Chapter 4 Random process. 4.1 Random process

Transcription

1 Random processes - Chapter 4 Random process 1 Random processes Chapter 4 Random process 4.1 Random process 4.1 Random process

2 Random processes - Chapter 4 Random process 2 Random process Random process, stochastic process The infinite set {X t, t T } of random variables is called a random process, where the index set T is an infinite set. In other words, a random vector with an infinite number of elements is called a random process. Discrete time process, continuous time process A random process is said to be discrete time if the index set is a countably infinite set. When the index set is an uncountable set, the random process is called a continuous time random process. 4.1 Random process

3 Random processes - Chapter 4 Random process 3 Discrete (alphabet) process, continuous (alphabet) process A random process is called a discrete alphabet, discrete amplitude, or discrete state process if all finite length random vectors drawn from the random process are discrete random vectors. A process is called a continuous alphabet, continuous amplitude, or continuous state process if all finite length random vectors drawn from the random process are continuous random vectors. A random process {X( )} maps an element ω of the sample space on a time function X(ω, t). A random process {X( )} is the collection of time functions X(t) called sample functions. This collection is called an ensemble. The value of the time function X(t) is a random variable at t. 4.1 Random process

4 Random processes - Chapter 4 Random process 4 A random process and sample functions 4.1 Random process

5 Random processes - Chapter 4 Random process 5 Since a random process is a collection of random variables with X(t) denoting a random variable at t, the statistical characteristics of the random process can be considered vi the cdf and pdf of X(t). For example, the first-order cdf, first-order pdf, second-order cdf, and nth-order cdf of the random process {X(t)} are and respectively. F X(t) (x) = Pr{X(t) x}, f X(t) (x) = df X(t)(x), dx F X(t1 ),X(t 2 )(x 1, x 2 ) = Pr{X(t 1 ) x 1, X(t 2 ) x 2 }, F X(t1 ),,X(t n )(x 1,, x n ) = Pr{X(t 1 ) x 1,, X(t n ) x n }, 4.1 Random process

6 Random processes - Chapter 4 Random process 6 Mean function The mean function m X (t) of a random process {X(t)} is defined by Autocorrelation function m X (t) = E{X(t)} = xf X(t) (x)dx. The autocorrelation function R X (t 1, t 2 ) of a random process {X(t)} is defined by R X (t 1, t 2 ) = E{X(t 1 )X (t 2 )}. 4.1 Random process

7 Random processes - Chapter 4 Random process 7 Known signal An extreme example of a random process is a known or deterministic signal. When X(t) = s(t) is a known signal, we have m(t) = E{s(t)} = s(t), R(t 1, t 2 ) = E{s(t 1 )s(t 2 )} = s(t 1 )s(t 2 ). Consider the random process {X(t)} with mean E{X(t)} = 3 and autocorrelation function R(t 1, t 2 ) = exp( 0.2 t 1 t 2 ). If Z = X(5), W = X(8), we can easily obtain E{Z} = E{X(5)} = 3, E{W } = 3, E{Z 2 } = R(5, 5) = 13, E{W 2 } = R(8, 8) = 13, Var{Z} = = 4, Var{W } = = 4, and E{ZW } = R(5, 8) = 9 + 4e In other words, the random variables Z and W have the variance σ 2 = 4 and covariance Cov(5, 8) = 4e Random process

8 Random processes - Chapter 4 Random process 8 The autocorrelation function of X(t) = Ae jωt defined with a random variable A can be obtained as R X (t 1, t 2 ) = E{Ae jωt 1 A e jωt 2 } = e jω(t 1 t 2 ) E{ A 2 }. Autocovariance function The autocovariance function K X (t 1, t 2 ) of a random process {X(t)} is defined by K X (t 1, t 2 ) = E{[X(t 1 ) m X (t 1 )][X (t 2 ) m X(t 2 )]}. In general, the autocovariance and autocorrelation functions are functions of t 1 and t 2. The autocovariance function can be expressed in terms of the autocorrelation and mean functions as K X (t 1, t 2 ) = R X (t 1, t 2 ) m X (t 1 )m X(t 2 ). 4.1 Random process

9 Random processes - Chapter 4 Random process 9 Uncorrelated random process A random process {X t } is said to be uncorrelated if R X (t, s) = E{X t }E{X s } or K X (t, s) = 0 for t s. If a random process {X(t)} is uncorrelated, the autocorrelation and autocovariance functions are and respectively. R X (t, s) = E{X t Xs } { E{ Xt 2 }, t = s, = E{X t }E{Xs }, t s, { σ 2 Xt, t = s, K X (t, s) = 0, t s, 4.1 Random process

10 Random processes - Chapter 4 Random process 10 Correlation coefficient function The correlation coefficient function ρ X (t 1, t 2 ) of a random process {X(t)} is defined by ρ X (t 1, t 2 ) = K X (t 1, t 2 ) KX (t 1, t 1 ) K X (t 2, t 2 ) = K X(t 1, t 2 ) σ(t 1 )σ(t 2 ), where σ(t i ) is the standard deviation of X(t i ). We can show that ρ X (t 1, t 2 ) 1, ρ X (t, t) = Random process

11 Random processes - Chapter 4 Random process 11 Crosscorrelation function The crosscorrelation function R XY (t 1, t 2 ) of random processes {X(t)} and {Y (t)} is defined by R XY (t 1, t 2 ) = E{X(t 1 )Y (t 2 )}. The autocorrelation and crosscorrelation functions satisfy R X (t, t) = E{X(t)X (t)} = σ 2 X(t) + E{X(t)} 2 0, R X (t 1, t 2 ) = R X(t 2, t 1 ), R XY (t 1, t 2 ) = R Y X(t 2, t 1 ). The autocorrelation function R X (t 1, t 2 ) is positive semi-definite. In other words, a i a j R(t i, t j ) 0 for non-negative constants {a k }. i j 4.1 Random process

12 Random processes - Chapter 4 Random process 12 Crosscovariance function The crosscovariance function K XY (t 1, t 2 ) of random processes {X(t)} and {Y (t)} is defined by K XY (t 1, t 2 ) = E{[X(t 1 ) m X (t 1 )][Y (t 2 ) m Y (t 2 )]} = R XY (t 1, t 2 ) m X (t 1 )m Y (t 2 ). Two random processes which are uncorrelated The random processes {X(t)} and {Y (t)} are said to be uncorrelated if R XY (t 1, t 2 ) = E{X(t 1 )}E{Y (t 2 )} or K XY (t 1, t 2 ) = 0 for all t 1 and t 2. Orthogonality The two random processes {X(t)} and {Y (t)} are said to be orthogonal if R XY (t 1, t 2 ) = 0 for all t 1 and t Random process

