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UCSD ECE153
Handout #18
Prof. YoungHan Kim
Thursday, April 28, 2011
Homework Set #5
Due: Thursday, May 5, 2011
1.
Neural net.
Let
Y
=
X
+
Z
, where the signal
X
∼
U[

1
,
1] and noise
Z
∼ N
(0
,
1) are
independent.
(a) Find the function
g
(
y
) that minimizes
MSE =
E
b
(sgn(
X
)

g
(
Y
))
2
B
,
where
sgn(
x
) =
±

1
x
≤
0
+1
x >
0
.
(b) Plot
g
(
y
) vs.
y
.
2.
Additive shot noise channel.
Consider an additive noise channel
Y
=
X
+
Z
, where
the signal
X
∼ N
(0
,
1), and the noise
Z
{
X
=
x
} ∼ N
(0
, x
2
), i.e., the noise power of
increases linearly with the signal squared.
(a) Find
E
(
Z
2
).
(b) Find the best linear MSE estimate of
X
given
Y
.
3.
Estimation vs. detection.
Let the signal
X
=
²
+1
,
with probability
1
2

1
,
with probability
1
2
,
and the noise
Z
∼
Unif[

2
,
2] be independent random variables. Their sum
Y
=
X
+
Z
is observed.
(a) Find the best MSE estimate of
X
given
Y
and its MSE.
(b) Now suppose we use a decoder to decide whether
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This note was uploaded on 07/07/2011 for the course EECS 153 taught by Professor Kim during the Spring '11 term at UCSD.
 Spring '11
 Kim

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