ECE 567
STATISTICAL SIGNAL PROCESSING
FALL 2013
Homework Assignment #6
Solutions
1. (a)
x(n) = + w(n)
1
p(w(n) = exp (w(n)) ,
2
1
.
x = . + w
.
1
0
.
E [w ] = .
.
IID
0
VAR[w] = C = 2 I
The BLUE is
T
1
. 2
. Ix
.
1
1
= T
=
N
1
1
. 2 .
. I .
.
.
ECE 567
STATISTICAL SIGNAL PROCESSING
FALL 2013
Homework Assignment #5
Solutions
1. The data model is
x(0)
.
.
=
.
x(N 1)
1
r1
.
.
.
1
r2
.
.
.
N
N
r1 1 r2 1
1
rp
.
.
.
N
rp 1
A1
A2
.
.
.
+
Ap
w(0)
.
.
.
w(N 1)
x
H
and then
= (H T H )1 H T x
C = 2 (H T H
ECE 567
STATISTICAL SIGNAL PROCESSING
FALL 2016
Homework Assignment #8
Due: 8 November 2016
1. (a) Suppose that you have N IID observations x(n) from a pdf within the exponential
family of pdfs. These have the form
p(x; ) = exp [A()B(x) + C(x) + D()]
wher
ECE 567
STATISTICAL SIGNAL PROCESSING
FALL 2016
Homework Assignment #5
Solutions
1. (a) We have x(n) N (0, 2 ) and
N 1
1 X 2
2
b
=
x (n)
N n=0
N 1
N 1
1 X
1 X 2
2
2
b
E[ ] =
E[x (n)] =
= 2
N n=0
N n=0
so its unbiased.
(b) The variance is
h
i
var[b2 ] =
ECE 567
STATISTICAL SIGNAL PROCESSING
FALL 2016
Homework Assignment #6
Solutions
1. This is a linear model with
x(0)
x(1)
x=
.
.
x(N 1)
,
w=
w(0)
w(1)
.
.
=A=
,
w(N 1)
A1
A2
.
.
Ap
We have
x = H + w,
where
H=
1
r1
r12
.
.
1
r2
r22
1
rp
rp2
.
.
r1N 1 r2N
ECE 567
STATISTICAL SIGNAL PROCESSING
FALL 2010
Exam #1
7 October 2010
Name:
This is an openbook, opennotes exam. The use of electronic calculators is permitted. The
exam lasts 75 minutes.
There are four questions on the exam. The first asks two short a
ECE 567
STATISTICAL SIGNAL PROCESSING
SPRING 2008
Exam #1
5 March 2008
Name:
This is an openbook, opennotes exam. The use of electronic calculators is permitted. The
exam lasts 75 minutes.
There are four questions on the exam. The first asks two short a
ECE 567
STATISTICAL SIGNAL PROCESSING
FALL 2009
Exam #2
12 November 2009
Name:
This is an openbook, opennotes exam. The use of electronic calculators is permitted. The
exam lasts 75 minutes.
There are four questions on the exam. The first is a short ans
ECE 567
STATISTICAL SIGNAL PROCESSING
SPRING 2015
Exam #1
26 February 2015
Name:
This is an openbook, opennotes exam. The use of electronic calculators is permitted. The
exam lasts 75 minutes.
There are three questions on the exam. Do all your work on t
ECE 567
STATISTICAL SIGNAL PROCESSING
SPRING 2015
Exam #2
14 April 2015
Name:
This is an openbook, opennotes exam. The use of electronic calculators is permitted. The
exam lasts 75 minutes.
There are three questions on the exam. Do all your work on thes
ECE 567
STATISTICAL SIGNAL PROCESSING
FALL 2016
Homework Assignment #5
Due: 13 October 2016
1. We observe data x(n) for n = 0, 1, . . . , N 1 with x(n) N (0, 2 ) (iid), and we want
to estimate 2 .