13 Random processes - Chapter 4 Random process 1 Random process Chapter 4 Random process 4.2 Properties of random processes 4.2 Properties of random processes

14 Random processes - Chapter 4 Random process 2 Stationary process and independent process A random process is said to be stationary if the probabilistic properties do not change under time shifts. Stationary process A random process {X(t)} is stationary, strict-sense stationary (s.s.s.), or stronglystationary if the joint cdf of {X(t 1 ), X(t 2 ),, X(t n )} is the same as the joint cdf of {X(t 1 + τ), X(t 2 + τ),, X(t n + τ)} for all n, τ, t 1, t 2,, t n. 4.2 Properties of random processes / Stationary process and independent process

15 Random processes - Chapter 4 Random process 3 Wide-sense stationary (w.s.s.) process A random process {X(t)} is w.s.s., weakly-stationary, or second-order stationary if (1) the mean function is constant and (2) the autocorrelation function R X (t, s) depends only on t s but not on t and s individually. The mean function m X and the autocorrelation function R X of a w.s.s. process {X(t)} are thus and respectively. m X (t) = m R X (t 1, t 2 ) = R(t 1 t 2 ), In other words, the autocorrelation function R X (t 1, t 2 ) of a w.s.s. process {X(t)} is a function of τ = t 1 t 2. For all t and τ, we have When τ = 0, R X (0) = E{ X(t) 2 }. R X (τ) = E{X(t + τ)x (t)}. 4.2 Properties of random processes / Stationary process and independent process

16 Random processes - Chapter 4 Random process 4 Consider two sequences of uncorrelated random variables A 0, A 1,, A m and B 0, B 1,, B m having mean zero and variance σi 2. Assume that the two sequences are uncorrelated with each other. Let ω 0, ω 1,, ω m be distinct frequencies in [0, 2π), and let X n = m {A k cos nω k +B k sin nω k } for n = 0, ±1, ±2,. Then we can obtain Thus, {X n } is w.s.s.. k=0 m E{X n X n+l } = σk 2 cos lω k, k=0 E{X n } = Properties of random processes / Stationary process and independent process

17 Random processes - Chapter 4 Random process 5 Properties of the autocorrelation function R X (τ) of a real stationary process {X(t)} R X ( τ) = R X (τ) : R X(τ) is an even function. R X (τ) R X (0) : R X (τ) is maximum at the origin. If R X (τ) is continuous τ = 0, then it is also continuous at every value of τ. If there is a constant T > 0 such that R X (0) = R X (T ), then R X (τ) is periodic. 4.2 Properties of random processes / Stationary process and independent process

18 Random processes - Chapter 4 Random process 6 Independent random process A random process is said to be independent if the joint cdf satisfies n F Xt1,X t2,,x (x) = t F n Xti (x i ) for all n and t 1, t 2,, t n, x 1, x 2,, x n. Independent and identically distributed (i.i.d.) process A random process is said to be i.i.d. if the joint cdf satisfies n F Xt1,X t2,,x (x) = t F n X (x i ) for all n and t 1, t 2,, t n, x 1, x 2,, x n. The i.i.d. process is sometimes called a memoryless process or a white noise. The i.i.d. process is the simplest process, and yet it is the most stochastic process in that past outputs do not have any information on the future. i=1 i=1 4.2 Properties of random processes / Stationary process and independent process

19 Random processes - Chapter 4 Random process 7 Bernoulli process An i.i.d. random process with two possible values is called a Bernoulli process. For example, consider the random process {X n } defined by X n = { 1, if the nth result is head, 0, if the nth result is tail, when we toss a coin infinitely. The random process {X n } is a discrete-time discreteamplitude random process. The success ( head ) and failure ( tail ) probabilities are P {X n = 1} = p and P {X n = 0} = 1 p, respectively. 4.2 Properties of random processes / Stationary process and independent process

20 Random processes - Chapter 4 Random process 8 We can easily obtain and m X (n) = E{X n } = p K X (n 1, n 2 ) = E{X n1 X n2 } m X (n 1 )m X (n 2 ) { p(1 p), n 1 = n 2, = 0, n 1 n 2. The mean function m X (n) of a Bernoulli process is not a function of time but a constant. The autocovariance K X (n 1, n 2 ) depends not on n 1 and n 2 individually, but only on the difference n 1 n 2. Clearly, the Bernoulli process is w.s.s Properties of random processes / Stationary process and independent process

21 Random processes - Chapter 4 Random process 9 Two random processes independent of each other The random processes {X(t)} and {Y (t)} are said to be independent of each other if the random vector (X t1, X t2,, X tk ) is independent of the random vector (Y s1, Y s2,, Y sl ) for all k, l, and t 1, t 2,, t k, s 1, s 2,, s l. 4.2 Properties of random processes / Stationary process and independent process

22 Random processes - Chapter 4 Random process 10 Normal process, Gaussian process A random process {X t } is said to be normal if (X t1, X t2,, X tk ) is a k dimensional normal random vector for all k and t 1, t 2,, t k. A stationary process is always w.s.s., but the converse is not always true. On the other hand, a w.s.s. normal process is s.s.s.. This result can be obtained from the pdf f X (x) = { 1 exp 1 } (2π) n/2 K X 1/2 2 (x m X) T K 1 X (x m X) of a jointly normal random vector. 4.2 Properties of random processes / Stationary process and independent process

23 Random processes - Chapter 4 Random process 11 Jointly w.s.s. processes Two random processes are said to be jointly w.s.s. if (1) the mean functions are constants and and (2) the autocorrelation functions and crosscorrelation function are all functions only of time differences. If two random processes {X(t)} and {Y (t)} are jointly w.s.s., then {X(t)} and {Y (t)} are both w.s.s.. The crosscorrelation function of {X(t)} and {Y (t)} is R XY (t + τ, t) = R XY (τ) = E{X(t + τ)y (t)} = ( E{Y (t)x (t + τ)} ) = R Y X( τ). 4.2 Properties of random processes / Stationary process and independent process

