UCSD ECE 153
Handout #20
Prof. YoungHan Kim
Thursday, November 20, 2008
Homework Set #6
Due: Thursday, December 4, 2008
1. Read Sections 7.1–7.4, 9.1–9.6, 10.1–10.2, 10.4 in the text. Try to work on all examples.
2.
Symmetric random walk.
Let
X
n
be a random walk defined as
X
0
=
0
X
n
=
n
summationdisplay
i
=1
Z
i
,
where
Z
1
, Z
2
, . . .
are i.i.d. with P(
Z
1
=

1) = P(
Z
1
= 1) =
1
2
.
(a) Find P
{
X
10
= 10
}
.
(b) Find P
{
max
1
≤
i<
20
X
i
= 10

X
20
= 0
}
.
(c) Find P
{
X
n
=
k
}
.
3.
Moving average process.
Let
Y
n
=
1
2
Z
n

1
+
Z
n
for
n
≥
1
,
where
Z
0
, Z
1
, Z
2
, . . .
are i.i.d.
∼
N
(0
,
1). Find the mean and autocorrelation function of
Y
n
.
4.
GaussMarkov process.
Let
X
0
= 0 and
X
n
=
1
2
X
n

1
+
Z
n
for
n
≥
1, where
Z
1
, Z
2
, . . .
are i.i.d.
∼
N
(0
,
1). Find the mean and autocorrelation function of
X
n
.
5.
Discretetime Wiener process.
Let
Z
n
,
n
≥
0 be a discrete time white Gaussian noise
(WGN) process, i.e.,
Z
1
, Z
2
, . . .
are i.i.d.
∼
N
(0
,
1). Define the process
X
n
,
n
≥
1 as
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.
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 Winter '09
 Eggers
 Signal Processing, Stochastic process, Autocorrelation, Wiener–Khinchin theorem

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