Homework 3 (Stochastic Processes)

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1 In the name of GOD. Sharif University of Technology Stochastic Processes CE 695 Dr. H.R. Rabiee Homework 3 (Stochastic Processes). Explain why each of the following is NOT a valid autocorrrelation function: { e τ τ (a) R X (τ) e 2τ τ < Real autocorrelation function must be odd. (b) R X (τ) e τ I τ τ : R() < R(τ). (c) R X (τ) +τ 2 +τ 4 τ : R() < R(τ). All of them is considered for a continous WSS process. 2. The discrete-time Linear system shown in below figure consists of one unit delay and a constant multiplier (a < ). The input to this system is a white noise with average power σ 2. Find the spectral density and average power of the output. H(e jω ) e jω R XX (n) σ 2 δ(n) S Y Y (e jω ) σ 2 ae jω 2 σ 2 +a 2 2a cos(ω) R Y Y () E[Y 2 (n)] δ h(t) Let ϕ, ϕ 2,..., ϕ n be idd sample from the uniform distribution U( π, π). Now consider an stochastic process X(t) n i a i sin ( 2πi n t + ϕ i), Where a i are constant coefficients. (a) find R X (t, t 2 )? R X (t, t 2 ) n i 2 a2 2πi i cos( n (t t 2 )) (b) is X(t) a WSS process? E[X(t)] E [ n i a i sin( 2πi n t + ϕ i) ]

2 (c) Consider an LTI system with impulse response h(t) e 2t u(t). If we apply X(t) as input to this system, Find R XY and R Y Y. H(ω) jω+2 S XX (ω) πδ(ω 2πi 2πi ) + πδ(ω + S XY (ω) S XX (ω)h(ω) S Y Y (ω) S XY (ω)h( ω) R XY (ω) F (S XY ) R Y Y (ω) F (S Y Y ) n n ) π + π j 2πi n +2 j 2πi n Consider an LTI system with system function: H(s) s 2 + 4s + 3 The input to this system is a WSS process X(t) with E{X 2 (t)}. Find S X (ω) such that the average power of output is maximum. R Y Y () E[y 2 ] 2π SXX (ω) H(ω) 2 dω 2π max H(ω) 2 S X (ω)dω max H(ω) 2 R XX () max H(ω) 2 44 R Y Y () E[y 2 ] < 44 S XX (ω) πδ(ω 5) + πδ(ω + 5) Solution is not unique! 5. Let {X(t)}, t R, be a continuous wide-sense-stationary process with unknown mean m and covariance function Cov(s) ae b s, t R, where a > and b >. For fixed T > set X T T X(s)ds. Show that (a) E[ X] m. That is, X is an unbiased estimator of m. (b) var( X) 2a[(bT ) (bt ) 2 ( e bt )] (a) E[X] E[ T T X(s)ds] T mds m E[X(s)]ds 2

3 (b) V ar(x) E[X 2 ] E[X] 2 E[ a a r ( R X (s r)dsdr m 2 C X (s r)dsdr a e bx dx + r X(s)X(r)dsds] m 2 e bx dx)dr ( b (e br + e b(t r) ))dr 2a b 2 (e bt ) + 2a bt C X (s r) + m 2 dsdr e b s r dsdr 6. Let X n (X+na mod ), n, ±, ±2,..., where X is an RV uniformly distributed on [, ] and a is a fixed irrational number. Show that X n is strictly stationary. Define I(p) where p is a composition as I(p) if p is true and I(p) if p is false. We have: F X (x,..., x n ; t,..., t n ) P (X(t ) x,..., X(t n ) x n ) P ((X + t a) mod x,..., (X + t n a) mod x n ) s s + I((X + t a) mod x,..., (X + t n a) mod x n )dx I((x + t a) mod x,..., (x + t n a) mod x n )dx I((x + t a) mod x,..., (x + t n a) mod x n )dx For any s. Setting s (ra) mod we have: F X (x,..., x n ; t,..., t n ) s + s I((x + ra + t a) mod x,..., (x + ra + t n a) mod x n )dx I((x + ra + t a) mod x,..., (x + ra + t n a) mod x n )dx I((x + (t + r)a) mod x,..., (x + (t n + r)a) mod x n )dx F X (x,..., x n ; t + r,..., t n + r) 7. Consider the random process X(t) as a stock index value at time t. Then S(t) X(t) X(t 5) is representative of the return if someone buys the stock at time t 5 and sells it at time t. Suppose S(t) is a normal WSS process with mean and autocorrelation function R(τ) 5exp( τ ). We 3

