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COL+UNIV+2012+SPRING-The+Joint+Characteristic+Function+forTwo+Gaussian+Random+Variables-1+JAN+2012

# COL+UNIV+2012+SPRING-The+Joint+Characteristic+Function+forTwo+Gaussian+Random+Variables-1+JAN+2012

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1 Random Signals and Noise ELEN E4815 Columbia University Spring Semester 2012 2 January 2012 Professor I. Kalet The Joint Characteristic Function, M(ju, jv), for two Gaussian random variables.

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2 Joint Probability Density Function, f(x, y), for Two Gaussian Random Variables, x and y f(x, y)= [1/(2 π σ x σ y )]exp{ -[(x-m x ) 2 /(2 σ x 2 )+ 2 ρ xy (x-m x ) (y-m y )/ ( σ x σ y ) + (y-m y ) 2 /(2 σ y 2 )] (1- ρ xy 2 ) σ x 2 =Variance {x} σ y 2 =Variance {y} m x =E{x} m y =E{y} ρ xy =E{(x-m x ) (y-m y )}/[ σ x σ y ]
3 If we have two jointly distributed Gaussian random variables with the joint pdf, f(x,y) (as shown on the previous page), then
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Unformatted text preview: the Joint Characteristic Function, M(ju,jv), is given by the expression below. (Papoulis, p.159) σ x 2 =Variance {x} σ y 2 =Variance {y} m x =E{x} m y =E{y} ρ xy =E{[(x-m x ) (y-m y )]} / ( σ x σ y ) If m x and m y = 0 and if σ x 2 and σ y 2 =1, then M(ju, jv)=exp{-(1/2)[u 2 + 2 ρ xy uv+ v 2 ]} The Joint Characteristic Function, M(ju, jv), for two Gaussian random variables. M(ju, jv)= exp{ j(m x u +m y v)} exp{-(1/2)[u 2 σ x 2 + 2 ρ xy σ x σ y uv+ v 2 σ y 2 ]}...
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COL+UNIV+2012+SPRING-The+Joint+Characteristic+Function+forTwo+Gaussian+Random+Variables-1+JAN+2012

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