/reader031/viewer/2022022808/5e2c3f2f540f9401ce4ccae2/html5/thumbnails/56.jpg)
11.7 Fourier Integral (푸리에 적분)Ex. 3 Laplace Integrals (라플라스 적분)
We shall derive the Fourier cosine and Fourier sine integrals of f (x) = e-kx, where x > 0 and k > 0. The result will be used to evaluate the so-called Laplace integrals.
kxkx
kvkv
kxkx
kvkv
edwwk
wxwdw
wk
wxwexf
wk
wwvwv
w
ke
wk
wdvwvewB
ek
dwwk
wxdw
wk
wxkexf
wk
kwvwv
k
we
wk
kdvwvewA
2
sin
sin2
2cossin
2sin
2
.2
2
cos
cos2
2cossin
2cos
2
.1
0
22
0
22
22
0
22
0
0
22
0
22
22
0
22
0
적분사인푸리에
적분코사인푸리에
![Page 57: ngineering Mathematics IIocw.snu.ac.kr/sites/default/files/NOTE/6060.pdf11.1 Fourier Series (푸리에급수) Orthogonality of the Trigonometric Systems: 삼각함수계는구갂-π](https://reader031.vdocument.in/reader031/viewer/2022022808/5e2c3f2f540f9401ce4ccae2/html5/thumbnails/57.jpg)
11.7 Fourier Integral (푸리에 적분)Ex. 3 Laplace Integrals (라플라스 적분)
We shall derive the Fourier cosine and Fourier sine integrals of f (x) = e-kx, where x > 0 and k > 0. The result will be used to evaluate the so-called Laplace integrals.
kxkx
kvkv
kxkx
kvkv
edwwk
wxwdw
wk
wxwexf
wk
wwvwv
w
ke
wk
wdvwvewB
ek
dwwk
wxdw
wk
wxkexf
wk
kwvwv
k
we
wk
kdvwvewA
2
sin
sin2
2cossin
2sin
2
.2
2
cos
cos2
2cossin
2cos
2
.1
0
22
0
22
22
0
22
0
0
22
0
22
22
0
22
0
적분사인푸리에
적분코사인푸리에
kxkx
kvkv
kxkx
kvkv
edwwk
wxwdw
wk
wxwexf
wk
wwvwv
w
ke
wk
wdvwvewB
ek
dwwk
wxdw
wk
wxkexf
wk
kwvwv
k
we
wk
kdvwvewA
2
sin
sin2
2cossin
2sin
2
integral sineFourier .2
2
cos
cos2
2cossin
2cos
2
integral cosineFourier .1
0
22
0
22
22
0
22
0
0
22
0
22
22
0
22
0
![Page 58: ngineering Mathematics IIocw.snu.ac.kr/sites/default/files/NOTE/6060.pdf11.1 Fourier Series (푸리에급수) Orthogonality of the Trigonometric Systems: 삼각함수계는구갂-π](https://reader031.vdocument.in/reader031/viewer/2022022808/5e2c3f2f540f9401ce4ccae2/html5/thumbnails/58.jpg)
11.7 Fourier Integral (푸리에 적분)Ex. 3 Laplace Integrals (라플라스 적분)
We shall derive the Fourier cosine and Fourier sine integrals of f (x) = e-kx, where x > 0 and k > 0. The result will be used to evaluate the so-called Laplace integrals.
