An allpass filter can be defined as any filter having a gain of
at all frequencies (but typically different delays at different
frequencies).
It is well known that the series combination of a feedforward and feedback comb filter (having equal delays) creates an allpass filter when the feedforward coefficient is the negative of the feedback coefficient.
Figure 2.30 shows a combination feedforward/feedback comb filter structure which shares the same delay line.3.13 By inspection of Fig.2.30, the difference equation is
This can be recognized as a digital filter in direct form II
[452]. Thus, the system of Fig.2.30 can be interpreted as
the series combination of a feedback comb filter (Fig.2.24) taking
to
followed by a feedforward comb filter (Fig.2.23)
taking
to
. By the commutativity of LTI systems, we can
interchange the order to get
Substituting the right-hand side of the first equation above for
in the second equation yields more simply
The coefficient symbols
and
here have been chosen to
correspond to standard notation for the transfer function
The frequency response is obtained by setting
An allpass filter is obtained when
, or, in the case
of real coefficients, when
. To see this, let
. Then we have