This chapter deals with the discrete Fourier transformation. Here, a periodic series in the time domain is mapped onto a periodic series in the frequency domain. Definitions of the discrete Fourier transformation and its inverse are given. Linearity, convolution, cross-correlation, and autocorrelation are treated as well as Parseval’s theorem. The sampling theorem is illustrated with a simple example. Data mirroring, cosine- and sine-transformations, as well as zero-padding are discussed. It concludes with the Fast Fourier Transformation algorithm by Cooley and Tukey.