This preview shows pages 1–2. Sign up to view the full content.
EE 562a
Final Solution
December 11, 2008
1
1.
Distortion 101
(a) Is the transformation of
s
(
u,t
) into
d
(
u,t
) a linear transformation?
YES
Is
d
(
u,t
) a widesense stationary random process?
YES
Is
d
(
u,t
) a strictly stationary random process?
YES
Is
d
(
u,t
) a Gaussian random process?
YES
(b) Use the fact that
d
(
u,t
) =
A
·
r
(
u,t
)

s
(
u,t
) and make input exp(
i
2
πft
) to determine
the system function (evalue).
G
(
f
) =
A
·
H
(
f
)

1
(c) Use
S
d
(
f
) =

G
(
f
)

2
S
s
(
f
) and substitute for
G
(
f
) to get the result.
S
d
(
f
) =

A
·
H
(
f
)

1

2
S
s
(
f
)
(d)
E
{
d
(
u,t
)

2
}
=
R
d
(0) =
Z
∞
∞
S
d
(
f
)
df
=
Z
∞
∞

A
·
H
(
f
)

1

2
S
s
(
f
)
df
=
A
2
Z
∞
∞

H
(
f
)

2
S
s
(
f
)
df

{z
}
a

2
A
Z
∞
∞
Re
{
H
(
f
)
}
S
s
(
f
)
df

{z
}
b
+
Z
∞
∞
S
s
(
f
)
df

{z
}
c
where
Re
{·}
is the realpart operator. (Many of you were stuck at this point because
you did not take the constant
A
outside the integral.) At this point one can complete the
square or diﬀerentiate with respect to
A
and equate to zero, or complete the square with
respect to
A
to get the answer. In completing the square, note that
a
·
A
2

2
b
·
A
+
c
=
a
(
A

b/a
)
2
+
c

b
2
/a
. This form gives the best choice of
A
and the minimum residual
distortion.
A
min
=
b
a
=
R
∞
∞
Re
{
H
(
f
)
}
S
s
(
f
)
df
R
∞
∞

H
(
f
)

2
S
s
(
f
)
df
(e) Evaluate
c

b
2
/a
in (d) to ﬁnd the minimum meansquared distortion.
E
{
d
(
u,t
)

2
}
min
=
Z
∞
∞
S
s
(
f
)
df

h
R
∞
∞
Re
{
H
(
f
)
}
S
s
(
f
)
df
i
2
R
∞
∞

H
(
f
)

2
S
s
(
f
)
df
Note
: Invoking the general LMMSE theory to solve this problem seems like an unnecessary
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview. Sign up
to
access the rest of the document.
 Fall '07
 ToddBrun

Click to edit the document details