This preview shows pages 1–13. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: 1 Chapter 9 General Concept Teacher: Office: 805 Tel: ext. 31822 Email: schen@mail.nctu.edu.tw 2 91 Definition ( ) t X RP1 && ( , ) t X is an outcome of a probability space S ,] X ] F t R continuoustime1 t I discretetime ( ) t X countable values1 discretestate ] F continuousstate ( ) 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 Firstorder distribution { } ( , ) ( ) F x t P t x = X for a specific t Firstorder density ( , ) ( , ) F x t f x t x = 6 2ndorder distribution { } 1 2 1 2 1 1 2 2 ( , , , ) ( ) , ( ) F x x t t P t x t x = X X n thorder distribution 1 1 , , n t t 1 ( ), , ( ) n t t X X 1 joint distribution F 2ndorder 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.93 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.94 ( ) 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.95 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...
View
Full
Document
This note was uploaded on 07/21/2009 for the course CM EM5102 taught by Professor Sinhorngchen during the Fall '08 term at National Chiao Tung University.
 Fall '08
 SinHorngChen

Click to edit the document details