1. Applications of Fourier Analysis I Patterns in Space and Time:
From Nano to Nirvana
Philip Moriarty, Room B403, ext
OUTLINE
Structure of module
Fourier analysis revisited: the basic premise
Fourier series
Time-frequency transformations and spectra
(Is Fourier analysis always applicable…?)

2. Module Structure 9 x 1 hr lectures 7 x 2 hr problems classes
Two class tests (40%, 60%)
Class tests will take place in the lecture slot (Monday 09:00 – 10:00) in the weeks beginning 26 February and 14 May.
- Downloadable lecture notes.
- Downloadable problems class question sheets.
- Downloadable worked solutions (after problems classes!).
- And, technology permitting, podcasts.

3. Module Timetable

4. Lectures and Problems Classes
Printed lecture notes handed out. Goal is to encourage your active involvement (albeit difficult with ~ 100 students!). Read notes BEFORE lecture + attempt to answer questions during lecture.
Synoptic module. (synoptic: presenting a summary or general view of a whole; "a synoptic presentation of a physical theory“)
NB The problems classes are an integral component of the module (NOT an optional extra). NO BOOKWORK in class tests.
Concepts covered in lectures.
Mathematical ability developed in problems classes.

5. Music and Tones ?
What determines the timbre of a musical note? Why does an ‘A’ note on the guitar sound so different to an ‘A’ note on the piano?
ANS: Harmonic content
..but the harmonic content of a note played on an instrument also depends on how the instrument is played.

6. Fourier Analysis in Science
Jean Baptiste Joseph Fourier ( )
“Mathematics compares the most diverse phenomena and discovers the secret analogies that unite them”
You have already encountered Fourier analysis in the “Elements of Mathematical Physics” module....
What is the fundamental purpose of Fourier analysis? ?
Keywords: decomposition, waves, sinusoids, signal, spectrum, vibrations, time  frequency, integrals, transforms

7. Fourier Analysis: Music and Tones
Harmonic content of a musical instrument can be rather complex (and – importantly – sometimes can include harmonics that aren’t multiples of the fundamental frequency (e.g. drums, bells, cymbals, and ‘non ideal’ stringed instruments)).)
What if we start ‘from scratch’ and construct a basic musical tone from a combination of harmonics....

8. Fourier Analysis: The Basics
Periodic functions: f(t)=f(t+T) (period=T). Of immense importance in a wide range of areas (....as you know)
Most fundamental periodic function is the sinusoid - any function that can be represented as a sine or cosine.
What is the period of the function f(x)=4 sin (2x)? ?
ANS: p radians
..and of the function f(x)=4 sin (2x) + 2 sin (4x)? ?
ANS: p radians

9. Fourier Analysis: The Basics
f(x)= 4 sin (2x)

10. Fourier Analysis: The Basics
f(x)=4 sin (2x), 2 sin (4x)

11. Fourier Analysis: The Basics
f(x)=4 sin (2x) + 2 sin (4x)

12. Fourier Analysis: Music and Tones?
What happens to the sound of the note as we increase the number of harmonics?
ANS: The note changes its tonal quality, in this case becoming ‘brighter’.
‘Wah-wah’ pedal – shifting centre frequency of band pass filter.

13. Fourier Analysis: The Basics
All the examples shown thus far have involved constructing or decomposing periodic waveforms using the methods of Fourier analysis described in the Elements of Mathematical Physics module.
We determine the frequencies, amplitudes, and/or phases of the harmonics and plot a spectrum.

14. Fourier Analysis: Spectra
Determining coefficients:
Just how these coefficients are derived is something you’ll revisit in the Problems Class.

15. Fourier Analysis: Spectra
Which coefficients (An or Bn) are plotted in the spectrum? ?
ANS: Bn (function is odd). Other function symmetries will be explored in the Problems Class
What can you say about the phases of alternate harmonic components of the sawtooth waveform? ?

16. Fourier Analysis: Gibbs phenomenon?
At what points in the waveform is the Fourier series representation of the function poorest?
At discontinuties.
Note that the Fourier series representation overshoots by a substantial amount.
..but we know that we get closer to the correct function if we include more harmonics. Can’t the approximation be improved by adding in more terms?
NO!

17. Fourier Analysis: Gibbs phenomenon
The inclusion of more terms does nothing to remove the overshoot – it simply moves it closer to the point of discontinuity.
Therefore, we need to be careful when applying Fourier analysis to consider the behaviour of a function near a discontinuity
N. D. Lang and W. Kohn, Phys. Rev. B (1970)
Gibbs phenomenon, however, is not just of mathematical interest. The behaviour of electrons near a sharp step in potential (e.g. at a surface) is fundamentally governed by Gibbs phenomenon.