11.2 Renewal Processes Let X k be i.i.d., with density f , where f is - PowerPoint PPT Presentation

11.2 Renewal Processes Let X k be i.i.d., with density f , where f is not necessarily exponential. Put and 1

11.2 Renewal Processes Let X k be i.i.d., with density f , where f is not necessarily exponential. Put and The mean function m ( t ) = E [ N t ] is called the renewal function. 2

Computing the Renewal Function The first fact we need is that 3

Computing the Renewal Function The first fact we need is that In general, we won’t know F k ( t ) . 4

Computing the Renewal Function The first fact we need is that In general, we won’t know F k ( t ) . In any case, it is now easy to see that since 5

Computing the Renewal Function A more useful formula that you will derive in the problems is the renewal equation 6

Computing the Renewal Function A more useful formula that you will derive in the problems is the renewal equation This integral equation can sometimes be solved in terms of moment generating functions by using Laplace transform methods. 7

11.3 The Wiener Process (Brownian Motion) 8

11.3 The Wiener Process (Brownian Motion) trouble follows... 9

11.3 The Wiener Process 10

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The Wiener Integral Suppose X τ is white noise applied to an LTI system 15

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The Random Walk Approximation of the Wiener Process 20

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