Another way to express the allpass condition is to write
This form generalizes by analytic continuation (see §D.2) to 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:
where denotes complex conjugation of the coefficients only of and not the powers of . For example, if , then . We can write, for example,
in which the conjugation of serves to cancel the outer conjugation.
We refrain from conjugating in the definition of the paraconjugate because is not analytic in the complex-variables sense. Instead, we invert , which is analytic, and which reduces to complex conjugation on the unit circle.
The paraconjugate may be used to characterize allpass filters as follows:
Theorem: A causal, stable, filter is allpass if and only if
Note that this is equivalent to the previous result on the unit circle since