kxkx
kvkv
kxkx
kvkv
edwwk
wxwdw
wk
wxwexf
wk
wwvwv
w
ke
wk
wdvwvewB
ek
dwwk
wxdw
wk
wxkexf
wk
kwvwv
k
we
wk
kdvwvewA
2
sin
sin2
2cossin
2sin
2
.2
2
cos
cos2
2cossin
2cos
2
.1
0
22
0
22
22
0
22
0
0
22
0
22
22
0
22
0
적분사인푸리에
적분코사인푸리에
kxkx
kvkv
kxkx
kvkv
edwwk
wxwdw
wk
wxwexf
wk
wwvwv
w
ke
wk
wdvwvewB
ek
dwwk
wxdw
wk
wxkexf
wk
kwvwv
k
we
wk
kdvwvewA
2
sin
sin2
2cossin
2sin
2
integral sineFourier .2
2
cos
cos2
2cossin
2cos
2
integral cosineFourier .1
0
22
0
22
22
0
22
0
0
22
0
22
22
0
22
0
Laplace integrals
![Page 59: ngineering Mathematics IIocw.snu.ac.kr/sites/default/files/NOTE/6060.pdf11.1 Fourier Series (푸리에급수) Orthogonality of the Trigonometric Systems: 삼각함수계는구갂-π](https://reader031.vdocument.in/reader031/viewer/2022022808/5e2c3f2f540f9401ce4ccae2/html5/thumbnails/59.jpg)
11.7 Fourier Integral (푸리에 적분)
PROBLEM SET 11.7
HW: 20
![Page 60: ngineering Mathematics IIocw.snu.ac.kr/sites/default/files/NOTE/6060.pdf11.1 Fourier Series (푸리에급수) Orthogonality of the Trigonometric Systems: 삼각함수계는구갂-π](https://reader031.vdocument.in/reader031/viewer/2022022808/5e2c3f2f540f9401ce4ccae2/html5/thumbnails/60.jpg)
11.8 Fourier Cosine and Sine Transforms(푸리에 코사인 및 사인 변환)
Integral Transform (적분변환):
주어짂 함수를 다른 변수에 종속하는 새로운 함수로 만드는 적분 형태의 변홖
예) Laplace Transform (Ch. 6)
• Fourier Cosine Transform (푸리에 코사인 변환)
0
0
00
cosˆ2 :
,cos2ˆ :
ˆ 2
cos2
,cos
wxdwwfxf
wxdxxfwff
wfwA
wvdvvfwAwxdwwAxf
c
cc
c
Transform CosineFourier Inverse
Transform CosineFourier F
dttfefsF st
0
L
![Page 61: ngineering Mathematics IIocw.snu.ac.kr/sites/default/files/NOTE/6060.pdf11.1 Fourier Series (푸리에급수) Orthogonality of the Trigonometric Systems: 삼각함수계는구갂-π](https://reader031.vdocument.in/reader031/viewer/2022022808/5e2c3f2f540f9401ce4ccae2/html5/thumbnails/61.jpg)
• Fourier Sine Transform (푸리에 사인 변환)
0
0
00
sinˆ2 :
,sin2ˆ :
ˆ 2
sin2
,sin
wxdwwfxf
wxdxxfwff
wfwB
wvdvvfwBwxdwwBxf
s
ss
s
Transform SineFourier Inverse
Transform SineFourier F
11.8 Fourier Cosine and Sine Transforms(푸리에 코사인 및 사인 변환)
![Page 62: ngineering Mathematics IIocw.snu.ac.kr/sites/default/files/NOTE/6060.pdf11.1 Fourier Series (푸리에급수) Orthogonality of the Trigonometric Systems: 삼각함수계는구갂-π](https://reader031.vdocument.in/reader031/viewer/2022022808/5e2c3f2f540f9401ce4ccae2/html5/thumbnails/62.jpg)
Ex. 1 Fourier Cosine and Fourier Sine Transforms
ax
axkxf
0
0
11.8 Fourier Cosine and Sine Transforms(푸리에 코사인 및 사인 변환)
![Page 63: ngineering Mathematics IIocw.snu.ac.kr/sites/default/files/NOTE/6060.pdf11.1 Fourier Series (푸리에급수) Orthogonality of the Trigonometric Systems: 삼각함수계는구갂-π](https://reader031.vdocument.in/reader031/viewer/2022022808/5e2c3f2f540f9401ce4ccae2/html5/thumbnails/63.jpg)
