Digital Signal Processing Reference
In-Depth Information
FIGURE 5.1.
Direct form I IIR filter structure.
this structure has 2
N
delay elements, represented by
z
-1
. For example, a second-
order filter with
N
=
2 will have four delay elements.
5.2.2 Direct Form II Structure
The direct form II structure shown in Figure 5.2 is one of the most commonly used
structures. It requires half as many delay elements as the direct form I. For example,
a second-order filter requires two delay elements
z
-1
, as opposed to four with the
direct form I. To show that (5.2) can be realized with the direct form II, let a delay
variable
U
(
z
) be defined as
()
()
Xz
Dz
()
=
Uz
(5.6)
where
D
(
z
) is the denominator polynomial of the transfer function in (5.4). From
(5.4) and (5.6),
Y
(
z
) becomes
() ()
()
NzXz
Dz
()
=
() ()
Yz
=
NzUz
()
(
)
-
1
-
2
-
N
=
Uz a
+
az
+
az
+◊◊◊+
a z
N
(5.7)
0
1
2
where
N
(
z
) is the numerator polynomial of the transfer function in (5.4). From (5.6)
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