EE 278
November 18, 2009
Statistical Signal Processing
Handout #18
Homework #8
Due: Wednesday, December 2
1.
Discretetime Wiener process.
Let
{
Z
n
:
n
≥
0
}
be a discretetime white Gaussian noise process;
that is,
Z
1
, Z
2
, Z
3
, . . .
are i.i.d.
N
(0
,
1).
Define the process
{
X
n
:
n
≥
0
}
by
X
0
= 0 and
X
n
=
X
n

1
+
Z
n
for
n
≥
1.
a. Is
X
n
an independent increment process? Justify your answer.
b. Is
X
n
a Gaussian process? Justify your answer.
c. Find the mean and autocorrelation functions of
X
n
.
d. Specify the firstorder pdf of
X
n
.
e. Specify the joint pdf of
X
3
,
X
5
, and
X
8
.
f.
Find E(
X
20

X
1
, X
2
, . . . , X
10
).
g. Given
X
1
= 4,
X
2
= 2, and 0
≤
X
3
≤
4, find the minimum MSE estimate of
X
4
.
2.
Sawtooth process.
Let
X
(
t
) =
g
(
t

T
), where
g
(
t
) is the periodic triangular waveform shown
in Figure 1, and the delay
T
is a random variable with
T
∼
U[0
,
1).
. . . .
. . . .
0
1
1
2
3
t
g
(
t
)
Figure 1: Periodic triangular waveform
Is
X
(
t
) a strictsense stationary random process? Justify your answer.
3.
Stationary GaussMarkov process.
(Bonus)
Consider the following variation on the Gauss
Markov process discussed in the lecture notes 8:
X
0
∼ N
(0
, a
)
X
n
=
1
2
X
n

1
+
Z
n
,
n
≥
1
,
where
Z
1
, Z
2
, Z
3
, . . .
are i.i.d.
N
(0
,
1) independent of
X
0
.
a. Find
a
such that
X
n
is stationary. Find the mean and autocorrelation functions of
X
n
.
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 Fall '09
 BalajiPrabhakar
 Signal Processing, LTI system theory, Stationary process, WSS process

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