24 Random processes - Chapter 4 Random process 12 The crosscorrelation function R XY of two jointly w.s.s. random processes has the following properties: 1. R Y X (τ) = R XY ( τ). 2. R XY (τ) R XX (0)R Y Y (0) 1 2 {R XX(0) + R Y Y (0)}. 3. R XY (τ) is not always maximum at the origin. Linear transformation and jointly normal process Two processes X(t) and Y (t) are w.s.s. if the linear combination Z(t) = ax(t) + by (t) is w.s.s. for all a and b. The converse is also true. 4.2 Properties of random processes / Stationary process and independent process

25 Random processes - Chapter 4 Random process 13 Moving average (MA) process Let a 1, a 2,, a l be a sequence of real numbers and W 0, W 1, W 2, be a sequence of uncorrelated random variables with mean E{W n } = m and variance Var{W n } = σ 2. Then the following process {X n } is called a moving average process. X n = a 1 W n + a 2 W n a l W n l+1 l = a i W n i+1. i=1 The mean and variance of X n are E{X n } = (a 1 + a a l )m, Var{X n } = (a a a 2 l )σ Properties of random processes / Stationary process and independent process

26 Random processes - Chapter 4 Random process 14 Since E{ ˆX 2 n} = σ 2 when ˆX n = W n m, {X n } is w.s.s. from Cov(X n, X n+k ) = E{ ( X n m { ( l = E = = i=1 l a i )(X n+k m i=1 a i ˆXn i+1 )( l ) } a i i=1 l ) } a j ˆXn+k j+1 j=1 E { a l a l k ˆX2 n+k l+1 + a l 1 a l k 1 ˆX n+k l a } k+1a 1 ˆX2 n, k l 1, 0, k l, { (a l a l k + + a k+1 a 1 )σ 2, k l 1, 0, k l. 4.2 Properties of random processes / Stationary process and independent process

27 Random processes - Chapter 4 Random process 15 Autoregressive (AR) process Let the variance of an uncorrelated zero-mean random sequence Z 0, Z 1, be { σ 2, n = 0, Var{Z n } = 1 λ 2 σ 2, n 1, where λ 2 < 1. Then the random process {X n } defined by X 0 = Z 0, X n = λx n 1 + Z n, n 1. is called the first order autoregressive process. We can obtain X n = λ(λx n 2 + Z n 1 ) + Z n = λ 2 X n 2 + λz n 1 + Z n. n = λ n i Z i. i=0 4.2 Properties of random processes / Stationary process and independent process

28 Random processes - Chapter 4 Random process 16 Thus the autocovariance function of {X n } is ( n Cov(X n, X n+m ) = Cov λ n i Z i, = i=0 n+m i=0 n λ n i λ n+m i Cov(Z i, Z i ) i=0 λ n+m i Z i ) ( ) = σ 2 λ 2n+m 1 n 1 λ + λ 2i 2 = σ2 λ m 1 λ 2. Now, from the result above and the fact that the mean of {X n } is E{X n } = 0, it follows that {X n, n 0} is w.s.s.. i=1 4.2 Properties of random processes / Stationary process and independent process

29 Random processes - Chapter 4 Random process 17 Square-law detector Let Y (t) = X 2 (t) where {X(t)} is a Gaussian random process with mean 0 and autocorrelation R X (τ). Then the expectation of Y (t) is E{Y (t)} = E{X 2 (t)} = R X (0). Since X(t+τ) and X(t) are jointly Gaussian with mean 0, the autocorrelation of Y (t) can be found as R Y (t, t + τ) = E{X 2 (t)x 2 (t + τ)} = E{X 2 (t + τ)}e{x 2 (t)} + 2E 2 {X(t + τ)x(τ)} = R 2 X(0) + 2R 2 X(τ). Thus E{Y 2 (t)} = R Y (0) = 3RX 2 (0) and σ2 Y = 3R2 X (0) R2 X (0) = 2R2 X (0). Clearly, {Y (t)} is w.s.s Properties of random processes / Stationary process and independent process

30 Random processes - Chapter 4 Random process 18 Limiter Let {Y (t)} = {g(x(t))} be a random process which is defined by a random process {X(t)} and a limiter { 1, x > 0, g(x) = 1, x < 0. Then we can easily obtain P {Y (t) = 1} = P {X(t) > 0} = 1 F X (0) and P {Y (t) = 1} = P {X(t) < 0} = F X (0). Thus the mean and autocorrelation of {Y (t)} are E{Y (t)} = 1 P {Y (t) = 1} + ( 1) P {Y (t) = 1} = 1 2F X (0), R Y (τ) = E{Y (t)y (t + τ)} = P {Y (t)y (t + τ) = 1} P {Y (t)y (t + τ) = 1} = P {X(t)X(t + τ) > 0} P {X(t)X(t + τ) < 0}. Now, if {X(t)} is a stationary Gaussian random process with mean 0, X(t + τ) and X(t) are jointly Gaussian with mean 0, variance R X (0), and correlation coefficient R X (τ)/r X (0). Clearly, F X (0) = 1/ Properties of random processes / Stationary process and independent process

31 Random processes - Chapter 4 Random process 19 We have (refer to (3.100), p. 156, Random Processes, Park, Song, Nam, 2004) { X(t) } P {X(t)X(t + τ) < 0} = P X(t + τ) < 0 = F Z (0) = π tan 1 rσ 1 σ 1 1 r 2 = π sin 1 r = π sin 1 R X(τ) R X (0), P {X(t)X(t + τ) > 0} = 1 P {X(t)X(t + τ) < 0} = π sin 1 R X(τ) R X (0). Thus the autocorrelation of the limiter output is R Y (τ) = 2 π sin 1 R X(τ) R X (0), from which we have E{Y 2 (t)} = R Y (0) = 1 and σ 2 Y = 1 {1 2F X(0)} 2 = Properties of random processes / Stationary process and independent process

32 Random processes - Chapter 4 Random process 20 Power of a random process Power spectrum The Fourier transform of the autocorrelation function of a w.s.s. random process is called the power spectrum, power spectral density, or spectral density. It is usually assumed that the mean is zero. When the autocorrelation is R X (τ), the power spectrum is S X (ω) = F{R X } R X (k)e jωk, k = R X (τ)e jωτ dτ, discrete time, continuous time. If the mean is not zero, the power spectral density is defined by the Fourier transform of the autocovariance instead of the autocorrelation. 4.2 Properties of random processes / Power of random process

