Allpass from Two Combs
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
[449]. 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
This can be recognized as direct form I [449], which requires


The coefficient symbols and
here have been chosen to
correspond to standard notation for the transfer function




![$ \omega\in[-\pi/T,\pi/T]$](http://www.dsprelated.com/josimages_new/pasp/img596.png)
An allpass filter is obtained when
, or, in the case
of real coefficients, when
. To see this, let
. Then we have

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Nested Allpass Filters
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FDN Stability