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Pseudo-QMF Cosine Modulation Filter Bank

Section 11.3.5 introduced two-channel quadrature mirror filter banks (QMF). QMFs were shown to provide a particular class of perfect reconstruction filter banks. We found, however, that the quadrature mirror constraint on the analysis filters,

$\displaystyle H_1(z) \eqsp H_0(-z),$ (12.97)

was rather severe in that linear-phase FIR implementations only exist in the two-tap case $ H_k(z) = h_{0k}+h_{1k}z^{-1}$ , $ k=0,1$ . In addition to relaxing this constraint, we need to be able to design an $ N$ -channel filter bank for any $ N$ .

The Pseudo-QMF (PQMF) filter bank is a ``near perfect reconstruction'' filter bank in which aliasing cancellation occurs only between adjacent bands [194,287]. The PQMF filters commonly used in perceptual audio coders employ bandpass filters with stop-band attenuation near $ 96$ dB, so the neglected bands (which alias freely) are not significant. An outline of the design procedure is as follows:

  1. Design a lowpass prototype window, $ h(n)$ , with length $ M=LN$ , $ L,M,N \in \mathbb{Z}.$
  2. The lowpass design is constrained to give aliasing cancellation in neighboring subbands:

\vert H(e^{j\omega})\vert^2 + \vert H(e^{j(\pi/N)-\omega})\vert^2 &=& 2, \hspace{.5cm}0 < \vert\omega\vert <
\pi/{2N} \\
\vert H(e^{j\omega})\vert^2 &=& 0, \hspace{.5cm}\vert w\vert > \pi/N

  3. The filter bank analysis filters $ h_k(n)$ are cosine modulations of $ h(n)$ :

    $\displaystyle h_k(n) \eqsp h(n)\hbox{cos}\left[\left(k+\frac{1}{2}\right)\left(n-\frac{M-1}{2}\right)\frac{\pi}{N} + \phi_k\right],$ (12.98)

    $ k=0,\ldots,N-1$ , where the phases are restricted according to

    $\displaystyle \phi_{k+1} - \phi_k \eqsp (2r+1)\frac{\pi}{2}$ (12.99)

    again for aliasing cancellation.
  4. Since it is an orthogonal filter bank by construction, the synthesis filters are simply the time-reverse of the analysis filters:

    $\displaystyle f_k(n) \eqsp h_k(M-1-n)$ (12.100)

This PQMF filter bank is reportedly used in MPEG audio, layers I and II with $ N=32$ bands and $ M=512$ taps ($ L=8$ ).

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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