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Spectral Audio Signal Processing
    Multirate Polyphase Filter Banks
       MPEG Filter Banks
          Perfect Reconstruction Cosine Modulated Filter Banks

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Perfect Reconstruction Cosine Modulated Filter Banks

By changing the phases $ \phi_k$, the pseudo-QMF filter bank can yield perfect reconstruction:

$\displaystyle \phi_k = \left(k+\frac{1}{2}\right)\left(L+1\right)\frac{\pi}{2} $

where $ L$ is the length of the polyphase filter ($ M=LN$).

If $ M=2N$, then this is the oddly-stacked Princen-Bradley filter bank, and the analysis filters are related by cosine modulations of the lowpass prototype:

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

However, the length of the filters $ M$ can be any even multiple of $ N$:

$\displaystyle M=LN, \quad (L/2) \in \cal{Z} $

The parameter $ L$ is called the overlapping factor. These filter banks are also referred to as extended lapped transforms, when $ K \ge 2$ [144].

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written by 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.


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