Digital Signal Processing Reference
In-Depth Information
(a) Length
N
is odd; (b) length
N
is even.
+
+
+
+
+
…
…
z
−1
z
−1
z
−1
z
−1
[
2
]
h
[
−1
]
N
−1
N
− 1
h
[1]
h
[2]
h
[3]
h
[0]
h
2
+
+
+
+
+
y
[
k
]
(a)
…
x
[
k
]
z
−1
z
−1
z
−1
z
−1
+
+
+
…
+
+
+
z
−1
…
z
−1
z
−1
z
−1
z
−1
h
[
−2
]
[
]
N
−1
N
h
h
[1]
h
[2]
h
[3]
h
[0]
h
−1
2
2
+
+
+
+
+
+
y
[
k
]
(b)
multipliers is roughly half that required in direct form or cascaded form. The
number of unit delay and addition elements, however, stays the same.
14.6.4 Transposed forms
Alternative flow graphs for implementations in Sections 14.6.1-14.6.3 can be
realized by applying the transpose operation. Transposition of a flow graph
is achieved by (i) interchanging the role of the input and output; (ii) revers-
ing the directions of all branches within a flow graph; and (iii) replacing the
source nodes by adders, and vice versa. Note that the number of fundamental
elements required to implement a filter does not change if the transposed form
is used for implementation. We explain the principle of transposition with an
example.
Example 14.4
Implement direct form and cascaded configurations of the flow graph for the
FIR filter with transfer function given by
−
1
+
1
.
4
z
−
2
−
0
.
4
z
−
3
−
0
.
8
z
−
4
.
H
(
z
)
=−
0
.
3
−
0
.
4
z
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