±
Massachusetts
Institute
of
Technology
Department
of
Electrical
Engineering
and
Computer
Science
6.432
Stochastic
Processes,
Detection
and
Estimation
Problem
Set
9
Spring
2004
Issued:
Thursday,
April
15,
2004
Due:
Thursday,
April
29,
2004
Reading:
Course
notes:
For
this
problem
set:
Chapter
6,
Sections
7.1
and
7.2
Next:
Chapter
7
Exam
#2
Reminder:
Our
second
exam
will
take
place
Thursday,
April
22,
2004,
9am

11am.
The
exam
will
cover
material
through
Lecture
16
(April
8)
as
well
as
the
associated
homework
through
Problem
Set
8.
You
are
allowed
to
bring
two
8
1
×
11
sheets
of
notes
(both
sides).
2
Problem
9.1
A
discretetime
random
process
y
[
n
]
is
observed
for
n
=
,
N
.
There
are
two
1
,
2
,
···
equally
likely
hypotheses
for
the
statistical
description
of
y
[
n
]:
H
0
:
y
[
n
] =
1
+
w
[
n
]
,
N
n
=
1
,
2
,
···
H
1
:
y
[
n
] =
−
1
+
w
[
n
]
where
w
[
n
]
is
a
nonstationary
zeromean
white
Gaussian
sequence,
i.e.
±
E
w
[
n
]
w
[
k
]
=
0
for
n
≥
=
k
(a)
Assume
that
E
w
2
[
n
]
±
=
nπ
2
,
n
=
1
,
2
,
···
,
N.
Find
the
minimum
probability
of
error
decision
rule
based
on
observation
of
[
N
],
and
determine
the
associated
error
probability
in
terms
of
Q
(
·
)
y
[1]
,
···
,
y
and
N
,
where
Q
(
x
)
=
1
²
±
e
−
2
/
2
d .
2
±
x
Find
the
asymptotic
detection
performance,
i.e.
,
compute
lim
Pr(error).
N
²±
(b)
For
this
part
assume
that
2
E
w
2
[
n
] =
n π
2
,
,
N.
n
=
1
,
2
,
···
Find
the
minimum
probability
of
error
decision
rule
based
on
observation
of
[
N
],
and
determine
the
associated
error
probability
in
terms
of
Q
(
·
)
y
[1]
,
···
,
y
and
N
.
What
is
the
asymptotic
performance
i.e.
,
what
is
lim
Pr(error)
in
this
N
²±
case?
1