Digital Signal Processing Reference
In-Depth Information
with
∆
ω
p
representing the passband frequency region(s), with
∆
ω
a
representing the stopband
frequency region(s), and with
τ
(
ω
)
representing the group-delay frequency response of the FRM
IIR digital filter. Here,
W
p
,
W
a
, and
W
gd
represent weighting factors for the passband and stopband
magnitude responses, and for the group-delay response, respectively. Moreover,
µ
τ
represents the
average group-delay over the passband region.
In [
48
], a convenient way to represent digital networks in terms of matrix representation is presented.
This technique can be used to find the magnitude and group delay frequency response of the digital
network in Fig.
12
. Let us consider the input to the digital network in Fig.
12
to be
x
D
and the
output of it to be
y
D
. In addition, let the output of the
i
-th time delay in Fig.
12
to be
x
i
and the
input to the
i
-th time delay to be
y
i
. The transfer function matrix of the network,
T
, can be found as
Y
=
TX
(68)
where
Y
= [
y
D
,
y
1
,
y
2
, . . . ,
y
2
m
+1
]
t
2
and
X
= [
x
D
,
x
1
,
x
2
, . . . ,
x
2
m
+1
]
t
, and
T
is a
(2
m
+ 2) ×
(2
m
+ 2)
matrix with the entries obtained as Eqn. (
69
).
2
4
3
5
0
1
−1
0
0
0
...
0
0
1
0
0
0
0
0
...
0
0
m
i
=
1
m
L
i
)
−
m
C
1
m
C
1
1
−
m
C
1
(
1
+
m
C
1
m
C
1
m
L
2
−
m
C
1
...
m
C
1
m
L
m
−
m
C
1
0
0
m
L
1
1
0
0
...
0
0
T
=
(69)
0
0
m
C
2
m
L
2
0
1−
m
C
2
m
L
2
m
C
2
...
0
0
0
0
m
L
2
0
−
m
L
2
1
...
0
0
.
.
.
.
.
.
.
.
.
0
0
m
C
m
m
L
m
0
0
0
1−
m
C
m
m
L
m
m
C
m
−
m
L
m
0
0
m
L
m
0
0
0
1
Since
x
i
=
z
−1
y
i
, the transfer function
G
(
z
) =
y
D
can be found as
x
D
G
(
z
) =
z
−1
E
[
I
−
z
−1
D
]
−1
C
(70)
where
E
is a row vector and
C
is a column vector of length
2
m
+
1
, and where
I
is the identity
matrix and
D
is a
(
2
m
+
1
) × (
2
m
+
1
)
matrix in accordance with
2
4
0
E
C D
3
5
T
=
(71)
2
X
t
denotes the transpose of the matrix
X
.
