Chapter 9 - 1 Chapter 9 General Concept Teacher: Office:...

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Unformatted text preview: 1 Chapter 9 General Concept Teacher: Office: 805 Tel: ext. 31822 Email: schen@mail.nctu.edu.tw 2 9-1 Definition ( ) t X RP1 && ( , ) t X is an outcome of a probability space S ,] X ] F t R continuous-time1 t I discrete-time ( ) t X countable values1 discrete-state ] F continuous-state ( ) t X 1 1 t 1 function (1) h fixed1 t 1 function, called sample function (2) h t fixed1 RV 3 h& Brownian Motion- ] F particle 1 1 function regular h ( ) cos( ) t t = + X random amplitude1 random phase1 ( , ) = 1 ( ) cos( ) t t = + X 1 time function 1 Regular 1 predictable process - ] F a general random process contains both regular and predictable components, 1 1 1 1 4 Equality1 RPs ( ) t X and ( ) t Y 1 equal1 if 2200 identical ( , ) ( , ) for all t t t = 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 = 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 properties1 F 7 Mean1 { } ( ) ( ) ( , ) t E t xf x t dx - = = X deterministic time function Autocorrelation1 { } 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 8 Average power1 { } 2 ( ) ( , ) E t R t t = X Autocovariance1 1 2 1 2 1 2 ( , ) ( , ) ( ) ( ) C t t R t t t t =- = { } 1 1 2 2 ( ( ) ( ))( ( ) ( )) E t t t t -- X X for 1 2 t t t = = ( , ) C t t 1 ( ) t X variance 9 Ex.9-3 S 1 RV defined by ( ) b a t dt = S X { } { } ( ) b s a E E t dt = = S X ( ) b a t dt = 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 = 10 Ex.9-4 ( ) cos( ) t t = + X 1 RP1 1 random amplitude 1 1 random phase and is uniformly distributed in ( , ) - , 1 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 =- + + + =- X X 11 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 1 2 ( ) ( ( , ) ) ! t k e t P t t k k - = = 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 12 h i t 1 form RP ( ) (0, ) t t = X n 1 staircase function1 1 Fig 9.3(a) 1 { } ( ) ( ) E t t t = = 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 + = + 1 2 1 2 1 2 1 2 1 2 ( , ) min( , ) ( ) ( ) C t t t t t u t t t u t t = =- +- 13...
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This note was uploaded on 07/21/2009 for the course CM EM5102 taught by Professor Sin-horngchen during the Fall '08 term at National Chiao Tung University.

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Chapter 9 - 1 Chapter 9 General Concept Teacher: Office:...

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