Introduction Definition the autocorrelation function of a real process x(t is: R(,12)=x(4)x(2)=了。了xx2/(x,x2x2(6-13) Where x=x(t1, and x2=x(t2), if the process is a second- order stationary the autocorrelation function is a function only of the time difference t=t2-tu R(τ)=x(1)x(2)(6-14) Properties of the autocorrelation function of a real wide sense stationary process are as follows: ()R()=x(t)=E(x(t)=average power (6-16) (2)R2(-t)=R3(τ) (6-17) (3)|R3(U)≤R2(0) (4)R (09=E Lx(o=dc power (5)R3(0)-R3(∞=0
6 Introduction • Definition : the autocorrelation function of a real process x(t) is: ( , ) ( ) ( ) ∫∫ ( , ) (6-13) ∞ -∞ 1 2 ∞ R t 1 t 2 x t 1 x t 2 -∞ x1 x2 f x1 x2 dx dx x = = x • Where x1=x(t1 ), and x2=x(t2 ), if the process is a secondorder stationary, the autocorrelation function is a function only of the time difference τ=t2 -t1 . (τ) = ( ) ( ) (6-14) 1 2 R x t x t x • Properties of the autocorrelation function of a real widesense stationary process are as follows: 2 2 2 2 (5) (0)- (∞) = σ (4) (∞) = [ ( )] = d c power (3)| (τ) |≤ (0) (6 -18) (2) (-τ) = (τ) (6 -17) (1) (0) = ( ) = { (t)} = a (6 -16) x x x x x x x x R R R E x t R R R R R x t E x verage power
Introduction Definition: the cross-correlation function for two real process x(t) and y(t is Rn(1,12)=x(1)y(t2)=」∞」myf(x1,y2k(6-19) s. ifx=x(t), and y=x(t2) are jointly stationary, the cross correlation function is a function only of the time difference t=t s Ryy(,t2)=R(r) Properties of the cross-correlation function of two real jointly stationary process are as follows: (1)R(-7)=R (6-20) (2)R()√R0R(O) (6-21) 3)|R2(r)[R2(0)+R1(O) (6-22
7 Introduction • Definition : the cross-correlation function for two real process x(t) and y(t) is: ( , ) ( ) ( ) ∫ ∫ ( , ) (6 -19) ∞ -∞ ∞ R t 1 t 2 x t 1 y t 2 -∞xyf x1 y2 dxdy x y = = x • if x=x(t1 ), and y=x(t2 ) are jointly stationary, the crosscorrelation function is a function only of the time difference τ=t2 -t1 . ( , ) ( ) 1 2 x y Rx y R t t = • Properties of the cross-correlation function of two real jointly stationary process are as follows: [ (0) (0)] (6 - 22) 2 1 (3)| ( ) | (2)| ( ) | (0) (0) (6 - 21) (1) ( ) ( ) (6 - 20) x x y x y x y x y y x R R R R R R R R + − =
Introduction Two random processes x(t) and y(t) are said to be uncorrelated if R3(r)=[x()y(t)=m2m (6-27) For all value of t, similarly, two random processes x(t) and y(t are said to be orthogonal if R,(z)=0 (6-28) For all value of t. If the random processes x(t)and y(t) are jointly ergodic, the time average may be used to replace the ensemble average. For correlation functions. this becomes: Rx(r)=[x(tI[y(]=[x(OIly( (6-29) 8
8 • Two random processes x(t) and y(t) are said to be uncorrelated if : ( ) [ ( )][ ( )] (6 - 27) x y mx my R = x t y t = • For all value of τ, similarly, two random processes x(t) and y(t) are said to be orthogonal if ( ) = 0 (6 - 28) Rx y • For all value of τ. If the random processes x(t) and y(t) are jointly ergodic, the time average may be used to replace the ensemble average. For correlation functions, this becomes: R ( ) [x(t)][ y(t)] [x(t)][ y(t)] (6 - 29) x y = = Introduction
Introduction Definition: a complex random process is g(t)=x()+ⅳ(t) (6-31) e where x(t) and y(t) are real random processes. Definition: the autocorrelation for complex random process Is. R2(41,t2)=g(1)g(t2) (6-33) Where the asterisk denotes the complex conjugate the autocorrelation for a wide-sense stationary complex random process has the hermitian symmetry property: Ro()=R(T) (6-34) 9
9 Introduction • Definition: a complex random process is: g(t) = x(t) + jy(t) (6 -31) Where x(t) and y(t) are real random processes. • Definition: the autocorrelation for complex random process is: ( , ) ( ) ( ) (6-33) 1 2 * 1 2 R t t g t g t g = Where the asterisk denotes the complex conjugate. the autocorrelation for a wide-sense stationary complex random process has the Hermitian symmetry property: ( ) ( ) (6 -34) * Rg − = Rg
Introduction For a Gaussian process, the one-dimension Pdf can be represented by: (x-mx) f(x)= expl 2π6 20 some properties of f(x)are (1)f(x)is a symmetry function about x-m (2)f(x)is a monotony increasing function at(- infinite, mx)and a monotony decreasing funciton at (mx,), the maximum value at mx is 1/(2r)(1/2)o]; .'f(xdx=1 and p f(x)dx=I f(x)dx=0.5 10
10 Introduction • For a Gaussian process, the one-dimension PDF can be represented by: ] 2σ ( -m ) exp[- 2πσ 1 ( ) = 2 2 x x f x • some properties of f(x) are: • (1) f(x) is a symmetry function about x=mx ; • (2) f(x) is a monotony increasing function at(- infinite,mx) and a monotony decreasing funciton at (mx, ), the maximum value at mx is 1/[(2π)(1/2)σ]; ∫ ( ) = 1 and∫ ( ) = ∫ ( ) = 0.5 ∞ m m -∞ ∞ -∞ x x f x d x f x d x f x d x