Chapter 6 Random Processes and Spectral Analysis
1 Chapter 6 Random Processes and Spectral Analysis
Introduction (chapter objectives) e. Power spectral density · Matched filters Recall former Chapter that random signals are used to convey information Noise is also described in terms of statistics. Thus, knowledge of random signals and noise is fundamental to an understanding of communication systems
2 Introduction (chapter objectives) • Power spectral density • Matched filters Recall former Chapter that random signals are used to convey information. Noise is also described in terms of statistics. Thus, knowledge of random signals and noise is fundamental to an understanding of communication systems
Introduction Signals with random parameter are random singals i All noise that can not be predictable are called random noise or noise Random signals and noise are called random process i Random process(stochastic process) is an indexed set of function of some parameter( usually time) that has certain statistical properties. a random process may be described by an indexed set of random variables. a random variable maps events into constants, whereas a random process maps events into functions of the parameter t
3 Introduction • Signals with random parameter are random singals ; • All noise that can not be predictable are called random noise or noise ; • Random signals and noise are called random process ; • Random process (stochastic process) is an indexed set of function of some parameter( usually time) that has certain statistical properties. • A random process may be described by an indexed set of random variables. • A random variable maps events into constants, whereas a random process maps events into functions of the parameter t
Introduction Random process can be classified as strictly stationary or wide-sense stationary; Definition: A random process x(t) is said to be stationary to the order n if, for any tu, t2,-., fx(x(1),x(2),…,x(tN)=fx(x(t1+o),x(t2+t0),…,x(tN+to)(6-3) Where to si any arbitrary real constant. Furthermore, the process is said to be strictly stationary if it is stationary to the order n-infinite Definition: a random process is said to be wide-sense stationary if 1 x(t)=constantan (6-15a) 2Rx(1,t2)=R()(6-15b) Whereτ=t2-t1
4 Introduction • Random process can be classified as strictly stationary or wide-sense stationary; • Definition: A random process x(t) is said to be stationary to the order N if , for any t1 ,t2 ,…,tN, : ( ( ), ( ),..., ( )) = ( ( + ), ( + ),..., ( + )) (6 -3) 1 2 1 0 2 0 0 f x t x t x t f x t t x t t x t t x N x N • Where t0 si any arbitrary real constant. Furthermore, the process is said to be strictly stationary if it is stationary to the order N→infinite • Definition: A random process is said to be wide-sense stationary if 2 ( , ) = (τ) (6 -15b) 1 ( ) = constant and (6 -15a) x 1 2 Rx R t t x t • Where τ=t2 -t1
Introduction Definition: A random process is said to be ergodic if all e time averages of any sample function are equal to the corresponding ensemble averages(expectations) Note: if a process is ergodic, all time and ensemble averages are interchangeable. Because time average cannot be a function of time, the ergodic process must be stationary, otherwise the ensemble averages would be a function of time. But not all stationary processes are ergodic. xdc([x([x(t)=mx (x(l)=lim F2 [x(ODt (6-6b) x(O)=x/(x)dx=m2(6-6c) 5 =V=o+m<(6-7)
5 Introduction • Definition: A random process is said to be ergodic if all time averages of any sample function are equal to the corresponding ensemble averages(expectations) • Note: if a process is ergodic, all time and ensemble averages are interchangeable. Because time average cannot be a function of time, the ergodic process must be stationary, otherwise the ensemble averages would be a function of time. But not all stationary processes are ergodic. = = σ + (6 - 7) [ ( )] = [ ] ( ) = (6 - 6c) [ ( )] (6 - 6b) 1 [ ( )] = lim [ ] [ ] (6 - 6a) 2 2 2 ∞ -∞ T/2 -T/2 ∫ ∫ rms x x x x T→→ d c x X x t m x t x f x dx m x t dt T x t x = x(t) = x(t) =m
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
