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

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1 Chapter 9 General Concept of Random Processes Teacher: h ‹ ª Office: 805 Tel: ext. 31822 Email: [email protected]
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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.
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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 = + 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 { 0 ( ), t t t X } can be completely predicted as we know 0 { ( ), } t t t < X .
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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 .
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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
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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.
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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 .
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8 Ex.9-3 Ex.9-3 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 = ∫ ∫
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9 Ex.9-4 Let ( ) cos( ) t t ϖ = + φ be a RP, where γ is the random amplitude1 φ is the random phase and is uniformly distributed in ( , ) π π - , and γ and φ are independent. Then { } { } { } { } 2 1 2 1 2 1 2 2 1 2 1 ( ) ( ) cos( ( )) cos( 2 ) 2 1 cos( ( )) 2 E t t E E t t t t E t t ϖ ϖ ϖ ϖ = - + + + = - X φ γ
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10 Ex.9-5 Poisson Process Let i t be Poisson Points. We define 1 2 ( , ) t t n as the number of point i t in 1 2 ( , ) t t of length 2 1 t
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