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21
CHAPTER 2
2.1
(a) Let
w
k
=
x
+
jy
p
(
k
) =
a
+
jb
We may then write
f
=
w
k
p
*(
k
)
= (
x
+
jy
)(
a

jb
)
= (
ax + by
) +
j
(
ay  bx
)
Let
f
=
u
+
jv
with
u
=
ax + by
v
=
ay  bx
Hence,
From these results we immediately see that
In other words, the product term
w
k
p
*(
k
) satisfies the CauchyRieman equations, and
so this term is analytic.
∂
u
∂
x

a
=
∂
u
∂
y
b
=
∂
v
∂
y
a
=
∂
v
∂
x
b
–
=
∂
u
∂
x
∂
v
∂
y
=
∂
v
∂
x
∂
u
∂
y
–
=
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(b) Let
f
=
w
k
*p
(
k
)
= (
x  jy
) (
a + jb
)
= (
ax + by
) +
j
(
bx  ay
)
Let
f
=
u + jv
with
u
=
ax + by
v
=
bx  ay
Hence,
From these results we immediately see that
In other words, the product term
w
k
*
p
(k) does not satisfy the CauchyRieman
equations, and so this term is
not
analytic.
2.2
(a) From the WienerHopf equation, we have
(1)
∂
u
∂
x

a
=
∂
u
∂
y
b
=
∂
v
∂
x
b
=
∂
v
∂
y
a
–
=
∂
u
∂
x
∂
v
∂
y
≠
∂
v
∂
x
–
∂
u
∂
y
=
w
o
R
1
–
p
=
23
We are given
Hence, the inverse matrix
R
1
is
Using Eq. (1), we therefore get
(b) The minimum meansquare error is
R
1
0.5
0.5
1
=
p
0.5
0.25
=
R
1
–
1
0.5
0.5
1
1
–
=
1
0.75

1
0.5
–
0.5
–1
=
w
o
1
0.75
1
0.5
–
0.5
0.5
0.25
=
1
3

1
0.5
–
0.5
2
1
=
1
3
1.5
0
=
0.5
0
=
J
min
σ
d
2
p
H
w
o
–
=
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(c) The eigenvalues of matrix
R
are roots of the characteristic equation
That is, the two roots are
The associated eigenvectors are defined by
R q
=
λ
q
For
λ
1
= 0.5, we have
Expanding
q
11
+ 0.5
q
12
= 0.5
q
11
0.5
q
11
+
q
12
= 0.5
q
12
Therefore,
q
11
= 
q
12
Normalizing the eigenvector
q
1
to unit length, we therefore have
σ
d
2
0.5, 0.25
0.5
0
–
=
σ
d
2
0.25
–
=
1
λ
–
()
2
0.5
2
–0
=
λ
1
0.5 and
λ
2
1.5
==
1
0.5
0.5
1
q
11
q
12
0.5
q
11
q
12
=
q
1
1
2

1
1
–
=
25
Similarly, for the eigenvalue
λ
2
= 1.5, we may show that
Accordingly, we may express the Wiener filter in terms of its eigenvalues and eigen
vectors as follows:
2.3
(a) From the WienerHopf equation we have
(1)
We are given
and
Hence, the use of these values in Eq. (1) yields
q
2
1
2

1
1
=
w
o
1
λ
i

q
i
q
i
H
i
=1
2
∑
p
=
1
1
–
1, 1
–
1
3

1
1
+
0.5
0.25
=
=
(
11
–
1
–1
1
3
+
)
0.5
0.25
1
λ
1

q
1
q
1
H
1
λ
2
q
2
q
2
H
p
w
o
R
1
–
p
=
R
1
0.5 0.25
0.5
1
0.5
0.25 0.5
1
=
p
0.5 0.25 0.125
T
=
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(b) The minimum meansquare error is
(c) The eigenvalues of matrix
R
are
The corresponding eigenvectors constitute the orthogonal matrix:
Accordingly, we may express the Wiener filter in terms of its eigenvalues and
eigenvectors as follows:
w
o
R
1
–
p
=
1
0.5 0.25
0.5
1
0.5
0.25 0.5
1
1
–
0.5
0.25
0.125
=
1.33
0.67
–0
0.67
–
1.67
0.67
–
0
0.67
–
1.33
0.5
0.25
0.125
=
0
.500
T
=
J
min
σ
d
2
p
H
w
o
–
=
σ
d
2
0.5 0.25 0.125
0.5
0
0
–
=
σ
d
2
0.25
–
=
λ
0.4069
,
0.75
,
1.8431
=
Q
0.4544
–
0.7071
–
0.5418
0.7662
0
0.6426
0.4544
–
0.7071 0.5418
=
27
2.4
By definition, the correlation matrix
where
w
o
1
λ
i

q
i
q
i
H
i
=1
3
∑
p
=
1
0.4069

0.4544
–
0.7662
0.4544
–
0.4544
–
0.7662
0.4554
–
=
1
0.75

0.7071
–
0
0.7071
0.7071
–
0 0.7071
+
1
1.8431
0.5418
0.6426
0.5418
0.5418 0.6426 0.5418
+
0.5
0.25
0.125
×
1
0.4069
0.2065
0.3482
–
0.2065
0.3482
–
0.5871
0.3482
–
0.2065
0.3482
–
0.2065
=
1
0.75
0.5 0
0.5
–
000
0.5
–
0 0.5
+
1
1.8431
0.2935 0.3482 0.2935
0.3482 0.4129 0.3482
0.2935 0.3482 0.2935
+
0.5
0.25
0.125
R
E
u
n
()
u
H
n
[]
=
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Invoking the ergodicity theorem,
Likewise, we may compute the crosscorrelation vector
as the time average
The tapweight vector of the Wiener filter is thus defined by
which is dependent on the length (
N
+1) of the time series.
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This note was uploaded on 09/11/2010 for the course EE EE245 taught by Professor Ujin during the Spring '10 term at YTI Career Institute.
 Spring '10
 Ujin
 Signal Processing

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