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].