Digital Signal Processing Reference
In-Depth Information
The quadratic minimization problem (3.7) can be readily solved. Let
ˆ
h
H
(
h
0
h
∗
M
0
−
1
···
000
k
)
ˆ
h
H
(
ω
h
0
h
∗
M
0
−
1
0
···
00
k
)
.
.
.
ˆ
h
H
(
ω
L
×
N
H
(
ω
k
)
=
=
∈ C
.
.
.
.
.
.
.
.
.
h
0
h
∗
M
0
−
1
···
ω
00 0
k
)
(3.8)
and
1
e
j
ω
k
.
e
j
ω
k
(
L
−
1)
L
×
1
η
(
ω
=
α
ˆ
ω
∈ C
.
k
)
(
k
)
(3.9)
Using this notation we can write the objective function in (3.7) as
2
y
0
.
y
N
−
1
K
−
1
−
H
(
ω
k
)
η
(
ω
k
)
.
(3.10)
k
=
0
Define the
L
×
g
and
L
×
(
N
−
g
) matrices
A
(
ω
k
) and
B
(
ω
k
)from
H
(
ω
k
) via the
following equality:
=
y
0
.
y
N
−
1
H
(
ω
k
)
A
(
ω
k
)
γ
+
B
(
ω
k
)
µ
.
(3.11)
Also, let
d
(
ω
k
)
=
η
(
ω
k
)
−
A
(
ω
k
)
γ
.
(3.12)
With this notation the objective function (3.10) becomes
−
K
1
2
0
B
(
ω
k
)
µ
−
d
(
ω
k
)
,
(3.13)
k
=
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