Chapter 7 Fourier Series (Fourier 급수 )

1

Chapter 7 Fourier Series (Fourier 급수 )

Mathematical methods in the physical sciences 3rd edition Mary L. Boas

Lecture1 Periodic function

Lecturer: Lee, Yunsang (Physics)Baird-Hall 01318 [email protected]

mailto:[email protected]

2

1. Introduction

Problems involving vibrations or oscillations occur frequently in physics and engineering. You can think of examples you have already met: a vibrating tuning fork, a pendulum, a weight attached to a spring, water waves, sound waves, alternating electric currents, etc. In addition, there are many more examples which you will meet as you continue to study physics. On the other hand, Some of them – for example, heat conduction, electric and magnetic fields, light – do not appear in elementary work to have anything oscillatory about them, but will turn out in your advanced work to involve the sine and cosines which are used in describing simple harmonic motion and wave motion. It is why we learn how to expand a certain function with Fourier series consisting of ‘infinite’ sines and cosines.

3

2. Simple harmonic motion and wave motion: periodic functions ( 단순 조화운동과 파동운동 ; 주기함수 )

locityangular ve:)22

( , fT

t

y coordinate of Q (or P): tAAy sinsin

The back and forth motion of Q simple harmonic motion

1) Harmonic motion ( 단순 조화 운동 )

- P moves at constant speed around a circle of radius A.- Q moves up and down in such a way that its y coordinate is always equal to that of P.

For a constant circular motion,

4

tAytAx sin ,cos

tiAe

titAiyxz

)sin(cosIn the complex plane,

imaginary part velocity of Q

)sin(cos)( titAieAiAedt

d

dt

dz titi

2) Using complex number ( 복소수의 사용 )

The x and y coordinates of P:

Then, it is often convenient to use the complex notation.

(Position of Q: imaginary part of the complex z)

Velocity:

5

)sin( ,cosor sin tAtAtA

cf. phase difference or different choice of the origin

i) Functional form of the simple harmonic motion:

3) Periodic function ( 함수의 주기 )

Time

Displacement

6

Displacement

Time

ii) Graph

amplitude

period:

a. Time (simple harmonic motion)

2

T

tBtAdt

dy

tAy

coscos

sin

Kinetic energy: tmBdt

dym 22

2

cos2

1

2

1

Total energy (kinetic+ potential = max of kinetic E) =

22222

2

1fAAmB

7

x

Ay2

sin

t

vxAvtxAy

22

sin)(2

sin

Wavelength: λ

c. Arbitrary periodic function (like wave)

ngth)(or wavele period :

)()(

p

xfpxf

b. Distance (wave)

distance

Tf

vTcf

1,.

8

3. Applications of Fourier Series (Fourier 급수의 응용 )

- Fundamental (first order):

- Higher harmonics (higher order):

- Combination of the fundamental and the harmonics complicated periodic

function. Conversely, a complicated periodic function the combination of the

fundamental and the harmonics (Fourier Series expansion).

tt cos ,sin

)cos( ),sin( tntn

9

-What a-c frequencies (harmonics) make up a given signal and in what proportions? We can answer the above question by expanding these various periodic functions with Fourier Series.

ex) Periodic function

10

0 5 10 15 200

0 5 10 15 200

Inte

nsity

0 5 10 15 200

0 5 10 15 200

0 5 10 15 200

0 5 10 15 200

xxx 3sin,2sin,sin xxx 3sin2sinsin3

1

xsin

xx 2sinsin2

1

xxx 3sin2sinsin3

1

xxxx 10sin3sin2sinsin10

1

11

4. Average value of a function ( 함수의 평균값 )

1) average value of a function

xn

xxfxfxfxf

n

xfxfxfxf

baxf

nn

)()()()()()()()(

),(on )( of average e'Approximat'

321321

ab

dxxfbaxf

xnabxn

b

a

)(),(on )( of Average

n)integratio ofconcept (,0&, ,When

With the interval n

abx

12

nxdxnxdx

dxdxnxnxnxnx

nxdxnxdx

n

xdxxdx

22

2222

22

22

cossin

2cossin ,1cossin Using

cossin

,)0(for Similarly

.cossin

2

1cos

2

1sin

2

1

)cos ( sin of period) a(over valueAverage

22

22

nxdxnxdx

nxornx

2) Average of sinusoidal functions ( 사인함수의 평균 )

2

2cos1cos,

2

2cos1sin

sincos2cos .

