chap 5 fourier series

中華大學資訊工程系

Fall 2002

Chap 5 Fourier Series Chap 5 Fourier Series

Fourier Analysis

FourierSeries

FourierSeries

FourierIntegral

FourierIntegral

DiscreteFourier

Transform

DiscreteFourier

Transform

FourierTransform

FourierTransform

FastFourier

Transform

FastFourier

Transform

Discrete Continuous

Outline

Periodic Function

Fourier Cosine and Sine Series

Periodic Function with Period 2L

Odd and Even Functions

Half Range Fourier Cosine and Sine

Series

Complex Notation for Fourier Series

Fourier, Joseph

Fourier, Joseph 1768-1830

Fourier, Joseph

In 1807, Fourier submitted a paper to the Academy of Sciences of Paris. In it he derived the heat equation and proposed his separation of variables method of solution. The paper, evaluated by Laplace, Lagrange, and Lagendre, was rejected for lack rigor. However, the results were promising enough for the academy to include the problem of describing heat conduction in a prize competition in 1812. Fourier’s 1811 revision of his earlier paper won the prize, but suffered the same criticism as before. In 1822, Fourier finally published his classic Theorie analytique de la chaleur, laying the fundations not only for the separation of variables method and Fourier series, but for the Fourier integral and transform as well.

Periodic Function

Definition: Periodic Function

A function f(x) is said to be periodic with

period T if for all x

)()( xfTxf

T

f(x)

x

Periodic Function

f(x+p)=f(x), f(x+np)=f(x) If f(x) and g(x) have period p, the the

function H(x)=af(x)+bg(x) , also has the period p

If a period function of f(x) has a smallest period p (p >0), this is often called the fundamental period of f(x)

Periodic Function

Example Cosine Functions: cosx, cos2x, cos3x, …

Sine Functions: sinx, sin2x, sin3x, …

eix, ei2x, ei3x, …

e-ix, e-i2x, e-i3x, …


Lemma: Trigonometric System is

Orthogonal

)( ,0coscos nmnxdxmx

)( ,0sinsin nmnxdxmx

),( ,0sincos nmanynxdxmx


A function f(x) is periodic with period 2

and

10

321

3210

sincos

3sin2sinsin

3cos2coscos)(

nnn nxbnxaa

xbxbxb

xaxaxaaxf

Fourier Cosine and Sine Series(Euler formulas)

Then

dxxfa )(

2

10

nxdxxfan cos)(

1

nxdxxfbn sin)(

1


Proof:

n

n

n

n

nnn

a

a

xdxnnxdxnna

nxdxnxa

nxdxnxbnxaa

nxdxxf

22

1.

)cos(2

1)cos(

2

1

coscos1

cossincos1

cos)(1

10


Example 5-1: Find the Fourier coefficients

corresponding to the function

Sol:

0)(2

10

dxxfa

)()2(

0 ,

0 ,)(

xfxf

xk

xkxf


Sol:

0

sinsin

coscos)(1

cos)(1

0

0

0

0

n

nx

n

nxk

nxdxknxdxk

nxdxxfan


Sol:

2,4,6,n ,0

1,3,5,n ,2cos1

cos12

coscos

sinsin)(1

sin)(1

0

0

0

0

n

nn

k

n

nx

n

nxk

nxdxknxdxk

nxdxxfbn


A periodic function f(x) with period 2L

2L

f(x)

x

10 sincos)(

nnn L

xnb

L

xnaaxf


Then

L

Ldxxf

La )(

2

10

L

Ln dxL

xnxf

La

cos)(

1

L

Ln dxL

xnxf

Lb

sin)(

1

Periodic Square Wave

2,2

21,0

11 ,12 ,0

)(

LLp

x

xkx

xf


A function f(x) is said to be even if

A function f(x) is said to be odd if

)()( xfxf

)()( xfxf


)()( xfxf )()( xfxf

f(x)

x

f(x)

x

Even Function Odd Function


Property

The product of an even and an odd function is odd.

evenisxfifdxxfdxxfLL

L )( ,)(2)(

0

oddisxfifdxxfL

L )( ,0)(


Fourier Cosine Series

Fourier Sine Series

1

0 )( ,cos)(n

n evenisxfifL

xnaaxf

1

)( ,sin)(n

n oddisxfifL

xnbxf

Sun of Functions

The Fourier coefficients of a sum f1+f2 are the sum of the corresponding Fourier coefficients of f1 and f2.

The Fourier coefficients of cf are c times the corresponding Fourier coefficients of f.

Examples

Rectangular Pulse The function f*(x) is the sum in Example 1

of Sec.10.2 and the constant k.

Sawtooth wave Find the Fourier series of the function

...)5sin5

13sin

3

1(sin

4kk )(* xxxxf

)()2(,

x )(

xfxf

xifxf

Half-Range Expansions

A function f is given only on half the range, half the interval of periodicity of length 2L. even periodic extention f1 of f

odd period extention f2 of f


inx

nnecxf

)(

dxexfc inx

n )(2

1

1

0 sincos)(n

nn nxbnxaaxf


A periodic function f(x) with period 2L

L

xin

nnecxf

)(

L

L

L

xin

n dxexfL

c

)(2

1

Complex Fourier Series

Find complex Fourier series

)()2(

)(

xfxf

xifexf x

Exercise

Section 10-4 #1,

Section 10-2 #5, #11

Section 10-3 #5, #9

Section 10-4 #1, #15

Fourier Cosine and Sine Integrals

Example 1

)()2(

1 ,0

11 ,1

1 ,0

)(

xfLxf

Lx

x

xL

xf


n

nn LLn

Ln

Ldx

L

xnxf

La

Ldxxf

La

sin2

/

)/sin(2cos)(

1

1)(

2

1

1

1

1

10

x

x

1

sin(x)


x

x

xsin


L

sin2

)( A

Gibb’s Phenomenon

Sine Integral u

duSi0

sin)(

Gibb’s Phenomenon

Gibb’s Phenomenon

Gibb’s Phenomenon

))1(())1((1

sin1sin1

)sin(1)sin(1

)sin()sin(1

sincos2)(

)1(

0

)1(

0

00

0

0

xaSixaSi

dtt

tdt

t

t

dx

dx

dxx

dx

xf

axax

aa

a

a


Fourier Cosine Integral of f(t)

0

)cos()( )( dtAtf

0

)cos()(2

)( dtttfA


Fourier Sine Integral of f(t)

0

)sin()( )( dtBtf

0

)sin()(2

)( dtttfB

Fourier Integrals

Fourier Integral of f(t)

dtttfB

dtttfA

)sin()(1

)(

)cos()(1

)(

0

)sin()()cos()( )( dtBtAtf

chap 5 fourier series

Documents