Lect3 - Stochastic Process 9/29/2006 Lecture 3 Jointly...

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Unformatted text preview: Stochastic Process 9/29/2006 Lecture 3 Jointly Gaussian Density Function NCTUEE Summary In this lecture, I will discuss: • Jointly Gaussian Random Variables • Joint Gaussian Density Function • Conditional Joint Gaussian Density • A Simple Detection Problem Notation We will use the following notation rules, unless otherwise noted, to represent symbols during this course. • Boldface upper case letter to represent MATRIX • Boldface lower case letter to represent vector • Superscript ( · ) T and ( · ) H to denote transpose and hermitian (conjugate transpose), respectively • Upper case italic letter to represent RANDOM VARIABLE 3-1 1 Joint Gaussian Density 1. (Recall) Any jointly Gaussian random vector x can be represented by a linear combination of the vector of i.i.d. standard normal random variables z ∼ N ( , I n ) . That is, if x ∼ N ( m x , K x ), we can write x = K x 1 / 2 z + m x , where K x 1 / 2 = EΛ 1 2 E H with E being the matrix of orthonormal eigen- vectors and Λ the diagonal matrix of eigenvalues of K x . 2. Let z ∼ N ( , I n ) and U be a unitary matrix. Then, Uz has an identical distribution as z , denoted by z = d Uz . (Justify) a. Uz is jointly Gaussian. b. Mean vector of Uz is a zero vector. c. Cov( Uz , Uz ) = U Cov( z , z ) U T = UU T = I 3-2 3. (General Expression) Let x = [ X 1 ,X 2 , ··· ,X n ] be a real jointly Gaus- sian random vector (Normal random vector) with mean vector m x and covariance matrix...
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This note was uploaded on 11/28/2010 for the course EE 301 taught by Professor Gfung during the Winter '10 term at National Chiao Tung University.

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Lect3 - Stochastic Process 9/29/2006 Lecture 3 Jointly...

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