CHAPTER THREE

FOURIER SERIES

After completing these tutorials, students should be able to:

determine whether the given function is even, odd or neither.

compute the Fourier Series for the given function

find the Fourier Series for the given function

determine the Fourier Cosine Series for the given function on the indicated interval

determine the Fourier Sine Series for the given function on the indicated interval

find the half-range Cosine and Sine expansions of the given function

apply the Fourier Series in solving engineering problems

51

Question 1

In problem I throught III:

a) Sketch the graph of the given function.

b) Determine whether the given function is even, odd or neither.

c) Compute the Fourier Series for the given function.

I ( )f x x ; x

( 2 ) ( )f x f x

Solution:

(a) The graph:

(b) ( )f x is symmetric with respect to the origin

- 0dd function

- Fourier Sine Series

(c) 0

2( )sin

L

n

n xb f x dx

L L

0

0

2

0

2sin

2sin

2 cos sin

2 cos0 (0 0)

2cos

n xx dx

x nxdx

x nx nx

n n

n

n

n

52

2( 1) ; 1,3,5,...

2(1) ; 2,4,6,...

2;

2;

nn

nn

n oddn

n evenn

1

( ) sinn

n

n xf x b

1

1

1

1

2( 1) sin

2( 1)sin

n

n

n

n

n x

n

nxn

II 1( )f x

x

;

;

2 0

0 2

x

x

( 4) ( )f x f x

Solution:

(a) The graph:

(b) ( )f x is neither even nor an odd function.

(c) 0

0

1( )

L

a f x dxL

2

2

2

2 0

1( )

2

11

2

n

f x dx

dx xdx

53

22

0

2

0

1

2 2

1(0 2 2 0)

2

2

x

x

xx

1

( )cos

L

n

L

n xa f x dx

L L

2

2

0 2

2 0

0 2

2 0

2

1( )cos

2 2

11cos cos

2 2 2

1 2 2 2sin sin cos

2 2 2 2

1 2 2 2. (0 0) 0 cos 0

2

2(cos 1)

2

n xf x dx

n x n xdx x dx

n x n x n xx x

n n n

nn n n

nn

2

2

2

( 1 1) ;

2(1 1) ;

2( 1) 1n

n oddn

n evenn

n

1

( )sin

L

n

L

n xb f x dx

L L

2

2

0 2

2 0

1( )sin

2 2

11sin sin

2 2 2

n xf x dx

n x n xdx x dx

54

20 2

2 0

1 2 2 2cos cos sin

2 2 2 2

1 2. 1 cos( ) 2cos 0 0 0

2

1( 1 cos )

n x n x n xx

n n n

n nn

nn

1( 1 ( 1)) ;

1( 1 1) ;

n oddn

n evenn

11 11 ( 1) ( 1 ( 1) )n n

n n

2 1

21

2 1( ) 1 ( 1) 1 cos ( 1 ( 1) )sin

( ) 2 2

n

n

n x n xf x

n n

III 2( )f t t ; 1 1t

( 2) ( )f t f x

Solution:

(a) The graph:

(b) ( )f x is symmetrical with y-axis

- Even function

- Cosine Fourier Series

(c) 2

0

0

2

1

L

a x dx

13

0

23

2(1 0)

3

x

55

1

2

0

2cos

1 1n

n xa x dx

12

0

1

2

2

0

2

2

sin 2 sin2

1 2 cos sin2 sin

( )

22 1cos 0

( )

4(cos )

x n x x n xdx

nx nx

x n x n xx n x

nx nx n n

nn

nn

2

2

2

4( 1) ; 1,3,5,...

4(1) ; 2,4,6,...

4( 1)n

nn

nn

n

2

1

1 4( ) ( 1) cos

3

n

n

f x n xn

Question 2

Find the Fourier Series for the given function.

