This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Chapter 9 General Concept of Random Processes 91 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 boldfaced letters, like X . If t R R , it is a continuoustime RP. If t I R , then it is a discretetime RP. If it has only countable values, it is a discrete state RP; while if it is realvalued, then it is a continuousstate 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. 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 R X } can be completely predicted as we know { ( ), } t t t < X . Regular and predictable processes are two extreme types of RPs. They have completely different properties. We will discuss the issue in latter chapters. Equalitye 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 . Statistics of Stochastic Process The firstorder distribution of a RP ( ) t X is defined by: { } ( , ) ( ) F x t P t x = X for a specific t Firstorder density ( , ) ( , ) F x t f x t x R = 2ndorder distribution { } 1 2 1 2 1 1 2 2 ( , , , ) ( ) , ( ) F x x t t P t x t x = X X The n thorder distribution is the joint distribution of 1 ( ), , ( ) n t t X X L for n points of time 1 , , n t t L . In many applications, only the 1st and 2ndorder 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. Autocorrelatione { } 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 R = = X X Average powere { } 2 ( ) ( , ) E t R t t = X Autocovariancee 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 . Ex.93 Let S be a RV defined by ( ) b a t dt = S X . Then, { } { } ( ) b s a E E t dt η = = S X ( ) b a t dt η = ; and 2 1 2 1 2 ( ) ( ) b b a a t t dt dt = S X X { } { } 2 1 2 1 2 ( ) ( ) b b a a E E t t dt dt = S X X { } 1 2 1 2 , b b a a R t t dt dt = Ex.94 Let ( ) cos( ) t t ϖ = + Xγ φ be a RP, where γ is the random amplitudee φ is the random phase and is uniformly distributed in ( , ) π π , and...
View
Full Document
 Fall '08
 SinHorngChen
 Autocorrelation, Stationary process, Rxx, Rxy

Click to edit the document details