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