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Paraunitary Filter Examples

The Haar filter bank is defined as

$\displaystyle \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

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

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
$

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].


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``Introduction to Digital Filters with Audio Applications'', by Julius O. Smith III, (September 2007 Edition).
Copyright © 2014-03-23 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
CCRMA