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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...
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 Fall '08
 SinHorngChen

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