50
D. Rocchesso: Sound Processing
parametric filters
be able to tune the bandwidth and the center frequency independently. To
construct such a filter, one of the two allpass filters is replaced by the identity
(i.e., a short circuit) while the other one is a second order allpass filter (see
fig. 31). Recall that, close to the frequency
0
that corresponds to the pole of
the filter, the phase response takes values that are very close to - (see fig. 23).
Therefore, the frequency
0
corresponds to a minimum in the overall frequency
response. In other words, it is the notch frequency. The closer is the pole to
the unit circumference, the narrower is the notch. The lattice implementation
of this allpass filter allows to tune the notch position and width independently,
since the two reflection coefficients have the form [76]
k
1
= - cos
0
(52)
k
2
=
1 - tan B/2
1 + tan B/2
,
where B is the bandwidth for 3dB of attenuation.
H (z)
1/2
x
a
y
Figure 31: Notch filter implemented by means of a second-order allpass filter
A structure that allows to convert a notch into a boost with a continuous
control is obtained by a weighted combination of the complementary outputs
and it is shown in fig 32. For values of k such that 0 < k < 1 the filter is a
notch, while for k > 1 the filter is a boost.
H (z)
-1
1/2
x
a
k
y
Figure 32: Notch/boost filter implemented by means of a second-order allpass
filter and a lattice section
Filters such as those of figures 31 and 32, whose properties can be controlled
by a few parameters decoupled with each other, are called parametric filters.
For thorough surveys on structures for parametric filtering, with analyses of
numerical properties in fixed-point implmentations, we refer the reader to a
book by Z¨olzer [109] and an article by Dattorro [29].