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Solutions Manual for Adaptive Filter Theory 5th Edition by Haykin IBSN 9780132671453
Full download: https://goo.gl/ZNQPPH
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Solutions Manual for Adaptive Filter Theory 5th Edition by Haykin IBSN 9780132671453
Full download: http://downloadlink.org/p/solutions-manual-for-adaptive-filter-theory-5th-edition-by-haykin-ibsn-9780132671453/
Chapter 2
Problem 2.1
a)
Let w
k
=
x
+
j y
p(−
k) =
a
+
j
b
We may then write
f
=
w
k
p
∗
(
−
k
)
=(x +
j y)(a
−
j
b
)
=(ax +
by)
+
j(ay
−
bx
)
Letting where
f =
u
+
j v
u
=
ax
+
by v
=
ay
−
bx Hence,
∂
u
∂
u
=
a
=
b
∂
x
∂
y
∂
v
∂
v
= a
∂
y
∂
x
=
−
b
From these results we can immediately see that
∂
u
∂
v
=
∂
x
∂
y
∂
v
∂
u
∂
x
=
−
∂
y In other words, the product term
w
k
p
∗
(−
k)
satisfies the Cauchy-Riemann equations, and so this term is analytic.
b)
Let Let
f
=
w
k
p
∗
(
−
k
)
=(x
−
j y)(a
+
j
b
)
=(ax +
by)
+
j(bx
−
ay
)
with
f =
u
+
j
v
u
=
ax
+
by v
=
bx
−
ay
Hence,
∂
u
∂
u
=
a
∂
x
∂
y
∂
v
∂
v
=
b
∂
x
∂
y
=
b
=
−
a
From these results we immediately see that
∂
u
∂
v
=
∂
x
∂
y
∂
v
∂
u
∂
x
=
−
∂
y In other words, the product term
w
∗
p(−
k)
does not satisfy the
Cauchy-Riemann
equations, and so this term is not analytic.
d
Problem 2.2
a)
From the Wiener-Hopf equation, we have w
0
=
R
−
1
p (1) We are given that
1
0
.
5
R
=
0.5 1
0
.
5
p
=
0.25 Hence the inverse of R is 1 0.5
−
1
R
−
1
=
=
0.5 1 1 1
−0.5
−
1
0
.
75
−0.5
1
Using Equation (1), we therefore get
1
1
−
0
.
5
0
.
5
w
0
=
0.75
−0.5
1 0.25
1
0
.
375
=
0.75 0
0
.
5
=
0
b)
The minimum mean-square error is
J
min
=
σ
2
−
p
H
w
0
=
σ
2
−
0
.
5
0
.
25
=
σ
2
−
0.25
0
.
5
0

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