22

22

xx

xx

xxxcf

13

-0.5

0.0

0.5

1.0

sin2x

-0.5

0.0

0.5

1.0

sin22x

-0.5

0.0

0.5

1.0

sin23x

-0.5

0.0

0.5

1.0sin24x

-0.5

0.0

0.5

1.0sin25x

Graph of sin2 nx

14

Chapter 7 Fourier Series


Lecture 2 Basic of Fourier series

Lecturer: Lee, Yunsang (Physics)Baird-Hall 01318 Email: [email protected]: 02-820-0404


15

5. Fourier coefficients (Fourier 계수 )

We want to expand a given periodic function in a series of sines and cosines.[First, we start with sin(nx) and cos(nx) instead of sin(nt) and cos(nt).]

,3sin2sinsin

3cos2coscos2

1)(

321

3210

xbxbxb

xaxaxaaxf

- Given a function f(x) of period 2,

We need to determine the coefficients!!

16

integer) :(0sincf.

0sinsin2

1

2

1

cossin2

1

integer) :,(period) a(over cossin of valueAverage 1)

ndxnx

dxxnmxnm

nxdxmx

nmnxmx

coscos2

1sinsin

coscos2

1coscos

sinsin2

1sincos

sinsin2

1cossin

In order to find formulas for an and bn, we need the following integrals on (-, )

17

.0interger for,0coscf.

0 ,0

0 ,2

1

,0

coscos2

1

2

1

sinsin2

1

period) a(over sinsin of valueAverage 2)

ndxnx

nm

nm

nm

dxxnmxnm

nxdxmx

nxmx

coscos2

1sinsin

coscos2

1coscos

sinsin2

1sincos

sinsin2

1cossin

18

0 ,1

0 ,2

1

,0

coscos2

1

2

1

coscos2

1

period) a(over coscos of valueAverage 3)

nm

nm

nm

dxxnmxnm

nxdxmx

nxmx

coscos2

1sinsin

coscos2

1coscos

sinsin2

1sincos

sinsin2

1cossin

19

Using the above integrals, we can find coefficients of Fourier series by calculating the average value.

,3sin2sinsin

3cos2coscos2

1)(

321

3210

xbxbxb

xaxaxaaxf

i-1) To find a_o, we calculate the average on (-,)

xdxbxdxb

xdxaxdxadxa

dxxf

2sin2

1sin

2

1

2cos2

1cos

2

1

2

1

2)(

2

1

21

210

21

0)(1

adxxf

20

i-2) To find a_1, we multiply cos x (n=1) and calculate the average on (-,).

.cos)(1

cos2sin2

1cossin

2

1

cos2cos2

1coscos

2

1cos

2

1

2cos)(

2

1

1

21

210

axdxxf

xdxxbxdxxb

xdxxaxdxxaxdxa

xdxxf

xcos

2

1

0 ,1

0 ,2

1

,0

coscos2

1.

nm

nm

nm

nxdxmxcf

21

.2cos)(1

2cos2sin2

12cossin

2

1

2cos2

12coscos

2

12cos

2

1

22cos)(

2

1

2

21

221

0

axdxxf

xdxxbxdxxb

xdxadxxaxdxa

xdxxf

i-3) To find a_2, we multiply cos 2x (n=2) and calculate the average on (-,).

x2cos

2

1

0 ,1

0 ,2

1

,0

coscos2

1.

nm

nm

nm

nxdxmxcf

22

i-4) To find a_n, we multiply cos nx and calculate the average on (-,).

.cos)(1

cossin2

1

cos2

1cos)(

2

1

1

2

n

n

anxdxxf

nxdxxb

nxdxanxdxxf

nxcos

2

1

0 ,1

0 ,2

1

,0

coscos2

1.

nm

nm

nm

nxdxmxcf

23

ii-1) To find b_1 and b_n, (cf. n=0 term is zero), we multiply the sin x (n=1) or sin nx and calculate the average on (-,).

nbnxdxxf

bxdxxf

xdxxbxdxb

xdxxadxxaxdxa

xdxxf

sin)(1

Similarly,

.2

1sin)(

2

1

sin2sin2

1sin

2

1

sin2cos2

1sincos

2

1sin

2

1

2sin)(

2

1

1

22

1

210

nxsin

2

1

0 ,0

0 ,2

1

,0

sinsin2

1.

nm

nm

nm

nxdxmxcf

24

. sin1

,cos1

.3sin2sinsin

3cos2coscos2

1)(

321

3210

nxdxxfbnxdxxfa

xbxbxb

xaxaxaaxf

nn

## Fourier series expansion

25

Example 1.

