[2008][12-2] engineering mathematics 2

2008_Fourier Transform(1)

Naval A

rch

itectu

re &

Ocean

En

gin

eerin

g

Engineering Mathematics 2

Prof. Kyu-Yeul Lee

Department of Naval Architecture and Ocean Engineering,

Seoul National University of College of Engineering

[2008][12-2]

November, 2008


Naval A

rch

itectu

re &

Ocean

En

gin

eerin

g

Fourier Transform(1) : Fourier Series and Transform

Fourier Series & Fourier Transform

Fourier Series- Review


Fourier Series and Fourier Transform

3/152


Fourier Series and Fourier Transform*

A real function is represented by a complex series ; a series in which the coefficients are complex numbers** ( )f x nc*Mary L. Boas, Mathematical Methods in the Physical Science., Second Edition, John Wiley & Sons, 1966, p647-648**Zill D.G., & Cullen M.R., Advanced Engineering Mathematics, Third Edition, Jones and Bartlett, 2006, p670

Function expansion

Fourier Series

4/152




Function expansion

Fourier Series

x

( )f x

5/152



/( ) in x pn

n

f x c e

/1( )

2

pin x p

np

c f x e dxp


Function expansion

Fourier Series

x

( )f x

2 22

2f

T p p

6/152



/( ) in x pn

n

f x c e

/1( )

2

pin x p

np

c f x e dxp


Function expansion

Fourier Series

x

( )f x

2 22

2f

T p p

7/152



/( ) in x pn

n

f x c e

/1( )

2

pin x p

np

c f x e dxp


Function expansion

-in series of sine, cosine, and complex**

Fourier Series

x

( )f x

2 22

2f

T p p

8/152



/( ) in x pn

n

f x c e

/1( )

2

pin x p

np

c f x e dxp


-expand a periodic function

Function expansion


Fourier Series

x

( )f x

2 22

2f

T p p

9/152



/( ) in x pn

n

f x c e

/1( )

2

pin x p

np

c f x e dxp



Function expansion


Fourier Series

x

( )f x

2 22

2f

T p p

10/152



/( ) in x pn

n

f x c e

/1( )

2

pin x p

np

c f x e dxp



Function expansion


-infinite but discrete set of frequencies

Fourier Series

x

( )f x

2 22

2f

T p p

11/152



/( ) in x pn

n

f x c e

/1( )

2

pin x p

np

c f x e dxp



Function expansion



Fourier Series

x

( )f x

2 22

2f

T p p

12/152



/( ) in x pn

n

f x c e

/1( )

2

pin x p

np

c f x e dxp



Function expansion



nc

2 3023

0, 1, 2,...n

......

Fourier Series

x

( )f x

2 22

2f

T p p

13/152



/( ) in x pn

n

f x c e

/1( )

2

pin x p

np

c f x e dxp



Function expansion



Q1. is it possible to represent a function which is not periodic by something analogues to a Fourier series?

nc

2 3023

0, 1, 2,...n

......

Fourier Series

x

( )f x

2 22

2f

T p p

14/152



/( ) in x pn

n

f x c e

/1( )

2

pin x p

np

c f x e dxp



Function expansion




nc

2 3023

0, 1, 2,...n

......

Q2. can we somehow extend or modify Fourier series to cover the case of a continuous set of frequencies?

Fourier Series

x

( )f x

2 22

2f

T p p

15/152



/( ) in x pn

n

f x c e

/1( )

2

pin x p

np

c f x e dxp



Function expansion




nc

2 3023

0, 1, 2,...n

......


Fourier Series

x

( )f x

2 22

2f

T p p

16/152



/( ) in x pn

n

f x c e

/1( )

2

pin x p

np

c f x e dxp



Function expansion




nc

2 3023

0, 1, 2,...n

......


n

recall, an ‘integral’ is a ‘limit of a sum’

Fourier Series

x

( )f x

2 22

2f

T p p

17/152



/( ) in x pn

n

f x c e

/1( )

2

pin x p

np

c f x e dxp



Function expansion




nc

2 3023

0, 1, 2,...n

......


n


Fourier Series

Fourier Series

x

( )f x

2 22

2f

T p p

18/152



/( ) in x pn

n

f x c e

/1( )

2

pin x p

np

c f x e dxp



Function expansion




nc

2 3023

0, 1, 2,...n

......


n


Fourier Series Fourier Integral

Fourier Series

x

( )f x

2 22

2f

T p p

19/152



/( ) in x pn

n

f x c e

/1( )

2

pin x p

np

c f x e dxp



Function expansion




nc

2 3023

0, 1, 2,...n

......


n



Fourier Series

x

( )f x

2 22

2f

T p p

20/152




Function expansion





nc

2 3023

2 2

2T p p

0, 1, 2,...n

......

t

( )f t

/( ) in x pn

n

f x c e

/1( )

2

pin x p

np

c f x e dxp

2 22

2f

T p p

21/152




Function expansion

Fourier Series Fourier Integralextend

n


Fourier Series Fourier Integraln


Fourier Series Fourier Integral-expand a periodic function



nc

2 3023

2 2

2T p p

0, 1, 2,...n

......

t

( )f t

/( ) in x pn

n

f x c e

/1( )

2

pin x p

np

c f x e dxp

2 22

2f

T p p

22/152




Function expansion

Fourier Series

1 ˆ( ) ( )2

i xf x f e d

1ˆ( ) ( )2

i xf f x e dx

Fourier Integralextend

n







nc

2 3023

2 2

2T p p

0, 1, 2,...n

......

t

( )f t

/( ) in x pn

n

f x c e

/1( )

2

pin x p

np

c f x e dxp

2 22

2f

T p p

23/152




Function expansion

Fourier Series

1 ˆ( ) ( )2

i xf x f e d

1ˆ( ) ( )2

i xf f x e dx


n







nc

2 3023

2 2

2T p p

0, 1, 2,...n

......

t

( )f t

/( ) in x pn

n

f x c e

/1( )

2

pin x p

np

c f x e dxp

2 22

2f

T p p

24/152




Function expansion

Fourier Series

1 ˆ( ) ( )2

i xf x f e d

1ˆ( ) ( )2

i xf f x e dx


-integral of sine, cosine, and complex**

n







nc

2 3023

2 2

2T p p

0, 1, 2,...n

......

t

( )f t

/( ) in x pn

n

f x c e

/1( )

2

pin x p

np

c f x e dxp

2 22

2f

T p p

25/152




Function expansion

Fourier Series

1 ˆ( ) ( )2

i xf x f e d

1ˆ( ) ( )2

i xf f x e dx


-represent nonperiodic functions


n







nc

2 3023

2 2

2T p p

0, 1, 2,...n

......

t

( )f t

/( ) in x pn

n

f x c e

/1( )

2

pin x p

np

c f x e dxp

2 22

2f

T p p

26/152




Function expansion

Fourier Series

1 ˆ( ) ( )2

i xf x f e d

1ˆ( ) ( )2

i xf f x e dx




n







nc

2 3023

2 2

2T p p

0, 1, 2,...n

......

t

( )f t

/( ) in x pn

n

f x c e

/1( )

2

pin x p

np

c f x e dxp

2 22

2f

T p p

27/152




Function expansion

Fourier Series

1 ˆ( ) ( )2

i xf x f e d

1ˆ( ) ( )2

i xf f x e dx




-continuous set of frequencies

n







nc

2 3023

2 2

2T p p

0, 1, 2,...n

......

t

( )f t

/( ) in x pn

n

f x c e

/1( )

2

pin x p

np

c f x e dxp

2 22

2f

T p p

28/152




Function expansion

Fourier Series

1 ˆ( ) ( )2

i xf x f e d

1ˆ( ) ( )2

i xf f x e dx





n







nc

2 3023

2 2

2T p p

0, 1, 2,...n

......

t

( )f t

/( ) in x pn

n

f x c e

/1( )

