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].