Examples
Consider the Haar filter bank discussed in §11.3.3, for which
![$\displaystyle \bold{H}(z) \eqsp \frac{1}{\sqrt{2}}\left[\begin{array}{c} 1+z^{-1} \\ [2pt] 1-z^{-1} \end{array}\right].$](http://www.dsprelated.com/josimages_new/sasp2/img2222.png) |
(12.92) |
The paraconjugate of
is
![$\displaystyle {\tilde {\bold{H}}}(z) \eqsp \frac{1}{\sqrt{2}}\left[\begin{array}{cc} 1+z & 1 - z \end{array}\right]$](http://www.dsprelated.com/josimages_new/sasp2/img2223.png) |
(12.93) |
so that
![$\displaystyle {\tilde {\bold{H}}}(z) \bold{H}(z) \eqsp \frac{1}{2} \left[\begin{array}{cc} 1+z & 1 - z \end{array}\right] \left[\begin{array}{c} 1+z^{-1} \\ [2pt] 1-z^{-1} \end{array}\right] \eqsp 1.$](http://www.dsprelated.com/josimages_new/sasp2/img2224.png) |
(12.94) |
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
[287].
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Polyphase Analysis of Portnoff STFT
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Properties of Paraunitary Filter Banks
