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### 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 [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

 (3.15)

This can be recognized as direct form I [452], 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

The frequency response is obtained by setting , where denotes radian frequency and denotes the sampling period in seconds [452]. For an allpass filter, the frequency magnitude must be the same for all .

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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