COL+UNIV+2012+SPRING-Multidimensional+Gaussian+Random+Variables-+1+JAN+2012

COL+UNIV+2012+SPRING-Multidimensional+Gaussian+Random+Variables-+1+JAN+2012

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Random Signals and Noise ELEN E4815 Columbia University Spring Semester 2012 2 January 2012 Professor I. Kalet Multidimensional Gaussian Random Variables and Joint Characteristic Function 1
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Multidimensional Gaussian Random Variable The equations below will be useful when discussing strict-sense stationary gaussian random processes. Let’s assume that we have N jointly distributed gaussian random variables. The joint pdf, f (x 1 , x 2 , x 3 ,. .x N )=f( x ) for these variables is shown below. All letters shown in blue italics represent vectors. x =(x 1 , x 2 , x 3 ,. .x N ) m = (m 1 , m 2 , m 3 ,…. m
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Unformatted text preview: N ) m i = E{x i } σ i 2 = Var {x i }= E{(x i-m i ) 2 } c ik = Covariance (x i , x j )= E{(x i-m i )( x k-m k )}= ρ ik σ i σ k ρ ik =Correlation Coefficient= E{(x i-m i )( x k-m k )}/ σ i σ k ]C x ]= The matrix of the covariances, c ik [C x –1 ] =The inverse of the matrix [C x ] The Joint Characteristic Function N N N M x (ju 1 , ju 2 , ju 3 , . .,ju N )=exp{ j Σ u i m i }exp{ -(1/2) Σ Σ c ik u i u k } i=1 i=1 k=1 2 f ( x ) = [exp{-(1/2) ( x-m ) T [C x ] –1 ( x-m ) }]/[ (2 π ) N/2 det [C x ] ]...
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This note was uploaded on 02/28/2012 for the course ELEN E4815 taught by Professor I during the Spring '12 term at Columbia.

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COL+UNIV+2012+SPRING-Multidimensional+Gaussian+Random+Variables-+1+JAN+2012

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