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Unformatted text preview: Stochastic Signals and Systems Random Processes Virginia Tech Fall 2008 Introduction • If we view a system as acting on an input random process to produce an output random process, we find that we need to develop a new “calculus” for random processes. • In particular we need to develop probabilistic methods for addressing the continuity , differentiability , and integrability of random processes, that is, of the ensemble of sample functions as a whole. Continuity • A natural way of viewing a random process is to imagine that each point ξ in S produces a deterministic sample function X ( t , ξ ) . Standard methods from calculus can be used to determine the continuity of the sample function at a point t for each ξ . The function X ( t , ξ ) is continuous at t if lim δ → [ X ( t + δ, ξ ) X ( t , ξ )] = • In some simple simple cases, we can establish that all sample functions of the random process are continuous at a point t , and so we can conclude that the random process is continuous at t . Example: X ( t , ξ ) = ξ cos ( 2 π t ) • In general, however, the question whether the sample function of a stochastic process X ( t ) is continuous at a particular time t can be answered only in a probabilistic sense. Continuity • A sample function of the Poisson counting process is continuous everywhere except at the jump times t 1 , t 2 , . . ....
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 Fall '08
 DASILVER
 Calculus, Derivative, Continuous function

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