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# Chapter 9 - Chapter 9 General Concept 9-1 Definition X(t RP...

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Chapter 9 General Concept 9-1 Definition ( ) t X - RPh ( , ) t ξ X - ξ is an outcome of a probability space S , h h ξ X i } È { t R S S continuous-time h t I S S discrete-time ( ) t X - countable valuesh discrete-state } { ( ) t X h ξ t function (1) h ξ fixedh t function, called sample function (2) h t fixedh RV Brownian Motion } { particle } function h } { regular ( ) cos( ) t t ϖ = + φ - γ - random amplitude h φ - random phaseh ( , ) ξ γ φ = + ( ) cos( ) t t γ ϖ φ = + X time function h h Regular h predictable process } { a general random process contains both regular and predictable components, } { Equality h RPs ( ) t X and ( ) t Y h equal h if ξ 2200 identical ( , ) ( , ) for all t t t ξ ξ = X Y i { relaxed h in MS sense h i.e., 2 { ( ) ( ) } 0 for E t t t - = 2200 X Y Note: Equal in MS sense do not set constraints to the relation of ( ) t X (and ( ) t Y ) in different t . So a predictable RP can be MS equal to a regular RP.

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Statistics of Stochastic Process First-order distribution { } ( , ) ( ) F x t P t x = X for a specific t First-order density ( , ) ( , ) F x t f x t x S = 2nd-order distribution { } 1 2 1 2 1 1 2 2 ( , , , ) ( ) , ( ) F x x t t P t x t x = X X n th-order distribution h 1 , , n t t L i G { 1 ( ), , ( ) n t t X X L joint distribution 8 )= { 2nd-order statistics properties7 )= { Meanh { } ( ) ( ) ( , ) t E t xf x t dx η - = = X deterministic time function Autocorrelationh { } 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 S - - = = �� X X Average powerh { } 2 ( ) ( , ) E t R t t = X Autocovarianceh 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 ( ) t X - variance Ex.9-3 S h RV defined by ( ) b a t dt = S X { } { } ( ) b s a E E t dt η = = S X ( ) b a t dt η =
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.9-4 ( ) cos( ) t t ϖ = + φ h RPh γ random amplitudeh φ random phase and is uniformly distributed in ( , ) π π - , γ h φ independenth 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 Ex.9-5 Poisson Process Poisson Points i t - 1 2 ( , ) t t n - number of point i t in 1 2 ( , ) t t of length 2 1 t t t = - Poisson RV with parameter t λ 1 2 ( ) ( ( , ) ) ! t k e t P t t k k λ λ - = = n   1 2 ( , ) t t 3 4 ( , ) t t overlaph 1 2 ( , ) t t n 3 4 ( , ) t t n independent   i t form RP ( ) (0, ) t t = X n h staircase functionh Fig 9.3(a) h { } ( ) ( ) E t t t η λ = = X 2 1 2 2 1 2 1 2 2 1 2 1 1 2 ( , ) t t t t t R t t t t t t t λ λ λ λ + = + 1 2 1 2 1 2 1 2 1 2 ( , ) min( , ) ( ) ( ) C t t t t t u t t t u t t λ λ λ = = - + - Γ∪♠ 1 2 t t 2 ( ) t X 1 2 ( ) ( ) t t - X X independenth

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{ } { } { } { } { } { } 2 1 2 2 1 2 2 1 2 2 2 2 1 2 2 2 2 1 2 2 2 2 ( )( ( ) ( )) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) E t t t E t E t t t t t t t t E t t E t E t t λ λ λ λ λ - = - = - = - = - = X X X X X X X X X X Q { } 2 2 2 2 2 1 2 1 2 (from variance of Poisson distribution) ( ) ( ) t E t t t t t λ λ λ + = + X X Ex.9-6 Telegraph signal h i t form RP ( ) t X s.t. ( ) t X 1 = if (0, ) t n h evenh ( ) 1 t = - X if (0, ) t n h oddh semirandom telegraph signalh see Fig.9.3(b)p P.379-380 h { } E ( ) t X 1 2 ( , ) R t t General Properties of RPs A real RP ( ) t X is completely determined in terms of its n th-order distribution { } 1 1 1 1 ( ; ) ( ) , , ( ) n n n n F x x t t P t x t x = X X L L L Joint statistics of h RPs ( ) t X and ( ) t Y h joint distribution h ' ' 1 1 ( ) ( ) ( ) ( ) n m t t t t X X Y Y L L
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Chapter 9 - Chapter 9 General Concept 9-1 Definition X(t RP...

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