This preview shows pages 1–2. Sign up to view the full content.
Final Exam of Stochastic Processes
Jan. 11, 2005
1.
(6 points) Let
( )
t
X
be a SSS process with mean
η
. Form an unbiased estimator of
the variance
V
and list the conditions that make
( )
t
X
be a varianceergodic
process.
2.
(5
points)
(a)
Point
out
the
draw
back
of
using
periodogram
2
2
( )
( )
T
j
T
T
S
e
d
ϖτ
ϖ
τ


=
T
R
as an estimate of the power spectrum of a process
( )
t
X
, where
( )
T
R
is an estimate of the autocorrelation
( )
R
of
( )
t
X
.
(10 points) (b) Briefly describe the principle of smoothed spectrum using lag
window
( )
w
and discuss the criteria of selecting a good window.
2
2
( )
( )
( )
T
j
w
T
T
S
w
R
e
d


=
3.
(15 points) An MA filter
1
( )
1
N
N
k
k
k
D z
a z

=
= 
can be implemented by a lattice
filter with forward prediction error filter
ˆ
( )
( )
N
E
z
D z
=
and backward prediction
error filter
( )
N
E
z
(
. We know that
ˆ
( )
N
E
z
and
( )
N
E
z
(
satisfy
ˆ
( )
N
E
z
1
1
1
ˆ
( )
( )
N
N
N
E
z
K z E
z



=

(
1
1
1
ˆ
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 '08
 SinHorngChen

Click to edit the document details