Fourier Transform

The Fourier transform of a real or complex function is a parallel description of the data in a separate “domain”. When a time series is Fourier transformed it moves to the frequency domain and vice versa.

Fourier transforms are performed to learn about the spectral characteristics of a data set. Thus in astronomy, when looking for periodicities in a time series (for example in pulsar data), we Fourier transform the data and look for peaks in the spectrum.

If the data are regularly sampled we can make use of the fast Fourier transform to decrease the computation time.

The Fourier transform of a function f (t) is F (u):

$  F(u) = \int_{-\infty}^{\infty} f(t) e^{-2\pi i u t} dt  $

You can think of the Fourier transform of a function as the amount of power a particular time series has in it at various frequencies. The most simple functions like sine and cosine have Fourier transforms that are just delta functions at the frequency of the sine/cosine waves.

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