2

pin x p

np

c f x e dxp

2 22

2f

T p p

29/152




Function expansion

Fourier Series

1 ˆ( ) ( )2

i xf x f e d

1ˆ( ) ( )2

i xf f x e dx





n







corresponding

nc

2 3023

2 2

2T p p

0, 1, 2,...n

......

t

( )f t

/( ) in x pn

n

f x c e

/1( )

2

pin x p

np

c f x e dxp

2 22

2f

T p p

30/152




Function expansion

Fourier Series

1 ˆ( ) ( )2

i xf x f e d

1ˆ( ) ( )2

i xf f x e dx





n







corresponding

interval-valued variable n

nc

2 3023

2 2

2T p p

0, 1, 2,...n

......

t

( )f t

/( ) in x pn

n

f x c e

/1( )

2

pin x p

np

c f x e dxp

2 22

2f

T p p

31/152




Function expansion

Fourier Series

1 ˆ( ) ( )2

i xf x f e d

1ˆ( ) ( )2

i xf f x e dx





n







corresponding


nc

2 3023

2 2

2T p p

0, 1, 2,...n

......

t

( )f t

/( ) in x pn

n

f x c e

/1( )

2

pin x p

np

c f x e dxp

2 22

2f

T p p

32/152




Function expansion

Fourier Series

1 ˆ( ) ( )2

i xf x f e d

1ˆ( ) ( )2

i xf f x e dx





n







corresponding


continuous variable

nc

2 3023

2 2

2T p p

0, 1, 2,...n

......

t

( )f t

/( ) in x pn

n

f x c e

/1( )

2

pin x p

np

c f x e dxp

2 22

2f

T p p

33/152




Function expansion

Fourier Series

1 ˆ( ) ( )2

i xf x f e d

1ˆ( ) ( )2

i xf f x e dx





n







corresponding


ncthe set of coefficients

continuous variable

nc

2 3023

2 2

2T p p

0, 1, 2,...n

......

t

( )f t

/( ) in x pn

n

f x c e

/1( )

2

pin x p

np

c f x e dxp

2 22

2f

T p p

34/152




Function expansion

Fourier Series

1 ˆ( ) ( )2

i xf x f e d

1ˆ( ) ( )2

i xf f x e dx





n







corresponding



continuous variable

nc

2 3023

2 2

2T p p

0, 1, 2,...n

......

t

( )f t

/( ) in x pn

n

f x c e

/1( )

2

pin x p

np

c f x e dxp

2 22

2f

T p p

35/152




Function expansion

Fourier Series

1 ˆ( ) ( )2

i xf x f e d

1ˆ( ) ( )2

i xf f x e dx





n







corresponding



continuous variable

become a function ˆ( )f

nc

2 3023

2 2

2T p p

0, 1, 2,...n

......

t

( )f t

/( ) in x pn

n

f x c e

/1( )

2

pin x p

np

c f x e dxp

2 22

2f

T p p

36/152




Function expansion

Fourier Series

1 ˆ( ) ( )2

i xf x f e d

1ˆ( ) ( )2

i xf f x e dx





n







corresponding



continuous variable


nc

2 3023

2 2

2T p p

0, 1, 2,...n

......

t

( )f t

/( ) in x pn

n

f x c e

/1( )

2

pin x p

np

c f x e dxp

2 22

2f

T p p

37/152




Function expansion

Fourier Series

1 ˆ( ) ( )2

i xf x f e d

1ˆ( ) ( )2

i xf f x e dx





n







corresponding

ˆ( )f



continuous variable


nc

2 3023

2 2

2T p p

0, 1, 2,...n

......

t

( )f t

/( ) in x pn

n

f x c e

/1( )

2

pin x p

np

c f x e dxp

2 22

2f

T p p

38/152




Function expansion

Fourier Series

1 ˆ( ) ( )2

i xf x f e d

1ˆ( ) ( )2

i xf f x e dx

extend




n







corresponding

ˆ( )f



continuous variable


nc

2 3023

2 2

2T p p

0, 1, 2,...n

......

t

( )f t

/( ) in x pn

n

f x c e

/1( )

2

pin x p

np

c f x e dxp

2 22

2f

T p p

39/152


/( ) in x pn

n

f x c e

/1( )

2

pin x p

np

c f x e dxp

1 ˆ( ) ( )2

i xf x f e d

1ˆ( ) ( )2

i xf f x e dx



Function expansion

Fourier Seriesextend




Fourier Transform




corresponding

ˆ( )f



continuous variable


nc

2 3023

2 2

2T p p

0, 1, 2,...n

......

t

( )f t

n





2 22

2f

T p p

40/152


/( ) in x pn

n

f x c e

/1( )

2

pin x p

np

c f x e dxp

1 ˆ( ) ( )2

i xf x f e d

1ˆ( ) ( )2

i xf f x e dx



Function expansion





Fourier Transform




corresponding

ˆ( )f



continuous variable


nc

2 3023

2 2

2T p p

0, 1, 2,...n

......

t

( )f t

n





Fourier Transform of ( )f x

2 22

2f

T p p

41/152


/( ) in x pn

n

f x c e

/1( )

2

pin x p

np

c f x e dxp

1 ˆ( ) ( )2

i xf x f e d

1ˆ( ) ( )2

i xf f x e dx



Function expansion





Fourier Transform




corresponding

ˆ( )f



continuous variable


nc

2 3023

2 2

2T p p

0, 1, 2,...n

......

t

( )f t

n






Inverse Fourier Transform of ˆ( )f 2 2

22

fT p p

42/152


1 ˆ( ) ( )2

i xf x f e d

1ˆ( ) ( )2

i xf f x e dx

Fourier Series and Fourier Transform*Function expansion

Fourier Series Fourier Transform


Inverse Fourier Transform of ˆ( )f

Try to represent a function which is not periodic by letting the period (-p,p) increase to (-∞, ∞)

/( ) in x pn

n

f x c e

/1( )

2

pin x p

np

c f x e dxp

/1( )

2

pin x p

np

c f x e dxp

2 22

2f

T p p

*Mary L. Boas, Mathematical Methods in the Physical Science., Second Edition, John Wiley & Sons, 1966, p647-64843/152


1 ˆ( ) ( )2

i xf x f e d

1ˆ( ) ( )2

i xf f x e dx






/( ) in x pn

n

f x c e

/1( )

2

pin x p

np

c f x e dxp

/( ) in x pn

n

f x c e

/1( )

2

pin x p

np

c f x e dxp

2 22

2f

T p p



1 ˆ( ) ( )2

i xf x f e d

1ˆ( ) ( )2

i xf f x e dx





n

n

p

If we call and 1n n

p


/( ) in x pn

n

f x c e

/1( )

2

pin x p

np

c f x e dxp

/( ) in x pn

n

f x c e

/1( )

2

pin x p

np

c f x e dxp

2 22

2f

T p p



1 ˆ( ) ( )2

i xf x f e d

1ˆ( ) ( )2

i xf f x e dx





n

n

p

If we call and 1n n

p


then, 1

2 2p

/( ) in x pn

n

f x c e

/1( )

2

pin x p

np

c f x e dxp

/( ) in x pn

n

f x c e

/1( )

2

pin x p

np

c f x e dxp

2 22

2f

T p p



1 ˆ( ) ( )2

i xf x f e d

1ˆ( ) ( )2

i xf f x e dx





n

n

p

If we call and 1n n

p


then, 1

2 2p

/( ) in x pn

n

f x c e

/1( )

2

pin x p

np

c f x e dxp

/( ) in x pn

n

f x c e

/1( )

2

pin x p

np

c f x e dxp

( ) ni x

nn

f x c e

( )2

np

i u

np

c f u e du

Integration value changed xu to avoid confusion

2 22

2f

T p p



1 ˆ( ) ( )2

i xf x f e d

1ˆ( ) ( )2

i xf f x e dx






/( ) in x pn

n

f x c e

/1( )