33 Random processes - Chapter 4 Random process 21 White noise, white process Suppose that a discrete time random process {X n } is uncorrelated so that R X (k) = σ 2 δ k. Then we have S X (ω) = k σ 2 δ k e jωk = σ 2. Such a process is called a white noise or white process. If {X n } is Gaussian in addition, it is called a white Gaussian noise. 4.2 Properties of random processes / Power of random process

34 Random processes - Chapter 4 Random process 22 Telegraph signal Consider Poisson points with parameter λ. Let N(t) be the number of points in the interval (0, t]. As shown in the figure above, consider the continuous-time random process X(t) = ( 1) N(t) with X(0) = 1. Here, W i is the time between adjacent Poisson points. Assuming τ > 0, the autocorrelation of X(t) is R X (τ) = E{X(t + τ)x(t)} = 1 P {the number of points in the interval (t, t + τ] is even} +( 1) P {the number of points in the interval (t, t + τ] is odd} { } } = e λτ 1 + (λτ)2 + e {λt λτ + (λτ)3 + 2! 3! = e λτ cosh λτ e λτ sinh λτ = e 2λτ. 4.2 Properties of random processes / Power of random process

35 Random processes - Chapter 4 Random process 23 We can obtain a similar result when τ < 0, Combining the two results, we have R X (τ) = e 2λ τ. Consider a random variable A which is independent of X(t) and takes on 1 or 1 with equal probability. Let Y (t) = AX(t). We then have E{Y (t)} = E{A}E{X(t)} = 0, E{Y (t 1 )Y (t 2 )} = E{A 2 }E{X(t 1 )X(t 2 )} = E{A 2 }R X (t 1 t 2 ) = e 2λ t 1 t 2 since E{A} = 0, E{A 2 } = 1. Thus {Y (t)} is w.s.s., and the power spectral density of {Y (t)} is S Y (ω) = F{e 2λ τ } 4λ = ω 2 + 4λ Properties of random processes / Power of random process

36 Random processes - Chapter 4 Random process 24 Band limited noise, colored noise When W > 0, let us consider a random process with the power spectral density { 1, ω [ W, W ], S X (ω) = 0, otherwise. Such a process is called a colored noise. The autocorrelation of a colored noise is thus R X (τ) = F 1 {S X (ω)} = sin(w τ). πτ 4.2 Properties of random processes / Power of random process

37 Random processes - Chapter 4 Random process 25 Power spectral density is not less than 0. That is, S X (ω) 0. Cross power spectral density The cross power spectral density S XY (ω) of jointly w.s.s. {Y (t)} is S XY (ω) = R XY (τ)e jωτ dτ. processes {X(t)} and Thus S XY (ω) = S Y X (ω), and the inverse Fourier transform of S XY (ω) is R XY (τ) = 1 2π S XY (ω)e jωτ dω. 4.2 Properties of random processes / Power of random process

38 Random processes - Chapter 4 Random process 26 Time delay process Consider a w.s.s. process {X(t)} of which the power spectral density is S X (ω). Letting Y (t) = X(t d), we have R Y (t, s) = E{Y (t)y (s)} = E{X(t d)x (s d)} = R X (t s). Thus the process {Y (t)} is w.s.s. and the power spectral density S Y (ω) equals to S X (ω). In addition, the crosscorrelation and cross power spectral density of {X(t)} and {Y (t)} are and R XY (t, s) = E{X(t)Y (s)} = E{X(t)X (s d)} = R X (t s + d) = R X (τ + d) S XY (ω) = F{R X (τ + d)} = = That is, {X(t)} and {Y (t)} are jointly w.s.s.. R X (τ + d)e jωτ dτ R X (u)e jωu e jωd du = S X (ω)e jωd. 4.2 Properties of random processes / Power of random process

39 Random processes - Chapter 4 Random process 27 Random process and linear systems If the input random process is two-sided and w.s.s., the output of a linear time invariant (LTI) filter is also w.s.s.. If the input random process is one-sided and w.s.s., however, the output of an LTI filter is not w.s.s. in general. 4.2 Properties of random processes / Random process and linear systems

40 Random processes - Chapter 4 Random process 28 Let h(t) be the impulse response of an LTI system and let H(ω) = F{h(t)} be the transfer function. Then the crosscorrelation R XY (t 1, t 2 ) of the input random process {X(t)} and output random process {Y (t)} and autocorrelation R Y (t 1, t 2 ) of the output are { } R XY (t 1, t 2 ) = E{X(t 1 )Y (t 2 )} = E X(t 1 ) X (t 2 α)h (α)dα = E{X(t 1 )X (t 2 α)}h (α)dα = R X (t 1, t 2 α)h (α)dα and R Y (t 1, t 2 ) = E{Y (t 1 )Y (t 2 )} = E{ } X(t 1 α)h(α)dα X (t 2 β)h (β)dβ = R X (t 1 α, t 2 β)h(α)h (β)dαdβ = R XY (t 1 α, t 2 )h(α)dα, respectively. 4.2 Properties of random processes / Random process and linear systems

41 Random processes - Chapter 4 Random process 29 If the input and output are jointly w.s.s., we can obtain R XY (τ) = R X (t + α)h (α)dα = R X (τ) h ( τ), R Y (τ) = R XY (τ) h(τ). since R X (t 1, t 2 α) = R X (t 1 t 2 + α) = R X (τ + α) and R XY (t 1 α, t 2 ) = R XY (t 1 t 2 α) = R XY (τ α). The cross power spectral density and power spectral density of output are respectively. S XY (ω) = S X (ω)h (ω), S Y (ω) = S XY (ω)h(ω), 4.2 Properties of random processes / Random process and linear systems

42 Random processes - Chapter 4 Random process 30 We can express the autocorrelation and power spectral density of the output in terms of those of the input. Specifically, we have and R Y (τ) = R X (τ) ρ h (τ) S Y (ω) = S X (ω) H(ω) 2, where ρ h (t) is called the deterministic autocorrelation of h(t) and is defined by ρ h (t) = F 1 ( H(ω) 2 ) = h(t) h ( t) = h(t + τ)h (τ)dτ. Let S Y (ω) be the power spectral density of the output process {Y t }. Then we can obtain R Y (τ) = F 1 {S Y (ω)} = F 1 { H(ω) 2 S X (ω)}. 4.2 Properties of random processes / Random process and linear systems