(a) Show that
N 1
1 X 2
2
b
x (n)
=
N n=0
is an unbiased
ECE 567
STATISTICAL SIGNAL PROCESSING
Homework Assignment #3
Solutions
1. We have
1 X 2
p(x; H0 ) = (2 )
exp 2
x (n)
2
1 X 2
N
x (n)
ln [p(x; H0 )] = ln(2 2 ) 2
2
2
d
N
1 X 2
[ln
(p(x;
x (n)
H
)]
=
+
0
d( 2 )
2 2 2( 2 )2
P 2
1
x (n)
= 2 N
2
2
2 (N/2)
whi
ECE 567
STATISTICAL SIGNAL PROCESSING
Homework Assignment #7
Due: 27 October 2016
1. Problem 5.3 from the text.
2. Problem 5.5 from the text.
3. Problem 5.9 from the text.
4. Show that the variance of the BLUE = aT x is
=
var[]
5. Problem 6.4 from the te
ECE 567
STATISTICAL SIGNAL PROCESSING
FALL 2016
Homework Assignment #7
Solutions
1.
p(x; ) =
N
1
Y
exp[ x(n)]u(x(n)
n=0
"
#
N
1
X
N
= exp
x(n)

cfw_z
g(T (x),)
Hence
T (x) =
N
1
X
u(x(n)
n=0
n=0

N
1
Y
cfw_z
h(x)
x(n).
n=0
2.
1
u (x(n) + ) u (x(n) +
ECE 567
STATISTICAL SIGNAL PROCESSING
FALL 2016
Homework Assignment #9
Due: 1 December 2016
1. In this problem you will construct an EMalgorithm to iteratively determine the MLE
in a problem setting in which you can analytically determine the MLE.
The pr
ECE 567
STATISTICAL SIGNAL PROCESSING
FALL 2016
Homework Assignment #3
Due: 22 September 2016
1. You observe iid data x(n) for n = 0, 1, . . . , N 1 which consist only of noise, but you
want to detect whether the noise is nonGaussian. The null hypothesis
ECE 567
STATISTICAL SIGNAL PROCESSING
FALL 2016
Homework Assignment #6
Due: 20 October 2016
1. Problem 4.1 from the text.
2. You make measurements x1 (n) and x2 (n) for n = 0, 1, . . . , N 1 that are different
noisy mixtures of the same pair of signals s1
ECE 567
STATISTICAL SIGNAL PROCESSING
FALL 2016
Homework Assignment #4
Due: 3 October 2016
1. We observe data x(n) for n = 0, 1, . . . , N 1 with
w(n),
under H0
x(n) =
A s(n) + w(n), under H1
The signal s(n) is known, cfw_w(n) is white Gaussian noise with
ECE 567
STATISTICAL SIGNAL PROCESSING
FALL 2013
Exam #1
3 October 2013
Name:
This is an openbook, opennotes exam. The use of electronic calculators is permitted. The
exam lasts 75 minutes.
There are three questions on the exam. Do all your work on these
ECE 567
STATISTICAL SIGNAL PROCESSING
FALL 2009
Exam #1
8 October 2009
Name:
This is an openbook, opennotes exam. The use of electronic calculators is permitted. The
exam lasts 75 minutes.
There are four questions on the exam. The first asks two short a
ECE 567
STATISTICAL SIGNAL PROCESSING
FALL 2013
Exam #2
12 November 2013
Name:
This is an openbook, opennotes exam. The use of electronic calculators is permitted. The
exam lasts 75 minutes.
There are three questions on the exam. Do all your work on the
ECE 567
STATISTICAL SIGNAL PROCESSING
Homework Assignment #4
Solutions
1. We have x(n) N (0, 2 ) and
N 1
1 2
2 =
x (n)
N n=0
N 1
N 1
1 2
2
2 ] = 1
E [
E [x (n)] =
= 2
N n=0
N n=0
so its unbiased.
The variance is
[
]
var[ 2 ] = E ( 2 2 )2
[
]
= E ( 2 )2 2
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ECE 567
STATISTICAL SIGNAL PROCESSING
FALL 2013
Homework Assignment #9
Solutions
1. We have
= E [  x]
=
p(  x)d
=
p(x  )p()
d
p(x  1 )p(1 )d1
We have
N 1
p(x  ) = exp
x(n) + N u (mincfw_x(n) )
n=0
N 1
= exp(N ) exp
x(n) u (mincfw_x(n) )
n=0
N 1
p(
EOE 56? STATISTICAL SIGNAL PROCESSING FALL 2013
Exam #1
3 October 2013
Name:
This is en openijook; open~11otes exam. The use of electronic calculators is permitted. The
exam lasts 75 minutes.
There are three questions on the exam. Do all your work