Paraunitary Filters^D.4

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

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

This form generalizes by analytic continuation (see §B.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:

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

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,

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

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

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

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

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