Examples

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

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

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

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

so that

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