2

pin x p

np

c f x e dxp

( ) ni x

nn

f x c e

( )2

np

i u

np

c f u e du

2 22

2f

T p p



1 ˆ( ) ( )2

i xf x f e d

1ˆ( ) ( )2

i xf f x e dx






/( ) in x pn

n

f x c e

/1( )

2

pin x p

np

c f x e dxp

( ) ni x

nn

f x c e

( )2

np

i u

np

c f u e du

( ) ( )2

n np

i u i x

pn

f x f u e du e

2 22

2f

T p p



1 ˆ( ) ( )2

i xf x f e d

1ˆ( ) ( )2

i xf f x e dx






/( ) in x pn

n

f x c e

/1( )

2

pin x p

np

c f x e dxp

( ) ni x

nn

f x c e

( )2

np

i u

np

c f u e du

( ) ( )2

n np

i u i x

pn

f x f u e du e

( )( )2

np

i x u

pn

f u e du

2 22

2f

T p p



1 ˆ( ) ( )2

i xf x f e d

1ˆ( ) ( )2

i xf f x e dx






/( ) in x pn

n

f x c e

/1( )

2

pin x p

np

c f x e dxp

( ) ni x

nn

f x c e

( )2

np

i u

np

c f u e du

( ) ( )2

n np

i u i x

pn

f x f u e du e

( )( )2

np

i x u

pn

f u e du

( )1( )

2n

pi x u

pn

f u e du

2 22

2f

T p p



1 ˆ( ) ( )2

i xf x f e d

1ˆ( ) ( )2

i xf f x e dx






/( ) in x pn

n

f x c e

/1( )

2

pin x p

np

c f x e dxp

( )1( ) ( )

2n

pi x u

pn

f x f u e du

2 22

2f

T p p



1 ˆ( ) ( )2

i xf x f e d

1ˆ( ) ( )2

i xf f x e dx






/( ) in x pn

n

f x c e

/1( )

2

pin x p

np

c f x e dxp

( )1( ) ( )

2n

pi x u

pn

f x f u e du

1( ) ( )

2n

n

f x F

( ), ( ) ( ) np

i x u

np

where F f u e du

2 22

2f

T p p



1 ˆ( ) ( )2

i xf x f e d

1ˆ( ) ( )2

i xf f x e dx






/( ) in x pn

n

f x c e

/1( )

2

pin x p

np

c f x e dxp

( )1( ) ( )

2n

pi x u

pn

f x f u e du

1( ) ( )

2n

n

f x F

( ), ( ) ( ) np

i x u

np

where F f u e du

0 ,as pp

2 22

2f

T p p



( ) ( )nn

F F d

n continuous variable

1 ˆ( ) ( )2

i xf x f e d

1ˆ( ) ( )2

i xf f x e dx






/( ) in x pn

n

f x c e

/1( )

2

pin x p

np

c f x e dxp

( )1( ) ( )

2n

pi x u

pn

f x f u e du

1( ) ( )

2n

n

f x F

( ), ( ) ( ) np

i x u

np

where F f u e du

0 ,as pp

2 22

2f

T p p



( ) ( )nn

F F d


1 ˆ( ) ( )2

i xf x f e d

1ˆ( ) ( )2

i xf f x e dx






/( ) in x pn

n

f x c e

/1( )

2

pin x p

np

c f x e dxp

( )1( ) ( )

2n

pi x u

pn

f x f u e du

1( ) ( )

2n

n

f x F

( ), ( ) ( ) np

i x u

np

where F f u e du

0 ,as pp

also,

( )( ) ( ) i x uF f u e du

2 22

2f

T p p



( ) ( )nn

F F d


1 ˆ( ) ( )2

i xf x f e d

1ˆ( ) ( )2

i xf f x e dx






/( ) in x pn

n

f x c e

/1( )

2

pin x p

np

c f x e dxp

1( ) ( )

2n

n

f x F

( ), ( ) ( ) n

pi x u

np

where F f u e du

0 ,as pp

also,

( )( ) ( ) i x uF f u e du

2 22

2f

T p p



( ) ( )nn

F F d


1 ˆ( ) ( )2

i xf x f e d

1ˆ( ) ( )2

i xf f x e dx






/( ) in x pn

n

f x c e

/1( )

2

pin x p

np

c f x e dxp

1( ) ( )

2n

n

f x F

( ), ( ) ( ) n

pi x u

np

where F f u e du

0 ,as pp

also,

( )( ) ( ) i x uF f u e du

1( ) ( )

2f x F d

( ), ( ) ( ) i x uwhere F f u e du

2 22

2f

T p p



1 ˆ( ) ( )2

i xf x f e d

1ˆ( ) ( )2

i xf f x e dx






/( ) in x pn

n

f x c e

/1( )

2

pin x p

np

c f x e dxp

1( ) ( )

2f x F d


2 22

2f

T p p



1 ˆ( ) ( )2

i xf x f e d

1ˆ( ) ( )2

i xf f x e dx






/( ) in x pn

n

f x c e

/1( )

2

pin x p

np

c f x e dxp

1( ) ( )

2f x F d


( )1( ) ( )

2

i x uf x f u e dud

2 22

2f

T p p



1 ˆ( ) ( )2

i xf x f e d

1ˆ( ) ( )2

i xf f x e dx






/( ) in x pn

n

f x c e

/1( )

2

pin x p

np

c f x e dxp

1( ) ( )

2f x F d


( )1( ) ( )

2

i x uf x f u e dud

1 1( )

2 2

i u i xf u e du e d

2 22

2f

T p p





1ˆ( ) ( )2

i xf f x e dx


1 ˆ( ) ( )2

i xf x f e d



/( ) in x pn

n

f x c e

/1( )

2

pin x p

np

c f x e dxp

1 1( ) ( )

2 2

i u i xf x f u e du e d

2 22

2f

T p p





1ˆ( ) ( )2

i xf f x e dx


1 ˆ( ) ( )2

i xf x f e d



/( ) in x pn

n

f x c e

/1( )

2

pin x p

np

c f x e dxp

1 1( ) ( )

2 2


If we define

1ˆ( ) ( ) )2

1(

2

i x i uf u e duf f x e dx

2 22

2f

T p p





1ˆ( ) ( )2

i xf f x e dx


1 ˆ( ) ( )2

i xf x f e d



/( ) in x pn

n

f x c e

/1( )

2

pin x p

np

c f x e dxp

1 1( ) ( )

2 2


If we define

1ˆ( ) ( ) )2

1(

2


1 ˆ( ) ( )2

i xf x f x e d

then,

2 22

2f

T p p



Fourier Cosine and Sine Transform*

Fourier Transform

Fourier Transform of : ( )f x

Inverse Fourier Transform of : ˆ( )f

1ˆ( ) ( )2

i xf f x e dx

1 ˆ( ) ( )2

i xf x f x e d

*Mary L. Boas, Mathematical Methods in the Physical Science., Second Edition, John Wiley & Sons, 1966, p647-648

is even, if f )()( xfxf

dxxfdxxfaa

a 0

)(2)(

is odd, if f )()( xfxf

0)( dxxfa

a

:

:

:

odd even

odd odd

even even

f f

f f

f f

odd

even

even

70/152



Fourier Transform



1ˆ( ) ( )2

i xf f x e dx

1 ˆ( ) ( )2

i xf x f x e d

cos sini xe x i x By the Euler formula



dxxfdxxfaa

a 0

)(2)(


0)( dxxfa

a

:

:

:

odd even

odd odd

even even

f f

f f

f f

odd

even

even

71/152



Fourier Transform



1ˆ( ) ( )2

i xf f x e dx

1 ˆ( ) ( )2

i xf x f x e d


1ˆ( ) ( )2

i xf f x e dx

1( ) cos sin

2f x x i x dx



dxxfdxxfaa

a 0

)(2)(


0)( dxxfa

a

:

:

