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
, the numerator
must be, in the
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
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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