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Introduction to Digital Filters
Allpass Filters
Multi-Input, Multi-Output (MIMO)
Allpass Filters
Paraunitary Filter Examples
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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]
$](http://www.dsprelated.com/josimages_new/filters/img1655.png)
is
![$\displaystyle {\tilde{\mathbf{H}}}(z) = \left[\begin{array}{cc} 1+z & 1 - z \end{array}\right] / \sqrt{2}
$](http://www.dsprelated.com/josimages_new/filters/img1656.png){kind=small}
![$\displaystyle {\tilde{\mathbf{H}}}(z) \mathbf{H}(z) = \left[\begin{array}{cc} 1...
...ight] \left[\begin{array}{c} 1+z^{-1} \\ [2pt] 1-z^{-1} \end{array}\right]
= 1
$](http://www.dsprelated.com/josimages_new/filters/img1657.png)
is 
,
power-complementary and paraunitary are the same property.
For more about paraunitary filter banks, see Chapter 6 of [98].
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Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.
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