:

odd even

odd odd

even even

f f

f f

f f

odd

even

even

72/152



Fourier Transform



1ˆ( ) ( )2

i xf f x e dx

1 ˆ( ) ( )2

i xf x f x e d


1ˆ( ) ( )2

i xf f x e dx

1( ) cos sin

2f x x i x dx

1 1( )cos ( )sin

2 2f x xdx i f x xdx



dxxfdxxfaa

a 0

)(2)(


0)( dxxfa

a

:

:

:

odd even

odd odd

even even

f f

f f

f f

odd

even

even

73/152



Fourier Transform



1ˆ( ) ( )2

i xf f x e dx

1 ˆ( ) ( )2

i xf x f x e d


1ˆ( ) ( )2

i xf f x e dx

1( ) cos sin

2f x x i x dx

1 1( )cos ( )sin


even odd



dxxfdxxfaa

a 0

)(2)(


0)( dxxfa

a

:

:

:

odd even

odd odd

even even

f f

f f

f f

odd

even

even

74/152



Fourier Transform



1ˆ( ) ( )2

i xf f x e dx

1 ˆ( ) ( )2

i xf x f x e d


1ˆ( ) ( )2

i xf f x e dx

1( ) cos sin

2f x x i x dx

1 1( )cos ( )sin


even odd

if, ( ) :oddf x



dxxfdxxfaa

a 0

)(2)(


0)( dxxfa

a

:

:

:

odd even

odd odd

even even

f f

f f

f f

odd

even

even

75/152



Fourier Transform



1ˆ( ) ( )2

i xf f x e dx

1 ˆ( ) ( )2

i xf x f x e d


1ˆ( ) ( )2

i xf f x e dx

1( ) cos sin

2f x x i x dx

1 1( )cos ( )sin


even odd

if, ( ) :oddf x

1 1( )cos ( )sin




dxxfdxxfaa

a 0

)(2)(


0)( dxxfa

a

:

:

:

odd even

odd odd

even even

f f

f f

f f

odd

even

even

76/152



Fourier Transform



1ˆ( ) ( )2

i xf f x e dx

1 ˆ( ) ( )2

i xf x f x e d


1ˆ( ) ( )2

i xf f x e dx

1( ) cos sin

2f x x i x dx

1 1( )cos ( )sin


even odd

if, ( ) :oddf x

1 1( )cos ( )sin




dxxfdxxfaa

a 0

)(2)(


0)( dxxfa

a

:

:

:

odd even

odd odd

even even

f f

f f

f f

odd

even

even

evenodd odd odd

77/152



Fourier Transform



1ˆ( ) ( )2

i xf f x e dx

1 ˆ( ) ( )2

i xf x f x e d


1ˆ( ) ( )2

i xf f x e dx

1( ) cos sin

2f x x i x dx

1 1( )cos ( )sin


even odd

if, ( ) :oddf x

1 1( )cos ( )sin


0

20 ( )sin

2i f x xdx



dxxfdxxfaa

a 0

)(2)(


0)( dxxfa

a

:

:

:

odd even

odd odd

even even

f f

f f

f f

odd

even

even

evenodd odd odd

78/152



Fourier Transform



1ˆ( ) ( )2

i xf f x e dx

1 ˆ( ) ( )2

i xf x f x e d


1ˆ( ) ( )2

i xf f x e dx

1( ) cos sin

2f x x i x dx

1 1( )cos ( )sin


even odd

if, ( ) :oddf x

1 1( )cos ( )sin


0

20 ( )sin

2i f x xdx

0

2ˆ( ) ( )sinf i f x xdx



dxxfdxxfaa

a 0

)(2)(


0)( dxxfa

a

:

:

:

odd even

odd odd

even even

f f

f f

f f

odd

even

even

evenodd odd odd

79/152



Fourier Transform



1ˆ( ) ( )2

i xf f x e dx

1 ˆ( ) ( )2

i xf x f x e d

0


if, ( ) :oddf x


dxxfdxxfaa

a 0

)(2)(


0)( dxxfa

a




Fourier Transform



1ˆ( ) ( )2

i xf f x e dx

1 ˆ( ) ( )2

i xf x f x e d

0


if, ( ) :oddf x


dxxfdxxfaa

a 0

)(2)(


0)( dxxfa

a


ˆ( ) :f cf, odd function?

81/152



Fourier Transform



1ˆ( ) ( )2

i xf f x e dx

1 ˆ( ) ( )2

i xf x f x e d

0


if, ( ) :oddf x

0

0

2ˆ( ) ( )sin( )

2 ˆ( )sin( ) ( )

f i f x x dx

i f x x dx f


dxxfdxxfaa

a 0

)(2)(


0)( dxxfa

a



82/152



Fourier Transform



1ˆ( ) ( )2

i xf f x e dx

1 ˆ( ) ( )2

i xf x f x e d

0


if, ( ) :oddf x

0

0

2ˆ( ) ( )sin( )

2 ˆ( )sin( ) ( )

f i f x x dx

i f x x dx f

odd function!


dxxfdxxfaa

a 0

)(2)(


0)( dxxfa

a



83/152



Fourier Transform



1ˆ( ) ( )2

i xf f x e dx

1 ˆ( ) ( )2

i xf x f x e d

0


if, ( ) :oddf x

0

0

2ˆ( ) ( )sin( )

2 ˆ( )sin( ) ( )

f i f x x dx

i f x x dx f

odd function!


dxxfdxxfaa

a 0

)(2)(


0)( dxxfa

a



84/152



Fourier Transform



1ˆ( ) ( )2

i xf f x e dx

1 ˆ( ) ( )2

i xf x f x e d

0


if, ( ) :oddf x

0

0

2ˆ( ) ( )sin( )

2 ˆ( )sin( ) ( )

f i f x x dx

i f x x dx f

odd function!


dxxfdxxfaa

a 0

)(2)(


0)( dxxfa

a

1 1ˆ ˆ( ) ( )cos ( )sin2 2

f x f x xd i f x xd



85/152



Fourier Transform



1ˆ( ) ( )2

i xf f x e dx

1 ˆ( ) ( )2

i xf x f x e d

0


if, ( ) :oddf x

0

0

2ˆ( ) ( )sin( )

2 ˆ( )sin( ) ( )

f i f x x dx

i f x x dx f

odd function!


dxxfdxxfaa

a 0

)(2)(


0)( dxxfa

a

1 1ˆ ˆ( ) ( )cos ( )sin2 2

f x f x xd i f x xd

oddodd



86/152



Fourier Transform



1ˆ( ) ( )2

i xf f x e dx

1 ˆ( ) ( )2

i xf x f x e d

0


if, ( ) :oddf x

0

0

2ˆ( ) ( )sin( )

2 ˆ( )sin( ) ( )

f i f x x dx

i f x x dx f

odd function!


dxxfdxxfaa

a 0

)(2)(


0)( dxxfa

a

1 1ˆ ˆ( ) ( )cos ( )sin2 2

f x f x xd i f x xd

evenodd oddodd



87/152



Fourier Transform



1ˆ( ) ( )2

i xf f x e dx

1 ˆ( ) ( )2

i xf x f x e d

0

2 ˆ0 ( )sin2

i f x xd

0


if, ( ) :oddf x

0

0

2ˆ( ) ( )sin( )

2 ˆ( )sin( ) ( )

f i f x x dx

i f x x dx f

odd function!


dxxfdxxfaa

a 0

)(2)(


0)( dxxfa

a

1 1ˆ ˆ( ) ( )cos ( )sin2 2

f x f x xd i f x xd

evenodd oddodd



88/152



Fourier Transform



1ˆ( ) ( )2

i xf f x e dx

1 ˆ( ) ( )2

i xf x f x e d

0

2 ˆ0 ( )sin2

i f x xd

0


if, ( ) :oddf x

0

0

2ˆ( ) ( )sin( )

2 ˆ( )sin( ) ( )

f i f x x dx

i f x x dx f

odd function!


