This course aims to introduce the basics of Digital Signal Processing using the mathematical tools presented in previous courses. We review some of these tools with additional new concepts to go more deeply in analyzing digital systems. The concepts and characteristics of Fourier series and their applications in discrete signals. We present the basic digital networks used to implement discrete systems including the various structures of FIR and IIR digital filters. The effect of coefficients quantization and finite precision arithmetic on the performance of LTI systems is also presented. We also present the main methods used in digital filters design. A special interest is given to the interpretation of the DFT of sinusoidal signals to understand the limits of this tool in spectral analysis. We conclude by introducing the discrete cosine transform which is used in many applications as in JPEG image compression.