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Gaussian Random Variables and Processes

Gaussian Random Variables and ProcessesSaravanan of Electrical EngineeringIndian Institute of Technology BombayAugust 1, 20121 / 33 Gaussian Random VariablesGaussian Random VariableDefinitionA continuous Random variable with pdf of the formp(x) =1 2 2exp( (x )22 2), <x< ,where is the mean and 2is the variance. 4 2024 (x)3 / 33 Notation N( , 2)denotes a Gaussian distribution with mean andvariance 2 X N( , 2) Xis a Gaussian RV with mean andvariance 2 X N(0,1)is termed a standard Gaussian RV4 / 33 Affine Transformations Preserve GaussianityTheoremIf X is Gaussian , then aX+b is Gaussianfor a,b IfX N( , 2), thenaX+b N(a +b,a2 2).

Gaussian Random Process Definition A random process fX(t) : t 2Tgis Gaussian if its samples X(t1);:::;X(tn) are jointly Gaussian for any n 2N. Properties The mean and autocorrelation functions completely characterize a Gaussian random process. Gaussian WSS processes are stationary. If the input to an LTI system is a Gaussian RP, the output is

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