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PART 4 Classification of Random Processes

PART 4 Classification of Random Processes. Huseyin Bilgekul Eeng571 Probability and astochastic Processes Department of Electrical and Electronic Engineering Eastern Mediterranean University. 1.7 Statistics of Stochastic Processes. n-th Order Distribution (Density) Expected Value

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PART 4 Classification of Random Processes

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  1. PART 4Classification of Random Processes Huseyin Bilgekul Eeng571 Probability and astochastic Processes Department of Electrical and Electronic Engineering Eastern Mediterranean University

  2. 1.7 Statistics of Stochastic Processes • n-th Order Distribution(Density) • Expected Value • Autocorrelation • Cross-correlation • Autocovariance • Cross-covariance

  3. First-Order and 2nd-Order Distribution • First-Order Dustribution For a specific t, X(t) is a random variable with first-order distribution function FX(t)(x) = P{X(t)  x}, The first-order density of X(t) is defined as • 2nd-Order Distribution FX(t1)X(t2)(x1, x2 ) = P{X(t1)  x1, X(t2)  x2}

  4. nth-Order Distribution • Definition The nth-order distribution of X(t) is the joint distribution of the random variables X(t1), X(t2), …, X(tn),i.e., FX(t1)…X(tn)(x1, …, xn) = P{X(t1)  x1,…, X(tn)  xn} • Properties

  5. Expected Value • Definition： The expected value of a stochastic process X(t) is the deterministic function X(t) = E[X(t)] • Example 1.10： If R is a nonnegative random variable, find the expected value of X(t)=R|cos2ft| . Sol：

  6. Autocorrelation Definition The autocorrelation function of the stochastic process X(t) is RX(t,) = E[X(t)X(t+ )] , The autocorrelation function of the random sequence Xn is RX[m,k] = E[XmXm+k]. P.S. If X(t) and Xn are complex, then RX(t,) = E[X(t)X*(t+ )] , RX[m,k] = E[XmX*m+k], where X* (t+ )andX*m+k are the conjugate.

  7. Complex Process and Vector Processes Definitions The complex process Z(t) = X(t) + jY(t) is specified in terms of the joint statistics of the real processes X(t) and Y(t). The vector process (n-dimensional process) is a family of n stochastic processes.

  8. Example 1.11 Find the autocorrelationRX(t,) of the process X(t) =rcos(t+), where the R.V. r and  are independent and  is uniform in the interval (, ). Sol：

  9. Cross-correlation Definition The cross-correlation function of two stochastic processes X(t) and Y(t) is RXY(t,) = E[X(t)Y*(t+ )] , The cross-correlation function of two random sequences Xn and Yn is RXY[m,k] = E[XmY*m+k] If , RXY(t,) = 0 then X(t) and Y(t) are called orthogonal.

  10. Autocovariance Definition The autocovariance function of the stochastic processes X(t) is CX(t, ) = Cov[X(t), X(t+)] = E{[X(t) X(t)][X*(t+) *X(t +)]} The autocovariance function of the random sequence Xn is CX[m, k] = Cov[Xm, Xm+k] = E{[XmX(m)][X*m+k*X(m+k)]}

  11. Example 1.12 The input to a digital filter is an iid random sequence …, X-1, X0,X1,X2,…with E[Xi] = 0 and Var[Xi] = 1. The output is a random sequence…, Y-1, Y0,Y1,Y2,…related to the input sequence by the formula Yn= Xn + Xn-1for all integers n. Find the expected valueE[Yn] and autocovarianceCY[m, k] . Sol：

  12. Example 1.13 Suppose that X(t) is a process with E[X(t)] = 3, RX(t,) = 9 + 4e0.2|| , Find the mean, variance, and the covariance of the R.V.Z = X(5) and W = X(8). Sol：

  13. Cross-covariance Definition The cross-covariance function of two stochastic process X(t) and Y(t) is CXY(t,) = Cov[X(t), Y(t+)] = E{[X(t) X(t)][Y*(t+) *Y(t+)]} The cross-covariance function of two random sequences Xn and Yn is CXY[m, k] = Cov[Xm, Ym+k] = E{[XmX(m)][Y*m+k*Y(m+k)]} Uncorrelated： CXY(t,) = 0, CXY[m,k] = 0.

  14. Correlation Coefficient, a-Dependent Definitions The correlation coefficient of the process X(t) is The process X(t) is called a-dependent if X(t) for t < t0 and for t > t0+a are mutually independent. Then we have, CX(t,) = 0, for| |>a. The process X(t) is called correlationa-dependent if CX(t,) satisfies CX(t,) = 0, for| |>a.

