Digital Signal Processing Reference
In-Depth Information
where vec(
) denotes the operation of stacking the columns of a matrix onto each
other. In (3.21),
y
l
1
,
l
2
is defined by
·
y
l
1
,
l
2
y
l
1
,
l
2
+
1
...
y
l
1
,
l
2
+
M
2
−
1
y
l
1
+
1
,
l
2
y
l
1
+
1
,
l
2
+
1
...
y
l
1
+
1
,
l
2
+
M
2
−
1
y
l
1
,
l
2
vec(
Y
l
1
,
l
2
)
vec
.
.
.
.
.
.
.
y
l
1
+
M
1
−
1
,
l
2
y
l
1
+
M
1
−
1
,
l
2
+
1
...
y
l
1
+
M
1
−
1
,
l
2
+
M
2
−
1
(3.22)
The APES spectrum estimate ˆ
α
(
ω
,ω
2
) and the filter coefficient matrix
1
H
(
ω
,ω
2
)are the minimizers of the following LS criterion:
1
L
1
−
1
L
2
−
1
x
l
1
,
l
2
2
)
e
j
(
ω
1
l
1
+
ω
2
l
2
)
min
2
−
α
(
ω
,ω
α
(
ω
,ω
2
)
,
H
(
ω
,ω
2
)
1
1
1
l
1
=
0
l
2
=
0
vec
H
(
H
(
s
.
t
.
ω
,ω
2
))
a
M
1
,
M
2
(
ω
,ω
2
)
=
1
.
(3.23)
1
1
Here
a
M
1
,
M
2
(
ω
,ω
2
)isan
M
1
M
2
×
1vector given by
1
a
M
1
,
M
2
(
ω
,ω
2
)
a
M
2
(
ω
2
)
⊗
a
M
1
(
ω
1
)
,
(3.24)
1
where
⊗
denotes the Kronecker matrix product and
e
j
ω
k
e
j
(
M
k
−
1)
ω
k
]
T
a
M
k
(
ω
k
)
[1
...
,
k
=
1
,
2
.
(3.25)
X
into (3.23), we have the following design objective for 2-D APES:
Substituting
vec(
H
(
2
)
2
ω
,ω
−
α
ω
,ω
ω
,ω
min
α
(
ω
1
,ω
2
)
,
H
(
ω
1
,ω
2
)
2
)
Y
)
(
2
)
a
L
1
,
L
2
(
1
1
1
vec
H
(
H
(
s
.
t
.
ω
,ω
2
))
a
M
1
,
M
2
(
ω
,ω
2
)
=
1
,
(3.26)
1
1
.
The solution to (3.26) can be readily derived. Define
ω
,ω
ω
,ω
where
a
L
1
,
L
2
(
2
)isdefined similar to
a
M
1
,
M
2
(
2
)
1
1
L
1
−
1
L
2
−
1
1
L
1
L
2
R
y
l
1
,
l
2
y
l
1
,
l
2
=
(3.27)
l
1
=
0
l
2
=
0
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