.0 ,1

,0 ,0)(

x

xxf

.0for 11

0for 0sin11

cos1

cos1cos01

cos)(1

0

0

0

0

n

nnxn

nxdx

nxdxnxdx

nxdxxfan

26

. oddfor 2

even for 0

1)1(1cos1

sin1

sin1sin01

sin)(1

00

0

0

nn

n

nn

nxnxdx

nxdxnxdx

nxdxxfb

n

n

.5

5sin

3

3sin

1

sin2

2

1)(

xxxxf

27

.0 ,1

,0 ,0)(

x

xxf .

5

5sin

3

3sin

1

sin2

2

1)(

xxxxf

9% overshoot: Gibbs phenomenon

28

Example 2.

.5

5sin

3

3sin

1

sin2

2

1)(

xxxxf

1

1

- case i

- case ii

5

5sin

3

3sin

1

sin4

12

xxx

xfxg

5

5cos

3

3cos

1

cos2

2

1

525sin

323sin

12sin2

2

1

2

xxx

xxx

xfxh

29

6. Dirichlet conditions (Dirichlet 조건 ) : convergence problem ( 수렴 문제 )

Does a Fourier series converge or does it converge to the values of f(x)?

-Theorem of Dirichlet: If f(x) is 1) periodic of period 2 2) single valued between - and

3) a finite number of Max., Min., and discontinuities4) integral of absolute f(x) is finite,

then, 1) the Fourier series converges to f(x) at all points where f(x) is continuous. 2) at jumps (e.g. discontinuity points), converges to the mid-point of the jump.

30

7. Complex form of Fourier series (Fourier 급수의 복소수 형태 )

.2

cos ,2

sininxinxinxinx ee

nxi

eenx

Using these relations, we can get a series of terms of the forms e^inx and e^-inx from the forms of sin nx and cos nx.

31

ixixixix

ixixixixixix

ixixixixixix

ei

bae

i

bae

i

bae

i

baa

i

eeb

i

eeb

i

eeb

eea

eea

eeaa

xbxbxb

xaxaxaaxf

22222211110

33

3

22

21

33

3

22

210

321

3210

222222222

1

222

2222

1

3sin2sinsin

3cos2coscos2

1)(

n

n

inxn

ixixixix

ec

ececececc 22

22110

32

.0cos ifonly zero-non is integral This

)sin()cos(

)(2

1 Here,

.)(

)(

xnmnm

xdxnmixnmdxedxee

dxexfc

ecxf

xnmiinximx

inxn

n

n

inxn

nxdxxfbnxdxxfacf nn sin)(

1 ,cos)(

1 .

33

Example.

.0 ,1

,0 ,0)(

x

xxf

.2

1

2

1

2

1

0,even ,0

odd ,1

12

1

2

1

12

10

2

1

00

0

0

dxdxxfc

n

nine

inin

e

dxedxec

ininx

o

inxinxn

Expanding f(x) with the e^inx series,

dxexfc inx

n )(2

1

34

Then,

xxi

ee

i

ee

eee

i

eee

iecxf

ixixixix

ixixixixixixinx

n

3sin3

1sin

2

2

1

23

1

2

2

2

1

functions) sinusoidal with converting(

531

1

531

1

2

1)(

33

5353

The same with the results of Fourier series with sines and cosines!!

35



Lecture 3 Fourier series



36

8. Other intervals ( 그 밖의 구간 )

2

0

2

0

2

0

)(2

1)(

2

1

sin)(1

sin)(1

cos)(1

cos)(1

dxexfdxexfc

nxdxxfnxdxxfb

nxdxxfnxdxxfa

inxinxn

n

n

- Same Fourier coefficients for the interval (-, ) and (0, 2 ).

1) (-, ) and (0, 2 ).

.2ith function w periodic :cos),(

cos)(cos)( Here,

cos)(cos)(cos)(

cos)(cos)(cos)(

Proof)(

20

2

0

2

0

0

0

nxxf

nxdxxfnxdxxf

nxdxxfnxdxxfnxdxxf

nxdxxfnxdxxfnxdxxf

37

(Caution)

Different periodic functions made from the same function, - same function f(x) = x^2 - different periodic with respect to the intervals, (-, ) and (0, 2 ).

