Stochastic Processes

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1 Elements of Lecture II Hamid R. Rabiee with thanks to Ali Jalali

2 Overview Reading Assignment Chapter 9 of textbook Further Resources MIT Open Course Ware S. Karlin and H. M. Taylor, A First Course in Stochastic Processes, nd ed., Academic Press, New York, 975.

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

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. X (t, β)= (X (β), X (β),..., X n (β),...) i.e. The mapping of outcomes to the real (complex) numbers changes with respect to index. 4

5 Basic Definitions (cont d) A stochastic process x(t) is a rule for assigning to every a function x(t, ). ensemble of functions: Family of all functions a random process generates 5

6 Basic Definitions (cont d) With fixed (zeta), we will have a time function called sample path. Are sample paths sufficient for estimating stochastic properties of a random process? Sometimes stochastic properties of a random process can be extracted just from a single sample path. (When?) 6

7 Basic Definitions (cont d) With fixed t, we will have a random variable. With fixed t and zeta, we will have a real (complex) number 7

8 Basic Definitions (cont d) Example I Brownian Motion Motion of all particles (ensemble) Motion of a specific particle (sample path) 8

9 Basic Definitions (cont d) Example II Voltage of a generator with fixed frequency Amplitude is a random variable V t, A. cos t 9

10 Basic Definitions (cont d) Equality Ensembles should be equal for each beta and t t, Y t, Equality (Mean Square Sense) If the following equality holds E Sufficient in many applications X X t, Y t, 0 0

11 Statistics of First-Order CDF of a random process F X x, t PrX t x First-Order PDF of a random process f x x, t F x t X X,

12 Statistics of (cont d) Second-Order CDF of a random process Second-Order PDF of a random process, ;,., ;, t t x x F x x t t x x f X X Pr, ;, x t X and x t X t t x x F X

13 Statistics of (cont d) n th order can be defined. (How?) Relation between first-order and secondorder can be presented as f X x, t f x, ; t, t Relation between different orders can be obtained easily. (How?) X 3

14 Statistics of (cont d) Mean of a random process X t x. f x t Autocorrelation of a random process R t E X, dx t t EX t. X t x. x. f X x, x; t, t, dx dx Fact: R t t, X ( t) X ( t) (Why?) 4

15 Statistics of (cont d) Autocovariance of a random process, t Rt, t t. t EX t t. X t t C t Correlation Coefficient C t, t r t, t Ct, t. Ct, t Example E X t X t Rt, t Rt, t Rt t, 5

16 End of Section Any Question? 6

17 Outline Basic Definitions (cont d) Stationary Processes Stochastic Analysis of Systems Power Spectrum Ergodic Processes 7

18 Basic Definitions (cont d) Example Poisson Process. t e K. k! t t k Mean t. t Autocorrelation R t, t. t. t. t. t. t. t t t t t Autocovariance C t, t.mint t, 8

19 Basic Definitions (cont d) Complex process Definition t X t i Y t Z. Specified in terms of the joint statistics of two real processes X t and Y t Vector Process A family of some stochastic processes 9

20 Basic Definitions (cont d) Cross-Correlation R XY Orthogonal Processes t t R XY t, t EX t Y t. 0, Cross-Covariance C XY Uncorrelated Processes t t C XY t, t R t, t t t XY X. 0, Y 0

21 Basic Definitions (cont d) a-dependent processes White Noise t t 0 t a C t, t t 0 t C t, Variance of Stochastic Process t, t t C X

for any For an arbitrary mean function t For an arbitrary covariance function C t,t

22 Basic Definitions (cont d) Normal Process t variables x t,..., x ( t and t,...,t. Existence Theorem x ) A Process is called Normal, if the random are jointly normal for any For an arbitrary mean function t For an arbitrary covariance function C t,t There exist a normal random process that its mean is t and its covariance is C t,t n

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

24 Stationary Processes Strict Sense Stationary (SSS) Statistical properties are invariant to shift of time origin First order properties should be independent of t or f X x, t f X x Second order properties should depends only on difference of times or t t f X x, x; t, t f X x, x; 4

