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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 in S produces a deterministic sample function X ( t , ) . Standard methods from calculus can be used to determine the continuity of the sample function at a point t for each . The function X ( t , ) is continuous at t if lim [ X ( t + , )- X ( t , )] = 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 , ) = cos ( 2 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