EE 511 Problem Set 6
Due on 5 November 2007
1. White noise of power spectral density
N
0
/
2 is filtered using an ideal low pass filter of
bandwidth
B
. What is the variance of the output noise process?
2. Suppose
X
t
is a Wiener process defined for
t
≥
0, i.e., a Gaussian process with
m
X
(
t
) =
mt
for
t
≥
0, where
m
is a constant, and
R
X
(
t, s
) =
σ
2
min (
t, s
) +
m
2
ts
for
t, s
≥
0.
Define a process
Y
t
=
X
t
+
D

X
t
for
t
≥
0, where
D
is a fixed positive number. a)
Find
m
Y
(
t
) and
R
Y
(
t, s
). b) Show that
Y
t
is stationary and find
S
Y
(
f
).
3. Let
X
t
be a zeromean stationary Gaussian process with autocorrelation function
R
X
(
τ
). This process is applied to a squarelaw device defined by
Y
t
=
X
2
t
. a) Show
that
E
[
Y
t
] =
R
X
(0). b) Show that the autocovariance function of
Y
t
,
C
Y
(
τ
) = 2
R
2
X
(
τ
).
4. A stationary Gaussian process
X
t
with zeromean and power spectral density
S
X
(
f
)
is applied to a linear filter with impulse response as shown in Figure 1. A sample
Y
is taken of the random process at the filter output at time
T
. a) Determine the mean
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '11
 TU
 Normal Distribution, Variance, Probability theory, Autocorrelation, Xt

Click to edit the document details