Consider the Haar filter bank discussed in §11.3.3, for which

$\displaystyle \bold{H}(z) \eqsp \frac{1}{\sqrt{2}}\left[\begin{array}{c} 1+z^{-1} \\ [2pt] 1-z^{-1} \end{array}\right].$ (12.92)

The paraconjugate of $ \bold{H}(z)$ is

$\displaystyle {\tilde {\bold{H}}}(z) \eqsp \frac{1}{\sqrt{2}}\left[\begin{array}{cc} 1+z & 1 - z \end{array}\right]$ (12.93)

so that

$\displaystyle {\tilde {\bold{H}}}(z) \bold{H}(z) \eqsp \frac{1}{2} \left[\begin{array}{cc} 1+z & 1 - z \end{array}\right] \left[\begin{array}{c} 1+z^{-1} \\ [2pt] 1-z^{-1} \end{array}\right] \eqsp 1.$ (12.94)

Thus, the Haar filter bank is paraunitary. This is true for any power-complementary filter bank, since when $ {\tilde {\bold{H}}}(z)$ is $ N\times 1$ , power-complementary and paraunitary are the same property. For more about paraunitary filter banks, see Chapter 6 of [287].

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Properties of Paraunitary Filter Banks