dxxfdxxfaa

a 0

)(2)(


0)( dxxfa

a

1 1ˆ ˆ( ) ( )cos ( )sin2 2

f x f x xd i f x xd

evenodd oddodd



0

2 ˆ( )sini f x xd

89/152



Fourier Transform



1ˆ( ) ( )2

i xf f x e dx

1 ˆ( ) ( )2

i xf x f x e d

0

2 ˆ( ) ( )sinf x i f x xd

0


if, ( ) :oddf x



dxxfdxxfaa

a 0

)(2)(


0)( dxxfa

a

90/152



Fourier Transform



1ˆ( ) ( )2

i xf f x e dx

1 ˆ( ) ( )2

i xf x f x e d

0


0


if, ( ) :oddf x



dxxfdxxfaa

a 0

)(2)(


0)( dxxfa

a

91/152



Fourier Transform



1ˆ( ) ( )2

i xf f x e dx

1 ˆ( ) ( )2

i xf x f x e d

0


0


if, ( ) :oddf x

2 2 2( )f x i i

The numerical factor



dxxfdxxfaa

a 0

)(2)(


0)( dxxfa

a

92/152



Fourier Transform



1ˆ( ) ( )2

i xf f x e dx

1 ˆ( ) ( )2

i xf x f x e d

0


0


if, ( ) :oddf x

2 2 2( )f x i i


Thus the imaginary factors are not needed, the factor may multiply

either of the two integrals, or each integral may be multiplied by .

Let us make the latter choice in giving the following definition

2

2



dxxfdxxfaa

a 0

)(2)(


0)( dxxfa

a

93/152



Fourier Transform



1ˆ( ) ( )2

i xf f x e dx

1 ˆ( ) ( )2

i xf x f x e d

0


0


if, ( ) :oddf x

2 2 2( )f x i i


Thus the imaginary factors are not needed, the factor may multiply

either of the two integrals, or each integral may be multiplied by .

Let us make the latter choice in giving the following definition

2

2



dxxfdxxfaa

a 0

)(2)(


0)( dxxfa

a

0

2 ˆ( ) ( ) (sin )f x f x d

0

2ˆ( ) ( )sinf f x xdx

94/152


Fourier Cosine and Sine Transform

Fourier Transform



1ˆ( ) ( )2

i xf f x e dx

1 ˆ( ) ( )2

i xf x f x e d


dxxfdxxfaa

a 0

)(2)(


0)( dxxfa

a

Fourier Sine Transform

The Fourier transform of an odd function on the interval is the sine

transform

Where,

),(

0

2 ˆ( ) ( )sinf x f xd

0


95/152



Fourier Transform



1ˆ( ) ( )2

i xf f x e dx

1 ˆ( ) ( )2

i xf x f x e d


dxxfdxxfaa

a 0

)(2)(


0)( dxxfa

a

In a similar way

Fourier Cosine Transform

The Fourier transform of an even function on the interval is the cosine

transform

Where,

),(

0

2 ˆ( ) ( )cosf x f xd

0

2ˆ( ) ( )cosf f x xdx

96/152


Fourier Transform

Convergence of a Fourier Transform

Conditions for Convergence

Let and be piecewise continuous on every finite interval, and let be

absolutely integrable on . Then the Fourier transform of on the

interval converges to at a point of continuity. At a point of discontinuity,

the Fourier integral will converge to the average

Where and denote the limit of at from the right and from

the left, respectively.

Theorem 15.1

2

)()( xfxf

f f f*),( f

)(xf

)( xf )( xf xf

* This means that the integral converges.

dxxf |)(|

97/152


1 if 0,

1 if ,1)(

x

xxf

Find the Fourier integral representation of the function

1

x

)(xf

1 0 1

Example 1 an impulse


1) Fourier Transform

1

1

1 1ˆ( ) ( ) 12 2

i x i xf f x e dx e dx

1

1

1 1 1 1( )

2 2

1 1( 2 sin )

2

i x i ie e ei i

ii

2 sin

cos sin

cos sin

i

i

e i

e i

By the Euler formula

1 ˆ( ) ( )2

i xf x f e d

1ˆ( ) ( )2

i xf f x e dx

0


0


0


0


When f : odd

When f : even

98/152


1 if 0,

1 if ,1)(

x

xxf


1

x

)(xf

1 0 1



1) Fourier Transform

2 sinˆ( )f

1 2 sin 1 sin

( ) cos sin2

i xf x e d x i x d

even

odd

sin

oddeven

odd

0

1 sin 2 sin cos( ) cos

xf x xd d

even

1 ˆ( ) ( )2

i xf x f e d

1ˆ( ) ( )2

i xf f x e dx

0


0


0


0


When f : odd

When f : even

99/152


1 if 0,

1 if ,1)(

x

xxf


1

x

)(xf

1 0 1



2) Fourier Sine Transform

1

1

1

1

ˆ( ) ( )sin 1 sin

cos cos cos( )0

f f x xdx xdx

x

1 ˆ( ) ( )2

i xf x f e d

1ˆ( ) ( )2

i xf f x e dx

0


0


0


0


When f : odd

When f : even

f : even can’t apply Fourier Sine Transform formula0


Bur if try to integrate f with sine over x

100/152


1 if 0,

1 if ,1)(

x

xxf

0 0

2 2 sin 2 sin cos( ) cos

xf x xd d


1

x

)(xf

1 0 1



3) Fourier Cosine Transform

11

0 00

2 2 2 sin 2 sinˆ( ) ( )cos 1 cosx

f f x xdx xdx

Because f(x) is even function, the result of Fourier transform and Fourier Cosine transform are identical

1 ˆ( ) ( )2

i xf x f e d

1ˆ( ) ( )2

i xf f x e dx

0


0


0


0


When f : odd

When f : even

101/152


0

2 cos sin( )

xf x d

1 if 0,

1 if ,1)(

x

xxf

1

x

)(xf

1 0 1


102/152


The average of the left- and right-hand limits of f (x) at x = 1 is equal to (1+0)/2, that is, 1/2. (Theorem 15.1)

Furthermore, multipling by we obtain by2/

0

2 cos sin( )

xf x d

1 if 0,

1 if ,1)(

x

xxf

1

x

)(xf

1 0 1


103/152



Furthermore, multipling by we obtain by

0

1 , 12 2

cos sin 1( ) , 1

2 2 2 4

0 0, 12

x

xf x d x

x

2/

0

2 cos sin( )

xf x d

1 if 0,

1 if ,1)(

x

xxf

1

x

)(xf

1 0 1


104/152



Furthermore, multipling by we obtain by

0

1 , 12 2

cos sin 1( ) , 1

2 2 2 4

0 0, 12

x

xf x d x

x

2/

We mention that this integral is called Dirichlet’s

discontinuous factor.

0

2 cos sin( )

xf x d

1 if 0,

1 if ,1)(

x

xxf

1

x

)(xf

1 0 1


105/152


Example 2 Cosine and Sine integral

Represent


( ) , 0xf x e x

(a) By a cosine integral

(b) By a sine integralx

y

1

1 ˆ( ) ( )2

i xf x f e d

1ˆ( ) ( )2

i xf f x e dx

0


0


0


0


When f : odd

When f : even

106/152



Represent


( ) , 0xf x e x


(b) By a sine integral


20

2 2 1ˆ( ) cos1

xf e x dx

x

y

1

1 ˆ( ) ( )2

i xf x f e d

1ˆ( ) ( )2

i xf f x e dx

0


0


0


0


When f : odd

When f : even

107/152



Represent


( ) , 0xf x e x




20

2 2 1ˆ( ) cos1

xf e x dx

x

y

1

0 00

0

2 200

2 2 0

20

sin sincos ( )

sin0

( cos ) ( cos )( )

1 1cos

1cos

1

x x x

x

x x

x

x

x xe x dx e e dx

xe dx

x xe e dx

e xdx

e xdx

1 ˆ( ) ( )2

i xf x f e d

1ˆ( ) ( )2

i xf f x e dx

0


0


0


0


When f : odd

When f : even

108/152



Represent


( ) , 0xf x e x




20

2 2 1ˆ( ) cos1

xf e x dx

20

2 cos( )

