##
Paraunitary
Filters^{C.4}

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:

*coefficients only*of

*and not the powers of*. For example, if , then . We can write, for example,

**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

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Multi-Input, Multi-Output (MIMO) Allpass Filters

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Allpass Examples