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Topic 7: Random Processes

Topic7: RandomProcesses De nition,discreteandcontinuousprocesses Specifyingrandomprocesses{Joint cdf's or pdf's{Mean,auto-covariance,auto-correlat ion{Cross-covariance,cross-correlation StationaryprocessesandergodicityES150{ Harvard SEAS1 Randomprocesses Arandomprocess, alsocalledastochasticprocess, is a familyof randomvariables,indexedby a parametertfromanindexingsetT. For eachexperiment outcome!2 ,we assigna functionXthatdependsontX(t; !)t2T; !2 {tis typicallytime,butcanalsobe a spatialdimension{tcanbe discreteor continuous{Therangeoftcanbe nite,butmoreoftenis in nite,which meanstheprocesscontainsanin nitenumber of randomvariables. Examples:{Thewirelesssignalreceivedby a cellphoneover time{Thedailystock price{Thenumber of packetsarrivingat a routerin 1-secondintervals{Theimageintensity over 1cm2regionsES150{ Harvard SEAS2 We areinterestedin specifyingthejoint behaviorof therandomvariableswithina family, or thebehaviorof a studying{Thedependenciesamongtherandomva riablesof theprocess( ){Long-termaverages{Extremeor boundaryevents ( ){Estimation/detectionof a signalcorruptedby noiseES150{ H}}}}}}}}}}}}}}}}}

{ For each n, Xn is a r.v., which can be continuous, discrete, or mixed. { Examples: Xn = Zn; n ‚ 1; Z » U[0;1]. Others: sending bits over a noisy channel, sampling of thermal noise. † When t comes from an uncountably inflnite set, the process is continuous-time. We then often denote the random process as X(t). At each t, X(t) is a random ...

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The autocorrelation function and the rate of change

www.ece.tufts.edu

White noise † Band-limited white noise: A zero-mean WSS process N(t) which has the psd as a constant N0 2 within ¡W • f • W and zero elsewhere. { Similar to white light containing all frequencies in equal amounts.

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Lecture 1: Entropy and mutual information

www.ece.tufts.edu

4.1 Non-negativity of mutual information In this section we will show that I(X;Y) ≥ 0, (27) and this is true for both the discrete and continuous cases. Before we get to the proof, we have to introduce some preliminary concepts like Jensen’s in-equality and the relative entropy.

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Topic 6: Convergence and Limit Theorems

www.ece.tufts.edu

Topic 6: Convergence and Limit Theorems ... – This is the Central Limit Theorem (CLT) and is widely used in EE. ES150 – Harvard SEAS 7 • Examples: 1. Suppose that cell-phone call durations are iid RVs with μ = 8 and σ = 2 (minutes). – Estimate the probability of 100 calls taking over 840 minutes.

  Limits, Theorem, Limit theorem

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