The general biquad transfer function was given in Eq.(B.8) to be

To specialize this to a second-order unity-gain allpass filter, we require

It is easy to show that, given any monic denominator polynomial , the numerator must be, in the real case,B.3

Thus, to obtain an allpass biquad section, the numerator polynomial is simply the flip'' of the denominator polynomial. To obtain unity gain, we set , , and .

In terms of the poles and zeros of a filter , an allpass filter must have a zero at for each pole at . That is if the denominator satisfies , then the numerator polynomial must satisfy . (Show this in the one-pole case.) Therefore, defining takes care of this property for all roots of (all poles). However, since we prefer that be a polynomial in , we define , where is the order of (the number of poles). is then the flip of .

For further discussion and examples of allpass filters (including muli-input, multi-output allpass filters), see Appendix C. Analog allpass filters are defined and discussed in §E.8.

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