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# Chapter 9 - Chapter 9 General Concept Te r ^ ache Office...

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1 Chapter 9 General Concept Teacher: ˆ ¤ ª Office: 805 Tel: ext. 31822 Email: [email protected]

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2 9-1 Definition ( ) t X γ RP 1 °° ( , ) t ξ X γ ξ is an outcome of a probability space S , ] ξ X ] F γ t R continuous-time 1 t I discrete-time ( ) t X γ γ countable values 1 discrete-state ] F continuous-state ( ) t X 1 ξ 1 t 1 function (1) h ξ fixed 1 t 1 function, called sample function (2) h t fixed 1 RV
3 Brownian Motion - ] F particle 1 1 function regular h ( ) cos( ) t t ϖ = + φ γ γ γ random amplitude 1 φ γ random phase 1 ( , ) ξ γ φ = 1 ( ) cos( ) t t γ ϖ φ = + X 1 time function 1 Regular 1 predictable process - ] F a general random process contains both regular and predictable components, 1 1 1 1

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4 Equality1 RPs ( ) t X and ( ) t Y 1 equal1 if ξ 2200 identical ( , ) ( , ) for all t t t ξ ξ = X Y F relaxed 1 in MS sense1 i.e., 2 { ( ) ( ) } 0 for E t t t - = 2200 X Y
5 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 =

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6 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 1 1 , , n t t 1 ( ), , ( ) n t t X X 1 joint distribution ¤ F 2nd-order statistics properties 1 F
7 Mean 1 { } ( ) ( ) ( , ) t E t xf x t dx η -∞ = = X deterministic time function Autocorrelation 1 { } 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

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8 Average power 1 { } 2 ( ) ( , ) E t R t t = X Autocovariance 1 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 1 ( ) t X γ variance
9 Ex.9-3 S 1 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 = ∫ ∫

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10 Ex.9-4 ( ) cos( ) t t ϖ = + φ 1 RP1 γ 1 random amplitude 1 φ 1 random phase and is uniformly distributed in ( , ) π π - , γ 1 φ 1 independent1 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
11 Ex.9-5 Poisson Process Poisson Points i t γ 1 2 ( , ) t t n γ 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 h 1 2 ( , ) t t 1 3 4 ( , ) t t 1 overlap1 1 2 ( , ) t t n 1 3 4 ( , ) t t n 1 independent

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12 h i t 1 form RP ( ) (0, ) t t = X n 1 staircase function 1 1 Fig 9.3(a) 1 { } ( ) ( ) 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 = = - + -
13 I ∕ ± h 1 2 t t 2 ( ) t X 1 1 2 ( ) ( ) t t - X X 1 independent1 { } { } { } { } { } { } 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 γ { } 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

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14 Ex.9-6 Telegraph signal h i t 1 form RP ( ) t X s.t. ( ) t X 1 = if (0, ) t n 1 even 1 ( ) 1 t = - X if (0, ) t n 1 odd 1 1 semirandom telegraph signal 1 see Fig.9.3(b) 1 1 P.379-380 1 - ] { } E ( ) t X 1 1 2 ( , ) R t t
15 General Properties of RPs A real RP ( ) t X is completely determined in terms of its n

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Chapter 9 - Chapter 9 General Concept Te r ^ ache Office...

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