[RP]Lecture Note XIII

# [RP]Lecture Note XIII - Kyung Hee University Department of...

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Kyung Hee University Random Processing Department of Electronics and Radio Engineering Prof. Hyundong Shin Communications and Coding Theory Laboratory (CCTLAB) XIII-1 C1002900 RP Lecture Handout VIII: Stochastic Processes in Linear Systems Reading: Chapter 9.2 In many applications, the stochastic processes of interest are the inputs or outputs of various kinds of linear and nonlinear systems:  Xt   Yt TXt T When a stochastic process is the input to a system, the output is also a process. In gen- eral, a complete characterization of the output process can be determined from com- plete characterizations of the system and input process. Ddifficult in practice. Linear time-invariant (LTI) systems and second-order characterization 13.1. Memoryless Systems A system is called memoryless if its output is only a function of the present input, i.e.,       Yt gXt . (XIII.1) Stationarity: Yt To determine the n th-order density of , we solve the system    ,, , nn gx ygx y y  12 , (XIII.2) which is assumed to have a unique solution.

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Kyung Hee University Random Processing Department of Electronics and Radio Engineering Prof. Hyundong Shin Communications and Coding Theory Laboratory (CCTLAB) XIII-2 Then,  ,, n n nn Yt Xt n fy y f x x Jx x 1 1 11 1 1  (XIII.3) where   x gx  . Since is SSS, (XIII.3) is invariant to a shift of the origin:  ,, . n n n n y f x x J fx x J y  1 1 1 1 1 1 (XIII.4) Similarly, if is stationary of order n , is also stationary of order n . If is WSS, then might not be stationary in any sense.
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## This note was uploaded on 06/10/2010 for the course ELECTRONIC C1002900 taught by Professor Hyungdongshin during the Spring '10 term at Kyung Hee.

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[RP]Lecture Note XIII - Kyung Hee University Department of...

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