Digital Signal Processing Reference
In-Depth Information
b
0
+
+
x
[
n
]
y
[
n
]
l
g
r
,
y
i
d
.
,
©
,
L
s
z
−
1
a
1
b
1
+
+
z
−
1
a
2
b
2
Figure 7.26
Direct Form II implementation of a second-order section.
The second order filter
b
1
z
−
1
b
2
z
−
2
1
+
+
H
(
z
)=
a
1
z
−
1
a
2
z
−
2
1
−
−
which gives rise to the second order section displayed in Figure 7.26, is par-
ticularly important in the case of cascade realizations. Consider the factored
form of
H
as in (7.27): if we combine the complex conjugate poles and
zeros, and group the real poles and zeros in pairs, we can create a modular
structure composed of second order sections. For instance, Figure 7.27 rep-
resents a 4th order system. Odd order systems can be obtained by setting
some of the
a
n
or
b
n
to zero.
(
z
)
b
1,0
b
2,0
[
]
+
+
+
+
[
]
x
n
y
n
z
−
1
z
−
1
a
1,1
b
1,1
a
2,1
b
2,1
+
+
+
+
z
−
1
z
−
1
a
1,2
b
1,2
a
2,2
b
2,2
Figure 7.27
4th order IIR: cascade implementation.
