EE 179 Introduction to Communications
Winter 2011
Homework 5
Due Thursday, Feb 10 4pm
(Total: 100 pts.)
1.
Random Cosine Signals
[15 pts] Given a random process
x
(
t
) =
a
cos(
ω
c
t
+ Θ)
where
ω
c
is a constant and
a
and Θ are independent RVs uniformly distributed in the ranges [

1
,
1]
and [0
,
2
π
, respectively.
(a) Sketch the ensemble of this process.
(b) Determine
x
(
t
).
(c) Determine
R
x
(
t
1
, t
2
).
(d) Is the process widesense stationary?
(e) Is the process ergodic?
(f) Find the average power
P
x
(that is, its mean square value
x
2
(
t
)) of the random process
x
(
t
).
2.
Stationarity
[10 pts] Let
X
and
Y
be statistically independent Gaussiandistributed random variables
with zero mean and unit variance. Define the Gaussian process
Z
(
t
) =
X
cos(2
πt
) +
Y
sin(2
πt
)
.
(a) Determine the joint probability density function of the random variables
Z
(
t
1
) and
Z
(
t
2
) obtained
by observing
Z
(
t
) at times
t
1
and
t
2
, respectively.
(b) Is the process
Z
(
t
) stationary? Why?
3.
PSD of Random Binary Wave:
[15 pts] Let
x
(
t
) be the infinite pulse train shown below with period
T
and random time shift
τ
, where
τ
is uniformly distributed between [0
, T
].
EE 179  Introduction to Communications  Spring 2009
Homework 4: Due Thursday 5/7 at 4pm (no late HWs).
1. (20 points) Let
x
(
t
) be the infinite pulse train shown below with period
T
and random time shift
τ
,
where
τ
is uniformly distributed between [0
, T
].
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 Spring '11
 Naragipour
 Probability theory, Stochastic process, probability density function, Stationary process, random process

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