Paraunitary Filter Examples

The Haar filter bank is defined as

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

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

$${\tilde{\mathbf{H}}}(z) = \left[\begin{array}{cc} 1+z & 1 - z \end{array}\right] / \sqrt{2}$$

so that

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

Thus, the Haar filter bank is paraunitary. This is true for any power-complementary filter bank, since when ${\tilde{\mathbf{H}}}(z)$ is $N\times 1$ , power-complementary and paraunitary are the same property.

For more about paraunitary filter banks, see Chapter 6 of [98].