The good life in the frequency domain

1. The good life in the frequency domain • Example: the Fermi-Pasta-Ulam-Tsingou experiment (the FFT is everywhere) • Tradeoff between time and frequency resolution – Heisenberg uncertainty principle • Windows, apodizing

2. Three birds with one stone • Here’s an example of how to set up an ode in Matlab, a peek at how the Fermi-Pasta-Ulam-Tsingou experiment works, and how handy the FFT is ( Dauxois & Ruffo 2008) • Their code works like this: the displacment at point j of the lattice is yj. The Matlab procedure ode45 is called for the ode , where y is a (2N )-vector.

4. So …etc. This is how the 2nd order ode is mapped to 1st order. See Dauxois & Ruffo 2008 for the FFT call in Matlab, to track the harmonics

5. Heisenberg uncertainty principleTime-Frequency duality • Nice animation: ipod.org • Another way to look at it: time and frequency are “dual”, so, intuitively brief duration  wide spectrum long duration  narrow spectrum • Or measuring: precise time  poor frequency precise frequency  poor time • Or: sharp change in time  wide spectrum slow change in time  narrow spectrum

6. Star image in a telescope = Fourier transform of the aperture The wider the aperture, the narrower the diffraction pattern  you need a big telescope to resolve close stars