25 Stationary Processes (cont d) Wide Sense Stationary (WSS) Mean is constant E X t Autocorrelation depends on the difference of times E X t. X t R First and Second order statistics are usually enough in applications. 5

26 Stationary Processes (cont d) Autocovariance of a WSS process C R Correlation Coefficient r C C 0 6

27 Stationary Processes (cont d) White Noise C q. If white noise is an stationary process, why do we call it noise? (maybe it is not stationary!?) a-dependent Process C 0 a a is called Correlation Time 7

28 Stationary Processes (cont d) Example SSS Suppose a and b are normal random variables with zero mean. X Why is it SSS? WSS Suppose has a uniform distribution in the interval, X Why is it WSS? t a. cos t b. sin t t a. cos t 8

29 Stationary Processes (cont d) Example Suppose for a WSS process X(8) and X(5) are random variables E R A. e X 8 X 5 E X 8 E X 5 R0 R0 R3 EX 5. X 8 9

30 End of Section Any Question? 30

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

32 Stochastic Analysis of Systems Linear Systems y t g xt a. yt b ga. xt b a, b Time-Invariant Systems y t g xt yt gxt Linear Time-Invariant Systems y t g x t y t h t * x t Where h(t) is called impulse response of the system 3

33 Stochastic Analysis of Systems (cont d) Memoryless Systems y t g x 0 t 0 Causal Systems y t g xt t 0 0 t Only causal systems can be realized. (Why?) 33

34 Stochastic Analysis of Systems (cont d) Linear time-invariant systems Mean yt Eht * xt ht Ext E * Autocorrelation R yy * t, t h t * R t, t h t xx * 34

35 Stochastic Analysis of Systems (cont d) Example I System: y t t 0 x( ). d Impulse response: Output Mean: E h t t t 0 ( t). dt yt Ext. 0 dt U t Output Autocovariance: R yy t t Rxx t, t. d. 0 0, d 35

36 Stochastic Analysis of Systems (cont d) Example II System: t d dt Impulse response: y xt h t d dt t Output Mean: Output Autocovariance: E d dt yt Ext R yy xx, t. t t, t R t t 36

37 Outline Basic Definitions Stationary Processes Stochastic Analysis of Systems Power Spectrum Ergodic Processes 37

38 Power Spectrum Definition WSS process X Autocorrelation t R Fourier Transform of autocorrelation S R. e j d 38

39 Power Spectrum (cont d) Inverse trnasform R S. e j d For real processes S R S.cos. 0 S.cos. 0 d d 39

40 Power Spectrum (cont d) For a linear time invariant system S yy * H. S. H H xx. S xx Fact (Why?) Var yt S. H xx d 40

41 Power Spectrum (cont d) Example I (Moving Average) System t tt Impulse Response Power Spectrum y T tt x. S d yy H sin T. T. S xx sin T T.. Autocorrelation R yy T T. Rxx T T. d 4

42 Power Spectrum (cont d) Example II System y t d dt xt Impulse Response Power Spectrum H i. S. S yy xx 4

43 Outline Basic Definitions Stationary Processes Stochastic Analysis of Systems Power Spectrum Ergodic Processes 43

44 Ergodic Processes Ergodic Process Equality of time properties and statistic properties. First-Order Time average T Defined as EX t X t t lim T X t Mean Ergodic Process E X t EX t Mean Ergodic Process in Mean Square Sense T E X t 0 lim C. d 0 t T T 0 T X 44 T dt T

45 Ergodic Processes (cont d) Slutsky s Theorem A process X(t) is mean-ergodic iff lim T T T 0 C d 0 Sufficient Conditions a) 0 C d b) lim T C 0 45

Elements of Stochastic Processes Lecture II Hamid R. Rabiee

Elements of Stochastic Processes Lecture II Hamid R. Rabiee Sochasic Processes Elemens of Sochasic Processes Lecure II Hamid R. Rabiee Overview Reading Assignmen Chaper 9 of exbook Furher Resources MIT Open Course Ware S. Karlin and H. M. Taylor, A Firs Course