Topic 2: The pendulum

PHY321F — cp 2005 36
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F=fft(f) # f is a Numeric array
g=inverse_fft(F)
Note that F and g will be complex.
Application of the FFT is complicated slightly by the conventions used to
structure the data (Note that these are not peculiar to Python).
Suppose that the initial data f(t) is in the time domain, with N points at
a spacing δt, with T = Nδt. The transformed data F (ω) is (a) complex,
and (b) runs over the range −(N/2)δω . . . (N/2)δω, with δω = 2π/T . In
addition, the values coresponding to positive frequencies are returned in the
first half of the array, while those at negative frequencies in the second half of
the array. If the initial data has negative times, the same convention should
be applied.
E.6 Uses of the FFT
There are a number of applications of the FFT, based on various theorems
on Fourier transforms.
E.6.1
Power spectrum
The power spectrum gives essentially the intensity of the time-domain data
as a function of frequency.
It is defined by
and may be computed by
P (ω) = |F (ω)| 2
transform=fft(data) # transform is complex
powerspectrum=abs(transform*conjugate(transform))/len(data)**2
A single frequency in the data will result in a delta function in the power
spectrum. However, the FFT is always applied to a finite domain, and this
causes ‘power’ to leak to adjacent frequencies. Hence the delta function is