### 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 delays instead of ; however, unlike direct-form II, direct-form I cannot suffer from ``internal'' overflow--overflow can happen only at the output.

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

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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