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Paraunitary Filter Banks

Paraunitary filter banks form an interesting subset of perfect reconstruction (PR) filter banks. We saw above that we get a PR filter bank whenever the synthesis polyphase matrix $ \bold{R}(z)$ times the analysis polyphase matrix $ \bold{E}(z)$ is the identity matrix $ \bold{I}$ , i.e., when

$\displaystyle \bold{P}(z) \isdefs \bold{R}(z)\bold{E}(z) \eqsp \bold{I}.$ (12.70)

In particular, if $ \bold{R}(z)$ is the paraconjugate of $ \bold{E}(z)$ , we say the filter bank is paraunitary.

Paraconjugation is the generalization of the complex conjugate transpose operation from the unit circle to the entire $ z$ plane. A paraunitary filter bank is therefore a generalization of an orthogonal filter bank. Recall that an orthogonal filter bank is one in which $ \bold{E}(\ejo )$ is an orthogonal (or unitary) matrix, to within a constant scale factor, and $ \bold{R}(\ejo )$ is its transpose (or Hermitian transpose).

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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