The document defines stochastic processes and their basic properties such as stationarity and ergodicity. It discusses analyzing systems using stochastic processes, including how the power spectrum represents the frequency content of a wide-sense stationary process. The power spectrum is the Fourier transform of the autocorrelation function, and the power spectrum of the output of a linear, time-invariant system is equal to the multiplication of the input power spectrum and the transfer function of the system.

1. Stochastic Processes Elements of Stochastic Processes By Mahdi Malaki

2. Outline Basic Definitions Stationary/Ergodic Processes Stochastic Analysis of Systems Power Spectrum

3. Basic Definitions Suppose a set of random variables indexed by a parameter Tracking these variables with respect to the parameter constructs a process that is called Stochastic Process . i.e. The mapping of outcomes to the real (complex) numbers changes with respect to index.

4. Basic Definitions (cont’d) In a random process, we will have a family of functions called an ensemble of functions

5. Basic Definitions (cont’d) With fixed “beta”, we will have a “time” function called sample path . Sometimes stochastic properties of a random process can be extracted just from a single sample path. (When?)

6. Basic Definitions (cont’d) With fixed “t”, we will have a random variable. With fixed “t” and “beta”, we will have a real (complex) number.

7. Basic Definitions (cont’d) Example I Voltage of a generator with fixed frequency Amplitude is a random variables

8. Basic Definitions (cont’d) Equality Ensembles should be equal for each “beta” and “t” Equality (Mean Square Sense) If the following equality holds Sufficient in many applications

9. Basic Definitions (cont’d) First-Order CDF of a random process First-Order PDF of a random process

10. Basic Definitions (cont’d) Second-Order CDF of a random process Second-Order PDF of a random process

11. Basic Definitions (cont’d) n th order can be defined. (How?) Relation between first-order and second-order can be presented as Relation between different orders can be obtained easily. (How?)

12. Basic Definitions (cont’d) Mean of a random process Autocorrelation of a random process Fact: (Why?)

13. Basic Definitions (cont’d) Autocovariance of a random process Correlation Coefficient Example

14. Basic Definitions (cont’d) Example Poisson Process Mean Autocorrelation Autocovariance

15. Basic Definitions (cont’d) Complex process Definition Specified in terms of the joint statistics of two real processes and Vector Process A family of some stochastic processes

16. Basic Definitions (cont’d) Cross-Correlation Orthogonal Processes Cross-Covariance Uncorrelated Processes

17. Basic Definitions (cont’d) a -dependent processes White Noise Variance of Stochastic Process

18. Basic Definitions (cont’d) Existence Theorem For an arbitrary mean function For an arbitrary covariance function There exist a normal random process that its mean is and its covariance is

19. Outline Basic Definitions Stationary/Ergodic Processes Stochastic Analysis of Systems Power Spectrum

20. Stationary/Ergodic Processes Strict Sense Stationary (SSS) Statistical properties are invariant to shift of time origin First order properties should be independent of “t” or Second order properties should depends only on difference of times or …

21. Stationary/Ergodic Processes (cont’d) Wide Sense Stationary (WSS) Mean is constant Autocorrelation depends on the difference of times First and Second order statistics are usually enough in applications.

22. Stationary/Ergodic Processes (cont’d) Autocovariance of a WSS process Correlation Coefficient

23. Stationary/Ergodic Processes (cont’d) White Noise If white noise is an stationary process, why do we call it “noise”? (maybe it is not stationary !?) a -dependent Process a is called “Correlation Time”

24. Stationary/Ergodic Processes (cont’d) Example SSS Suppose a and b are normal random variables with zero mean. WSS Suppose “ ” has a uniform distribution in the interval

25. Stationary/Ergodic Processes (cont’d) Example Suppose for a WSS process X(8) and X(5) are random variables

26. Stationary/Ergodic Processes (cont’d) Ergodic Process Equality of time properties and statistic properties. First-Order Time average Defined as Mean Ergodic Process Mean Ergodic Process in Mean Square Sense

27. Stationary/Ergodic Processes (cont’d) Slutsky’s Theorem A process X(t) is mean-ergodic iff Sufficient Conditions a) b)

28. Outline Basic Definitions Stationary/Ergodic Processes Stochastic Analysis of Systems Power Spectrum

29. Stochastic Analysis of Systems Linear Systems Time-Invariant Systems Linear Time-Invariant Systems Where h(t) is called impulse response of the system

30. Stochastic Analysis of Systems (cont’d) Memoryless Systems Causal Systems Only causal systems can be realized. (Why?)

31. Stochastic Analysis of Systems (cont’d) Linear time-invariant systems Mean Autocorrelation

32. Stochastic Analysis of Systems (cont’d) Example I System: Impulse response: Output Mean: Output Autocovariance:

33. Stochastic Analysis of Systems (cont’d) Example II System: Impulse response: Output Mean: Output Autocovariance:

34. Outline Basic Definitions Stationary/Ergodic Processes Stochastic Analysis of Systems Power Spectrum

35. Power Spectrum Definition WSS process Autocorrelation Fourier Transform of autocorrelation

36. Power Spectrum (cont’d) Inverse trnasform For real processes

37. Power Spectrum (cont’d) For a linear time invariant system Fact (Why?)

38. Power Spectrum (cont’d) Example I (Moving Average) System Impulse Response Power Spectrum Autocorrelation

39. Power Spectrum (cont’d) Example II System Impulse Response Power Spectrum

40. Question?