Multirate Filter Banks

The preceding chapters have been concerned essentially with the short-time Fourier transform and all that goes with it. After developing the overlap-add point of view in Chapter 8, we developed the alternative (dual) filter-bank point of view in Chapter 9. This chapter is concerned more broadly with filter banks, whether they are implemented using an FFT or by some other means. In the end, however, we will come full circle and look at the properly configured STFT as an example of a perfect reconstruction (PR) filter bank as defined herein. Moreover, filter banks in practice are normally implemented using FFT methods.

The subject of PR filter banks is normally considered only in the context of systems for audio compression, and they are normally critically sampled in both time and frequency. This book, on the other hand, belongs to a tiny minority which is not concerned with compression at all, but rather useful time-frequency decompositions for sound, and corresponding applications in music and digital audio effects.

Perhaps the most important new topic introduced in this chapter is the polyphase representation for filter banks. This is both an important analysis tool and a basis for efficient implementation. We will see that it can be seen as a generalization of the overlap-add approach discussed in Chapter 8.

The polyphase representation will make it straightforward to determine general conditions for perfect reconstruction in any filter bank. The STFT will provide some special cases, but there will be many more. In particular, the filter banks used in perceptual audio coding will be special cases as well. Polyphase analysis is used to derive classes of PR filter banks called ``paraunitary,'' ``cosine modulated,'' and ``pseudo-quadrature mirror'' filter banks, among others.

Another extension we will take up in this chapter is multirate systems. Multirate filter banks use different sampling rates in different channels, matched to different filter bandwidths. Multirate filter banks are very important in audio work because the filtering by the inner ear is similarly a variable resolution ``filter bank'' using wider pass-bands at higher frequencies. Finally, the related subject of wavelet filter banks is briefly introduced, and further reading is recommended.