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CE00998-3Coding and
Transformations
Sept - Nov 2010
Teaching Staff
Module Leader: Dr Pat Lewis (K219, [email protected], 353549)
Other Teaching Staff: Prof Brian Burrows (K221, [email protected], 353420)
Dr Martin Paisley (K219, [email protected], 353549)
Mr Chris Mann (K219, [email protected], 353549)
Lectures/Tutorials• Large lecture (all together)
Mon 10.00-11.00 (D109)
• Small lecture/tutorial
Group 1: Weds 11.00-13.00 (C307)
Group 2: Mon 13.00-15.00 (C307)
Group 3: Tues 09.00-11.00 (C328)
• Lab session
Group 1: Fri 11.00-12.00 (KC001)
Group 2: Weds 10.00-11.00 (KC001)
Group 3: Weds 10.00-11.00 (KC001)
Module Aims
To introduce key mathematical techniques for modern digital signal processing, and coding/information theory
• Decomposition of a periodic function into its harmonics (frequency analysis)
• Mapping from the time domain to the frequency domain (filtering for image enhancement, noise reduction etc)
• Algorithms for high-speed computing
• Efficient coding algorithms
Topics Covered
• Maple Mathematical Software• Fourier Series• Fourier Transforms• Discrete Fourier Analysis and
Fast Fourier Transform• Coding
ScheduleWeek Grande Lecture Petite Lecture Tutorial Lab
6 Sep Introduction Intro to MAPLE Intro MAPLE Integration
13 Sep Integration by Parts Step Functions Matrices Programming
20 Sep Fourier Series Fourier Series Examples MAPLE
27 Sep FS Odd & Even Functions Examples MAPLE
4 Oct FS Complex Form Examples Assignment 1
11 Oct Class Test 1 Fourier Transforms Examples MAPLE
18 Oct FT Properties Examples MAPLE
25 Oct FT Generalised Functions Examples Assignment 2
1 Nov Class Test 2 Discrete FT Examples MAPLE
8 Nov DFT Fast FT Examples Assignment 3
15 Nov DFT Huffman Coding Examples MAPLE
22 Nov Class Test 3
1. Introduction2. Sine and Cosine3. What is a Fourier Series?4. Some Demonstrations
Week 3Introduction to Fourier Series
• Fourier methods and their generalisations lie at the heart of modern digital signal processing
• Fourier analysis starts by– representing complicated periodicity by harmonics of
simpler periodic functions: sine and cosine– “frequency domain representation”
Introduction
Joseph Fourier (1768-1830)• Born in Auxerre• Scientific advisor to
Napoleon during invasion of Egypt in 1798
• Introduced Fourier Series in “Theorie Analytique de la Chaleur “ for heat flow analysis in 1822.
• Discovered the ‘greenhouse effect’
Fourier’s discovery
• Any periodic function…
• …can be represented as a sum of harmonics of sine and/or cosine
3 6 9 -3 -9 -6
2
0
2 4 6 -2 -4 -6 0
1
Sine and Cosine
xsin (x)
cos (x)
sin (x)
cos (x)
cos(x) and sin(x) are periodic with period 2
1
Other Periods?
sin(2x)
Periodic with period 1
cos(2x)
Other Periods?
sin(2x/T)
cos(2x/T)
T
T
Periodic with period Teg T=13.2
Harmonics
T
xn2cos
T
xn2sin
n=1 n=2 n=3
n=1 n=2 n=3
What is a Fourier Series?
• The representation of a periodic function as a sum of harmonics of sine and/or cosine
10
2sin
2cos
2
1)(
nnn T
xnb
T
xnaaxf
• An infinite series but usually only a few terms are needed for a reasonable approximation
Finding the Fourier Series
The coefficients are given by
10
2sin
2cos
2
1)(
nnn T
xnb
T
xnaaxf
T
dxxfT
a0
0 )(2
T
n dxT
xnxf
Ta
0
2cos)(
2
T
n dxT
xnxf
Tb
0
2sin)(
2
(so is…? 02
1a …the mean value of f(x))
)...1( n
)...1( n
Square Wave Demo
• Find the Fourier series for
2.x0when1
0x2-when0)(xf
Square Wave Demo
• More integration for the other coefficients shows that the series is
...
2
5sin5
1
2
3sin3
1
2sin
2
2
1)(
xxxxf
• Easy integration for
T
dxxfT
a0
0 )(2
2
0
4
2
014
2dxdx 0
2
1 20 x 2
2
1 1
0a
Square Wave Demo• What does it look like?
2
sin2
2
1)(
xxf
2
3sin3
1
2sin
2
2
1)(
xxxf
Square Wave Demo• What does it look like?
2
5sin5
1
2
3sin3
1
2sin
2
2
1)(
xxxxf
2
17sin
17
1...
2
5sin5
1
2
3sin3
1
2sin
2
2
1)(
xxxxxf
Square Wave Demo• What does it look like?
2
47sin
47
1...
2
5sin5
1
2
3sin3
1
2sin
2
2
1)(
xxxxxf
2
199sin
199
1...
2
5sin5
1
2
3sin3
1
2sin
2
2
1)(
xxxxxf
• Search on youtube for “square wave Fourier series”
Square Wave Demo
(For music lovers: when the frequency doubles the pitch of the note rises by one octave)
Saw Tooth Wave Demo
• Find the Fourier series for the function of period 4 given by
40for)( xxxf
Saw Tooth Wave Demo
• More integration for the other coefficients shows that the series is
....
2
3sin3
1
2
2sin2
1
2sin1
142)(
xxxxf
• Easy integration for
T
dxxfT
a0
0 )(2
4
04
2xdx
4
0
2
22
1
x
2
0
2
4
2
1 22
4
0a
Saw Tooth Wave Demo• What does it look like?
2
sin4
2)(x
xf
2
2sin2
1
2sin
42)(
xxxf
Saw Tooth Wave Demo• What does it look like?
2
3sin3
1
2
2sin2
1
2sin1
142)(
xxxxf
2
4sin4
1
2
3sin3
1
2
2sin2
1
2sin1
142)(
xxxxxf
Saw Tooth Wave Demo• What does it look like?
2
24sin
24
1...
2
2sin2
1
2sin1
142)(
xxxxf
2
99sin
99
1...
2
2sin2
1
2sin1
142)(
xxxxf
• More on youtube for “square wave Fourier series”
• How many terms in the series are need for a ‘good’ representation?– It depends on the function
Saw Tooth Wave Demo
• Fourier analysis starts with the representation of periodic behaviour by sums of harmonics of sine and cosine functions
• The Fourier series tells you which harmonics (frequencies) are present, and their relative amplitudes
• “Frequency domain representation”• The technique relies heavily on integration• There are some short cuts for ‘odd’ and even’
functions – see Week 4
Summary
• The Fourier Series can be written in ‘complex form’ (where the sines and cosines are replaced by exponentials) – see Week 5. This is the form that will be used later in the module.
• Following discussion of the theory we will do some examples by hand calculation
• We will also use MAPLE to remove some of the hard work
Summary
