1. Problem sets will be posted once a week. The solution will be posted a week after.
2. Two midterms will be given during the semester. The problems will be based on ones that appeared in the lectures, recitations and HW assignments.
3. During the course we will publish up to four Matlab assignments which are related to the topics covered in class. The submission of these exercises will be done in pairs, and is accompanied by an oral exam on the exercises.
4. A Final exam which covers all the course's material.
Part A: Random Variables and Operations
1. Probability and Random Variables (about two weeks)
• Engineering motivation, probability space, axioms.
• Revisiting basic concepts for a single random variable.
• The characteristic function, moments.
• Functions of random variables.
2. Random vectors (about two weeks)
• Two random variables: joint, conditional and marginal distributions.
• Vectors of random variables.
• Gaussian vectors.
3. Estimation (about two weeks)
• Optimal estimation, error criteria.
• Minimum mean squared-error (MMSE) estimation.
• Linear minimum mean squared-error (LMMSE) estimation.
Part B: Random Processes
4. Introduction, definitions and properties. (about one week)
• The formation of random processes.
• Joint distribution, autocorrelation function.
• Strict-sense and wide-sense stationarity (SSS and WSS).
5. Basic discrete –time and continuous-time processes (about one week)
• Autoregressive process (stationarity conditions, Markovity).
• Random walk, discrete-time white-noise.
• Gaussian random processes.
• Wiener process \ Brownian motion.
6. Markov chains (about one week)
• Transition matrix, stationary distribution.
• Characterizing a Markov chains its state diagram.
7. Ergodicity (about one week)
• The law of large numbers.
• Mean-ergodicity, correlation-ergodicity.
8. Power spectrum density (about one week)
• Definitions, periodogram, continuous-time white-noise.
9. Linear time-invariant (LTI) processing of WSS processes (about two weeks)
• Joint stationarity of random processes.
• Random processes passing LTI systems.
• Optimal linear MMSE estimation (Wiener filtering).
• Multiple-input multiple-output (MIMO) systems.
• Parallel processing of frequency-bands.
10. Advanced random processes (about one week)
• Poisson process.