Ex. 1 Fourier Cosine and Fourier Sine Transforms
ax
axkxf
0
0
w
awkwxdxkwf
w
awkwxdxkwf
a
s
a
c
cos12 sin
2ˆ
sin2cos
2ˆ
0
0
11.8 Fourier Cosine and Sine Transforms(푸리에 코사인 및 사인 변환)
![Page 64: ngineering Mathematics IIocw.snu.ac.kr/sites/default/files/NOTE/6060.pdf11.1 Fourier Series (푸리에급수) Orthogonality of the Trigonometric Systems: 삼각함수계는구갂-π](https://reader031.vdocument.in/reader031/viewer/2022022808/5e2c3f2f540f9401ce4ccae2/html5/thumbnails/64.jpg)
Linearity
Cosine and Sine Transforms of Derivatives
f(x)가 연속이고, x축 상에서 절대적분 가능
f’ (x)가 모든 유한구갂에서 구분 연속
gbfabgaf
gbfabgaf
sss
ccc
FFF
FFF
02
'' ,0'2
''
' ,02
'
0 ,
22 wfxfwxffxfwxf
xfwxffxfwxf
xfx
sscc
cssc
FFFF
FFFF
11.8 Fourier Cosine and Sine Transforms(푸리에 코사인 및 사인 변환)
Prove!
![Page 65: ngineering Mathematics IIocw.snu.ac.kr/sites/default/files/NOTE/6060.pdf11.1 Fourier Series (푸리에급수) Orthogonality of the Trigonometric Systems: 삼각함수계는구갂-π](https://reader031.vdocument.in/reader031/viewer/2022022808/5e2c3f2f540f9401ce4ccae2/html5/thumbnails/65.jpg)
11.8 Fourier Cosine and Sine Transforms(푸리에 코사인 및 사인 변환)
Ex. 3 An Application of the Operational Formula (9)
Find the Fourier cosine transform 0 aexf ax in two ways
![Page 66: ngineering Mathematics IIocw.snu.ac.kr/sites/default/files/NOTE/6060.pdf11.1 Fourier Series (푸리에급수) Orthogonality of the Trigonometric Systems: 삼각함수계는구갂-π](https://reader031.vdocument.in/reader031/viewer/2022022808/5e2c3f2f540f9401ce4ccae2/html5/thumbnails/66.jpg)
11.8 Fourier Cosine and Sine Transforms(푸리에 코사인 및 사인 변환)
Ex. 3 An Application of the Operational Formula (9)
Find the Fourier cosine transform 0 aexf ax
0 2
2
20'
2''
'' '' ,
22
22
222
22
awa
ae
afwa
afwffwffa
xfxfaeae
axc
c
cccc
axax
F
F
FFFF
이므로의하여미분에
in two ways
![Page 67: ngineering Mathematics IIocw.snu.ac.kr/sites/default/files/NOTE/6060.pdf11.1 Fourier Series (푸리에급수) Orthogonality of the Trigonometric Systems: 삼각함수계는구갂-π](https://reader031.vdocument.in/reader031/viewer/2022022808/5e2c3f2f540f9401ce4ccae2/html5/thumbnails/67.jpg)
PROBLEM SET 11.8
HW: 4, 18
11.8 Fourier Cosine and Sine Transforms(푸리에 코사인 및 사인 변환)
![Page 68: ngineering Mathematics IIocw.snu.ac.kr/sites/default/files/NOTE/6060.pdf11.1 Fourier Series (푸리에급수) Orthogonality of the Trigonometric Systems: 삼각함수계는구갂-π](https://reader031.vdocument.in/reader031/viewer/2022022808/5e2c3f2f540f9401ce4ccae2/html5/thumbnails/68.jpg)
xixeix sincos
11.9 Fourier Transform. Discrete and Fast Fourier Transforms(푸리에 변환. 이산 및 고속 푸리에 변환)
dwwxwBwxwAxf
0
sincos
wvdvvfwB
wvdvvfwA
sin1
cos1
dwdvwvwxvfxf
)cos()(
2
1
0)sin()(2
1
dwdvwvwxvf
dwdvevfxf vxiw
2
1
Prove!
