Lecture 9Lecture 9Fourier TransformsFourier Transforms

Remember homework 1 for submission 31/10/08

http://www.hep.shef.ac.uk/Phil/PHY226.htmRemember Phils Problems and your notes = everything

Today• Introduction to Fourier Transforms• How to work out Fourier Transforms• Examples

http://uk.youtube.com/watch?v=tUcOaGawIW0

Fourier seriesFourier seriesWe have seen in the last couple of lectures how a periodically repeating function can be represented by a Fourier series

1

0

2sin

2cos

2

1)(

nnn L

xnb

L

xnaaxf

• Representing the sum of special solutions to wave equations such as standing waves on a string or multiple eigenfunctions in a potential well

What is the Fourier Series great at ???What is the Fourier Series great at ???

Compare with Half range sine series

dxandxwhenn 00

11

sin)()(n

nn

n d

xnBxx

• Replacing non continuous functions such as square wave digital signals with sine and cosine series that can be worked on mathematically in IODEs

Applying a square wave driver to mechanical oscillators is crazy but we do this to digital electronics all the time

1

sin)(n

n d

xnbxf

What is the Fourier Series rubbish at ???What is the Fourier Series rubbish at ???• Providing frequency information

Fourier series are designed to express AMPLITUDE in terms of sine and cosine harmonics

1

0

2sin

2cos

2

1)(

nnn T

tnb

T

tnaatf

1

0

2sin

2cos

2

1)(

nnn L

xnb

L

xnaaxf

The fact that they do this with a sum of harmonics only works because we can use an infinite number of terms.

Choosing discrete harmonic frequencies allows direct application to standing wave problems in which boundary conditions state that each wave function must agree with the boundary conditions ψ = 0 when x = 0 and x = L

But if we are only interested in the frequency distribution we can ask….

Fourier Transforms Fourier Transforms

Can we somehow modify the series to display a continuous spectrum rather than discrete harmonics?

Since an integral is the limit of a sum, you may not be surprised to learn that the Fourier series (sum) can be manipulated to form the Fourier transform which describes the frequencies present in the original function.

Fourier transforms, can be used to represent a continuous spectrum of frequencies, e.g. a continuous range of colours of light or musical pitch.

They are used extensively in all areas of physics and astronomy.

What is the best device to perform FTs ???What is the best device to perform FTs ???The human ear can instantly deconvolve multiple summed pressure waves from:-

Can you resolve the following 6 songs?

into…

time

amplitude

frequency

intensity

Imagine developing a device which could transform such complex pressures wave into the frequency domain instantly !!!

Fourier Transforms Fourier Transforms http://uk.youtube.com/watch?v=fsKvtjjY3A0http://uk.youtube.com/watch?v=4iruQlZicuUhttp://uk.youtube.com/watch?v=IPjMl9u3qechttp://uk.youtube.com/watch?v=sXSMcmnDlwY&feature=related

Fourier Transforms on TV Fourier Transforms on TV

We have the tigerPlay the message !!!

Meow!!!!

How do we find out if tiger is still alive ?? How do we find out if tiger is still alive ??

This is the amplitude vs time plot for the composite sounds

time

amplitude

frequency

intensity

This is the frequency vs time plot for the composite sounds between 3.5 and 6.5 s

Note log scales on X and Y axes

Note big peak at 100Hz, background noise, and spikes around 2000Hz

How do we find out if tiger is still alive ?? How do we find out if tiger is still alive ??

frequency

intensity

This is the original intensity vs frequency plot for the composite sounds between 3.5 and 6.5 s

This is the high pass filtered intensity vs frequency plot for the composite sounds between 3.5 and 6.5 s

We’ve boosted f > 1000Hz and attenuated f < 1000Hz

Fourier Transforms Fourier Transforms

deFtf ti)(

2

1)(

dtetfF ti

)(

2

1)(

dkekFxf ikx)(

2

1)(

dxexfkF ikx)(2

1)(

where

where

The functions f(x) and F(k) (similarly f(t) and F(w)) are called a pair of Fourier transforms

k is the wavenumber, (compare with ).2

kT

2

Fourier Transforms Fourier Transforms

pxandxp

pxpxf

0

1)(

Example 1: A rectangular (‘top hat’) function

Find the Fourier transform of the function

ikpikpp

p

ikxikx eeik

dxedxexfkF

1

2

1

2

1)(

2

1)(

given that iAiA eei

A 2

1sin

kp

kpp

k

kpee

ikkF ikpikp sin

2

2sin

2

21

2

1)(

This function occurs so often it has a name: it is called a sinc function.

A

AA

sinsinc

Fourier Transforms Fourier Transforms Example 2: The Gaussian

Find the Fourier transform of the Gaussian function

2 / 2( )2

axaf x e

dxea

dxeea

kFikx

axikx

ax

22

22

222

1)(

Using the formula above,

n

jjxnx e

ndxe 4

2

2

2

an ikj

This integral is pretty tricky. It can be shown that

Here and

a

k

a

kikx

ax

eea

adxe

akF 222

222

2

12

22)(

So

Hence we have found that the Fourier transform of a Gaussian is a Gaussian!

a

k

ekF 2

2

2

1)(

2 / 2( )2

axaf x e

Gaussian distributions Gaussian distributions

We define 1 (sigma) as the error in the mean when 68% of the data set is within ±1

Let the half-width when drops

to of its max value, be defined

as and

)(xf

1e

ak 2

21

ax

2

2a

x k

The value a is chosen such that

12

)( 2

2

dxea

xfax

So error in position of particle is given as xx

So error in wave number of the particle is given as kk

We find the following important result: 222

aa

kx

The product of the widths of any Gaussian and its Fourier transform is a constant, independent of a, its exact value determined by how the width is defined.

Heisenberg’s Uncertainty Principle Heisenberg’s Uncertainty Principle

The narrower the function, the wider the transform, and vice versa. The broader the function in real space (x space), the narrower the transform in k space. Or similarly, working with time and frequency, .

constant t

2

px

In quantum physics, the Heisenberg uncertainty principle states that the position and momentum of a particle cannot both be known simultaneously. The more precisely known the value of one, the less precise is the other.

Remember that momentum is related to wave number by

Thus and so

p k

kp 2 pxkx

One can understand this by thinking about ‘wavepackets’. A pure sine wave

has uniform intensity throughout all space and comprises a

single frequency, i.e. .

Heisenberg’s Uncertainty Principle Heisenberg’s Uncertainty Principle

kxxf sin)(

x 0k

If we add together two sine waves of similar k, ,

the sines add together constructively at the origin but begin to cancel each other out (interfere destructively) further away. As one adds together more functions with a wider range of k’s (Δk increases), the waves add constructively over an increasingly narrow region (Δx decreases), and interfere destructively everywhere else. Eventually

xdkkkxxg )sin(sin)(

0x k