46
D. Rocchesso: Sound Processing
signal flowgraph
Direct Form I
Direct Form II
transposition of a signal
flowgraph
of allpass filters in sec. 2.3.
2.2.4
Realizations of IIR Filters
So far, we have studied the IIR filters by analysis of transfer functions or im-
pulse responses. In this section we want to face the problem of implementing
these filters as computational structures that can be directly coded using sound
processing languages or real-time sound processing environments.
Consider a second-order filter with two poles and two zeros, which is rep-
resented by the transfer function (25) with N = M = 2. This can be realized
by the signal flowgraph of fig. 24, where the nodes having converging edges
are considered as points of addition, and the nodes having diverging edges are
considered as branching points. Such a realization is called Direct Form I.
z
-1
z
-1
z
-1
z
-1
x
y
b
b
b
0
1
2
-a
1
-a
2
Figure 24: Second-order filter, Direct Form I
Signal flowgraphs can be manipulated in several ways, thus leading to al-
ternative realizations having different numerical properties and, possibly, more
computationally efficient. For instance, if we want to implement a filter as a
cascade of second-order cells such as that of fig. 24, we can share, between two
contiguous cells, the unit delays that are on the output stage of the first cell,
with the unit delays that are on the input stage of the second cell, thus saving
a number of memory accesses.
We are going to show some other kind of manipulation of signal flowgraphs,
in the special case of the realization of the second-order allpass filter, which has
the property
b
i
= a
2-i
, i = 0, 1, 2 .
(50)
A first transformation comes from the observation that the structure of fig. 24
is formed by the cascade of two blocks, each being linear and time invariant.
Therefore, the two blocks can be commuted without altering the input-output
behavior. Moreover, from the block exchange we get a flowgraph with two side-
to-side stages of pure delays, and these stages can be combined in one only. The
realization of these transformations is shown in fig. 25 and it is called Direct
Form II.
Another transformation that can be done on a signal flowgraph without
altering its input-output behavior is the transposition [65]. The transposition of
a signal flowgraph is done with the following operations:
Inversion of the direction of all the edges
Transformation of the nodes of addition into branching nodes, and vice
versa
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