Solution to EE351k fall 2007 hw 13
1.
)
2
cos(
))
(
2
2
cos(
))
(
2
sin(
)
2
sin(
0
))
(
2
cos(
)
2
cos(
))}
(
2
sin(
)
2
sin(
))]
(
2
cos(
)
2
sin(
))
(
2
sin(
)
2
[cos(
))
(
2
cos(
)
2
cos(
{
))]}
(
2
sin(
)))
(
2
cos(
)][
2
sin(
)
2
cos(
{[
)]
(
)
(
[
)
,
(
2
2
2
2
2
2
τ
π
σ
τ
π
π
σ
τ
π
π
σ
τ
π
π
σ
τ
π
π
τ
π
π
τ
π
π
τ
π
π
τ
π
τ
π
π
π
τ
τ
f
t
f
ft
t
f
ft
t
f
ft
t
f
ft
Y
t
f
ft
t
f
ft
XY
t
f
ft
X
E
t
f
Y
t
f
X
ft
Y
ft
X
E
t
W
t
W
E
t
R
W
=
+

=
+
+
+
+
=
+
+
+
+
+
+
+
=
+
+
+
+
=
+
=
0
)
2
sin(
)
(
)
2
cos(
)
(
)]
2
sin(
)
2
cos(
[
))
(
(
=
+
=
+
=
ft
Y
E
ft
X
E
ft
Y
ft
X
E
t
W
E
π
π
π
π
A stochastic process is widesense stationary if its mean is constant and its autocorrelation
depends only on
τ
.
So W(t) is wss.
2.
a.
There are at least three approaches we can follow to compute the mean.
We can use the
definition of the expectation of a random variable X whose PDF is computed in last homework, or
use the definition of a function of a random variable W whose PDF is given here, or use the linear
property of a expectation operator.
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 Fall '07
 BARD
 Probability theory, Sin, Cos

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