Lecture 9 Fourier Transforms

1. Lecture 9Fourier Transforms • Today • Introduction to Fourier Transforms • How to work out Fourier Transforms • Examples Professor David Mowbray

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. Fourier Transforms What happens when a signal is no longer periodic? As T the fundamental frequency 1/T0 and the harmonics get closer together. The Fourier series is replaced with a continuous distribution of frequencies with the discrete summations replaced with an integral. Fourier transforms are used extensively in all areas of physics and astronomy.

4. 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 ). You don’t need to remember the above formulae – they will be given in any exam question!

5. f(x) 1 -p +p x 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.

6. f(x) 1 -p +p x Width x=2p Widths of f(x) and F(k) Widths of f(x) and F(k) are inversely related xk=4 This inverse relationship is a general result

7. V(t) V0 - + t What does a Fourier transform tell us? Fourier Transform Voltage pulse width 2 • The Fourier transform tells us how much of each frequency component we would need to use to construct the original voltage pulse • The Fourier transform of any time varying signal tells us the frequency components present in that signal

8. Conclusions • Definition of Fourier transforms • Calculation for the Fourier transform of a top-hat function