ECE 330:541, Stochastic Signals and Systems
Midterm Exam 2
Fall 2002
Instructions: There are three problems on this exam, each with multiple parts. On the booklet(s)
provided, please answer the problems and clearly state on each page of the booklet which problem you
are solving.
Note: The exam will be graded out of 100 points, and will be scaled later to contribute a total of 25
points towards your final grade.
1.
Random Phase Signals:
Consider the continuous time process
X
(
t
) =
A
cos(
ω
0
t
+
φ
), where
ω
0
is a known frequency,
φ
is uniformly distributed in (0
,
2
π
) and independently of the random
variable
A
.
(a) (
10 points
) Is
X
(
t
) is a wide sense stationary (WSS) process? Hint: You may want to use
the fact that cos(
A
) cos(
B
) = (cos(
A

B
) + cos(
A
+
B
))
/
2.
(b) (
10 points
) If we take
φ
to be a known constant, is
X
(
t
) WSS?
2.
Stationary Autoregressive1 Process:
Let
e
(
t
), for
t
= 0
,
±
1
,
±
2
,
· · ·
be a realvalued white
noise process (i.e.
a sequence of uncorrelated realvalued random variables with zero mean and
variance
σ
2
e
). We may define a process
X
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 Spring '08
 Ferry
 Autocorrelation, Stationary process, RM vj

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