Paraunitary FiltersC.4

Another way to express the allpass condition $\left\vert H(e^{j\omega})\right\vert=1$ is to write

$\displaystyle \overline{H(e^{j\omega})} H(e^{j\omega}) = 1, \quad\forall\omega.$

This form generalizes by analytic continuation (see §D.2) to ${\tilde H}(z)H(z)$ over the entire the

plane, where ${\tilde H}(z)$ 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:

$\displaystyle {\tilde H}(z) \isdef \overline{H}(z^{-1})$

where $\overline{H}(z)$ denotes complex conjugation of the coefficients only of

and not the powers of . For example, if $H(z)=1+jz^{-1}$ , then $\overline{H}(z) = 1-jz^{-1}$ . We can write, for example,

$\displaystyle \overline{H}(z) \isdef \overline{H\left(\overline{z}\right)}$

in which the conjugation of

serves to cancel the outer conjugation.

Examples:

$H(z)=1+z^{-1}\quad\Rightarrow\quad {\tilde H}(z)=1+z$
$H(z)=1+2jz^{-1}+3z^{-2}\quad\Rightarrow\quad {\tilde H}(z)=1-2jz+3z^2$

We refrain from conjugating in the definition of the paraconjugate because $\overline{z}$ 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

$\displaystyle {\tilde H}(z) H(z) = 1$

Note that this is equivalent to the previous result on the unit circle since

$\displaystyle {\tilde H}(e^{j\omega}) H(e^{j\omega}) \isdef \overline{H}(1/e^{j\omega})H(e^{j\omega}) = \overline{H(e^{j\omega})}H(e^{j\omega})$

Previous: Allpass Examples
Next: Multi-Input, Multi-Output (MIMO) Allpass Filters

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About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.

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Introduction to Digital Filters
Allpass Filters
Paraunitary FiltersC.4