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IIR Filter Structures 215
FIGURE 6.2
Direct form I structure
which can be implemented as shown in Figure 6.2. This block diagram is
called direct form I structure.
The direct form I structure implements each part of the rational
function H(z) separately with a cascade connection between them. The
numerator part is a tapped delay line followed by the denominator part,
which is a feedback tapped delay line. Thus there are two separate delay
lines in this structure, and hence it requires eight delay elements. We
can reduce this delay element count or eliminate one delay line by interchanging
the order in which the two parts are connected in the cascade.
Now the two delay lines are close to each other, connected by a unity
gain branch. Therefore one delay line can be removed, and this reduction
leads to a canonical structure called direct form II structure, shown in
Figure 6.3. It should be noted that both direct forms are equivalent from
the input-output point of view. Internally, however, they have different
signals.
x(n)
−a 1
−a 2
−a 3
−a 4
z −1
z −1
z −1
z −1
b 0
b 1
b 2
b 3
b 4
y (n)
x(n)
b 0
b 1
z −1 −a 1
b 4
z −1 −a 4
b 2
b 3
z −1
z −1
−a 2
−a 3
y (n)
(a) Normal
FIGURE 6.3
Direct form II structure
(b) Transposed
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