Notes on Digital Signal Processing: Practical Recipes for Design, Analysis and Implementation

Note 10. Fourier Series

The Fourier series forms a mathematical link between periodic continuous-time signals and their discrete-frequency spectra. In many books and engineering courses, the Fourier series is often introduced merely as a stepping stone on the way to the Fourier transform. However, within DSP, the Fourier series plays an important role in the design of finite-impulse-response (FIR) digital filters, which are discussed in Note 32.

10.1. Classical Form

The “classical” presentation of the Fourier series (FS) is given in a form that emphasizes that the time signal is being represented by a weighted sum of discrete sinusoids:

10.1

where

Using a few trigonometric identities, the FS can be put in the magnitude-and-phase form given ...