Transcript using the FFT

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
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-UlamTsingou 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 y  fpu1( y ) , where y is a (2N )-vector.
y (1)  y (1  N )
y ( 2)  y ( 2  N )
...
y ( N )  y ( 2 N )
y ( N  1)  y ( 2)  2 y (1)  NL
y ( N  2)  y (3)  y (1)  2 y ( 2)  NL
...
y ( 2 N )  y ( N  1)  2 y ( N )  NL
So
y (1)  y (1  N )  y ( 2)  2 y (1)  NL
…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
Heisenberg uncertainty principle
Time-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
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