Topic 2: The pendulum

PHY321F — cp 2005 35
The Fourier transform can be approximated by a discrete Fourier transform
(DFT). The time or frequency axes are replaced by a finite set of N points
on a grid and the integral performed using the trapezoidal rule:
F (n∆ω) ≈
=
N−1
∑
m=0
N−1
∑
m=0
f(m∆t) e −in∆ωm∆t
f(m∆t) e −i2πnm/N
The inverse transform is then
f(m∆t) = 1 N
N−1
∑
n=0
F (n∆ω) e i2πmn/N
Aliasing occurs for frequencies higher than the Nyquist frequency ω ny =
N∆ω/2 = (2π/∆t)/2
E.4 The fast Fourier transform
The importance of Fourier methods in computation derives from the existence
of the fast Fourier transform (FFT) algorithm, which is, well, a fast
way of computing the DFT. (What does fast mean? The DFT requires
O(N 2 ) calculations to compute the transform of N values; the FFT requires
O(N log N) — when N is of the order of 10 3 , a typical value, the FFT is
about 1000 times faster than the DFT. . . )
The FFT is a non-trivial algorithm and we won’t discuss it here — we’ll just
make use of a packaged version.
E.5 The fast Fourier transform in Python
In Python we can use:
from Numeric import *
from FFT import * # In windows : from Fft