43 Random processes - Chapter 4 Random process 31 Coherence function A measure of the degree to which two w.s.s. processes are related by an LTI transformation is the coherence function ρ(ω) defined by ρ XY (ω) = S XY (ω) [S X (ω)s Y (ω)] 1/2. Here, ρ(ω) = 1 if and only if {X(t)} and {Y (t)} are the linearly related, that is, Y (t) = X(t) h(t). Note the similarity between the coherence function ρ(ω) and the correlation coefficient ρ, a measure of the degree to which two random variables are linearly related. The coherence function exhibits the property ρ(ω) 1 similar to the correlation coefficient between two random variables. 4.2 Properties of random processes / Random process and linear systems

44 Random processes - Chapter 4 Random process 32 Ergodic theorem* If there exists a random variable ˆX such that 1 T T 0 1 n n 1 i=0 X i n ˆX, discrete time random process, X(t)dt T ˆX, continuous time random process. {X n, n I} is said to satisfy ergodic theorem. When a process satisfies an ergodic theorem, the sample mean converges to something, which may be different from the expectation. In some cases, a random process with time-varying mean satisfies an ergodic theorem as shown in the example below. 4.2 Properties of random processes / Ergodic theorem*

45 Random processes - Chapter 4 Random process 33 Suppose that nature at the beginning of time randomly selects one of two coins with equal probability, one having bias p and the other having bias q. After the coin is selected it is flipped once per second forever. The output random process is a one-zero sequence depending on the face of a coin. Clearly, the time average will converge: it will converge to p if the first coin was selected and to q if the second coin was selected. That is, the time average will converge to a random variable. In particular, it will not converge to the expected value p/2 + q/2. If lim n E{(Y n Y ) 2 } = 0, Y n, n = 1, 2, is said to converge to Y in mean square, which is denoted as where l.i.m. denotes limit in the mean. l.i.m. n Y n = Y, 4.2 Properties of random processes / Ergodic theorem*

46 Random processes - Chapter 4 Random process 34 Mean ergodic theorem Let {X n } be an uncorrelated discrete time random process with finite mean E{X n } = m and finite variance σx 2 n = σx 2. Then the sample mean S n = n 1 X i /n converges to the expected value E{X n } = m in mean square. That is, l.i.m. n 1 n n 1 i=0 X i = m. A sufficient condition for a w.s.s. discrete time random process {X n, n I} to satisfy a mean ergodic theorem is K X (0) < and lim K X(n) = 0. n i=0 4.2 Properties of random processes / Ergodic theorem*

47 Random processes - Chapter 4 Random process 35 Let {X n } be a discrete time random process with mean E{X n } and autocovariance function K X (i, j). The process need not be stationary in any sense. A necessary and sufficient condition for is and lim n l.i.m. n lim n 1 n 2 1 n 1 n n 1 i=0 n 1 i=0 n 1 i=0 n 1 k=0 X i = m E{X i } = m K X (i, k) = 0. That is, if and only if a process is asymptotically uncorrelated and its sample averages converge, the sample mean converge in mean square. 4.2 Properties of random processes / Ergodic theorem*

48 Random processes - Chapter 4 Random process 36 Mean square ergodic theorem Let Xn = n i=1 X i /n where {X n, n 1} is a second order stationary process with mean m and autocovariance K(i) = Cov(X n, X n+i ). Then lim E{( X n m) 2 } = 0 n if and only if lim n n 1 i=0 K(i)/n = 0. Let K(i) be the autocovariance of {X n }, a second order stationary Gaussian process with mean 0. If then lim T 1 T T K 2 (i) = 0, i=1 lim E{ ˆK T (i) K(i) 2 } = 0 T for i = 1, 2,, where ˆK T (i) = T X l X l+i /T is the sample autocovariance. l=1 4.2 Properties of random processes / Ergodic theorem*

49 Random processes - Chapter 4 Random process 37 Ergodicity* Shift operator An operator T for which T ω = {x t+1, t I}, where ω = {x t, t I} is an infinite sequence in the probability space (R I, B(R) I, P ), is called a shift operator. Stationary process A discrete time random process with process distribution P P (T 1 F ) = P (F ) for any element F in B(R) I. is stationary if 4.2 Properties of random processes / Ergodicity*

50 Random processes - Chapter 4 Random process 38 Invariant event An event F is said to be invariant with respect to the shift operator T if and only if T 1 F = F. Ergodicity, ergodic process A random process is said to be ergodic if P (F ) = 0 or P (F ) = 1 for any invariant event F. Consider a two-sided process with distribution P (, x 1 = 1, x 0 = 0, x 1 = 1, x 2 = 0, ) = p, P (, x 1 = 0, x 0 = 1, x 1 = 0, x 2 = 1, ) = 1 p. Clearly, F = {sequence of alternating 0 and 1} is an invariant event, and has probability P (F ) = 1. Any other invariant event - for example, the all 1 sequence - that does not include F has probability 0. Thus the random process is ergodic. Ergodicity has nothing to do with stationarity or convergence of sample averages. 4.2 Properties of random processes / Ergodicity*

51 Random processes - Chapter 4 Random process 1 Random processes Chapter 4 Random process 4.3 Process with i.s.i. 4.3 Process with i.s.i.

52 Random processes - Chapter 4 Random process 2 Process with i.s.i. Process with independent increments A random process {Y t,t I} is said to have independent increments or to be an independent increment process if for all choices of k =1, 2, and all choices of ordered sample times {t 0,t 1,,t k }, the k increments Y ti Y ti 1,i =1, 2,,k are independent random variables. Process with stationary increments When the increments {Y t Y s } are stationary, the random process {Y t } is called a stationary increment random process. Process with i.s.i. A random process is called an independent and stationary increment (i.s.i.) process if its increments are independent and stationary. 4.3 Process with i.s.i.

53 Random processes - Chapter 4 Random process 3 A discrete time random process is an i.s.i. process if and only if it can be represented as the sum of i.i.d. random variables. Mean and autocovariance of i.s.i. process The mean and autocovariance of a discrete time i.s.i. process are and E{Y t } = te(y 1 ), t 0 K Y (t, s) = σ 2 Y 1 min(t, s), t,s 0. Clearly, an i.s.i. process itself is not stationary. 4.3 Process with i.s.i.

