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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’