Topic 2: The pendulum

PHY321F — cp 2005 37
spread into a peak of finite width. The spread can be shaped by ‘windowing’
the data. See, e.g. Numerical Recipes for further information.
E.6.2
Convolution
The convolution f ∗ g of f(t) and g(t) is defined by
f ∗ g =
∫ ∞
−∞
f(τ) g(t − τ)dτ
By the convolution theorem,
F[f ∗ g] = F[f]F[g]
This provides a fast convolution algorithm via the FFT.
E.6.3
Correlation
The correlation corr(f, g) of f(t) and g(t) is defined by
corr(f, g) = 1 √
2π
∫ ∞
−∞
f ∗ (τ) g(t + τ)dτ
From the Fourier transforms F (ω) = F[f(t)] and G(ω) = F[g(t)],
F[corr(f, g)] = F ∗ (ω)G(ω)
The Wiener-Khinchine theorem relates the autocorrelation of a function to
the power spectrum:
F[corr(f, f)] = |F (ω)| 2