Prove!
Complex Form of the Fourier Integral (푸리에 적분의 복소형식)
![Page 69: ngineering Mathematics IIocw.snu.ac.kr/sites/default/files/NOTE/6060.pdf11.1 Fourier Series (푸리에급수) Orthogonality of the Trigonometric Systems: 삼각함수계는구갂-π](https://reader031.vdocument.in/reader031/viewer/2022022808/5e2c3f2f540f9401ce4ccae2/html5/thumbnails/69.jpg)
xixeix sincos
11.9 Fourier Transform. Discrete and Fast Fourier Transforms(푸리에 변환. 이산 및 고속 푸리에 변환)
dwwxwBwxwAxf
0
sincos
wvdvvfwB
wvdvvfwA
sin1
cos1
dwdvwvwxvfxf
)cos()(
2
1
0)sin()(2
1
dwdvwvwxvf
dwdvevfxf vxiw
2
1
Prove!
Prove!
Complex Form of the Fourier Integral (푸리에 적분의 복소형식)
![Page 70: ngineering Mathematics IIocw.snu.ac.kr/sites/default/files/NOTE/6060.pdf11.1 Fourier Series (푸리에급수) Orthogonality of the Trigonometric Systems: 삼각함수계는구갂-π](https://reader031.vdocument.in/reader031/viewer/2022022808/5e2c3f2f540f9401ce4ccae2/html5/thumbnails/70.jpg)
Complex Form of the Fourier Integral (푸리에 적분의 복소형식)
dwedvevfxf iwxiwv
]2
1[
2
1
11.9 Fourier Transform. Discrete and Fast Fourier Transforms(푸리에 변환. 이산 및 고속 푸리에 변환)
dxexfwf iwx
2
1ˆ
dwewfx f iwx
ˆ
2
1
dttfefsF st
0
L
![Page 71: ngineering Mathematics IIocw.snu.ac.kr/sites/default/files/NOTE/6060.pdf11.1 Fourier Series (푸리에급수) Orthogonality of the Trigonometric Systems: 삼각함수계는구갂-π](https://reader031.vdocument.in/reader031/viewer/2022022808/5e2c3f2f540f9401ce4ccae2/html5/thumbnails/71.jpg)
Complex Form of the Fourier Integral (푸리에 적분의 복소형식)
• Complex Fourier Integral:
• Fourier Transform:
• Inverse Fourier Transform:
Existence of the Fourier Transform (푸리에 변환의 존재)
dwdvevfxf vxiw
2
1
dxexfwff iwx
2
1ˆF
dwewfx f iwx
ˆ
2
1
dxexfwfwfxf
xxf
iwx
2
1ˆ , ˆ
존재하며는변환푸리에의
구분연속구간에서유한모든가능이고적분절대상에서축가
11.9 Fourier Transform. Discrete and Fast Fourier Transforms(푸리에 변환. 이산 및 고속 푸리에 변환)
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otherwise 0 and 1 if 1 xfxxf
11.9 Fourier Transform. Discrete and Fast Fourier Transforms(푸리에 변환. 이산 및 고속 푸리에 변환)
Ex. 1 Fourier Transform
![Page 73: ngineering Mathematics IIocw.snu.ac.kr/sites/default/files/NOTE/6060.pdf11.1 Fourier Series (푸리에급수) Orthogonality of the Trigonometric Systems: 삼각함수계는구갂-π](https://reader031.vdocument.in/reader031/viewer/2022022808/5e2c3f2f540f9401ce4ccae2/html5/thumbnails/73.jpg)
otherwise 0 and 1 if 1 xfxxf
w
w
iw
wi
eeiwiw
edxewf iwiw
iwxiwx
sin2
2
sin2
2
1
2
1
2
1ˆ1
1
1
1
11.9 Fourier Transform. Discrete and Fast Fourier Transforms(푸리에 변환. 이산 및 고속 푸리에 변환)
Ex. 1 Fourier Transform
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Linearity. Fourier Transform of Derivatives (도함수의 푸리에 변환)
• Linearity of the Fourier Transform
• Fourier Transform of Derivatives
함수 f(x)가 x축 상에서 연속
f’ (x)가 x축 상에서 절대적분 가능
gbfabgaf FFF
xfwxfxfiwxf
xfx
FFFF 2'' ,'
0 ,
11.9 Fourier Transform. Discrete and Fast Fourier Transforms(푸리에 변환. 이산 및 고속 푸리에 변환)
Prove!
