Improving the Octave Band Filters

To see the true filter-bank frequency response corresponding to Fig.9.33, we may feed an impulse to the filter bank and take a long FFT (for zero padding) of the channel signals to produce the interpolated response shown in Fig.9.35.

The horizontal line along 0 dB in Fig.9.35 was obtained by summing the channel responses, indicating that it is a perfect-reconstruction filter-bank, as expected. However, the stopband performance of the channels is quite poor, being comparable to the side-lobes of a rectangular window transform (an aliased sinc function). In fact, the stopband is identical to the rectangular-window side-lobes for the two lowest bands. Notice that the original eight samples of Fig.9.33 still lie along the 0-dB line, and there are still zeros in each channel response beneath the unity-gain point of every other channel's response, so Fig.9.33 can be obtained by sampling Fig.9.35 at those eight points. However, the interpolated response shows that the filter bank is quite poor by audio standards.

Figure 9.36 shows an octave filter bank (again for complex signals) that is better designed for audio usage. Instead of basing the channel filter prototype on the rectangular window, a Dolph-Chebyshev window was used (using the matlab function call chebwin(127,80)). The FFT size is about twice the filter length, thereby allowing for a data frame of equal length (to the filter) without incurring time aliasing, in the usual way for STFTs (Chapter 7). The data window is chosen to overlap-add to a constant, as typical in STFTs, so its choice does not affect the quality of the filter-bank output channel signals. We therefore may continue to use a rectangular data window that advances by its full length each frame. Choosing an odd filter length facilitates zero-phase offline STFT processing, in which the middle FIR sample is treated as occurring at time zero, so that there is no delay in any filter-bank channel [248].

The filter bank is PR in the full-rate case because the underlying STFT is PR, in the absense of modifications, and because the channel filter-bank is constant-overlap-add in the frequency domain (again in the absence of modifications) according to STFT theory (Chapter 7).

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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.