Another way to express the allpass condition is to write
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:
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
Multi-Input, Multi-Output (MIMO) Allpass Filters