This book is an introduction to Fourier transformation with a focus on signal analysis, based on the first edition. It is well suited for undergraduate students in physics, mathematics, electronic engineering, as well as for scientists in research and development. It provides illustrations and recommendations when using existing Fourier programs, helping to avoid frustrations. Moreover, it is entertaining, and you will learn a lot unconsciously. Fourier series, as well as continuous and discrete Fourier transformation, are discussed with particular emphasis on window functions. Filter effects of digital data processing are illustrated. Two new chapters are devoted to modern applications. The first deals with data streams and fractional delays, and the second with the back-projection of filtered projections in tomography. There are many figures and mostly easy to solve exercises with solutions.