1. Fast Fourier Transform (FFT)

1.1. Purpose

Perform Fourier transform of a signal.

1.2. Description

Given a time series \(x(t)\), the (continuous) Fourier transform of \(x\) is then defined as

\[ \hat{x}(f) = \int x(t)\exp(–2\pi i ft) \, dt,\]

where \(f\) is frequency and \(i=\sqrt{-1}\) is the imaginary unit. The Fast Fourier Transform (FFT) is an algorithm to perform the discrete Fourier transform. The FFT implementation is based on:
Cooley, James W. and Tukey, John W. (1965). An algorithm for the machine calculation of complex Fourier series. Math. Comput. 19: 297-301.

1.3. Input

Any equidistant signal (see Signal Types ) with time on the \(x\)-axis.

1.4. Output

The output is the modulus of the Fourier transform, \(|\hat{x}|\), as a function of period of frequency, depending on the user’s unit preferences.