38

- Other period 2l [(0, 2l) or (-l, l)], not 2 [(0, 2) ]

l

l

lxin

l

lel

xn

l

xn

nx

2

1

2

1

2),cosor (sin

2sin

/

2) period 2 vs. 2l

39

For f(x) with period 2l,

.sincos2

2sinsin

2coscos

2)(

1

0

21210

l

xnb

l

xna

al

xb

l

xb

l

xa

l

xa

axf

nn

.sin)(1

,cos)(1

l

ln

l

ln dxl

xnxf

lbdx

l

xnxf

la

l

l

lxinn dxexf

lc .)(

2

1 /.)( /

lxinnecxf

i) sinusoidal

ii) complex

40

Example.

lxl

lxxf

2 ,1

,0 ,0)( - period 2l

.2

1

2

1

, odd ,1

,0even ,01

2

1

2

1

/2

1

12

10

2

12

1

2

0

2

2/

2 /

0

2

0

/

l

l

in

inin

l

l

lxin

l

l

lxinl

l lxinn

dxl

cn

in

ne

in

eeinlin

e

l

dxel

dxl

dxexfl

c

Using the complex functions as Fourier series,

41

Then,

.3

sin3

1sin

2

2

1

3

1

3

11

2

1)( /3/3//

l

x

l

x

eeeei

xf lxilxilxilxi

42

9. Even and odd functions ( 짝함수 , 홀함수 )

)()( ifeven is )( xfxfxf

)()( if odd is )( xfxfxf

1) definition

43

- Even powers of x even function, and odd powers of x odd function.

- Any functions can be written as the sum of an even function and an odd function.

)()(2

1)()(

2

1)( xfxfxfxfxf

ex. odd)( sinheven)( cosh)(2

1)(

2

1xxeeeee xxxxx

44

Integral over symmetric intervals like (-, ) or (-l, l)

l

l

l xfdxxf

xfdxxf

0even. is )( if )(2

odd, is )( if 0)(

2) Integration

45

- In order to represent a f(x) on interval (0, l) by Fourier series of period 2l, we need to have f(x) defined on (-l, 0), too. - We can expand the function on (-l, 0) to be even or odd on (-l, 0). Anything is OK!!

46

1

0 sincos2

)(l

xnb

l

xna

axf nn

- Cosine function: even, Sine function: odd.

- If f(x) is even, the terms in Fourier series should be even. b_n should be zero.

- If f(x) is odd, the terms in Fourier series should be odd. a_n should be zero.

.sin)(2

0 odd, is )( If

0

l

n

n

dxl

xnxf

lb

axf

.0

,cos)(2

even, is )( If 0

n

l

n

b

dxl

xnxf

la

xf

3) Fourier series

47

- How to represent a function on (0, 1) by Fourier series 1) sine-cosine or exponential (ordinary method) (period 1, l=1/2) 2) odd or even function (period 2, l=1) (caution) different period!!

48

Example

1 ,0

0 ,1)(

21

21

x

xxf

(c) original function (period 1) Ordinary sine-cosine, or exponential

(b) even function (period 2) Fourier cosine series.

(a) odd function (period 2) Fourier sine series.

49

(a) Fourier sine series (using odd function with period 2, l = 1)

,4

0 ,

3

2 ,

2

4 ,

2

,4,3,2,1for ,0,1,2,112

cos

,12

cos2

cos2

sin2sin)(2

4321

2/1

0

2/1

0

1

0

bbbb

nn

n

nxn

n

dxxndxxnxfl

bn

6

6sin2

5

5sin

3

3sin

2

2sin2sin

2)(

xxxxxxf

.sin)(2

0 odd, is )( If

0

l

n

n

dxl

xnxf

lb

axf

50

(b) Fourier sine series (using odd function with period 2, l = 1)

5

5cos

3

3cos

1

cos2

2

1)(

.2

sin2

sin2

cos)(2

,12)(2

2/1

0

1

0

2/1

0

1

00

xxxxf

n

nxn

nxdxnxfa

dxdxxfa

n

.0

,cos)(2

even, is )( If 0

n

l

n

b

dxl

xnxf

la

xf

51

(c) Ordinary Fourier series

.

