Digital Signal Processing Reference
In-Depth Information
Minimize
M
πn
1
−
M
d
πn
1
b
N
1
N
2
N
1
,
πn
2
N
1
,
πn
2
J
=
(8.4)
N
2
N
2
n
1
=
0
n
2
=
0
subject to the constraints
|
q
k
+
r
k
|−
1
<s
k
,k
=
1
,
2
,...,N,
(8.5)
s
k
<
1
−|
q
k
−
r
k
|
,k
=
1
,
2
,...,N
(8.6)
where
N
1
,
N
2
, and
N
are positive integers.
Without loss of generality, we consider the case of
N
=
2. Thus
H(z
1
,z
2
)
in
(
8.1
) can be simplified as shown in (
8.7
):
H
0
p
00
+
p
02
z
2
+
p
20
z
1
+
p
12
z
1
z
2
H(z
1
,z
2
)
=
p
01
z
2
+
p
10
z
1
+
p
11
z
1
z
2
+
p
22
z
1
z
2
/
(
1
p
21
z
1
z
2
+
+
+
q
1
z
1
+
r
1
z
2
+
s
1
z
1
z
2
)(
1
+
q
2
z
1
s
2
z
1
z
2
)
.
+
r
2
z
2
+
(8.7)
Now, transforming the variables
z
1
and
z
2
to the frequency domain terms
ω
1
and
ω
2
, we obtain the following expression for
M(ω
1
,ω
2
)
asshownin(
8.8
):
H
0
{
M(ω
1
,ω
2
)
=
p
00
+
p
01
f
01
+
p
02
f
02
+
p
10
f
10
+
p
20
f
20
+
p
11
f
11
+
p
12
f
12
+
p
21
f
21
+
p
22
f
22
}
/D
−
j
{
p
01
g
01
+
p
02
g
02
+
p
10
g
10
/D
+
p
20
g
20
+
p
11
g
11
+
p
12
g
12
+
p
21
g
21
+
p
22
g
22
}
(8.8)
where
=
(
1
s
1
g
11
)
·
(
1
+
q
1
f
10
+
r
1
f
01
+
−
j(q
1
g
10
+
r
1
g
01
+
+
D
s
1
f
11
)
q
2
f
10
s
2
g
11
)
,
+
r
2
f
01
+
−
j(q
2
g
10
+
r
2
g
01
+
s
2
f
11
)
(8.9)
=
cos
(xω
1
+
f
xy
(ω
1
,ω
2
)
yω
2
),
(8.10)
=
sin
(xω
1
+
=
g
xy
(ω
1
,ω
2
)
yω
2
), x,y
0
,
1
,
2
.
(8.11)
In a more compact form,
M(ω
1
,ω
2
)
can be expressed as follows:
A
R
−
jA
I
M(ω
1
,ω
2
)
=
H
0
(8.12)
(B
1
R
−
jB
1
I
)(B
2
R
−
jB
2
I
)
where
A
R
=
p
00
+
p
01
f
01
+
p
02
f
02
+
p
10
f
10
+
p
20
f
20
+
p
11
f
11
+
p
12
f
12
+
p
21
f
21
+
p
22
f
22
,
(8.13)
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