4. Write an expression for the crosscorrelation vector
r
ud
= E
{
u
[
n
]
d
*
[
n
]
}
using
H
,
σ
2
s
, and
i
Δ
.
5. Write an expression for
σ
2
d
, the variance of
{
d[
n
]
}
, using
σ
2
s
.
6. Write an expression for the minimum MSE
J
mse
,⋆
using
H
,
σ
2
a
,
σ
2
s
,
I
, and
i
Δ
.
7. Write an expression for the Wiener Flter
w
⋆
using
H
,
σ
2
a
,
σ
2
s
,
I
, and
i
Δ
.
8. Using Matlab, compute both
J
mse
,⋆
and
w
⋆
assuming that
h
=
b
1

0
.
5
B
,
Δ = 0
, L
= 4
, σ
2
a
= 0
.
01
, σ
2
s
= 1
.
Hint:
Useful Matlab commands include
convmtx
,
eye
,
pinv
.
9. ±or the system parameters given in problem 8, what is the allowed range for the stepsize
μ
of
gradient descent?
Hint:
Use the Matlab commands
eig
and
max
.
10. ±or the system parameters in problem 8, adapt the Flter
w
using gradient descent. Start from the
initialization
w
=
0
, use the stepsize
μ
= 1
/λ
max
(where
λ
max
denotes the maximum eigenvalue of
R
u
), and use 50 iterations.
(a) Plot the evolution of the Flter coe²cients
w
in Matlab using
plot(W.’,’+’)
, where the
n
th
column of matrix
W
contains the coe²cient vector
w
at iteration
n
. Do the Flter coe²cients
converge to the Wiener solution
w
⋆
?
(b) Plot the evolution of the MSE cost
J
mse
[
n
]. Does it converge to the minimum MSE
J
mse
,⋆
?
Your plots should look something like below. Note that the Wiener solution
wstar
is plotted using
squares at the Fnal iteration
N
via
plot(N,wstar,’s’)
.
0
5
10
15
20
25
30
35
40
45
50
0
0.2
0.4
0.6
0.8
1
iteration
filter coefficients
gradient descent
0
5
10
15
20
25
30
35
40
45
50
0
0.2
0.4
0.6
0.8
1
1.2
1.4
iteration
Jmse
11. ±or the parameters in problem 8, adapt the Flter
w
using LMS. Start from the initialization
w
=
0
,
use the stepsize
μ
= 0
.
1
/λ
max
, and use 500 iterations. Use the realizations of
{
u[
n
]
}
and
{
d[
n
]
}
provided in the Fle
u.txt
and
d.txt
on the course webpage.
(a) Plot the evolution of the Flter coe²cients
w
in Matlab as above. Do the Flter coe²cients
converge near to the Wiener solution
w
⋆
?
(b) Plot the evolution of the MSE cost
J
mse
[
n
] as above. Does it converge near to the minimum
MSE
J
mse
,⋆
?
Hint:
Remember the correct (backwards) ordering of the samples within the
u
[
n
] vector.
P. Schniter, 2012
2