even. 0,

odd, ,1

2

)1(1

2

1

)(

21

2/1

00

2/1

0

21

0

2

dxc

n

nin

inin

e

dxedxexfc

nin

xinxinn

).3

6sin2(sin

2

2

1

)(1

2

1)( 6

316

3122

xx

eeeei

xf inininin

i) exponential

52

,0 ,3

2 ,0 ,

2

.)1(11

)cos1(1

2sin2

.02cos2

12)(2

4321

2/1

0

2/1

0

2/1

0

1

00

bbbb

nn

nxdxnb

xdxna

dxdxxfa

nn

n

ii) sine-cosine

53

10. Application to sound ( 소리에 대한 응용 )

- odd function- period = 1/262

7

)7524sin(

6

)6524sin(30

5

)5524sin(

3

)3524sin(

2

)2524sin(30

1

524sin

4

1)(

cos8

71

2cos

8

152 524sin)()524(2

524/1

0

ttt

ttttp

nn

ntdtntpbn

54

- Intensity of a sound wave is proportional to the average of the square of amplitude, A2.

n = 1 2 3 4 5 6 7 8 9 10

relative intensity

= 1 225 1/9 0 1/25 25 1/49 0 1/81 9

- Second harmonics is dominant!!

55



Lecture 4 Fourier Transform



56

11. Parseval’s theorem (completeness relation) (Parseval’s 정리 ; 완전성 관계 )

?)(2

1)( of Average

sincos)(

22

102

1

dxxfxf

nxbnxaaxf nn

Hint 1) average of (1/2a_0)^2 = (1/2a_0)^2 2) average of (a_n cos nx)^2) = a_n^2* 1/2 3) average of (b_n sin nx)^2 = b_n^2 * 1/2. 4) average of all cross product terms, a_n*b_m*cos nx*sin mx, = 0.

1

2

1

2202

12

2

1

2

1)( of Average nn baaxf

22)( of average ncxfSimilarly,

- Parseval’s theorem or completeness relation- The set of cos nx and sin nx is a complete set!!

57

12. Fourier Transforms (Fourier 변환 )

expansion) series(Fourier 2

1, //

l

l

lxinn

lxinn dxexf

lcecxf

ransform)(Fourier t.2

1,

dxexfgdegxf xixi

- Periodic function Fourier series with discrete frequencies

- What happens for ‘non-periodic function’?

Fourier transform with continuous frequencies

ddxll

nd

l

l

2

1

2

1,,

cf. Fourier series vs. Fourier transform

58

confusion) theavoid to(.22

1

),(,

l

l

uil

l

xin

nxi

n

dueufdxexfl

c

ll

necxf

nn

n

- Conversion of the Fourier series to the Fourier transform

.where,2

1

2

2

l

l

uxinn

l

l

uxi

xil

l

ui

dueufFFdueuf

edueufxf

nn

nn

.

,2

1

2

1

.2

1

2

1

2

1

degxf

dueufdxexfg

dueufdedudeufdFxf

dueufF

xi

uixi

uixiuxi

l

l

uxi

59

.sin2 Similarly,

. odd,

.sinsin2

1

, oddFor

sincos2

1

sincos

0

0

xdgidxegxf

ggg

xdxxfi

dxxixfg

xf

dxxixxfg

xixe

xi

xi

- Fourier sine/cosine transforms

60

.cos2

,cos2

Transform, CosineFourier ii)

.sin2

,sin2

Transform, SineFourier i)

0

0

0

0

xdxxfxg

xdgxf

xdxxfxg

xdgxf

cc

cc

ss

ss

61

Example 1.

,1,0

,11,1

x

xxf

.sin

2

1

2

1

2

1

2

1

1

1

1

1

i

ee

i

edxe

dxexfg

iixixi

xi

0

cossin2sincossin1sin

d

xdx

xixdxexf xi

62

Example 2.

,1,0

,11,1

x

xxf

.1for 0

1for 4

,1for 2

2

cossin0

x

xxxfd

x

.2

sin,0For

0

dx

63

- Parseval’s Theorem for Fourier integrals

.2

1

.~

2

1~

~

2

1~

2

1

.~

2

1~

.~

2

1~

22

2121

2121

2121

11

dxxfdg

dxxfxfdagg

dxxfxfdegdxxf

dgdxexfdgg

dxexfg

xi

xi

xi

22)( of averagecf. ncxf

64

- Various Fourier transforms

65

- Michelson interferometer

66

HW

Chapter 7

2-3, 9, 13, 18 (G1)5-1, 7 (G2)7-1 (G3)9-1,6,7 (G4)

Documents

Chapter 7 Fourier Series (Fourier 급수 )