131A_1_Monday_June_2_lecture1

# 131A_1_Monday_June_2_lecture1 - Monday June 2nd Lecture...

This preview shows pages 1–8. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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
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 ⎧⎫ ⎡⎤ ⎛⎞⎛⎞ ⎪⎪ ⎢⎥ −+ ⎨⎬ ⎜⎟⎜⎟ −⎢ ⎝⎠⎝⎠ ⎣⎦ ⎩⎭ ∞∞

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
UCLA EE131A (KY) 4
UCLA EE131A (KY) 5

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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 " ##
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 .

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 20

131A_1_Monday_June_2_lecture1 - Monday June 2nd Lecture...

This preview shows document pages 1 - 8. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online