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,

$$H_1(z) \eqsp H_0(-z),$$ (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:
   $$\begin{eqnarray*} \vert H(e^{j\omega})\vert^2 + \vert H(e^{j(\pi/N)-\omega})\vert^2 &=& 2, \hspace{.5cm}0 < \vert\omega\vert < \pi/{2N} \\ \vert H(e^{j\omega})\vert^2 &=& 0, \hspace{.5cm}\vert w\vert > \pi/N \end{eqnarray*}$$
   

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

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

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

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

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

   
   

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