1

xf x d

x

y

1

x

y

1

Cosine integral

0 00

0

2 200

2 2 0

20

sin sincos ( )

sin0

( cos ) ( cos )( )

1 1cos

1cos

1

x x x

x

x x

x

x

x xe x dx e e dx

xe dx

x xe e dx

e xdx

e xdx

1 ˆ( ) ( )2

i xf x f e d

1ˆ( ) ( )2

i xf f x e dx

0


0


0


0


When f : odd

When f : even

109/152


Example 4 Cosine and Sine integral Representation

Represent


( ) , 0xf x e x


(b) By a sine integralx

y

1

Fourier Cosine and Sine Integrals

(i) The Fourier Integral of an even function on the interval is the cosine integral

Where,

(ii) The Fourier Integral of an odd function on the interval is the sine integral

Where,

Definition 15.2

),(

),(

0( ) ( ) (cos )f x A x d

0

2( ) ( )cos ,A f x xdx

0( ) ( ) (sin )f x B x d

0

2( ) ( )sinB f x xdx

110/152



Represent


( ) , 0xf x e x




20

2 2ˆ( ) sin1

xf e xdx

x

y

1

0 00

2 200

2 0

20

( cos ) ( cos )sin ( )

1 sin (sin )( )

1 10 sin

sin1

x x x

x x

x

x

x xe x dx e e dx

x xe e dx

e xdx

e xdx



Where,


Where,

Definition 15.2

),(

),(

0( ) ( ) (cos )f x A x d

0


0( ) ( ) (sin )f x B x d

0


111/152



Represent


( ) , 0xf x e x




20

2 2ˆ( ) sin1

xf e xdx

20

2 sin( )

1

xf x d

x

y

1

x

y

1

sine integral

0 00

2 200

2 0

20

( cos ) ( cos )sin ( )

1 sin (sin )( )

1 10 sin

sin1

x x x

x x

x

x

x xe x dx e e dx

x xe e dx

e xdx

e xdx



Where,


Where,

Definition 15.2

),(

),(

0( ) ( ) (cos )f x A x d

0


0( ) ( ) (sin )f x B x d

0


112/152


Background of Fourier Series

113/152


Orthogonal Set/ Weight Function

A set of real-valued functions is said to be

orthogonal with respect to a weight function on an interval if

Definition 12.4

nmdxxxxwb

anm ,0)()()(

),(),(),( 210 xxx

],[ ba)(xw

Sturm-Liouville Problem

Solve(Equation):

0)]()([])([ yxpxqyxrdx

d

Subject to (Boundary Condition): 0)()(

0)()(

22

11

byBbyA

ayBayA

Boundary Value Problem

Background of Fourier Series①

② Solutions of Sturm-Liouville equation are

Eigenfunctions (All of the solutions are

linearly independent and Orthogonal)

③ Solutions are Basis functions.

ex)

0 yy

kxBkxAxy sincos)(

)()(

)()(

yy

yyboundary

condition

,2,1,0k

solution

‘Basis functions’ and ‘Orthogonal Set’

1

0 )sincos()(n

nn nxbnxaaxf

④ Series solution using Orthogonal Set

Fourier Sine/Cosine Series

By utilizing the inner product

Properties of the Regular Sturm-Liouville Problem

(a) There exist an infinite number of real eigenvalues that can be arranged in increasing

order such that as

(b) For each eigenvalues there is only one eigenfunction (except for nonzero constant multiples)

(c) Eigenfunctions corresponding to different eigenvalues are linearly independent

(d) The set of eigenfunctions corresponding to the set of eigenvalues is orthogonal with

respect to the weight function on interval

Theorem 12.3

)(xp

n 321 n n

],[ ba

( )f x :a given function

Fourier-Bessel Series

Fourier-Legendre Series

1

)()(i

ini xJcxf

0

)()(n

nn xPcxf

Sturm-Liouville Problem Boundary Value Problem

114/152


Fourier Series

Fourier Series

The Fourier series of a function defined on the interval

Is given by

Definition 12.5

p

pn

p

pn

p

p

nnn

xdxp

nxf

pb

xdxp

nxf

pa

dxxfp

a

where

xp

nbx

p

na

axf

sin)(1

cos)(1

)(1

,

sincos2

)(

0

1

0

f ),( pp

115/152


Complex Fourier Series

1

0 sincos2

)(n

nn xp

nbx

p

na

axf

Fourier Series

For some engineering problem, it is actually more convenient to representa real function in an infinite series of complex-valued function of a real variables such as x

xixe

xixeix

ix

sincos

sincos

Recall,

i

eex

eex

ixixixix

2sin,

2cos

Then,

n

pxinnecxf /)(

A real function is represented by a complex series ; a series in which the coefficients are complex numbers*

( )f x

nc

*Zill D.G., & Cullen M.R., Advanced Engineering Mathematics, Third Edition, Jones and Bartlett, 2006, p670

,( 1,2,3..)inxe n

2 22

2f

T p p

116/152




The complex Fourier series of functions defined on an interval

is given by

Definition 12.7

),( pp

,...2,1,0,)(2

1

,

/ ndxexf

pc

where

p

p

pxinn

f

n

pxinnecxf /)(

2 22

2f

T p p

117/152



1

0 sincos2

)(n

nn xp

nbx

p

na

axf

i

eex

p

n

eex

p

n

pxinpxin

pxinpxin

2sin

2cos

//

//

1

/

1

/0

1

//0

1

////0

][2

1][

2

1

2

222

n

pxinn

n

pxinn

n

pxinnn

pxinnn

n

pxinpxin

n

pxinpxin

n

ececc

eibaeibaa

i

eeb

eea

a

0 0

,

1

2

1[ ]

2

1[ ]

2

n n n

n n n

where

c a

c a ib

c a ib

Derivation of complex Fourier series


( )f x

nc

*Zill D.G., & Cullen M.R., Advanced Engineering Mathematics, Third Edition, Jones and Bartlett, 2006, p670118/152



1

0 sincos2

)(n

nn xp

nbx

p

na

axf

1

/

1

/0

n

pxinn

n

pxinn ececc

p

pdxxf

pac )(

1

2

1

2

100

p

p

pxin

p

p

p

p

p

pn

dxexfp

dxxp

nix

p

nxf

p

xdxp

nxf

pixdx

p

nxf

pc

/)(2

1

sincos)(1

2

1

sin)(1

cos)(1

2

1

p

pn xdx

p

nxf

pb

sin)(

1

p

pdxxf

pa )(

10

p

pn xdx

p

nxf

pa

cos)(

1

xixe

xixeix

ix

sincos

sincos

Recall,

,

1[ ]

2n n n

where

c a ib


( )f x

nc




1

0 sincos2

)(n

nn xp

nbx

p

na

axf

1

/

1

/0

n

pxinn

n

pxinn ececc

p

pdxxf

pac )(

1

2

1

2

100

p

p

pxin

p

p

p

p

p

pn

dxexfp

dxxp

nix

p

nxf

p

xdxp

nxf

pixdx

p

nxf

pc

/)(2

1

sincos)(1

2

1

sin)(1

cos)(1

2

1

xixe

xixeix

ix

sincos

sincos

Recall,

p

pn xdx

p

nxf

pb

sin)(

1

p

pdxxf

pa )(

10

p

pn xdx

p

nxf

pa

cos)(

1

,

1[ ]

2n n n

where

c a ib


( )f x

nc




1

0 sincos2

)(n

nn xp

nbx

p

na

axf

1

/

1

/0

n

pxinn

n

pxinn ececc

0 0

,

1

2

1 1[ ], [ ]

2 2n n n n n n

where

c a

c a ib c a ib

p

p

pxinn dxexf

pc /)(

2

1

p

p

pxinn dxexf

pc /)(

2

1

p

pdxxf

pc )(

2

10


/ /1, ( )

