Polyphase Decomposition of Haar Example

Let's look at the polyphase representation for this example. Starting with the filter bank and its reconstruction (see Fig.10.18), the polyphase decomposition of is

Thus, , and therefore

We may derive polyphase synthesis filters as follows:

The polyphase representation of the filter bank and its reconstruction can now be drawn as in Fig.10.19. Notice that the reconstruction filter bank is formally the transpose of the analysis filter bank [247]. A filter bank that is inverted by its own transpose is said to be an orthogonal filter bank, a subject to which we will return §10.3.8.

figure[htbp]

figure[htbp]

Commuting the downsamplers (using the noble identities from §10.2.5), we obtain Figure 10.20. Since , this is simply the OLA form of an STFT filter bank for , with , and rectangular window . That is, the DFT size, window length, and hop size are all 2, and both the DFT and its inverse are simply sum-and-difference operations.

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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.