Digital Signal Processing Reference
In-Depth Information
FIGURE 5.2.
Direct form II IIR filter structure.
(
)
()
=
() ()
=
()
+
Xz
UzDz
Uz
1
bz
-
1
+
bz
-
2
+◊◊◊+
b z
N
-
N
(5.8)
1
2
Taking the inverse
z
-transform of (5.8) yields
()
=
()
+
(
)
+
(
)
+◊◊◊+
(
)
xn
un
bun
-
1
bun
-
2
b un N
N
-
(5.9)
1
2
Solving for
u
(
n
) in (5.9) gives us
()
=
()
-
(
)
-
(
)
-◊◊◊-
(
)
un
xn
bun
-
1
bun
-
2
b un N
N
-
(5.10)
1
2
Taking the inverse
z
-transform of (5.7) yields
()
=
()
+
(
)
+
(
)
+◊◊◊+
(
)
yn
aun
aun
-
1
aun
-
2
a un N
N
-
(5.11)
0
1
2
The direct form II structure can be represented by (5.10) and (5.11). The delay
variable
u
(
n
) at the middle top of Figure 5.2 satisfies (5.10), and the output
y
(
n
) in
Figure 5.2 satisfies (5.11).
Equations (5.10) and (5.11) are used to program an IIR filter. Initially,
u
(
n
-
1),
u
(
n
2),...are set to zero. At time
n
, a new sample
x
(
n
) is acquired, and (5.10) is
used to solve for
u
(
n
). The filter's output at time
n
then becomes
-
()
=
()
+
yn
aun
0
0
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