Stochastic Signals

The course focuses on stochastic signals / random processes covering the basic mathematical tools of their qualitative and quantitative characterization as well as their processing. In the first part of the course, an overview of basic concepts of linear algebra (norms, vector spaces, eigenvalues ​​and eigenvectors of a matrix), probabilities (definitions and properties) and random variables (statistical functions and metrics, transformation of random variables, multiple random variables, correlation, orthogonality, well known distributions, estimators). In the second part of the course, we study random processes (power spectrum, stationarity, ergodicity), their time and frequency analysis (periodogram, filtering, principal component analysis, parametric methods AR, MA, ARMA), Wiener filters, maximum likelihood estimation, Kalman filtering, and the reduced rank minimum variance filter as well as their applications (processing in noisy environments, linear prediction, system identification, multi-antenna systems) and basic adaptive filtering methods (steepest descent method as well as algorithms LMS, NLMS, and RLS). The course's theory and exercises contain example applications of the methods for analyzing and processing stochastic signals in fifth generation (5G) communications systems.