(a)

2

0( )f x

x

;

;

0

0

x

x

Solution:

0

1( )a f x dx

0

2

0

3

0

3 2

10

1

3

1

3 3

dx x dx

x

56

1( )cosn

n xa f x dx

0

2

0

2

00

2

0 0

2

0

10cos cos

1 sin 2sin

1 2 cos 1sin cos

1 2 cos 1 sinsin

nxdx x nxdx

x nxx nxdx

n n

x nxn nxdx

n n n n

x nx nxn

n n n n n

2

2 2

1 2 cos 2sinsin

n nn

n n n

2

2

2

1 2 cos

2cos

2( 1)n

n

n

nn

n

1( )sinn

n xb f x dx

0

2

0

2

00

2

0 0

2

2 2

0

2

2

10sin sin

1 cos 2cos

1 2 sin sincos

1 2 2 coscos sin

1 2cos s

nxdx x nxdx

x nxx nxdx

n n

x nx nxn dx

n n n n

nxn n

n n n n

nn n

3 3

3 3

2 cos 2in

2( 1) 2( 1)

nn

nn

n n

n n n

2 1

2 31

2 ( 1) 2( ) ( 1) cos ( 1) 1 sin

6

nn n

n

f x nx nxn n n

57

Question 3

Determine the Fourier Cosine Series for the following functions on the indicated interval.

(a)

0( )

1f x

;

;

0

2

x

x

Solution:

2

0

0

2( )

2a f x dx

2

0

2

10 1

1

dx dx

x

12

1

2

0

2( )cos

2n

n xa f x dx

2

0

2

10cos 1.cos

2 2

1 2sin

2

1 2 2sin sin

2

1 2sin

2

nx nxdx dx

nx

n

nn

n n

n

n

1

1 2( ) sin cos

2 2 2n

n nxf x

n

1

1 2sin cos

2 2 2n

n nx

n

(b)

( ) xf x e ; 0 1x

Solution: 1

0

0

2( )

1a f x dx

1

0

2 xe dx

58

1

02 xe

1 0

1

2( )

12 1

e e

e

1

0

2( )cos

1 1n

n xa f x dx

1

0

2 cosxe n xdx

11 1

0 00

cos sin sinx x

x e ee n xdx n x n xdx

n n

1 1 1 1

0 0

1cos cos sin cos

xx e e e

e n xdx n xdx n nn n n n

1 1

0

1 11 cos cosx e

e n xdx nn n n

1 1

0

1 ( 1)cos

1

nx e

e n xdxn

1

0

2 cosx

na e n xdx

12 2 ( 1)

1

ne

n

1

1

1 2 2 ( 1)( ) 1 cos

1

n

n

ef x n x

e n

1

1

1 1 ( 1)1 2 cos

1

n

n

en x

e n

xu e ' cosv n x

' xu e sin n x

vn

xu e ' sinv n x

' xu e cos n x

vn

59

Question 4

Determine the Fourier Sine Series for the following functions on the indicated interval.

(a) ( ) 1f x x ; 0 1x

Solution:

1

0

2( )sin

1 1n

n xb f x dx

1

0

1 1

0 0

1 11

0 00

1

2

0

2 ( 1)sin

2 sin sin

cos cos cos2

cos sin cos 12

x n xdx

x n xdx n xdx

x n x n x n x

n n n

n n x n

n n nn

1 2cos

2n

n n

1

2( ) 1 2( 1) sinn

n

f x n xn

(b) ( )

2

xf x

x

;

; 0 1

1 2

x

x

Solution:

2

0

2( )sin

2 2n

n xb f x dx

1 2

0 1

1 2 2

0 1 1

2 31

sin (2 )sin2 2

sin 2sin sin2 2 2

case casecase

n nx xdx x xdx

n n nx xdx xdx x xdx

' 1

u x

u

' sin2

2cos

2

n xv

n xv

n

60

Case 1:

