This chapter presents the discrete Fourier transform DFT with applications and examples. The alias effect is studied in detail with its disadvantages, but also with its great advantages for low-cost signal processing. The connection of the DFT with interpolation by Chebyshev polynomials is deduced. Further applications worked out are: trigonometric interpolation and interpolation with Chebyshev polynomials. The use of the discrete cosine transform DCT in numerical Clenshaw-Curtis integration is shown as well as the 2D-Cosine transform in image processing like JPEG. The principle of the Fast Fourier Transform FFT is demonstrated with a programmable algorithm. The exercises treat approximation error estimates of trigonometric interpolations, dependent on the number of nodes, DFT frequency assignments, low-cost subsampling, comparison of interpolations on an interval with equidistant nodes versus Chebyshev abscissae. As practice tasks, a Chebyshev lowpass filter can be designed with the help of the Joukowsky transformation, and characteristic values like DC gain, distortion, or RMS value for a transmitter in emitter circuit can be computed with a DFT.