lectr14 - 14 Stochastic Processes Introduction Let denote...

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1 14. Stochastic Processes Let denote the random outcome of an experiment. To every such outcome suppose a waveform is assigned. The collection of such waveforms form a stochastic process. The set of and the time index t can be continuous or discrete (countably infinite or finite) as well. For fixed (the set of all experimental outcomes), is a specific time function. For fixed t , is a random variable. The ensemble of all such realizations over time represents the stochastic ξ ) , ( ξ t X } { k ξ S i ξ ) , ( 1 1 i t X X ξ = ) , ( ξ t X PILLAI/Cha t 1 t 2 t ) , ( n t X ξ ) , ( k t X ξ ) , ( 2 ξ t X ) , ( 1 ξ t X # # # Fig. 14.1 ) , ( ξ t X 0 ) , ( ξ t X Introduction
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2 process X ( t ). (see Fig 14.1). For example where is a uniformly distributed random variable in represents a stochastic process. Stochastic processes are everywhere: Brownian motion, stock market fluctuations, various queuing systems all represent stochastic phenomena. If X ( t ) is a stochastic process, then for fixed t , X ( t ) represents a random variable. Its distribution function is given by Notice that depends on t , since for a different t , we obtain a different random variable. Further represents the first-order probability density function of the process X ( t ). ), cos( ) ( 0 ϕ ω + = t a t X ϕ } ) ( { ) , ( x t X P t x F X = ) , ( t x F X (14-1) (14-2) PILLAI/Cha (0,2 ), π dx t x dF t x f X X ) , ( ) , ( =
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3 For t = t 1 and t = t 2 , X ( t ) represents two different random variables X 1 = X ( t 1 ) and X 2 = X ( t 2 ) respectively. Their joint distribution is given by and represents the second-order density function of the process X ( t ). Similarly represents the n th order density function of the process X ( t ). Complete specification of the stochastic process X ( t ) requires the knowledge of for all and for all n . (an almost impossible task in reality). } ) ( , ) ( { ) , , , ( 2 2 1 1 2 1 2 1 x t X x t X P t t x x F X = (14-3) (14-4) ) , , , , , ( 2 1 2 1 n n t t t x x x f X " " ) , , , , , ( 2 1 2 1 n n t t t x x x f X " " n i t i , , 2 , 1 , " = PILLAI/Cha 2 1 2 1 2 1 2 1 2 1 2 ( , , , ) ( , , , ) X X F x x t t f x x t t x x =
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4 Mean of a Stochastic Process: represents the mean value of a process X ( t ). In general, the mean of a process can depend on the time index t . Autocorrelation function of a process X ( t ) is defined as and it represents the interrelationship between the random variables X 1 = X ( t 1 ) and X 2 = X ( t 2 ) generated from the process X ( t ). Properties: 1. 2. (14-5) (14-6) * 1 * 2 1 2 * 2 1 )}] ( ) ( { [ ) , ( ) , ( t X t X E t t R t t R XX XX = = (14-7) . 0 } | ) ( {| ) , ( 2 > = t X E t t R XX PILLAI/Cha (Average instantaneous power) ( ) { ( )} ( , ) X t E X t x f x t dx µ +∞ −∞ = = * * 1 2 1 2 1 2 1 2 1 2 1 2 ( , ) { ( ) ( )} ( , , , ) XX X R t t E X t X t x x f x x t t dx dx = = ∫∫
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5 3. represents a nonnegative definite function, i.e., for any set of constants Eq. (14-8) follows by noticing that The function represents the autocovariance function of the process X ( t ). Example 14.1 Let Then . ) ( for 0 } | {| 1 2 = = n i i i t X a Y Y E ) ( ) ( ) , ( ) , ( 2 * 1 2 1 2 1 t t t t R t t C X X XX XX µ µ = (14-9) . ) ( = T T dt t X z = = T T T T T T T T dt dt t t R dt dt t X t X E z E XX 2 1 2 1 2 1 2 * 1 2 ) , ( )} ( ) ( { ] | [| (14-10) n i i a 1 } { = ) , ( 2 1 t t R XX ∑∑ = = n i n j j i j i t t R a a XX 1 1 * . 0 ) , ( (14-8) PILLAI/Cha
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6 Similarly , 0 } {sin sin } {cos cos )} {cos( )} ( { ) ( 0 0 0 = = + = = ϕ ω ϕ ω ϕ ω µ E t a E t a t aE t
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