![$\displaystyle \mathbf{H}(z) = \frac{1}{\sqrt{2}}\left[\begin{array}{c} 1+z^{-1} \\ [2pt] 1-z^{-1} \end{array}\right]
$](img1672.png)
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The Haar filter bank is defined as
The paraconjugate of
is
![$\displaystyle {\tilde{\mathbf{H}}}(z) = \left[\begin{array}{cc} 1+z & 1 - z \end{array}\right] / \sqrt{2}
$](img1673.png){kind=small}
so that
![$\displaystyle {\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
$](img1674.png)
Thus, the Haar filter bank is paraunitary. This is true for any power-complementary filter bank, since when
is 
,
power-complementary and paraunitary are the same property.
For more about paraunitary filter banks, see Chapter 6 of [98].
