呀 /936 Chapter 1 Preparations UESTC 16
UESTC 何子述等 16 Chapter 1 Preparations
1.1 Chapter highlights ·Random variable. PDF of Gaussian random variable. Discrete-time stochastic processes. Complex representation of narrowband signal and noise K-L expansion of signal plus noise. UESTC 17
UESTC 何子述等 1.1 Chapter highlights • Random variable. • PDF of Gaussian random variable. • Discrete-time stochastic processes. • Complex representation of narrowband signal and noise . • K-L expansion of signal plus noise. 17
1.2 Random variable >Definition of real random variable >Complex random variable Given two real random variables x and y,a complex random variable is defined by z=x+jy (1.1) Theoretically,when the pdfs of x and y known, the pdf of z can be computed UESTC 18
UESTC 何子述等 1.2 Random variable 18 ➢Definition of real random variable ➢Complex random variable Given two real random variables x and y, a complex random variable is defined by z x jy = + (1.1) Theoretically, when the pdfs of x and y known, the pdf of z can be computed
1.3 PDF of Gaussian random variable >Probability density function(pdf)p(x) Pdf of a real Gaussian random variable wim (1.2) Joint pdf of two real Gaussian random variables p(x,) 2mo-4pt20- (1.3) o 2pX-4x-4+3,-]} 0102 UESTC 19
UESTC 何子述等 1.3 PDF of Gaussian random variable 19 ➢Probability density function(pdf) p x( ) Pdf of a real Gaussian random variable Joint pdf of two real Gaussian random variables 2 2 1 ( ) ( ) exp[ ] (1.2) 2 2 x p x − = − 2 1 1 1 2 2 1/2 2 2 1 2 1 2 1 1 2 2 2 2 2 1 2 2 1 1 ( ) ( , ) exp{ [ (1.3) 2 (1 ) 2(1 ) 2 ( )( ) ( ) ]} x p x x x x x − = − − − − − − − +