1. Elements of Stochastic Processes Lecture II

2. Overview Reading Assignment Further Resources Chapter 9 of textbook
MIT Open Course Ware
S. Karlin and H. M. Taylor, A First Course in Stochastic Processes, 2nd ed., Academic Press, New York, 1975.

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

4. 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.

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

6. 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?)

7. 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.

8. Basic Definitions (cont’d)
Example I
Brownian Motion
Motion of all particles (ensemble)
Motion of a specific particle (sample path)
Example II
Voltage of a generator with fixed frequency
Amplitude is a random variables

9. 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

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

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

12. Basic Definitions (cont’d)
nth 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?)

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

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

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

16. 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

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

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

19. 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

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

21. 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 …

22. 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.

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

24. 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”

25. 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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

41. Markov Processes & Markov Chains
Next Lecture
Markov Processes
&
Markov Chains