CE Coding and Transformations Sept - Nov 2010

Teaching Staff Module Leader: Dr Pat Lewis (K219, ) Other Teaching Staff: Prof Brian Burrows (K221, ) Dr Martin Paisley (K219, ) Mr Chris Mann (K219, )

Lectures/Tutorials Large lecture (all together) Mon (D109) Small lecture/tutorial Group 1: Weds (C307) Group 2: Mon (C307) Group 3: Tues (C328) Lab session Group 1: Fri (KC001) Group 2: Weds (KC001) Group 3: Weds (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

Schedule WeekGrande LecturePetite LectureTutorialLab 6 Sep IntroductionIntro to MAPLEIntro MAPLEIntegration 13 Sep Integration by PartsStep FunctionsMatricesProgramming 20 Sep Fourier Series ExamplesMAPLE 27 Sep FSOdd & Even FunctionsExamplesMAPLE 4 Oct FSComplex FormExamplesAssignment 1 11 Oct Class Test 1Fourier TransformsExamplesMAPLE 18 Oct FTPropertiesExamplesMAPLE 25 Oct FTGeneralised FunctionsExamplesAssignment 2 1 Nov Class Test 2Discrete FTExamplesMAPLE 8 Nov DFTFast FTExamplesAssignment 3 15 Nov DFTHuffman CodingExamplesMAPLE 22 Nov Class Test 3

1. Introduction 2. Sine and Cosine 3. What is a Fourier Series? 4. Some Demonstrations Week 3 Introduction 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 ( ) 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 Discovered the ‘greenhouse effect’

Fourier’s discovery Any periodic function… …can be represented as a sum of harmonics of sine and/or cosine

Sine and Cosine x sin (x) cos (x) sin (x) cos (x) cos(x) and sin(x) are periodic with period 2 1

Other Periods? sin(2x) Periodic with period 1 cos(2x)

Other Periods? sin(2x/T) cos(2x/T) T T Periodic with period T eg T=13.2

Harmonics 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 An infinite series but usually only a few terms are needed for a reasonable approximation

Finding the Fourier Series The coefficients are given by (so is…? …the mean value of f(x))

Square Wave Demo Find the Fourier series for

Square Wave Demo More integration for the other coefficients shows that the series is Easy integration for

Square Wave Demo What does it look like?

Square Wave Demo What does it look like?

Square Wave Demo What does it look like?

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

Saw Tooth Wave Demo More integration for the other coefficients shows that the series is Easy integration for

Saw Tooth Wave Demo What does it look like?

Saw Tooth Wave Demo What does it look like?

Saw Tooth Wave Demo What does it look like?

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