4 have observed that S(6), S(7) are below 3. That is we know there is negative return if some buys at time 6 5, and sells at time 6 and 7, respectively. We want to estimate what is the probability that S() is positive. In other terms we want to see whether we have any chance to get profit if we buy the stock at time 5 5 and sell it at time. we have to compute P (S() > S(6) < 3, S(7) < 3). e e 4 S(6), S(7), S() N([,, ], Σ) where Σ 5 e e 3 e 4 e 3 P (S() > S(6) < 3, S(7) < 3) 3 3 x x 2 x 3 f x x 2 x 3 (x,x 2,x 3 )dx dx 2 dx x 2 x 3 f x 2,x 3 (x 2,x 3 )dx 2 dx 3 8. Suppose V is a uniform random variable with U(, ) as its PDF. Define two stochastic processes X(t) u(t V ) and Y (t) δ(t V ). (Assume t ) a. Explain intuitively what X(t) and Y (t) describe. X(t) represents whether V t or V > t, i.e. if V t, then X(t) and otherwise X(t). Y(t) represents wether V y or not, i.e. if v t then X(t) and otherwise X(t). b. Evaluate the mean of X(t) and Y (t). (Assume t ) For a constant t, X(t) is a discrete random variable and its pmf is as following: t α f X(t) (α) t α otherwise E[X] ( t) + timest t For a constant t, Y (t) is a random variable which is a function of V, thus let s represent it as Y t (V ) E[Y t ] Y t(v )f V (α)dα δ(t α)dα Set β t α, then E[Y t ] t t δ(β)dβ c. Evaluate R XX (t, t 2 ), R XY (t, t 2 ) and R Y Y (t, t 2 ). (Assume t and t 2 ) R XX (t, t 2 ) P {X(t ), X(t 2 ) } +... p{x(max{t, t2}) } f X(max{t,t 2 })() max{t, t2} R XY (t, t2) E[X(t )Y (t 2 )] E[u(t V )δ(t 2 V )] u(t V )δ(t 2 V )f V (α)dα 4

5 Set β t α, then R XY (t, t 2 ) t t u(β)δ(β (t t 2 ))dβ t { δ(β t t (t t 2 ))dβ 2 t 2 otherwise if t t 2 then R Y Y (t, t 2 ) E[δ(t V )δ(t 2 V )] E[]. if t t 2 then R Y Y (t, t 2 ) E[δ 2 (t V )] δ2 (t α)f V (α)dα δ2 (t α)dα t t δ2 (β)dβ 9. Show that if the processes x(t) and y(t) are WSS and E{ x() y() 2 }, then R xx (τ) R xy (τ) R xz (τ). Hint: Set z x(t + τ), w x (t) y (t) and use Schwartz s inequality. E[zw] E[ z e iθ z w e iθ w ] z w ei(θ +θ 2 ) f(z, w)dzdw z w e i(θ z +θ w )f(z,w) dzdw z w f(z, w)dzdw E[ z w ] E[zw] E[ z w ] E[zw] 2 E 2 [ z w ] E[ z 2 ]E[ w 2 ] (Cauchy-Schwarz inequality) E[zw] 2 E[x(t + τ)x (t)] E[x(t + τ)y (t)] R XX (τ) R Y Y (τ) E[ z 2 ]E[ w 2 ] E[ z 2 ]E[ x (t) y (t) 2 ] E[ z 2 ]E[ x(t) y(t) 2 ] if we set t, then E[ x(t) y(t) 2 ], therefor: R XX (τ) R Y Y (τ) R XX (τ) R Y Y (τ) We can do the same to demonstrate R Y Y (τ) R XY (τ) 5