The textbook is intended for university students studying in the areas of training bachelors and masters "Applied Mathematics," "Computer Science and Computer Engineering," and is aimed at studying the theoretical foundations of digital signal processing (DSP). In addition to the basic sections of the DSP related to digital signal representation, analysis and synthesis of digital filters, the manual discusses the issues of multi-rate signal processing, adaptive filtering, discrete spectral analysis. Considerable attention is paid to such special sections of the DSP as efficient signal representation (data compression), theory and applications of discrete wavelet transformations. The manual can also be recommended as an additional educational material for other engineering areas of training related to DSP (radio engineering, electronics, telecommunications and communications, automatic control, etc.). At the same time, the basic part of the material of the first chapters is the basis for the introductory course (bachelor's level), and in full, the textbook is focused on a more in-depth study of the theory of DSP in the magistracy.