  15. Example 1.14 (a) Find the autocorrelation RX(t,) if X(t) = aejt. (b) Find the autocorrelation RX(t,) and autocovarianceCX(t,), if , where the random variables ai are uncorrelated with zero mean and variancei2. Sol：

  16. Theorem 1.2 The autocorrelation and autocovariance functions of a stochastic process X(t) satisfy CX(t,) = RX(t,) X(t) *X(t+) The autocorrelation and autocovariance functions of the random sequence Xn satisfy CX[m,k] = RX[m,k] X(m) *X(m+k) The cross-correlation and cross-covariance functions of a stochastic process X(t) and Y(t) satisfy CXY(t,) = RXY(t,) X(t) *Y(t+) The cross-correlation and cross-covariance functions of two random sequences Xn and Yn satisfy CXY[m,k] = RXY[m,k] X(m) *Y(m+k)

  17. 1.8 Stationary Processes Definition A stochastic process X(t)is stationaryif and only if for all sets of time instants t1, t2, …, tm, and any time difference , A random sequence Xnis stationaryif and only if for any set of integer time instants n1, n2, …, nm, and integer time difference k, Also called as Strict Sense Stationary (SSS).

  18. Theorem 1.3 Let X(t)be a stationary random process. For constants a > 0 and b, Y(t) = aX(t)+b is also a stationary process. Pf： Y(t)is stationary. X(t)is stationary

  19. Theorem 1.4(a) For a stationary process X(t), the expected value, the autocorrelation, and the autocovariance have the following properties for all t： (a) X(t) = X (b) RX(t,) = RX(0,) = RX() (c) CX(t,) = RX()  X2= CX() Pf：

  20. Theorem 1.4(b) For a stationary random sequence Xn, the expected value, the autocorrelation, and the autocovariance satisfy for all n： (a) E[Xn] = X (b) RX[n,k] = RX[0,k] = RX[k] (c) CX[n,k] = RX[k]  X2= CX[k] Pf：D.I.Y.

  21. Example 1.15 At the receiver of an AM radio, the received signal contains a cosine carrier signal at the carrier frequency fc with a random phase  that is a sample value of the uniform (0,2) random variable. The received carrier signal is X(t) = A cos(2 fc t +). What are the expected value and autocorrelation of the process X(t)? Sol：

  22. Wide Sense Stationary Processes Definition X(t)is a wide sense stationary (WSS) stochastic process if and only if for all t, E[X(t)] = X , andRX(t,) = RX(0,) = RX(). (ref: Example 1.15) Xnis a wide sense stationary random sequence if and only if for all n, E[Xn] = X , andRX[n,k] = RX[0,k] = RX[k]. Q：SSS implies WSS? WSS implies SSS?

  23. Theorem 1.5 For a wide sense stationary stochastic process X(t), the autocorrelation function RX() has the following properties： RX(0)  0, RX() = RX() andRX(0)  | RX()| . If Xnis a wide sense stationary random sequence： RX[0]  0, RX[n] = RX[n]andRX[0]  |RX[0]|. Pf：

  24. Average Power Definition The average power of a wide sense stationary processX(t) is RX(0) = E[X2(t)]. The average power of a wide sense stationary random sequenceXnis RX[0] = E[Xn2]. Example：use x(t) = v2(t)/R=i2(t)Rto model the instantaneous power.

  25. Ergodic Processes Definition For a stationary random process X(t), if the ensemble average equals the time average, it is called ergodic. For a stationary processX(t), we can view the time average X(T), as an estimate of X .

  26. Theorem 1.6 Let X(t) be a stationary process with expected value X and autocovariance CX() . If , then is an unbiased, consistent sequence of estimates of X . Pf： unbiased. consistent sequence: we must show that

  27. Jointly Wide Sense Stationary Processes Continuous-time random process X(t) and Y(t) are jointly wide sense stationary if X(t) and Y(t) are both wide sense stationary, and the cross-correlation depends only on the time difference between the two random variables： RXY(t,) = RXY(). Random Sequences Xnand Ynare jointly wide sense stationary if Xnand Ynare both wide sense stationary, and the cross-correlation depends only on the index difference between the two random variables：RXY[m,k] = RXY[k].

  28. Theorem 1.7 If X(t) and Y(t) are jointly wide sense stationary continuous-time processes,then RXY() = RYX(). If Xnand Ynare jointly wide sense stationary random sequences, then RXY[k] = RYX[k]. Pf：

  29. Example 1.16 Suppose we are interested in X(t) but we can observe only and Y(t) = X(t) + N(t), where N(t) is a noise process. Assume X(t) and N(t) are independent WSS with E[X(t)] = Xand E[N(t)] = N = 0. Is Y(t) WSS? Are X(t) and Y(t)jointly WSS ?Are Y(t) and N(t)jointly WSS ? Sol：

  30. Example 1.17 Xn is a WSS random sequence with RX[k]. The random sequence Yn is obtained from Xn by reversing the sign of every other random variable in Xn：Yn= (1)nXn. (a) Find RY[n,k] in terms of RX[k]. (b) Find RXY[n,k] in terms of RX[k]. (c) Is Yn WSS? (d) Are Xn and Yn jointly WSS? Sol：

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