11 1

00 0

2 2sin cos cos

2 2 2

n n x n xx xdx dx

n n

2 1

0

2

2 2cos sin

2 2

2 4cos sin

2 2

n x n x

n n

n x n

n n

Case 2: 22

11

22sin 2 cos

2 2

n n xxdx

n

4

cos cos2

nn

n

Case 3: 2 2 22

1 11

2 2sin cos sin

2 2 2

n n x n xx xdx

n n

2

4 2 4cos cos sin sin

2 2

n nn n

n n n

2

2 4 4 4 4 2cos sin cos cos cos cos

2 2 2 2n

n n n nb n n

n n n n nn

2 2

4 4sin sin

2

nn

n n

2

8sin

2

n

n

2

1

8( ) sin sin

2 2n

n n xf x

n

u x ' sin2

n xv

' 1u 2

cos2

n xv

n

61

Question 5

Expand the given function in an appropriate Cosine or Sine Series.

(a) 2( )f x x ; 1 1x

Solution:

( )f x is an even function, 0nb

1

0

0

2( )

1a f x dx

1

2

0

13

0

2

23

12

3

2

3

x dx

x

1

0

2( )cos

1 1n

n xa f x dx

1

2

0

2 cosx n xdx

2u x ' cosv n x

' 2u x sin n x

vn

62

1 12

00

sin sin2 2n

x n x x n xa dx

n n

1 1

0 0

sin 2 cos cos2n

n x n x n xa

n n n n

1

2

0

2 2 cos sin2

n n

n n n n

2 3

2

2

2 2cos 2sin2

2 2( 1)2

4 ( 1) 1

n

n

n n

n n n

n n

n n

21

1 4 ( 1) 1( ) cos

3

n

n

f x n xn n

(b)

3( )f x x ; x

Solution:

( )f x is an odd function, 0 0na a

0

2( )sinn

n xb f x dx

u x ' sinv n x

' 1u cos n x

vn

63

3

0

2sin

nxx dx

32

00

2 cos 3cosn

x nxb x nxdx

n n

3 2

00

2 cos 3 sin 2 sinn

n x nx x nxb dx

n n n n

3 2

0

2 cos 3 sin 2sin

n nx nxdx

n n n n

32

2 2

0 0

2 cos 3 6 sin 1cos sinn

n x nxb n xdx

n n n n n

3

2 2 0

3

3 2 2

3

2 2

3

2 2

2 cos 6 sin 1cos

2 cos 6 sin cos 1

2 cos cos 1

2 ( 1) ( 1) 1n n

n nnx

n n n n

n n n

n n n n

n n

n n n

n n n

3

2 21

2 ( 1) ( 1) 1( ) sin( )

n n

n

f x nxn n n

3u x ' sinv nx

2' 3u x cos nx

vn

2u x ' cosv nx

' 2u x sin nx

vn

u x ' sinv nx

' 1u sin nx

vn

64

Question 6

Find the half-range Cosine and Sine expansions of the given function.

( )

x

f x

x

;

;

02

2

x

x

Solution:

Half-range cosine expansion:

0

0

2( )a f x dx

2

0

2

2( )x dx x dx

2 22

02

2 2 2 22

2

2

2 2

2

8 2 2 8

2

4 2

x

x xx

0

2( )cosn

n xa f x dx

2

0

2

2

0

2 21

2 3

2cos cos

2cos cos cos

case

case case

x nxdx x nxdx

x nxdx nxdx x nxdx

65

Case 1:

2 22

00 0

sin( ) sin( )cos

x nx nxx nxdx dx

n n

22 0

2

1sin cos( )

2 2

1sin cos 1

2 2 2

nnx

n n

n n

n n

Case 2:

2

2

cos sinnxdx nn

sin sin2

nn

n

Case 3:

2

2 2

sin 1cos sin

x nx nxdx n dx

n n

22

2 2

1sin sin cos

2 2

1 1sin sin cos cos

2 2 2

nn nx

n n n

n nn n

n n n n

2 2 2 2

2 1 1 1 1sin cos sin sin cos cos

2 2 2 2 2 2 2n

n n n n na n

n n n n n n n

2 2 2

2

2 2 1 1cos cos

2

2coscos ( ) 12 2

n

nn

n n n

n

n

21

2coscos ( ) 13 2( ) cos

4 4

n

n

n

f x nxn

66

Half-Range Sine Expansion:

