131B_1_Noise - NOISE : Random Proceses The theory of Random...

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Unformatted text preview: NOISE : Random Proceses The theory of Random Processes is quite rich aith many applications. The use in Communication Systems is quite limited in scope and we prefer tha name Noise indicative of this. Noise is a special case of Stationary Random Processes. We need only the concept of Stationary Gaussian Random Processes and even at that we shall see that it is an evolution of the theory of signals of Finite Power that we have already covered. A Random Process is a generaliztion of the notion of a finite dimensional random variable H ' vector' variable L indexed by the components. Here we allow the components to be nonfinite- a sequence or a continuum-- a Discrete Time random process or a Continuous Time random process. Sample ' points' now become sample paths but the descritption is the same : we have an ensemble of functions defined on time denoted by x H t, Ω L- ¥ < t < ¥ and Ω ˛ W , the set of sample paths. For each Ω we have a sample path ~ a realization. If we fix t , we have the ' scatter'- a random variable whose distribution is given. If we take a finite number of instants of time , we are given the corresponding finite dimensional distribution. If all the distributions are Gaussian, we have a ' Gaussian' Process The importance of the Gaussian is that it is completely determined by the mean and covariance. Moreover it is easier to specify the characteristic function than the density. There is also the deeper reason that any continuous time ' white' noise process has to be Gaussian ! A fact not understood by many engineers ! Of course this is not true in the discrete time case- we can define IID nonGaussian sequences which is not specified completely by the covariance....
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This note was uploaded on 02/20/2012 for the course EE 131B taught by Professor Balakrishnan during the Spring '11 term at UCLA.

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131B_1_Noise - NOISE : Random Proceses The theory of Random...

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