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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