Chapter 9_English - 1 Chapter 9 General Concept of Random...

Info iconThis preview shows pages 1–9. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 1 Chapter 9 General Concept of Random Processes Teacher: h ‹ ª Office: 805 Tel: ext. 31822 Email: [email protected] 2 9-1 Definition Let ( ) t X denote a random process (RP). It is a function of two variables and is supposed to be denoted as ( , ) t ξ X . But we usually neglect the variable ξ and denote it by bold-faced letters, like X . If t R ∈ , it is a continuous-time RP. If t I ∈ , then it is a discrete-time RP. If it has only countable values, it is a discrete-state RP; while if it is real-valued, then it is a continuous-state RP. ( ) t X is a function of two variables, i.e. ξ and t . (1) When ξ is fixed, it is a time function. (2) When t is fixed, it is a random variable. 3 For instances, Brownian Motion is a RP. It represents the location of a very small moving particle driving by molecule of 2 H O . Since it is unpredictable, we call it a regular RP . ( ) cos( ) t t ϖ = + Xγ φ is another example. Here, γ is the random amplitude and φ is the random phase. For a specific ξ (i.e., a pair of specific γ and φ ), ( ) cos( ) t t γ ϖ φ = + X is a time function and is a predictable process. “ Predictable ” means that { ( ), t t t ≥ X } can be completely predicted as we know { ( ), } t t t < X . 4 Regular and predictable processes are two extreme types of RPs. They have completely different properties. General RP is a mixture of them. We will discuss the issue in latter chapters. Equality 1 Two RPs ( ) t X and ( ) t Y are equal, if for ξ 2200 they are identical, i.e. ( ) ( ) for all t t t = X Y . This definition can be relaxed to define “ equal in MS sense ” by letting 2 { ( ) ( ) } 0 for E t t t- = 2200 X Y . 5 Statistics of Stochastic Process The first-order distribution of a RP ( ) t X is defined by: { } ( , ) ( ) F x t P t x = ≤ X for a specific t First-order density ( , ) ( , ) F x t f x t x ∂ = ∂ 2nd-order distribution { } 1 2 1 2 1 1 2 2 ( , , , ) ( ) , ( ) F x x t t P t x t x = ≤ ≤ X X 6 The n th-order distribution is the joint distribution of 1 ( ), , ( ) n t t X X for n points of time 1 , , n t t . In many applications, only the 1st- and 2nd-order statistics of RPs are used. We need to know them well. Mean : The mean of a RP is defined by { } ( ) ( ) ( , ) t E t xf x t dx η ∞-∞ = = ∫ X . It is a deterministic time function. 7 Autocorrelation : { } 1 2 1 2 1 2 1 2 1 2 1 2 ( , ) ( ) ( ) ( , , , ) R t t E t t x x f x x t t dx dx ∞ ∞-∞-∞ = = ∫ ∫ X X Average power : { } 2 ( ) ( , ) E t R t t = X Autocovariance : 1 2 1 2 1 2 ( , ) ( , ) ( ) ( ) C t t R t t t t η η = - { } 1 1 2 2 ( ( ) ( ))( ( ) ( )) E t t t t η η = - - X X For 1 2 t t t = = , ( , ) C t t is the variance of ( ) t X . 8 Ex.9-3 Ex.9-3 Let S be a RV defined by ( ) b a t dt = ∫ S X ....
View Full Document

This note was uploaded on 07/21/2009 for the course CM EM5102 taught by Professor Sin-horngchen during the Fall '08 term at National Chiao Tung University.

Page1 / 81

Chapter 9_English - 1 Chapter 9 General Concept of Random...

This preview shows document pages 1 - 9. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online