Lecture 7 Fourier Series Skim through notes before lecture Ask questions in lecture After lecture read notes and try some problems See me in E47 office.

Lecture 7Lecture 7Fourier SeriesFourier Series

•Skim through notes before lecture•Ask questions in lecture•After lecture read notes and try some problems•See me in E47 office if you have any questions

Best way to handle course

Remember homework 1 for submission 31/10/08

•ALL tutorial, problems class, homework in notes and web•Completed solutions version of notes at Phils Problems•All .ppt presentations from lectures at Phils Problems•Loads of questions with worked answers at Phils Problems

Where can I find stuff?

http://www.hep.shef.ac.uk/Phil/PHY226.htm

Fourier SeriesFourier SeriesLast lecture we learned how a Fourier series formed from sine and cosine harmonics can represent any periodically repeating function.

What’s a harmonic? Below are the first three harmonics of sine and cosine F(x) = a1cos x

F(x) = b3sin 3x

F(x) = b2sin 2x

F(x) = b1sin x

F(x) = a3cos 3x

F(x) = a2cos 2x

1

0

2sin

2cos

2

1)(

nnn L

xnb

L

xnaaxf

For these cases L is taken to be

2to simplify

expressions

n = 1

n = 2

n = 3

SummarySummary

1

0

2sin

2cos

2

1)(

nnn L

xnb

L

xnaaxf

L

dxxfL

a00 )(

2L

n dxL

xnxf

La

0

2cos)(

2 L

n dxL

xnxf

Lb

0

2sin)(

2

The Fourier series can be written with period L as

The Fourier series coefficients can be found by:-

Fourier SeriesFourier Series

http://www.falstad.com/fourier/

Sine terms

Cosine terms

http://www.univie.ac.at/future.media/moe/galerie/fourier/fourier.html

A key point to notice is that the summed output will repeat with the period of the 1st harmonic

We must decide on the amplitude of each harmonic term. This amplitude may be zero, positive, negative, big or small……..




