A signal f(x) can be described as a sum of complex sinusoids with varying frequencies ω.

Fourier transform:

The Fourier transform transforms f(x) into a complex function g(ω).
$$ g(\omega)=\frac{1}{T}\int_{-T/2}^{T/2}f(t) e^{-2i\pi\omega t}dt $$

Fourier frequencies ω correspond to \(1/n,2/n,3/n,... (n-1)/n\) where n is the length of the signal.

In this application the Fourier transform is computed through a Fast Discrete Fourier Transform (FFT). The function used internally is the 'fft' function of the R package 'stats'

Fourier spectrum:

The Fourier spectrum describes the amplitude of all the sinusoids with Fourier frequencies. The amplitude can be represented either according to Fourier frequencies ω or (as is the case here), according to the corresponding periods \(T=\frac{\Delta x}{\omega}\).

The Fourier spectrum h(T), hence corresponds to: $$h(T)=h\left(\frac{\Delta x}{\omega}\right)=\mid g(\omega)\mid$$

Limitation:

Contrary to wavelet spectra, Fourier spectra cannot adequately describe non-stationary signals.

Reference

For more information about the Fourier transform, please refer to the first part of this very helpful tutorial:

Polikar, Robi. 1995. "The Wavelet Tutorial." The Engineer's Ultimate Guide to Wavelet Analysis: The Wavelet Tutorial