54 Random processes - Chapter 4 Random process 4 Let the process {X t,t T } be an i.s.i. process. If we have m 0 = E{X 0 }, m 1 = E{X 1 } m 0, σ0 2 = E{(X 0 m 0 ) 2 }, σ1 2 = E{(X 1 m 1 ) 2 } σ0, 2 E{X t } = m 0 + m 1 t, Var{X t } = σ0 2 + σ1t Process with i.s.i.

55 Random processes - Chapter 4 Random process 5 Point process and counting process A sequence T 1 T 2 T 3 of ordered random variables is called a point process. For example, the set of times defined by Poisson points is a point process. A counting process Y (t) can be defined as the number of points in the interval [0,t). We have, with T 0 =0,. Y (t) =i, T i t<t i+1, i =0, 1,. 4.3 Process with i.s.i.

56 Random processes - Chapter 4 Random process 6 A counting process A process constructed by summing the outputs of a Bernoulli process Let {X n,n =1, 2, } be a Bernoulli process with parameter p. random process {Y n,n=1, 2, } as Define the Y 0 = 0, n Y n = i=1 X i = Y n 1 + X n,n=1, 2,. Since the random variable Y n represents the number of 1 s in {X 1,X 2,,X n }, we have Y n = Y n 1 or Y n = Y n 1 +1, n =2, 3,. In general, a discrete time process satisfying this relation is called a counting process since it is nondecreasing, and when it jumps, it is always with an increment of Process with i.s.i.

57 Random processes - Chapter 4 Random process 7 Properties of the random process {Y n } E{Y n } = np, Var{Y n } = np(1 p), K Y (k, j) =p(1 p)min(k, j) Marginal pmf for Y n p Yn (y) = Pr{ there are exactly y ones in X 1,X 2,,X n.} ( ) n = p y (1 p) n y, y =0, 1,,n. y Since the marginal pdf is binomial, the process {Y n } is called a binomial counting process. The process {Y n } is not stationary since the marginal pmf depends on the time index n. 4.3 Process with i.s.i.

58 Random processes - Chapter 4 Random process 8 Random walk process One dimensional random walk, random walk Consider the modified Bernoulli process for which the event failure is represented by 1 instead of 0. { +1, for success in the nth trial, Z n = 1, for failure in the nth trial. Let p = Pr{Z n =1}, and consider the sum n W n = of the variables Z n. The process {W n } is referred to as one dimensional random walk or random walk. i=1 Z i 4.3 Process with i.s.i.

59 Random processes - Chapter 4 Random process 9 Since Z i = 2X i 1, it follows that the random walk process {W n } is related to the binomial counting process {Y n } by W n = 2Y n n. Using the results on the mean and autocorrelation functions of the binomial counting process and the linearity of expectation, we have m W (n) = (2p 1)n, K W (n 1,n 2 ) = 4p(1 p)min(n 1,n 2 ), σ 2 W n = 4p(1 p)n. 4.3 Process with i.s.i.

60 Random processes - Chapter 4 Random process 1 Random process Chapter 4 Random process 4.4 Discrete time process with i.s.i. 4.4 Discrete time process with i.s.i.

61 Random processes - Chapter 4 Random process 2 Discrete time process with i.s.i. Discrete time discrete alphabet process {Y n } with i.s.i. As mentioned before, {Y n } can be defined by the sum of i.i.d. random variables {X i }. Let us consider the following conditional pmf. p Yn Y n 1(y n y n 1 ) = p Yn Y n 1(y n y n 1,,y 1 ) = Pr(Y n = y n Y n 1 = y n 1 ), where Y n =(Y n,y n 1,,Y 1 ) and y n =(y n,y n 1,,y 1 ). The conditioning event {Y i = y i,i=1, 2,,n 1} above is the same as the event {X 1 = y 1,X i = y i y i 1, i =2,,n 1}. In addition, under the conditioning event, we have Y n = y n if and only if X n = y n y n Discrete time process with i.s.i.

62 Random processes - Chapter 4 Random process 3 Assuming y 0 =0, p Yn Y n 1(y n y n 1 ) = Pr(Y n = y n X 1 = y 1,X i = y i y i 1,i=2, 3,,n 1) = Pr(X n = y n y n 1 X i = y i y i 1,i=1, 2,,n 1) = p Xn X n 1(y n y n 1 y n 1 y n 2,,y 2 y 1,y 1 ), where X n 1 =(X n 1,X n 2,,X 1 ). If {X n } are i.i.d., p Yn Y n 1(y n y n 1 )=p X (y n y n 1 ) since X n is independent of X k for k<n. Thus the joint pmf is p Y n(y n ) = p Yn Y n 1(y n y n 1 ) p Y n 1(y n 1 ). n = p Y1 (y 1 ) p Yi Y i 1,,Y 1 (y i y i 1,,y 1 )= i=2 n p X (y i y i 1 ). i=1 4.4 Discrete time process with i.s.i.

63 Random processes - Chapter 4 Random process 4 Applying the result above to the binomial counting process, we obtain p Y n(y n )= n p (y i y i 1 ) (1 p) 1 (y i y i 1 ), where y i y i 1 =0or 1 for i =1, 2,,n and y 0 =0. i=1 Properties of processes with i.s.i. We can express the conditional pmf of Y n given Y n 1 as follows: p Yn Y n 1 (y n y n 1 ) = Pr(Y n = y n Y n 1 = y n 1 ) = Pr(X n = y n y n 1 Y n 1 = y n 1 ). The conditioning event {Y n 1 = y n 1 } depends only on X k for k<n,andx n is independent of X k for k<n. Thus, this conditioning event does not affect X n. Consequently, p Yn Y n 1 (y n y n 1 )=p X (y n y n 1 ). 4.4 Discrete time process with i.s.i.

64 Random processes - Chapter 4 Random process 5 Discrete time i.s.i. processes (such as the binomial counting process and discrete random walk) has the following property: p Yn Y n 1(y n y n 1 ) = p Yn Y n 1 (y n y n 1 ), Pr{Y n = y n Y n 1 = y n 1 } = Pr{Y n = y n Y n 1 = y n 1 }. Roughly speaking, given the most recent past sample (or the current sample), the other past samples do not affect the probability of what happens next. A discrete time discrete alphabet random process with this property is called a Markov process. Thus all i.s.i. processes are Markov processes. 4.4 Discrete time process with i.s.i.