2

pin x p in x p

n np

n

c e c f x e dxp

in a more compact manner

121/152



1

0 sincos2

)(n

nn xp

nbx

p

na

axf

1

/

1

/0

n

pxinn

n

pxinn ececc

0 0

,

1

2

1 1[ ], [ ]

2 2n n n n n n

where

c a

c a ib c a ib

p

p

pxinn dxexf

pc /)(

2

1

p

p

pxinn dxexf

pc /)(

2

1

p

pdxxf

pc )(

2

10

/ / / /00

1 1 1 1

0

1

1 1( ) [ ] [ ]

2 2 2

1[ ](cos sin ) [ ](cos sin )

2 2

in x p in x p in x p in x pn n n n n n

n n n n

n n n nn

af x c c e c e a ib e a ib e

a n n n na ib x i x a ib x i x

p p p p


/ /1, ( )

2

pin x p in x p

n np

n

c e c f x e dxp


122/152



1

0 sincos2

)(n

nn xp

nbx

p

na

axf

1

/

1

/0

n

pxinn

n

pxinn ececc

0 0

,

1

2

1 1[ ], [ ]

2 2n n n n n n

where

c a

c a ib c a ib

p

p

pxinn dxexf

pc /)(

2

1

p

p

pxinn dxexf

pc /)(

2

1

p

pdxxf

pc )(

2

10

/ / / /00

1 1 1 1

0

1

1 1( ) [ ] [ ]

2 2 2


2 2


n n n n

n n n nn



p p p p


0

1

1cos sin sin cos cos sin sin cos

2 2n n n n n n n n

n

a n n n n n n n na x b x i a x b x a x b x i a x b x

p p p p p p p p

/ /1, ( )

2

pin x p in x p

n np

n

c e c f x e dxp


123/152



1

0 sincos2

)(n

nn xp

nbx

p

na

axf

1

/

1

/0

n

pxinn

n

pxinn ececc

0 0

,

1

2

1 1[ ], [ ]

2 2n n n n n n

where

c a

c a ib c a ib

p

p

pxinn dxexf

pc /)(

2

1

p

p

pxinn dxexf

pc /)(

2

1

p

pdxxf

pc )(

2

10

/ / / /00

1 1 1 1

0

1

1 1( ) [ ] [ ]

2 2 2


2 2


n n n n

n n n nn



p p p p


0

1


2 2n n n n n n n n

n


p p p p p p p p

imaginary partimaginary part

/ /1, ( )

2

pin x p in x p

n np

n

c e c f x e dxp


124/152



1

0 sincos2

)(n

nn xp

nbx

p

na

axf

1

/

1

/0

n

pxinn

n

pxinn ececc

0 0

,

1

2

1 1[ ], [ ]

2 2n n n n n n

where

c a

c a ib c a ib

p

p

pxinn dxexf

pc /)(

2

1

p

p

pxinn dxexf

pc /)(

2

1

p

pdxxf

pc )(

2

10

/ / / /00

1 1 1 1

0

1

1 1( ) [ ] [ ]

2 2 2


2 2


n n n n

n n n nn



p p p p


0

1


2 2n n n n n n n n

n


p p p p p p p p

0

1

( ) cos sin2

n nn

a n nf x a x b x

p p


/ /1, ( )

2

pin x p in x p

n np

n

c e c f x e dxp


125/152



1

0 sincos2

)(n

nn xp

nbx

p

na

axf

1

/

1

/0

n

pxinn

n

pxinn ececc

0 0

,

1

2

1 1[ ], [ ]

2 2n n n n n n

where

c a

c a ib c a ib

p

p

pxinn dxexf

pc /)(

2

1

p

p

pxinn dxexf

pc /)(

2

1

p

pdxxf

pc )(

2

10


( )f x

nc


/ / / /00

1 1 1 1

0

1

1 1( ) [ ] [ ]

2 2 2


2 2


n n n n

n n n nn



p p p p


0

1


2 2n n n n n n n n

n


p p p p p p p p

0

1

( ) cos sin2

n nn

a n nf x a x b x

p p

real part


/ /1, ( )

2

pin x p in x p

n np

n

c e c f x e dxp


126/152



1

0 sincos2

)(n

nn xp

nbx

p

na

axf

1

//

1

// )(2

1)(

2

1)(

2

1

n

pxinp

p

pxin

n

pxinp

p

pxinp

pedxexf

pedxexf

pdxxf

p

1

/

1

/0

n

pxinn

n

pxinn ececc


127/152



1

0 sincos2

)(n

nn xp

nbx

p

na

axf

1

//

1

// )(2

1)(

2

1)(

2

1

n

pxinp

p

pxin

n

pxinp

p

pxinp

pedxexf

pedxexf

pdxxf

p

1

/

1

/0

n

pxinn

n

pxinn ececc

0

0

//)(2

1

n

pxinp

p

pxin edxexfp

1

//)(2

1

n

pxinp

p

pxin edxexfp

nn , 0when n


128/152



1

0 sincos2

)(n

nn xp

nbx

p

na

axf

1

//

1

// )(2

1)(

2

1)(

2

1

n

pxinp

p

pxin

n

pxinp

p

pxinp

pedxexf

pedxexf

pdxxf

p

1

/

1

/0

n

pxinn

n

pxinn ececc

0

0

//)(2

1

n

pxinp

p

pxin edxexfp

1

//)(2

1

n

pxinp

p

pxin edxexfp

nn , 0when n

n

pxinp

p

pxin edxexfp

//)(2

1


129/152



1

0 sincos2

)(n

nn xp

nbx

p

na

axf

1

//

1

// )(2

1)(

2

1)(

2

1

n

pxinp

p

pxin

n

pxinp

p

pxinp

pedxexf

pedxexf

pdxxf

p

1

/

1

/0

n

pxinn

n

pxinn ececc

0

0

//)(2

1

n

pxinp

p

pxin edxexfp

1

//)(2

1

n

pxinp

p

pxin edxexfp

nn , 0when n

n

pxinp

p

pxin edxexfp

//)(2

1

,...2,1,0,)(2

1, //

ndxexfp

cecp

p

pxinn

n

pxinn



130/152



Example 1 Complex Fourier SeriesExpandin a complex Fourier series

xexf x ,)(

pWith

][)1(2

1

2

1

2

1

)1()1(

)1(

inin

xininxxn

eein

dxedxeec

eninee

enineenin

nin

)1()sin(cos

)1()sin(cos)1(

)1(

0sin)1(cos nandn nsince

2

( ) ( ) 1 1( 1) ( 1)

2( 1) 2 ( 1)

sinh 1( 1)

1

n nn

n

e e e ec

in in

in

n

then

inx

n

n en

inxf

1

1)1(

sinh)(

2

/ /1( ) , ( ) , 0, 1, 2,...

2

pin x p in x p

n np

n

f x c e c f x e dx np

xixe

xixeix

ix

sincos

sincos

131/152



1

0 sincos2

)(n

nn xp

nbx

p

na

axf

1

/

1

/0

n

pxinn

n

pxinn ececc

A real function is represented by a complex series ; a series in which the coefficients are complex numbers* ( )f x nc


/ /1, ( )

2

pin x p in x p

n np

n

c e c f x e dxp



Example)

xexf x ,)(

Expand in a complex Fourier series

132/152



1

0 sincos2

)(n

nn xp

nbx

p

na

axf

1

/

1

/0

n

pxinn

n

pxinn ececc



/ /1, ( )

2

pin x p in x p

n np

n

c e c f x e dxp



Example)

xexf x ,)(

sinh( ) inx

nn

f x c e


2

1( 1)

1

nn

inc

n

133/152



1

0 sincos2

)(n

nn xp

nbx

p

na

axf

1

/

1

/0

n

pxinn

n

pxinn ececc



/ /1, ( )

2

pin x p in x p

n np

n

c e c f x e dxp



Example)

xexf x ,)(

sinh( ) inx

nn

f x c e


:complex

2

1( 1)

1

nn

inc

n

134/152



1

0 sincos2

)(n

nn xp

nbx

p

na

axf

1

/

1

/0

n

pxinn

n

pxinn ececc



/ /1, ( )

2

pin x p in x p

n np

n

c e c f x e dxp



Example)

xexf x ,)(

sinh( ) inx

nn

f x c e


:complex

2

1( 1)

1

nn

inc

n

:complex

135/152



1

0 sincos2

)(n

nn xp

nbx

p

na

axf

1

/

1

/0

n

pxinn

n

pxinn ececc



/ /1, ( )

2

pin x p in x p

n np

n

c e c f x e dxp



Example)

xexf x ,)(

sinh( ) inx

nn

f x c e


:complex

2

1( 1)

1

nn

inc

n

:complex( )f x : complex too?

