[RP]Lecture Note V

# [RP]Lecture Note V - Kyung Hee University Department of...

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Kyung Hee University Random Processing Department of Electronics and Radio Engineering Prof. Hyundong Shin Communications and Coding Theory Laboratory (CCTLAB) V-1 C1002900 RP Lecture Handout V: Gaussian Random Vectors Reading: Chapter 7.1-7.2 5.1. Jointly Gaussian Distribution The joint characteristic function of X and Y is    , ,. jX Y XY e   12 Inverse: ,, . jx y fx y e d d       1 2 2 1 4 Let   T n XX X 1 . Then,   T j e  ω X X ω where   . T n  ω 1 Definition: Let , n X be real random variables defined on the same sample space and n X X X 1 . Then, , n X are jointly Gaussian if their joint characteristic function is given by exp TT j    X ωω μ ω Σ ω 1 2 where Σ is positive semidefinite.

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Kyung Hee University Random Processing Department of Electronics and Radio Engineering Prof. Hyundong Shin Communications and Coding Theory Laboratory (CCTLAB) V-2 If Σ is positive definite, ,, , n XX X 12 are jointly Gaussian if and only if the joint densi- ty is of the form   exp . T n f     X xx μΣ x μ Σ 1 21 2 11 2 2 which is denoted by n X μ , Σ . (Gaussian random vector) Proof:   Let Y Σ X μ . Then One-to-one nonsingular linear transformation    det J  yA xb xA det J x Σ 1 2 det exp exp , T n T n nn f Y y Σ yy Σ YI 2 2 2 2 2 2 0    / exp exp . T T TT T j j jj jT e e ee e j     ω X X ωΣ Y μ ωμ Y ω ωΣ Σ ω ωΣω 1 2 1 2 The converse follows from the uniqueness of Fourier inversion. Mean of X :
Kyung Hee University Random Processing Department of Electronics and Radio Engineering Prof. Hyundong Shin Communications and Coding Theory Laboratory (CCTLAB) V-3      X Σ Y μμ 12  Covariance matrix of X :      n nn n T XX X X X X T X X X X           Σ YY ΣΣ 11 2 1 2 22 2 2 2 Bivariate Gaussian X X X X X X X X Σ 2 1 1 2 21 2 1 2 2 Since det   Σ 222 1 , the covariance Σ is positive definite if .  1 X X       Σ 2 1 2 1 2 1 1 . T T xx x x x x             2 2 1 2 11 2 1 2 21 1 1 22 2 2 12 1 1 1 2 2 1 21 1 2 2 12 2 2 Therefore, , ,e x p , .

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## This note was uploaded on 06/10/2010 for the course ELECTRONIC C1002900 taught by Professor Hyungdongshin during the Spring '10 term at Kyung Hee.

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[RP]Lecture Note V - Kyung Hee University Department of...

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