65 Random processes - Chapter 4 Random process 6 Gambler s ruin problem A person wants to buy a new car of which the price is N won. The person has k (0 <k<n) won, and he intends to earn the difference from gambling. The game this person is going to play is that if a toss of a coin results in a head, he will earn 1 won, and if it results in a tail, he will lose 1 won. Let p represent the probability of heads, and q =1 p. Assuming the man continues to play the game until he earns enough money for a new car or lose all the money he has, what is the probability that the man loses all the money he has? Let A k be the event that the man loses all the money when he has started with k won and B be the event the man wins a game. Then, P (A k ) = P (A k B)P (B)+P (A k B c )P (B c ). Since the game will start again with k +1won if a toss of a coin results in a head and k 1 won if a toss of a coin results in a tail, it is easy to see that P (A k B) = P (A k+1 ), P (A k B c ) = P (A k 1 ). 4.4 Discrete time process with i.s.i.

66 Random processes - Chapter 4 Random process 7 Let p k = P (A k ), then p 0 =1, p N =0,and p k = pp k+1 + qp k 1, 1 k N 1. Assuming p k = θ k, we get from the equation above which gives θ 1 =1and θ 2 = q/p. pθ 2 θ + q =0, If p 0.5, p k = A 1 θ1 k + A 2 θ2. k Thus using the boundary conditions p 0 =1and p N =0, we can find p k = (q/p)k (q/p) N. 1 (q/p) N If p =0.5, wehavep k =(A 1 + A 2 k)θ k 1 since q/p =1. Thus using the boundary conditions p 0 =1and p N =0,wecanfind p k =1 k N. 4.4 Discrete time process with i.s.i.

67 Random processes - Chapter 4 Random process 8 Discrete time Wiener process Discrete time Wiener process Let {X n } be an i.i.d. zero-mean Gaussian process with variance σ 2. The discrete time Wiener process {Y n } is defined by Y 0 = 0, n Y n = i=1 X i = Y n 1 + X n, n =1, 2,. The discrete time Wiener process is also called the discrete time diffusion process or discrete time Brownian motion. Since the discrete time Wiener process is formed as sums of an i.i.d. process, it has i.s.i.. Thus we have E{Y n } =0and K Y (k, j) =σ 2 min(k, j). The Wiener process is a first-order autoregressive process. 4.4 Discrete time process with i.s.i. / discrete time Wiener process

68 Random processes - Chapter 4 Random process 9 The discrete time Wiener process is a Gaussian process with mean function m(t) =0and autocovariance function K X (t, s) =σ 2 min(t, s). Since the discrete time Wiener process is an i.s.i. process, we have f Yn Y n 1 (y n y n 1 ) = f X (y n y n 1 ), f Yn Y n 1(y n y n 1 ) = f Yn Y n 1 (y n y n 1 ) = f X (y n y n 1 ). As in the discrete alphabet case, a process with this property is called a Markov process. Markov process A discrete time random process {Y n } is called a first order Markov process if it satisfies for all n, y n,y n 1,y n 2,. Pr{Y n y n y n 1,y n 2, }= Pr{Y n y n y n 1 }. 4.4 Discrete time process with i.s.i. / discrete time Wiener process

69 Random processes - Chapter 4 Random process 1 Random processes Chapter 4 Random process 4.5 Continuous time i.s.i. process 4.5 Continuous time i.s.i. process

70 Random processes - Chapter 4 Random process 2 Continuous time i.s.i. process Continuous time i.s.i. process When we deal with a continuous time process with i.s.i. we need to consider more general collection of sample times than in the case of discrete time process. In the continuous time case, we assume that we are given the cdf of the increments as F Yt Y s (y) = F Y t s Y 0 (y) = F Y t s (y), t > s. 4.5 Continuous time i.s.i. process

71 Random processes - Chapter 4 Random process 3 The joint probability functions of a continuous time process Define the random variable {X n } by X n = Y tn Y tn 1. Then {X n } are independent and Y tn = n X i, i=1 Pr{Y tn y n Y tn 1 = y n 1,Y tn 2 = y n 2, }= F Xn (y n y n 1 ) = F Yt n Y t n 1 (y n y n 1 ). As in the case of discrete time processes, these can be used to find the joint pmf or pdf as n p Yt1,,Y (y t n 1,,y n ) = p Yti Y ti 1 (y i y i 1 ), f Yt1,,Y t n (y 1,,y n ) = i=1 n f Yti Y ti 1 (y i y i 1 ). i=1 4.5 Continuous time i.s.i. process

72 Random processes - Chapter 4 Random process 4 If a process {Y t } has i.s.i., and the cdf pdf, or pmf for Y t = Y t Y 0 is given, the process can be completely described as shown above. As in the discrete time case, a continuous time random process {Y t } is called a Markov process if we have or Pr{Y tn y n Y tn 1 = y n 1,Y tn 2 = y n 2, }= Pr{Y tn y n Y tn 1 = y n 1 }, f Yt n Y t n 1,,Y t1 (y n y n 1,,y 1 )=f Yt n Y t n 1 (y n y n 1 ), p Yt n Y t n 1,,Y t1 (y n y n 1,,y 1 )=p Yt n Y t n 1 (y n y n 1 ) for all n, y n,y n 1,,andt 1,t 2,,t n. A continuous time i.s.i. process is a Markov process. 4.5 Continuous time i.s.i. process

73 Random processes - Chapter 4 Random process 5 Wiener process Wiener process AprocessiscalledWienerprocessifitsatisfies The initial position is zero. That is, W (0) = 0. The mean is zero. That is, E{W (t)} =0,t 0. The increments of W (t) are independent, stationary, and Gaussian. Wiener process is a continuous time i.s.i. process. The increments of Wiener process are Gaussian random variables with zero mean. The first order pdf of Wiener process is f Wt (x) = { } 1 exp x2 2πtσ 2 2tσ Continuous time i.s.i. process / Wiener process

74 Random processes - Chapter 4 Random process 6 Wiener process, Brownian motion Wiener process is the limit of the random-walk process. Properties of Wiener process The distribution of X t2 X t1,t 2 >t 1, depends only on t 2 t 1,nott 1 and t 2 individually. 4.5 Continuous time i.s.i. process / Wiener process

