Lecture 9 Fourier Transforms Remember homework 1 for submission 31/10/08 Remember Phils Problems and your notes = everything Today Introduction to Fourier Transforms How to work out Fourier Transforms Examples

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

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 ??? Compare with Half range sine series 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

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 ψ = 0 when x = 0 and x = L But if we are only interested in the frequency distribution we can ask….

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 ??? 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 on TV We have the tiger Play the message !!! Meow!!!!

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 ?? 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 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 ).

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.

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!

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 The value a is chosen such that So error in position of particle is given as So error in wave number of the particle is given as

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. 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,. 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

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 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