0

2( )sinn

n xb f x dx

2

0

2

2

0

2 21

2 3

2sin( ) ( )sin

2sin( ) sin sin

case

case case

x nx dx x nx dx

x nx dx nx dx x nx dx

Case 1:

2 22

0 00

cos cossin( )

x n nx nx dx dx

n n

22 0

2

1cos sin( )

2 2

1cos sin

2 2 2

nnx

n n

n n

n n

Case 2:

0

2

2

sin( ) cosnx dx x nn

cos cos2

nn

n

67

Case 3:

2

2 2

cos cossin

x nx nxx nx dx dx

n n

2

2

2 2

cos cos sin2 2

sin2cos cos sin

2 2

n nxn

n n n

nn n

nn n n n

2 2 2

2 1 1 2 1cos sin cos cos cos cos cos sin

2 2 2 2 2 2 2n

n n n n nb n n n

n n n n n n n

2

2 2sin

2

n

n

21

4 1( ) sin sin( )

2n

nf x nx

n

Question 7

A mass-spring system, in which a piston is attached to the end of the spring and drives the

system.The driving force acts on the piston is represented by ( )f t .Find the Fourier series of

( )f t .

( )k

f tk

,

,

0

0

t

t

tk

tktf

0,

0,)(

( )f t

k

t

k

68

Fourier sines series

1

2 1 1( )( ) sin

n

n

kf t nt

np

Question 8

Find the Fourier series of the given input signal.

0

( )sin

u tE t

,

,

02

02

Tt

Tt

Where E is the constant and 2

T

20,sin

02

,0)(

TttE

tT

tu

2T

0

0

0

2

2

2

2

2 1 1

( )sin

sin

cos

cos

( )

L

n

n

n tb f x dx

L L

k nt dx

k nt

n

k n k

n

k

n

p

p

p

p

p

p

p

p

00

2

0

20

1

1

( )

sin

cos

L

T

T

a f x dxL

E t dxT

Et

T

E

w

ww

p

69

0

2

0

1( ) cos

sin cos

L

n

T

n xa f x dt

L L

Et n t dt

T

p

w w

2

0cossin

T

dttnt = tsin

n

tnsin- tcos

2

cos

n

tn+

2

2

n 2

0cossin

T

dttnt

1

2

2

n 2

0cossin

T

dttnt = tsin

n

tnsin- tcos

2

cos

n

tn

2

0cossin

T

dttnt = ( tsin

n

tnsin- tcos

2

cos

n

tn) 2

0

T

/

1

2

2

n

=

22coscos

1

nn

n/

2

2 1

n

n

=

1

1)1(2n

n

n

Ea

T

1

1)1(2n

n

2

E

p

1

1)1(2n

n

2

2 cossin

sincos

cossin

n

tnt

n

tnt

tnt

70

0

2

0

1( ) sin

sin sin

L

n

T

n xb f x dt

L L

Et n t dt

T

p

w w

2

0

2

0

1 2

2 2

1 2

2 4

1

2 2 4

cos

sin

T

T

E tdx

T

E tt

T

EE

w

w

w

p

p w

w

21

1 1 1

2 1 4

( )( ) cos sin

n

n

E Eu t n t E t

nw w

p p

Question 9

Find the Fourier series of the given function.

0

( )

0

f t k

,

,

,

2 1

1 1

1 2

t

t

t

Period, 4T

Solution:

Even function => Fourier Cosine Series, 0nb

71

00

1

0

1

0

2

2

2

( )L

a f x dxL

k dx

kt

k

0

1

0

1

0

2

2

2 2

2

2

2

2

( ) cos

cos

sin

sin

L

n

n xa f x dt

L L

n tk dt

n tk

n

k n

n

p

p

p

p

p

p

1 2

sin2

2)(

n

n

n

kktf

2cos

tn