131A_1_Monday_June_2_lecture1

131A_1_Monday_June_2_lecture1 - Monday June 2nd Lecture...

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UCLA EE131A (KY) 1 Monday June 2 nd Lecture One-dimensional Gaussian pdf Two-dimensional Gaussian pdf N-dimensional Gaussian pdf Linear transformation of random Gaussian vectors Linear mean-squares estimation and its modern applications in signal processing and communication
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UCLA EE131A (KY) 2 Higher-dimensional Gaussian pdf’s Since Gaussian rv is one of the most important rv’s in probability, applied science, and engineering, we want to study the higher dim. Gaussian random vectors. 1. One-dimensional case The mean μ is a translation parameter and the variance σ 2 is a width scaling parameter. 22 2- ( x - μ )/ (2 σ ) X f( x ) = ( 1 /2 πσ )e , <x< . ∞∞ -4 -2 0 2 4 6 8 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 μ=0,σ=1 μ=0,σ=0.5 μ=4,σ=1 μ=4,σ=0.5 x
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UCLA EE131A (KY) 3 Two-dimensional Gaussian pdf 2. Two dimensional case with X 1 and X 2 . μ 1 is the mean of X 1 , μ 2 is the mean of X 2 . σ 2 1 is the variance of X 1 and σ 2 2 is the variance of X 2 . ρ is the correlation coefficient between X 1 and X 2 . 22 11 2 2 2 2 2 12 -1 x - μ x- μ μ μ 2 ρ σσ σ σ 2(1 ρ ) 1 XX 1 2 2 2 - < x < , e , f( x , x ) = - < x < . 2 πσ σ 1 ρ ⎧⎫ ⎡⎤ ⎛⎞⎛⎞ ⎪⎪ ⎢⎥ −+ ⎨⎬ ⎜⎟⎜⎟ −⎢ ⎝⎠⎝⎠ ⎣⎦ ⎩⎭ ∞∞
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UCLA EE131A (KY) 6 N-dimensional case (1) Denote the n-dimensional Gaussian column vector X and its mean vector μ as The nxn dimensional covariance matrix R is [] 11 22 T 2 2 n n nn X μ X μ = E{( )( ) } E X μ X μ X μ X μ ⎡⎤ ⎢⎥ −− = ⎣⎦ ⎩⎭ RX μ X μ " # T 12 n X μ X μ = , = X X X , = = E {} . X μ XX μ X " ##
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UCLA EE131A (KY) 7 We also denote the covariance matrix R by The n-dimensional Gaussian pdf is given by | R | is the determinant of R .
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131A_1_Monday_June_2_lecture1 - Monday June 2nd Lecture...

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