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.