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
bracketleftbig
(sgn(
X
)

g
(
Y
))
2
bracketrightbig
,
where
sgn(
x
) =
braceleftBigg

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
=
braceleftbigg
+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
X
= +1 or
X
=

1 so that the
probability of error is minimized. Find the optimal decoder and its probability of
error. Compare the optimal decoder’s MSE to the minimum MSE.
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 Spring '11
 Kim
 Normal Distribution, Probability theory, minimum MSE, MSE estimate, linear MSE estimate, best linear MSE

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