Notes on discrete-time Fourier series and transform

A detailed mathematical walkthrough of discrete-time Fourier series (DTFS) and the discrete-time Fourier transform (DTFT), building from discrete complex exponentials to derive the DTFS coefficient formula, illustrated with a worked example using a sampled triangular function. The piece then derives the DTFT as a limiting case of DTFS for non-periodic finite-duration signals, establishes the DTFT/inverse DTFT integral pair, and closes with three appendices proving supporting lemmas (periodicity of discrete exponentials, sums of consecutive complex exponentials, and a sine-series decomposition). Framed as theoretical groundwork for a future post on the DFT used in digital signal processing.

Source: https://eli.thegreenplace.net/2026/notes-on-discrete-time-fourier-series-and-transform. 8 Sync News only summarizes and links out; content copyright belongs to the authors and original sources.