EE 805, Random Processes and Linear Systems
Feb. 13, 2012
OSU, Winter 2012
Due: Feb. 24, 2011
Problem Set 4
Problem 1
Let the input
X
(
t
) to a LTI system be white noise with
C
X
(
τ
) =
σ
2
δ
(
τ
). The LTI system
has impulse response
h
(
t
) =
u
(
t
)

u
(
t

T
) =
(
1
0
≤
t
6
T
0
otherwise
.
Find
C
Y
(
τ
) for the system’s output
Y
(
t
).
Problem 2
Consider a discretetime system, whose realvalued input
X
[
n
] is white noise with
C
X
[
n
] =
σ
2
δ
[
n
]. The LTI system has impulse response
h
[
n
] =
a
n
u
[
n
], where
a
is real and

a

<
1.
Find
C
Y X
[
k
],
C
XY
[
k
], and
C
Y
[
k
]. Plot these in Matlab using the
stem
command.
Problem 3
The input
X
(
t
) to a LTI system with transfer function
H
(
s
) =
s
+ 2 has mean
m
X
= 4
and autocovariance
C
X
(
τ
) = 2
e

τ

. Find
m
Y
,
R
Y X
(
τ
), and
R
Y
(
τ
). Also, find
S
Y
(
ω
).
Problem 4
Consider a random telegraph signal
S
(
t
) with amplitude
A
and Poisson switching times
with parameter
λ
. A measured signal
X
(
t
) =
S
(
t
) +
N
(
t
), where
N
(
t
) is uncorrelated
with
X
(
t
). Assume
N
(
t
) is zero mean white noise with
C
N
(
τ
) =
σ
2
N
δ
(
τ
).
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 Spring '12
 Eryilmaz
 Digital Signal Processing, Signal Processing, LTI system theory, Lowpass filter

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