Another way to express the allpass condition
This form generalizes by analytic continuation
over the entire the plane, where
denotes the paraconjugate of :
Definition: The paraconjugate of a transfer function may be defined as the
analytic continuation of the complex conjugate from the unit circle to
the whole plane:
denotes complex conjugation of the
coefficients only of and not the powers of .
For example, if
. We can
write, for example,
in which the conjugation of serves to cancel the outer
We refrain from conjugating in the definition of the paraconjugate
is not analytic in the complex-variables sense.
Instead, we invert , which is analytic, and which
reduces to complex conjugation on the unit circle.