75 Random processes - Chapter 4 Random process 7 Let us show that the Wiener process is Gaussian. From the definition of the Wiener process, the random variables W (t 1 ), W (t 2 ) W (t 1 ), W (t 3 ) W (t 2 ),, W (t k ) W (t k 1 ) are independent Gaussian random variables. Thus the random variables W (t 1 ), W (t 2 ), W (t 3 ),, W (t k ) can be obtained from the following linear transformation of W (t 1 ) and the increments W (t 1 ) = W (t 1 ), W (t 2 ) = W (t 1 )+{W(t 2 ) W (t 1 )}, W (t 3 ) = W (t 1 )+{W(t 2 ) W (t 1 )} + {W (t 3 ) W (t 2 )},. W (t k ) = W (t 1 )+{W(t 2 ) W (t 1 )} + +{W (t k ) W (t k 1 )}. Since W (t 1 ), W (t 2 ), W (t 3 ),, W (t k ) is jointly Gaussian, {W (t)} is Gaussian. 4.5 Continuous time i.s.i. process / Wiener process

76 Random processes - Chapter 4 Random process 8 Poisson counting process Poisson counting process A continuous time counting process {N t,t 0} with the following properties is called the Poisson counting process. N 0 =0. The process has i.s.i.. Hence, the increments over non-overlapping time intervals are independent random variables. In a very small time interval, the probability of an increment of 1 is proportional to the length of the interval, and the probability of an increment larger than 1 is 0. Thus, Pr{N t+ t N t =1} = λ t + o( t), Pr{N t+ t N t 2} = o( t), and Pr{N t+ t N t =0} =1 λ t + o( t), where λ is a proportionality constant. 4.5 Continuous time i.s.i. process / Poisson counting process

77 Random processes - Chapter 4 Random process 9 The Poisson counting process is a continuous time discrete alphabet i.s.i. process. We have obtained the Wiener process as the limit of a discrete time discrete amplitude random-walk process. Similarly, the Poisson counting process can be derived as the limit of a binomial counting process using the Poisson approximation. 4.5 Continuous time i.s.i. process / Poisson counting process

78 Random processes - Chapter 4 Random process 10 The probability mass function p Nt N 0 (k) =p Nt (k) of the increment N t N 0 between the starting time 0 and t>0 Let us use the notation p(k, t) =p Nt N 0 (k), t > 0. Using the independence of increments and the third property of the Poisson counting process, we have k p(k, t + t) = Pr{N t = n}pr{n t+ t N t = k n N t = n} which yields = n=0 k Pr(N t = n)pr(n t+ t N t = k n) n=0 p(k, t)(1 λ t)+p(k 1,t)λ t, p(k, t + t) p(k, t) t = p(k 1,t)λ p(k, t)λ. 4.5 Continuous time i.s.i. process / Poisson counting process

79 Random processes - Chapter 4 Random process 11 When t 0, the equation above becomes the following differential equation d p(k, t)+λp(k, t) = λp(k 1,t), t > 0, dt where the initial conditions are { 0, k 0, p(k, 0) = 1, k =0 since Pr{N 0 =0} =1. Solving the differential equation gives p Nt (k) = p(k, t) = (λt)k e λt, k =0, 1, 2,, t 0. k! The pmf for k jumps in an arbitrary interval (s, t), t s p Nt N s (k) = (λ(t s))k e λ(t s), k =0, 1,, t s. k! 4.5 Continuous time i.s.i. process / Poisson counting process

80 Random processes - Chapter 4 Random process 12 Martingale Martingale property An independent increment process {X t } with zero mean satisfies E {X(t n ) X(t n 1 ) X(t 1 ),X(t 2 ),,X(t n 1 )} =0 for all t 1 <t 2 < <t n and integer n 2. This property can be rewritten as E {X(t n ) X(t 1 ),X(t 2 ),,X(t n 1 )} = X(t n 1 ), which is called the martingale property. Martingale A process {X n,n 0} with the following properties is called a martingale process. E{ X n } <. E{X n+1 X 0,,X n } = X n. 4.5 Continuous time i.s.i. process / Martingale

81 Random processes - Chapter 4 Random process 1 Random processes Chapter 4 Random process 4.6 Compound process* 4.6 Compound process*

82 Random processes - Chapter 4 Random process 2 Discrete time compound process* Discrete time compound process Let {N k,k =0, 1, } be a discrete time counting process such as the binomial counting process, and let {X k,k=0, 1, }be an i.i.d. random process. Assume that the two processes are independent of each other. Define the random process {Y k,k =0, 1, } by Y 0 = 0, N k Y k = X i, k =1, 2,, i=1 where we assume Y k =0if N k =0. The process {Y k } is called a discrete time compound process. The process is also referred to as a doubly stochastic process because of the two sources {N k } and {X k } of randomness. 4.6 Compound process* / Discrete time compound process*

83 Random processes - Chapter 4 Random process 3 The expectation of Y k can be obtained using the conditional probability as { N k E{Y k } = E i=1 X i } = E{E{Y k N k }} = p Nk (n)e{y k N k = n} n (1) = n p Nk (n)ne{x} = E{N k }E{X}. Let X 1,X 2, be an i.i.d. sequence of random variables and G X be their common moment generating function. Let the random variable N be independent of X i and G N be the moment generating function of N. Then the moment generating function of S = N i=1 X i is G S (t) =G N (G X (t)). 4.6 Compound process* / Discrete time compound process*

84 Random processes - Chapter 4 Random process 4 Continuous time compound process* When a continuous time counting process {N t,t 0} and an i.i.d. process {X i,i=1, 2, }are independent of each other, the process Y t = Y (t) = N t is called a continuous time compound process. Here, we put Y (t) =0 when N t =0. We have i=1 X i E{Y t } = E{N t }E{X}, M Yt (u) = E{M N t X (u)}. 4.6 Compound process* / Continuous time compound process*

85 Random processes - Chapter 4 Random process 5 A compound process is continuous and differentiable except at the jumps, where a new random variable is added. If {N t } is a Poisson counting process, we have E{Y t } = λte{x}, (λt) k e λt M Yt (u) = M k k! X(u) k=0 = e λt (λtm X (u)) k k! k=0 = e λt(1 M X(u)). 4.6 Compound process* / Continuous time compound process*