Digital Signal Processing Reference
In-Depth Information
finite precision multiplication be truncated, rather than rounded, as they are
accumulated. Other, more detailed, analysis of limit cycles is included in the
references.
8.2 C CODE FOR IIR FILTER IMPLEMENTATION
When discussing the implementation of IIR filters, we assume that the filter is
described by a set of quadratic coefficients of the form determined in Chapter 7.
As we have seen in the previous section, a cascaded sequence of quadratic
structures is the recommended method of implementation. The basic quadratic
building block is shown in the system diagram of Figure 8.2.
Figure 8.2
System diagram for a single quadratic factor.
We can generate the transfer function for this section by determining the
expressions for the intermediate signal
w
(
n
) and the output signal
y
(
n
).
w
(
n
)
=
x
(
n
)
+
b
⋅
w
(
n
−
1
+
b
⋅
w
(
n
−
2
(8.7)
1
2
y
(
n
)
=
w
(
n
)
+
a
⋅
w
(
n
−
1
+
a
⋅
w
(
n
−
2
(8.8)
1
2
These equations can be
z
-transformed to give
−
1
−
2
W
(
z
)
=
X
(
z
)
+
b
⋅
z
⋅
W
(
z
)
+
b
⋅
z
⋅
W
(
z
)
(8.9)
1
2
−
1
−
2
Y
(
z
)
=
W
(
z
)
+
a
⋅
z
⋅
W
(
z
)
+
a
⋅
z
⋅
W
(
z
)
(8.10)
1
2
Equations (8.9) and (8.10) can be rewritten and combined to determine the
transfer function for this section of the filter, as shown in (8.11). This formulation
matches the quadratic terms we developed for IIR filters. We will be able to match
