1. DEPARTMENT OF HEALTH SCIENCE AND TECHNOLOGY www.hst.aau.dk STOCHASTIC SIGNALS AND PROCESSES Lecture 1 WELCOME

2. DEPARTMENT OF HEALTH SCIENCE AND TECHNOLOGY www.hst.aau.dk Introduction to probability and random variables A deterministic signal can be derived by mathematical expressions. A deterministic model (or system) will always produce the same output from a given starting condition or initial state. Stochastic (random) signals or processes Counterpart to a deterministic process Described in a probabilistic way Given initial condition, many realizations of the process

3. DEPARTMENT OF HEALTH SCIENCE AND TECHNOLOGY www.hst.aau.dk Introduction to probability

4. DEPARTMENT OF HEALTH SCIENCE AND TECHNOLOGY www.hst.aau.dk Axiomatic Definition of Probability A probability law (measure or function) that assigns probabilities to events such that: o P(A) ≥ 0 o P(S) =1 o If A and B are disjoint events (mutually exclusive), i.e. A ∩ B = ∅, then P(A ∪ B) = P(A) + P(B) That is if A happens, B cannot occur.

5. DEPARTMENT OF HEALTH SCIENCE AND TECHNOLOGY www.hst.aau.dk Some Useful Properties

6. DEPARTMENT OF HEALTH SCIENCE AND TECHNOLOGY www.hst.aau.dk Joint and marginal probability Joint probability: is the likelihood of two events occurring together. Joint probability is the probability of event A occurring at the same time event B occurs. It is P(A ∩ B ) or P(AB). Marginal probability: is the probability of one event, ignoring any information about the other event. Thus P(A) and P(B) are marginal probabilities of events A and B

7. DEPARTMENT OF HEALTH SCIENCE AND TECHNOLOGY www.hst.aau.dk Conditional probability

8. DEPARTMENT OF HEALTH SCIENCE AND TECHNOLOGY www.hst.aau.dk Random variables A random variable is a function, which maps events or outcomes (e.g., the possible results of rolling two dice: {1, 1}, {1, 2}, etc.) to real numbers (e.g., their sum). A random variable can be thought of as a quantity whose value is not fixed, but which can take on different values. A probability distribution is used to describe the probabilities of different values occurring.

9. DEPARTMENT OF HEALTH SCIENCE AND TECHNOLOGY www.hst.aau.dk Random variables Notations: Random variables with capital letters: X, Y,..., Z Real value of the random variable by lowercase letters (x, y, …, z) Types: Continuous random variables: maps outcomes to values of an uncountable set. the probability of any specific value is zero. Discrete random variable: maps outcomes to values of a countable set. Each value has probability ≥ 0. P(x i ) = P(X = x i ) Mixed random variables

10. DEPARTMENT OF HEALTH SCIENCE AND TECHNOLOGY www.hst.aau.dk Continuous random variables

11. DEPARTMENT OF HEALTH SCIENCE AND TECHNOLOGY www.hst.aau.dk Distribution functions Cumulative distribution function (CDF) for a normal distribution

12. DEPARTMENT OF HEALTH SCIENCE AND TECHNOLOGY www.hst.aau.dk Probability density function (pdf)

13. DEPARTMENT OF HEALTH SCIENCE AND TECHNOLOGY www.hst.aau.dk Expectation operator

14. DEPARTMENT OF HEALTH SCIENCE AND TECHNOLOGY www.hst.aau.dk Definition of moments Mean of X

15. DEPARTMENT OF HEALTH SCIENCE AND TECHNOLOGY www.hst.aau.dk Central moments Variance of X

16. DEPARTMENT OF HEALTH SCIENCE AND TECHNOLOGY www.hst.aau.dk Two random variables

17. DEPARTMENT OF HEALTH SCIENCE AND TECHNOLOGY www.hst.aau.dk Useful Properties

18. DEPARTMENT OF HEALTH SCIENCE AND TECHNOLOGY www.hst.aau.dk Useful Properties

19. DEPARTMENT OF HEALTH SCIENCE AND TECHNOLOGY www.hst.aau.dk Indenpendent and uncorrelated random variables

20. DEPARTMENT OF HEALTH SCIENCE AND TECHNOLOGY www.hst.aau.dk Expected value The correlation

21. DEPARTMENT OF HEALTH SCIENCE AND TECHNOLOGY www.hst.aau.dk

22. DEPARTMENT OF HEALTH SCIENCE AND TECHNOLOGY www.hst.aau.dk Correlation coefficient Covariance

23. DEPARTMENT OF HEALTH SCIENCE AND TECHNOLOGY www.hst.aau.dk Random processes A random process may be viewed as a collection of random variables, with time t as a parameter running through all real numbers.

24. DEPARTMENT OF HEALTH SCIENCE AND TECHNOLOGY www.hst.aau.dk Example

25. DEPARTMENT OF HEALTH SCIENCE AND TECHNOLOGY www.hst.aau.dk First and second order statistics Expected value Autocorrelation function Properties: Stationarity, Ergodicity, Power spectrum

26. DEPARTMENT OF HEALTH SCIENCE AND TECHNOLOGY www.hst.aau.dk Example

27. DEPARTMENT OF HEALTH SCIENCE AND TECHNOLOGY www.hst.aau.dk Homework Workshop: Get familiar with the following terms Probability density function, independency Autocorrelation, Stationarity, Ergodicity, Power spectrum. Exercises on the web (click here)click here