Lecture 9 Fourier Transforms

1
Lecture 9Fourier Transforms
Remember homework 1 for submission 31/10/08

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

http//uk.youtube.com/watch?vtUcOaGawIW0
Remember Phils Problems and your notes
everything
http//www.hep.shef.ac.uk/Phil/PHY226.htm
2
Fourier series
We have seen in the last couple of lectures how a
periodically repeating function can be
represented by a Fourier series
3
What is the Fourier Series great at ???

• 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

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

Compare with Half range sine series
4
What is the Fourier Series rubbish at ???

• Providing frequency information

Fourier series are designed to express AMPLITUDE
in terms of sine and cosine harmonics
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
? 0 when x 0 and x L
But if we are only interested in the frequency
distribution we can ask.
5
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.


6
What is the best device to perform FTs ???
The human ear can instantly deconvolve multiple
summed pressure waves from-
amplitude
time
into
intensity

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

Can you resolve the following 6 songs?
7
Fourier Transforms
http//uk.youtube.com/watch?vfsKvtjjY3A0 http//u
k.youtube.com/watch?v4iruQlZicuU http//uk.youtub
e.com/watch?vIPjMl9u3qec http//uk.youtube.com/wa
tch?vsXSMcmnDlwYfeaturerelated


8
Fourier Transforms on TV
We have the tiger Play the message !!!
Meow!!!!


9
How do we find out if tiger is still alive ??
amplitude
time
This is the amplitude vs time plot for the
composite sounds
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

intensity
frequency

Note big peak at 100Hz, background noise, and
spikes around 2000Hz
10
How do we find out if tiger is still alive ??
This is the original intensity vs frequency plot
for the composite sounds between 3.5 and 6.5 s
intensity
frequency

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

Weve boosted f gt 1000Hz and attenuated f lt 1000Hz
11
Fourier Transforms
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 ).


12
Fourier Transforms
Example 1 A rectangular (top hat) function
Find the Fourier transform of the function
given that
This function occurs so often it has a name it
is called a sinc function.


13
Fourier Transforms
Example 2 The Gaussian Find the Fourier
transform of the Gaussian function
Using the formula above,
This integral is pretty tricky. It can be shown
that
Here
and
So

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

14
Gaussian distributions
The value a is chosen such that
We define 1s (sigma) as the error in the mean
when 68 of the data set is within 1s.
Let the half-width when drops to
of its max value, be defined as and
So error in position of particle is given as


So error in wave number of the particle is given
as
15
Heisenbergs Uncertainty Principle
We find the following important result
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.
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,
.
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

16
Heisenbergs Uncertainty Principle
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.
.
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 ks (?k increases), the waves add
constructively over an increasingly narrow region
(?x decreases), and interfere destructively
everywhere else. Eventually