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 $ z$ plane, where $ {\tilde H}(z)$ denotes the paraconjugate of $ H(z)$:


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 $ z$ plane:

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

where $ \overline{H}(z)$ denotes complex conjugation of the coefficients only of $ H(z)$ and not the powers of $ z$. 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 $ z$ 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 $ z$ in the definition of the paraconjugate because $ \overline{z}$ is not analytic in the complex-variables sense. Instead, we invert $ z$, 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 $ H(z)$ 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})
$


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Allpass Examples