CE00998-3Coding and

Transformations

ScheduleWeek Grande Lecture Petite Lecture Tutorial Lab

7 Sep Introduction Intro to MAPLE Intro MAPLE Integration

14 Sep Integration by Parts Step Functions Matrices Programming

21 Sep Fourier Series Fourier Series Examples MAPLE

28 Sep FS Odd & Even Functions Examples MAPLE

5 Oct FS Complex Form Examples Assignment 1

12 Oct Class Test 1 Fourier Transforms Examples MAPLE

19 Oct FT Properties Examples MAPLE

26 Oct FT Generalised Functions Examples Assignment 2

2 Nov Class Test 2 Discrete FT Examples MAPLE

9 Nov DFT Fast FT Examples Assignment 3

16 Nov DFT Huffman Coding Examples MAPLE

23 Nov Class Test 3

Fourier Series Class Test

• 9.00 next Monday (12th October)

• It will take 40 mins

• What will I need to do?

- Section A (20 marks)

10 multi-choice questions (2 mins each)

- Section B (20 marks)

1 long question (eg Tutorial questions)

- No Maple

Fourier Series Maple Assignment

• Submit by 3.30 Monday 19th October

– to Faculty Reception (Octagon L2)

– do not email to me

• Include

– an Assignment Submission Form (available from Faculty Reception)

– an electronic copy on disc

Week 5

Fourier SeriesHome Work Exercises 2

(see p15 of notes)

Finding the Fourier Series

The coefficients are given by

10

2sin

2cos

2

1)(

nnn T

xnb

T

xnaaxf

2/

2/

0 )(2T

T

dxxfT

a

2/

2/

2cos)(

2T

T

n dxT

xnxf

Ta

2/

2/

2sin)(

2T

T

n dxT

xnxf

Tb

(so is…? 02

1a …the mean value of f(x))

)...1( n

)...1( n

Exercise (i)

• Find the Fourier series for

x

xxf

0when1

0when1)(

T=2

Exercise (i)

• This is an ODD function, so….

0na )...0( n

2/

0

2sin)(

4T

n dxT

xnxf

Tb

)...1( n

Exercise (i)

• Find

2/

0

2sin)(

4T

n dxT

xnxf

Tb

0

sin)1(2

dxnx

0

)cos(12

nxn

0coscos2

nn

nb

0

2

2sin)(

2

4dx

xnxf

n

n ]1)1[(2

n

n ])1(1[2

Exercise (i)

• So the series is

1

sin)1(12

)(n

n

nxn

xf

)sin(1

22)( xxf

• First few terms are

)2sin(2

0x )3sin(

3

2x

)4sin(4

0x ...)5sin(

5

2 x

Exercise (i)

• What does it look like?

...)5sin(

5

1)3sin(

3

1)sin(

4)( xxxxf

Exercise (i)

• Rate of convergence

Magnitude of nb

n

• Size of terms decreases slowly

• Terms for ‘large’ n are still important

• Convergence rate is ‘slow’

Exercise (ii)

• Find the Fourier series for

xx

xxxf

0when

0when)(

T=2

Exercise (i)

• This is an EVEN function, so….

0nb )...1( n

2/

0

2cos)(

4T

n dxT

xnxf

Ta

)...1( n

,)(4

2/

0

0 T

dxxfT

a

Exercise (ii)

• Easy integration for

2/

0

0 )(4T

dxxfT

a

0

2

4xdx

0

2

2

2

x

2

0

2

2 22 2

2 2

0a

T=2

Exercise (ii)• Find

2/

0

2cos)(

4T

n dxT

xnxf

Ta

0

0

sin1

1sin12

nxn

nxnx

0sin

10sin

12n

nn

n

na

0

cos2

4dxnxx

02cos

1nx

n

)0cos()cos(122

nn

2

1)1(2

n

n

2])1(1[2

n

n

Exercise (ii)

• So the series is

12

cos)1(12

2)(

n

n

nxn

xf

)cos(1

22

2)( xxf

• First few terms are

)2cos(4

0x )3cos(

9

2x

)4cos(16

0x ...)5cos(

25

2 x

Exercise (ii)

• What does it look like?

...)cos(4

2)( xxf

Exercise (ii)

• What does it look like?

...)5cos(

25

1)3cos(

9

1)cos(

4

2)( xxxxf

Exercise (ii)

• Rate of convergence

Magnitude of nb

n

• Size of terms decreases rapidly

• Terms for ‘large’ n are not important

• Convergence is ‘rapid’