In the name of GOD.
Sharif University of Technology
Stochastic Processes CE 695 Fall 2015 Dr. H.R. Rabiee
Homework 2 (LTI Systems, Power Spectrum &
Ergodicity)
(140 + 20 points)
1. Reading assignment:
• Sections 9-2,9-3,9-4,12-1 from Papoulis.
2. (9 pts) Answer briefly to each of the following questions:
(a) Define stochastic and deterministic systems with stochastic inputs.
(b) Give an informal definition of an ergodic process.
(c) Do you see any relation between the notion of ergodicity and the law
of large numbers?
3. (9 pts) Which of the followings can be autocorrelation function of a WSS
process? Explain.
(a) Rx (τ ) = (1 + τ 2 )e−τ
2
(b) Rx (τ ) = e−α|τ | where α > 0 and 0 < c < 1.
c
(c)
{
Rx (τ ) =
2 |τ | < 2
0 |τ | ≥ 2
4. (12 pts) Show that the output of a BIBO stable LTI system with bounded
WSS input is also WSS. Are input and output jointly WSS?
5. (27 pts) Prove following properties about PSDs:
(a) Sxx (ω) is a real function of ω.
(b) if x(t) is real then Sxx (ω) is even.
∗
(c) Sxy (ω) = Syx
(ω) where ∗ denotes complex conjugate.
(d) Syy (ω) = Sxx (ω)|H(ω)|2
(e) if Sxx (ω) is a real and nonnegative function of ω whose integral over
the entire frequency axis is finite, then its inverse transform Rxx (τ ) is
the autocorrelation function of a WSS process, with finite variance.
(f) prove converse of above statement.
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6. (20 pts) Consider LTI system with following impulse response:
h(t) = e−2t u(t)
(a) Find mean of output if the input is x(t) = 5sin(ωt + Φ) + 2 where Φ
is uniform in [−π, π].
(b) Find Rxy and Ryx if the input is a stationary white noise.
√
(c) If the input mean is 2 and Rx (τ ) = 3δ(τ ) + 2 find output mean,
Rxy , Ry , Sx and Sy .
7. (14 pts) Let x(t) be a Gaussian process with Sx (ω) = 2πe−π|ω| . Let y(t)
be a process governed by
y(t) = x(t) + 3x′ (t) − 2x′ (t − 1)
Find P[y(t) > 2].
8. (14 pts) Prove following statements about ergodicity of x(t) which is a
WSS process with autocorrelation C(τ ):
(a) It’s mean-ergodic iff(if and only if)
∫
1
T
2T
C(α)(1 −
0
α
T →∞
)dα −−−−→ 0
2T
(b) It’s mean-ergodic iff
1
T
(c) It’s mean-ergodic if
∫
T
T →∞
C(τ )dτ −−−−→ 0
0
∫
∞
C(τ )dτ < ∞
0
(d) It’s mean-ergodic if
τ →∞
C(τ ) −−−−→ 0
9. (6 pts) Consider a LTI system with impulse response:
H(ω) =
−ω 2
1
+ 2jω + 5
System input is x(t) with average power 10. Find Sx (ω) such that E[y 2 (t)]
is maximum where y(t) is output of the system.
10. (24 pts) Solve problems 29,32,42,43 from chapter 9 of Papoulis’s book.
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11. (5 pts) x(t) is a white noise with average intensity q. It’s given to two
LTI systems with impulse response h1 (t) and h2 (t) with outputs y1 (t) and
y2 (t) as shown below. Find Sy1 y2 (ω) and Sy2 y1 (ω).
12. (20 extra pts) A discrete time process y[n] is defined by:
y[n] = αy[n − 1] + x[n]
where x[n] is a zero-mean white noise with average power σ 2 . y[n] can be
viewed as the output of the system shown below. Find the power spectral
density and autocorrelation of y[n].
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