Decomposition of a time-dependent free-surface profile into a double Fourier series using a two-dimensional Discrete Fourier Transform algorithm (e.g., the fft2 function in MATLAB)

This technical note presents the derivation of the formulae in order to compute the power spectral density of a real-valued signal using a Discrete Fourier Transform (DFT) algorithm such as the fft function in MATLAB. In particular, we will show how to decompose a discrete signal into a Fourier series. This note is intended to help non-specialists in signal processing understand how to use DFT algorithms for spectral analysis. For this purpose, the mathematical derivations are detailed step-by-step without consideration for conciseness.