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Properties of Paraunitary Filter Banks

Paraunitary systems are essentially multi-input, multi-output (MIMO) allpass filters. Let $ \bold{H}(z)$ denote the $ p\times q$ matrix transfer function of a paraunitary system. In the square case ($ p=q$ ), the matrix determinant, $ \det[\bold{H}(z)]$ , is an allpass filter. Therefore, if a square $ \bold{H}(z)$ contains FIR elements, its determinant is a simple delay: $ \det[\bold{H}(z)]=z^{-K}$ for some integer $ K$ .

An $ N$ -channel analysis filter bank can be viewed as an $ N\times 1$ MIMO filter:

$\displaystyle \bold{H}(z) \eqsp \left[\begin{array}{c} H_1(z) \\ [2pt] H_2(z) \\ [2pt] \vdots \\ [2pt] H_N(z)\end{array}\right]$ (12.86)

A paraunitary filter bank must therefore satisfy

$\displaystyle {\tilde {\bold{H}}}(z)\bold{H}(z) \eqsp 1.$ (12.87)

More generally, we allow paraunitary filter banks to scale and/or delay the input signal:

$\displaystyle {\tilde {\bold{H}}}(z)\bold{H}(z) \eqsp c_K z^{-K}$ (12.88)

where $ K$ is some nonnegative integer and $ c_K\neq 0$ .

We can note the following properties of paraunitary filter banks:

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``Spectral Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2011, ISBN 978-0-9745607-3-1.
Copyright © 2022-02-28 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University