ABSTRACT
This chapter describes the discrete version of Fourier analysis. It shows how Fourier series can be discretized, resulting in the Discrete Fourier Transform (DFT). The chapter describes discretization of the Fourier sine and Fourier cosine series. It discusses some basic properties of the DFT. These properties are linearity, periodicity, and inversion. This last property will allow people to define the inverse DFT and remove the asymmetry between the original sequence versus the transformed sequence. An important application of the DFT is to calculations with sampled Fourier series. A Fourier sine series and a Fourier cosine series can be approximated using a Discrete Sine Transform and a DCT, respectively. The chapter explains further the approximation involved in replacing Fourier coefficients by the DFT. The approximation described in must be interpreted carefully.
