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.
Examples:
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