ECE 534 RANDOM PROCESSES SPRING 2005
PROBLEM SET 6 Due Wednesday, November 16
Random Processes in Linear Systems and Spectral Analysis
Assigned Reading: Chapter 6 of the course notes.
Reminder: Exam 2 will be given Monday, November 7, 7-8:15 p.m. in Room 151 Everitt Laboratory.
Problems to be handed in:
1. On filtering a WSS random process
Suppose Yis the output of a linear time-invariant system with WSS input X, impulse response
function h, and transfer function H. Indicate whether the following statements are true or false.
Justify your answers.
(a) If |H(ω)| ≤ 1 for all ωthen the power of Yis less than or equal to the power of X.
(b) If Xis periodic (in addition to being WSS) then Yis WSS and periodic.
(c) If Xhas mean zero and strictly positive total power, and if ||h||2>0, then the output power
is strictly positive.
2. Slight smoothing
Suppose Yis the output of the linear time-invariant system with input Xand impulse response
function h, such that Xis WSS with RX(τ) = exp(−|τ|), and h(τ) = 1
aI{|τ|≤ a
2}for a > 0. If a
is small, then happroximates the delta function δ(τ), and consequently Yt≈Xt. This problem
explores the accuracy of the approximation.
(a) Find RY X (0), and use the power series expansion of euto show that RYX (0) = 1 −a
4+o(a) as
a→0. Here, o(a) denotes any term such that o(a)/a →0 as a→0.
(b) Find RY(0), and use the power series expansion of euto show that RY(0) = 1 −a
3+o(a) as
a→0.
(c) Show that E[|Xt−Yt|2] = a
6+o(a) as a→0.
3. The accuracy of approximate differentiation
Let Xbe a WSS baseband random process with power spectral density SX, and let ωobe the
one-sided band limit of X. The process Xis m.s. differentiable and X0can be viewed as the
output of a time-invariant linear system with transfer function H(ω) = jω.
(a) What is the power spectral density of X0?
(b) Let Yt=Xt+a−Xt−a
2a, for some a > 0. We can also view Y= (Yt:t∈R) as the output of
a time-invariant linear system, with input X. Find the impulse response function kand transfer
function Kof the linear system. Show that K(ω)→jω as a→0.
(c) Let Dt=X0
t−Yt.Find the power spectral density of D.
(d) Find a value of a, depending only on ωo, so that E[|Dt|2]≤(0.01)E[|X0
t|]2. In other words, for
such a, the m.s. error of approximating X0
tby Ytis less than one percent of E[|X0
t|2]. You can use
the fact that 0 ≤1−sin(u)
u≤u2
6for all real u. (Hint: Find aso that SD(ω)≤(0.01)SX0(ω) for
|ω| ≤ ωo.)
4. Sampling a cubed Gaussian process
Let X= (Xt:t∈R) be a baseband mean zero stationary real Gaussian random process with
one-sided band limit foHz. Thus, Xt=P∞
n=−∞ XnT sinc t−nT
Twhere 1
T= 2fo. Let Yt=X3
tfor
each t.
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