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Embedded SystemsFPGAElectronics

Spectral Audio Signal Processing
    Multirate Filter Banks
       MPEG Filter Banks
          Pseudo-QMF Cosine Modulation Filter Bank

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Pseudo-QMF Cosine Modulation Filter Bank

Section 10.3.5 introduced two-channel quadrature mirror filter banks (QMF). We found that the quadrature mirror constraint on the analysis filters

$\displaystyle H_1(z) = H_0(-z)\quad \hbox{(QMF Symmetry Constraint)} \protect$

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$.

Quadrature Mirror Filters (QMF), defined in §10.3.5, provide a particular class of perfect reconstruction filter banks. The Pseudo-QMF (PQMF) filter bank is a ``near perfect reconstruction'' filter bank in which aliasing cancellation occurs only between adjacent bands [183,266]. 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. The design procedure is as follows:

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

    \begin{eqnarray*} \vert H(e^{j\omega})\vert^2 + \vert H(e^{j(\pi/N)-\omega})\ve... ...t H(e^{j\omega})\vert^2 &=& 0, \hspace{.5cm}\vert w\vert > \pi/N \end{eqnarray*}

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

    $\displaystyle h_k(n) = h(n)\hbox{cos}\left[\left(k+\frac{1}{2}\right)\left(n-\frac{M-1}{2}\right)\frac{\pi}{N} + \phi_k\right], \quad k=0,\ldots,N-1 $

    where the phases are restricted according to

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

    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) = h_k(M-1-n) $

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

Previous: MPEG Filter Banks
Next: Perfect Reconstruction Cosine Modulated Filter Banks

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About the Author: 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.


Comments

 

nataliahf wrote:

4/6/2009
 
Hello Julius Orion my name is Natalia Haro; I am a student of telecommunication engineering and I am doing my Master.
I have to make a graph in matlab tutorial on MPEG layer III and I did not know QMF filter program, but when I've found on the web DSPRelated.com a review of his conduct as a sub-band filter, I set as you specified in the web and finally I have solved the problem.
I wish to thanks for their hard work and help

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