CE Coding and Transformations April – June 2011

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 – second session this week Summary

The Fourier Series can be written in ‘complex form’ (where the sines and cosines are replaced by exponentials) – see Week 4. 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