136/152



1

0 sincos2

)(n

nn xp

nbx

p

na

axf

1

/

1

/0

n

pxinn

n

pxinn ececc



/ /1, ( )

2

pin x p in x p

n np

n

c e c f x e dxp



Example)

xexf x ,)(

sinh( ) inx

nn

f x c e


:complex

2

1( 1)

1

nn

inc

n


2

sinh 1( ) ( 1)

1

n inx

n

inf x e

n

from

137/152



1

0 sincos2

)(n

nn xp

nbx

p

na

axf

1

/

1

/0

n

pxinn

n

pxinn ececc



/ /1, ( )

2

pin x p in x p

n np

n

c e c f x e dxp



Example)

xexf x ,)(

sinh( ) inx

nn

f x c e


:complex

2

1( 1)

1

nn

inc

n


(1 )(cos sin ) cos sin (sin cos )in nx i nx nx n nx i nx n nx

2

sinh 1( ) ( 1)

1

n inx

n

inf x e

n

from

138/152



1

0 sincos2

)(n

nn xp

nbx

p

na

axf

1

/

1

/0

n

pxinn

n

pxinn ececc



/ /1, ( )

2

pin x p in x p

n np

n

c e c f x e dxp



Example)

xexf x ,)(

sinh( ) inx

nn

f x c e


:complex

2

1( 1)

1

nn

inc

n



2

sinh 1( ) ( 1)

1

n inx

n

inf x e

n

from

For positive integer : n=m (m>0): cos sin (sin cos )mx m mx i mx m mx

139/152



1

0 sincos2

)(n

nn xp

nbx

p

na

axf

1

/

1

/0

n

pxinn

n

pxinn ececc



/ /1, ( )

2

pin x p in x p

n np

n

c e c f x e dxp



Example)

xexf x ,)(

sinh( ) inx

nn

f x c e


:complex

2

1( 1)

1

nn

inc

n



2

sinh 1( ) ( 1)

1

n inx

n

inf x e

n

from


For negative integer : n=-m (m>0): cos sin ( sin cos )mx m mx i mx m mx

140/152



1

0 sincos2

)(n

nn xp

nbx

p

na

axf

1

/

1

/0

n

pxinn

n

pxinn ececc



/ /1, ( )

2

pin x p in x p

n np

n

c e c f x e dxp



Example)

xexf x ,)(

sinh( ) inx

nn

f x c e


:complex

2

1( 1)

1

nn

inc

n



2

sinh 1( ) ( 1)

1

n inx

n

inf x e

n

from


For negative integer : n=-m (m>0): cos sin ( sin cos )mx m mx i mx m mx :opposite in sign

141/152



1

0 sincos2

)(n

nn xp

nbx

p

na

axf

1

/

1

/0

n

pxinn

n

pxinn ececc



/ /1, ( )

2

pin x p in x p

n np

n

c e c f x e dxp



Example)

xexf x ,)(

sinh( ) inx

nn

f x c e


:complex

2

1( 1)

1

nn

inc

n



2

sinh 1( ) ( 1)

1

n inx

n

inf x e

n

from

2

1( 1)

1

n inxine

n

real part of will be remained in the summation


For negative integer : n=-m (m>0): cos sin ( sin cos )mx m mx i mx m mx :opposite in sign

142/152



1

0 sincos2

)(n

nn xp

nbx

p

na

axf

1

/

1

/0

n

pxinn

n

pxinn ececc



/ /1, ( )

2

pin x p in x p

n np

n

c e c f x e dxp



Example)

xexf x ,)(

sinh( ) inx

nn

f x c e


:complex

2

1( 1)

1

nn

inc

n



2

sinh 1( ) ( 1)

1

n inx

n

inf x e

n

from

2

1( 1)

1

n inxine

n




( ) :f x real function

:opposite in sign

143/152



1

0 sincos2

)(n

nn xp

nbx

p

na

axf

1

/

1

/0

n

pxinn

n

pxinn ececc



/ /1, ( )

2

pin x p in x p

n np

n

c e c f x e dxp



Example)

xexf x ,)(

sinh( ) inx

nn

f x c e


:complex

2

1( 1)

1

nn

inc

n



2

sinh 1( ) ( 1)

1

n inx

n

inf x e

n

from

2

1( 1)

1

n inxine

n




( ) :f x real function

:opposite in sign

144/152



Fundamental Frequency

n

pxinn ecxf /)(

1

0 sincos2

)(n

nn xp

nbx

p

na

axf

145/152




2T pFundamental period of the function

n

pxinn ecxf /)(

1

0 sincos2

)(n

nn xp

nbx

p

na

axf

146/152





Fundamental angular frequency

n

pxinn ecxf /)(

1

0 sincos2

)(n

nn xp

nbx

p

na

axf

,2

whereT

147/152






n

pxinn ecxf /)(

1

0 sincos2

)(n

nn xp

nbx

p

na

axf

,2

whereT

2 2

2T p p

148/152




n

xinn ecxf )(



n

pxinn ecxf /)(

1

0 sincos2

)(n

nn xnbxnaa

xf

1

0 sincos2

)(n

nn xp

nbx

p

na

axf

,2

whereT

2 2

2T p p

149/152



Frequency Spectrum

nc

2 3023

If is periodic and fundamental period

the plot of the points is called

frequency spectrum

where is the fundamental angular frequency,

are the coefficient.

f T

( , )nn c

( ) in x

nnx cf e

nC

150/152



Example 2 Frequency SpectrumIn Example 1, so that take on the values Using

22 2

2 2 2

sinh 1 sinh 1

1 1 1n

nc

n n n

1 n,2,1,0

22 i

162.1644.1599.2676.3599.2644.1162.1nc

n 3 2 1 0 1 2 3

xexf x ,)(

2

sinh 1( ) ( 1)

1

n inx

n

inf x e

n

Example 1

2 21

2T p p

nc

1 2 30123

0.5

1.0

1.5

2.0

2.5

3.0

3.5

sinh3.676

xexf x ,)(

/ /1( ) , ( ) , 0, 1, 2,...

2

pin x p in x p

n np

n


p

151/152



Example 3 Frequency Spectrum

Find the frequency spectrum of the periodic square wave

2

1

4

1,0

4

1

4

1,1

4

1

2

1,0

)(

x

x

x

xf

2

121 psopT

1/2 1/42 2

1/2 1/4

1/42 /2 /2 /2 /2

1/4

( ) 1

1 1

2 2 2

in x in xn

in x in in in in

c f x e dx e dx

e e e e e

in n i n i

2sin

1

n

ncn

1/4

01/4

11

2c dx

5

10

3

10

1

2

110

3

10

5

1nc

n 3 2 1 0 1 2 345 4 5x

y

11

/ /1( ) , ( ) , 0, 1, 2,...

2

pin x p in x p

n np

n


2 2

2T p p

nc

2 40 3 5

2435

0.1

0.2

0.3

0.4

0.5

/2

/2

cos / 2 sin / 2

cos / 2 sin / 2

in

in

e n i n

e n i n

2 22

2T p p

152/152

[2008][12-2] engineering mathematics 2

Documents