http://www.univie.ac.at/future.media/moe/galerie/fourier/fourier.html

Fourier Series - QUIZFourier Series - QUIZTeam A questions in white

1. What is when n = 3 ?




n)1(1

n)1(1

)2(cos1 n

)2(cos1 n

5. What is when n = 52 ?)2(cos1 n

6. What is when n = 1 ?)(cos1 n

7. What is when n = 4 ?)(cos1 n

0)1(1 3

2)1(1 52

2)1(1

2)1(1

2)1(1

0)1(1)(cos1

2)1(1)4(cos1

Team B questions in red

Fourier Series - QUIZFourier Series - QUIZ

8. Team B: What is ?

-40

-30

-20

-10

0

10

20

30

40

-20 -15 -10 -5 0 5 10 15 20x axis

y ax

is

y=4x

2002002410

10

10

10

2

xdxxI

10

10

4 dxxI


-40

-30

-20

-10

0

10

20

30

40

50

60

-30 -20 -10 0 10 20 30x axis

y ax

isy=2x+5

9. Team A: What is ? 1505)52(10

0

10

0

2 xxdxxI 10

0

)52( dxxI


-40

-20

0

20

40

60

80

100

-30 -20 -10 0 10 20 30

x axis

y ax

isstep

10. Team B: Describe the following step function in terms of f(x) and x ?

50)(0

0)(0

xfxwhen

xfxwhen


-40

-20

0

20

40

60

80

100

-30 -20 -10 0 10 20 30

x axis

y ax

isstep

11. Team A: What is ?

10

10

0

10

10

0

100

010 500500500)( xxdxdxdxxfI

10

10

)( dxxfI


-40

-20

0

20

40

60

80

100

-30 -20 -10 0 10 20 30x axis

y ax

isstep periods

12. Team B: Describe the following step function over one period in terms of f(x) and x ?

50)(510

0)(05

xfxwhen

xfxwhen


-40

-20

0

20

40

60

80

100

-30 -20 -10 0 10 20 30x axis

y ax

isstep periods

13. Team A: What is the integral of f(x) over one period ?

10

0

5

0

10

5

105

50 250500500)( xxdxdxdxxfI


-40

-20

0

20

40

60

80

100

-30 -20 -10 0 10 20 30x axis

y ax

is

step period raised

14. Team B: Describe the following step function over one period in terms of f(x) and x ?

70)(510

20)(05

xfxwhen

xfxwhen


-40

-20

0

20

40

60

80

100

-30 -20 -10 0 10 20 30x axis

y ax

is

step period raised

15. Team A: What is the integral of f(x) over one period ?

10

0

5

0

10

5

105

50 45070207020)( xxdxdxdxxfI


-40

-20

0

20

40

60

80

100

-30 -20 -10 0 10 20 30x axis

y ax

is

step period raised

16. Team B: If we were to represent the function below as a Fourier series what could you say about the value of a0 ?

1

0

2sin

2cos

2

1)(

nnn L

xnb

L

xnaaxf

Fourier series

a0 is baseline shifter. Half way between 20 and 70 is 45. So ao = 90

907020]70[5

1]20[

5

170

10

220

10

2)(

2 105

50

10

5

5

000 xxdxdxdxxfperiod

aperiod


-40

-20

0

20

40

60

80

100

-30 -20 -10 0 10 20 30x axis

y ax

is

step period raised

17. Team A: If we were to represent the function below as a Fourier series what could you say about the values of the an terms ?

1

0

2sin

2cos

2

1)(

nnn L

xnb

L

xnaaxf

Fourier series

odd function so all an terms are zero


-40

-20

0

20

40

60

80

100

-30 -20 -10 0 10 20 30x axis

y ax

is

step period raised

18. Team B: If we were to represent the function below as a Fourier series what could you say about the sign of the b1 term ?

1

0

2sin

2cos

2

1)(

nnn L

xnb

L

xnaaxf

Fourier series


18. Team B: If we were to represent the function below as a Fourier series what could you say about the value of the b1 term ?

1

0

2sin

2cos

2

1)(

nnn L

xnb

L

xnaaxf

Fourier series

It would have a negative amplitude

-40

-20

0

20

40

60

80

100

-30 -20 -10 0 10 20 30x axis

y ax

is

step period raised

1st sine harmonic (fundamental)

Finding coefficients of the Fourier SeriesFinding coefficients of the Fourier SeriesFind Fourier series to represent this repeat pattern.

0 x

1

20

01)(

x

xxf

Steps to calculate coefficients of Fourier series

1. Write down the function f(x) in terms of x. What is period?

2. Use equation to find a0?1][

10

11

1)(

10

2

0

2

00

xdxdxdxxfa

Period is 2

3. Use equation to find an?

0sin1

cos1

cos)0(1

cos)1(1

cos)(1

00

2

0

2

0

n

nxdxnxdxnxdxnxdxnxxfan

4. Use equation to find bn?

00

2

0

2

0

cos1sin

1sin)0(

1sin)1(

1sin)(

1

n

nxdxnxdxnxdxnxdxnxxfbn

nn

n

nn

n

n

nxbn

1cos10coscos1cos1

0

Finding coefficients of the Fourier SeriesFinding coefficients of the Fourier Series

5. Write out values of bn for n = 1, 2, 3, 4, 5, ….

4. Use equation to find bn?

n

n

nnn

n

n

nxbn

cos110coscos1cos1

0

2

1

)1(1

1

1

1cos

1

111

b 0

2

)1(

2

11

2

2cos

2

112

b

3

2

3

)1(

3

11

3

3cos

3

113

b 0

4

)1(

4

11

4

4cos

4

114

b

5

2

5

)1(

5

11

5

5cos

5

115

b

6. Write out Fourier series

with period L, an, bn in the generic form replaced with values for our example

1

0

2sin

2cos

2

1)(

nnn L

xnb

L

xnaaxf

...5sin5

23sin

3

2sin

2

2

1

2

2sin

2

2cos

2

1)(

10

xxxxn

bxn

aaxfn

nn

Finding coefficients of the Fourier SeriesFinding coefficients of the Fourier Series

So what does this Fourier series look like if we only use first few terms?

