Digital Signal Processing Reference
In-Depth Information
At time
n
+
1, a newer sample
x
(
n
+
1) is acquired and the delay variables in (5.10)
are updated, or
(
)
=+
(
)
-
()
-
un
+
1
xn
1
bun
0
1
where
u
(
n
-
1) is updated to
u
(
n
). From (5.11), the output at time
n
+
1 is
(
)
=
(
)
+
()
+
yn
+
1
aun
+
1
aun
0
0
1
and so on, for time
n
3,...,when, for each specific time, a new input sample
is acquired and the delay variables and then the output are calculated using (5.10)
and (5.11), respectively.
+
2,
n
+
5.2.3 Direct Form II Transpose
The direct form II transpose structure is a modified version of the direct form II
and requires the same number of delay elements. The following steps yield a trans-
pose structure from a direct form II version:
1.
Reverse the directions of all the branches.
2.
Reverse the roles of the input and output (input
output).
3.
Redraw the structure such that the input node is on the left and the output
node is on the right (as is typically done).
ยด
The direct form II transpose structure is shown in Figure 5.3. To verify this, let
u
0
(
n
) and
u
1
(
n
) be as shown in Figure 5.3. Then, from the transpose structure,
()
=
()
-
()
un axn byn
0
(5.12)
2
2
()
=
()
-
()
+-
(
)
un axn byn un
1
1
(5.13)
1
1
0
()
=
()
+-
(
)
yn
axn
u n
1
(5.14)
0
1
Equation (5.13) becomes, using (5.12) to find
u
0
(
n
-
1),
()
=
()
-
()
+
[
(
)
-
(
)
]
u n
axn
byn
axn
-
1
byn
-
1
(5.15)
1
1
1
2
2
FIGURE 5.3.
Direct form II transpose IIR filter structure.
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