![Page 75: ngineering Mathematics IIocw.snu.ac.kr/sites/default/files/NOTE/6060.pdf11.1 Fourier Series (푸리에급수) Orthogonality of the Trigonometric Systems: 삼각함수계는구갂-π](https://reader031.vdocument.in/reader031/viewer/2022022808/5e2c3f2f540f9401ce4ccae2/html5/thumbnails/75.jpg)
11.9 Fourier Transform. Discrete and Fast Fourier Transforms(푸리에 변환. 이산 및 고속 푸리에 변환)
Ex. 3 Application of the Operational Formula (9)
Find the Fourier transform of
)0( 2
14
22
aea
e aw
axF
2xxe
![Page 76: ngineering Mathematics IIocw.snu.ac.kr/sites/default/files/NOTE/6060.pdf11.1 Fourier Series (푸리에급수) Orthogonality of the Trigonometric Systems: 삼각함수계는구갂-π](https://reader031.vdocument.in/reader031/viewer/2022022808/5e2c3f2f540f9401ce4ccae2/html5/thumbnails/76.jpg)
44
22
2222
222
1
2
1
2
1'
2
1'
2
1
ww
xxxx
eiw
eiw
eiweexe
FFFF
11.9 Fourier Transform. Discrete and Fast Fourier Transforms(푸리에 변환. 이산 및 고속 푸리에 변환)
Ex. 3 Application of the Operational Formula (9)
Find the Fourier transform of
)0( 2
14
22
aea
e aw
axF
2xxe
![Page 77: ngineering Mathematics IIocw.snu.ac.kr/sites/default/files/NOTE/6060.pdf11.1 Fourier Series (푸리에급수) Orthogonality of the Trigonometric Systems: 삼각함수계는구갂-π](https://reader031.vdocument.in/reader031/viewer/2022022808/5e2c3f2f540f9401ce4ccae2/html5/thumbnails/77.jpg)
Convolution (합성곱)
• Convolution:
• Convolution Theorem (합성곱 정리):
함수 f(x)와 g(x)가 구분연속이고, 유한하며, x축 상에서 절대적분가능
dppgpxfdppxgpfxgf
gfgf FFF 2
11.9 Fourier Transform. Discrete and Fast Fourier Transforms(푸리에 변환. 이산 및 고속 푸리에 변환)
Prove using x-p=q
gffg FFF
![Page 78: ngineering Mathematics IIocw.snu.ac.kr/sites/default/files/NOTE/6060.pdf11.1 Fourier Series (푸리에급수) Orthogonality of the Trigonometric Systems: 삼각함수계는구갂-π](https://reader031.vdocument.in/reader031/viewer/2022022808/5e2c3f2f540f9401ce4ccae2/html5/thumbnails/78.jpg)
PROBLEM SET 11.9
HW: 14
11.9 Fourier Transform. Discrete and Fast Fourier Transforms(푸리에 변환. 이산 및 고속 푸리에 변환)