Discrete Fourier transform (DFT) is a stepping stone to Fourier series, discussed in the next chapter. It is also immensely important in data analysis, where it is practically synonymous with digital signal processing (DSP). After reviewing orthogonal expansions, we show that DFT is one, and that it is also a way of approximating functions with sums of real harmonics or, equivalently, complex exponentials. The role of DFT in data analysis is illustrated by its use for filtering out noise, comparing spectra of different data sets, and predicting periodic events, such as solar activity. The most important use of DFT, as far as the subject-matter of this book is concerned, is approximation of solutions of linear nonhomogeneous ODE with constant coefficients.