Paraunitary Filters

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

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

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