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.
This can be recognized as a digital filter in direct form II . 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 , 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
Nested Allpass Filters