Use Fourier_checker on Phils problems website

...5sin5

23sin

3

2sin

2

2

1)( xxxxf

Fourier series (5 terms only for sine and cosine)

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 60 120 180 240 300 360

degrees

tota

l am

plit

ud

e

0 x

1

Finding coefficients of the Fourier Series - QUIZFinding coefficients of the Fourier Series - QUIZ

Find Fourier series to represent this repeat pattern.

20

0)(

x

xxxf

Steps to calculate coefficients of Fourier series

1. Write down the function f(x) in terms of x. What is period?

2. Use equation to find a0?

22

10

11)(

2

2)(

2

0

22

0

2

000

xdxxdxdxxfdxxf

La

L

3.

4.

Team A find coefficients an?

Team B find coefficients bn?

Period is 2



20

0)(

x

xxxf

3. Team A find coefficients an?Period is 2

vduuvudv

nxn

nxdxv sin1

cos

220

20

10cos

1sin

1cos

1sin

nn

nn

nnx

nnx

n

xan

Integrate by parts so set u = x and cos (nx) dx = dv

and du = dx

L

n dxL

xnxf

La

0

2cos)(

2

2

0

2

00cos0

1cos

1cos)(

2

22cos)(

2dxnxdxnxxdxnxxfdx

L

xnxf

La

L

n

000

sin11

sin1

cos1

nxdxn

nxn

xdxnxxan

n=1n=1 n=2n=2 n=3n=3 n=4n=4 n=5n=5

211

01

a 0

4

1

4

102

a

9

2

9

1

9

103

a 04 a 25

2

25

1

25

105

a



20

0)(

x

xxxf

4.

Period is 2

vduuvudv

nxn

nxdxv cos1

sin

nn

nn

nxn

nxn

xbn sin

1cos

1sin

1cos

20

20

Integrate by parts so set u = x and sin (nx) dx = dv

du = dx

L

n dxL

xnxf

Lb

0

2sin)(

2

2

0

2

00sin0

1sin

1sin)(

2

22sin)(

2dxnxdxnxxdxnxxfdx

L

xnxf

Lb

L

n

000

cos11

cos1

sin1

nxdxn

nxn

xdxnxxbn

n=1n=1 n=2n=2 n=3n=3 n=4n=4 n=5n=511 b

2

12 b

Team B find coefficients bn?

3

13 b

4

14 b

5

15 b



5. Write out the first few terms of Fourier series

20

a

1

0

2sin

2cos

2

1)(

nnn L

xnb

L

xnaaxf

n=1n=1 n=2n=2 n=3n=3 n=4n=4 n=5n=511 b

2

12 b

3

13 b

4

14 b

5

15 b

L is the period = 2

n=1n=1 n=2n=2 n=3n=3 n=4n=4 n=5n=5

211

01

a 0

4

1

4

102

a

9

2

9

1

9

103

a 04 a 25

2

25

1

25

105

a

So .....5sin5

14sin

4

13sin

3

12sin

2

11sin15cos

25

23cos

9

21cos

2

4)( xxxxxxxxxf


Can we check our Fourier series using Fourier_checker.xls at Phils Problems ???

Yes!!

.....5sin5

14sin

4

13sin

3

12sin

2

11sin15cos

25

23cos

9

21cos

2

4)( xxxxxxxxxf

Lecture 7 Fourier Series Skim through notes before lecture Ask questions in lecture After lecture read notes and try some problems See me in E47 office.

Documents

fourier series quiz

fourier series http

fourier series coefficients

fourier series skim

fourier series odd function

team b questions

n terms

terms of fx