Homework 6
These are all practice problems – you do not need to submit anything.
The solutions will be provided later in the week.
SS1. Problem 5.4 in
Stark and Woods
(
Stark and Woods
problems are at the end of this PDF.)
SS2. Problem 5.8 in
Stark and Woods
SS3. Problem 5.9 in
Stark and Woods
1. Problem 5.10 in
Stark and Woods
2. Problem 5.15 in
Stark and Woods
SS4. Problem 6.55 in
LeonGarcia
3. Problem 6.57 in
LeonGarcia
4. Problem 6.61 in
LeonGarcia
SS5. Problem 5.21 in
Stark and Woods
5. Problem 6.85 in
LeonGarcia
6. Do Problem 5.22 in
Stark and Woods
with the following clariﬁcations: After ﬁnding the
whitening transform, use the inverse transform to create correlated random variables with
the given covariance matrix. On the same graph as the scatter plot of the correlated random
variables, plot lines showing the eigenvectors of the covariance matrix.
7. Generate 5000 pairs
(
X
,
Y
)
of zeromean, correlated Gaussian RVs with the covariance
matrix given in Problem 5.22. Consider the performance of the following two data compression
approaches:
(a) Keep all the values of
X
, but replace all the values of
Y
by their mean, 0. Thus, the data
size is approximately reduced by half. Plot the compressed data on top of the original
data, using a different marker and color. Compute the average squareerror between
the original data and the compressed version.
(b) Use the KLT to transform
(
X
,
Y
)
into uncorrelated random variables
(
W
,
Z
)
such that
Var
[
W
]
is maximized (under the unitary transform of the KLT). Replace all of the values
of
Z
by their mean, 0. Again, the data size is approximately reduced by half. Apply
the inverse transform to the compressed data to create
(
X
0
,
Y
0
)
, which are compressed
versions of
(
X
,
Y
)
. Plot
(
X
0
,
Y
0
)
on top of the original data using a different marker and
color. Compute the average squareerror between
(
X
0
,
Y
0
)
and the original data
(
X
,
Y
)
.
SS6. For each of the following matrices, determine if it is a valid covariance matrix for some
random vector. In the case that it is a valid covariance matrix, determine if the random
variables with that covariance matrix are linearly independent.
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K
1
=
4
1
.
6

1
.
6
1
.
6
1

0
.
5

1
.
6

0
.
5 0
.
69
4
(b)
K
2
=
4
1
.
6

1
.
6
1
.
6
1

0
.
6

1
.
6 0
.
6
1
(c)
K
3
=
4
1
.
6

1
.
6
1
.
6
1
0
.
5

1
.
6 0
.
5
1
(d)
K
4
=
4
1
.
6

1
.
6
1
.
6
1

0
.
5

1
.
6

0
.
5
1
8. Let
X
be a zeromean random vector with covariance matrix
K
=
4
1

1
1
2
0
.
5

1 0
.
5
1
(a) Specify a vector
b
with
k
b
k
2
=
1 such that the variance of
b
T
X
is
minimized
, and give
the value of that variance.
(b) Specify a vector
a
with
k
a
k
2
=
1 such that the variance of
a
T
X
is
maximized
, and give
the value of that variance.
(c) Give an equation to whiten
X
. In other words, give an equation for a random vector
Y
in terms of
X
,
V
, and
D
such that
Y
is a vector of uncorrelated random variables with
unit variance.
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 Spring '11
 Wang
 Variance, Probability theory, Covariance matrix, LeonGarcia

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