Digital Filters
49
phase opposition
crossover filter
complementary filters
2.3
Complementary filters and filterbanks
In sec. 2.2.4 we have presented several different realizations of allpass filters
because they find many applications in signal processing [76]. In particular, a
couple of allpass filters is often combined in a parallel structure in such a way
that the overall response is not allpass. If H
a1
and H
a2
are two different allpass
filters, their parallel connection, having transfer function H
l
(z) = H
a1
(z) +
H
a2
(z) is not allpass. To figure this out, just think about frequencies where the
two phase responses are equal. At these points the signal will be doubled at
the output of H(z). On the other hand, at points where the phase response are
different by (i.e., they are in phase opposition), the outputs of the two branches
cancel out at the output. In order to design a lowpass filter it is sufficient to
connect in parallel two allpass filters having a phase response similar to that of
fig. 29.The same parallel connection, with a subtraction instead of the addition
at the output, gives rise to a highpass filter H
h
(z), and it is possible to show
that the highpass and the lowpass transfer functions are complementary, in the
sense that |H
l
()|
2
+ |H
h
()|
2
is constant in frequency. Therefore, we have the
a
Figure 29: Phase responses of two allpass filters that, if connected in parallel,
give a lowpass filter
compact realization of a crossover filter, as depicted in fig. 30, which is a device
with one input and two outputs that conveys the low frequencies to one outlet,
and the high frequencies to the other outlet. Devices such as this are found not
only in loudspeakers, but also in musical instrument models. For instance, the
bell of woodwinds transmits to the air the high frequencies and reflects the low
frequencies back to the bore.
H (z)
H (z)
a1
a2
-1
1/2
x
y
y
1
2
Figure 30: Crossover implemented as a parallel of allpass filters and a lattice
junction
The idea of connecting two allpass filters in parallel can be applied to the
realization of resonant complementary filters. In particular, it is interesting to