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Spectral Audio Signal Processing
    Multirate Polyphase Filter Banks
       Paraunitary Filter Banks
          Paraunitary Examples

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Paraunitary Examples

Consider the Haar filter bank discussed previously, for which

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

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

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

so that

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

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

For more about paraunitary filter banks, see Chapter 6 of [264].

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Next: Filter Banks Equivalent to STFTs

written by Julius Orion Smith III
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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