Digital Filters
47
Transposed Form II
Transposed Form I
lattice structure
z
-1
z
-1
-a
1
-a
2
x
y
2
a
a
1
Figure 25: Second-order allpass filter, Direct Form II
· Exchange of the roles of the input and output edges
The transposition of a realization in Direct Form II leads to the Transposed
Form II, which is shown in fig. 26. Similarly, the Transposed Form I is obtained
by transposition of the Direct Form I.
z
-1
z
-1
a
1
2
a
-a
2
-a
1
x
y
Figure 26: Second-order allpass filter, Transposed Form II
By direct manipulation of the graph, we can also take advantage of the
properties of special filters. For instance, in an allpass filter, the coefficients of
the numerator are the same of the denominator, in inverted order (see (50)).
With simple transformations of the graph of the Direct Form II it is possible
to obtain the realization of fig. 27, which is interesting because it only has two
multiplies. In fact, the multiplications by -1 can be avoided by replacing two
additions with subtractions.
z
-1
z
-1
-a
1
x
2
a
z
-2
-1
-1
y
Figure 27: Second-order allpass filter, realization with two multipliers and four
state variables
A special structure that plays a very important role in signal processing is
the lattice structure, which can be used to implement FIR and IIR filters [65].