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.

- Upsampling and Downsampling

- Polyphase Decomposition
- Two-Channel Case
- N-Channel Polyphase Decomposition
- Type II Polyphase Decomposition
- Filtering and Downsampling, Revisited
- Multirate Noble Identities

- Critically Sampled PR Filter Banks
- Two-Channel Critically Sampled Filter Banks
- Amplitude-Complementary 2-Channel Filter Bank
- Haar Example
- Polyphase Decomposition of Haar Example
- Quadrature Mirror Filters (QMF)
- Linear Phase Quadrature Mirror Filter Banks
- Conjugate Quadrature Filters (CQF)
- Orthogonal Two-Channel Filter Banks

- Perfect Reconstruction Filter Banks
- Simple Examples of Perfect Reconstruction
- Sliding Polyphase Filter Bank
- Hopping Polyphase Filter Bank
- Sufficient Condition for Perfect Reconstruction (PR)
- Necessary and Sufficient Conditions for PR
- Polyphase View of the STFT
- Polyphase View of the Overlap-Add STFT
- Polyphase View of the Weighted-Overlap-Add STFT

- Paraunitary Filter Banks

- Filter Banks Equivalent to STFTs

- MPEG Filter Banks
- Pseudo-QMF Cosine Modulation Filter Bank
- PR Cosine Modulated Filter Banks
- MPEG Layer III Filter Bank

- Review of STFT Filterbanks
- STFT, Rectangular Window, No Overlap
- STFT, Rectangular Window, 50% Overlap
- STFT, Triangular Window, 50% Overlap
- STFT, Hamming Window, 75% Overlap
- STFT, Kaiser Window, Beta=10, 90% Overlap
- Sliding FFT (Maximum Overlap), Any Window, Zero-Padded by 5

- Wavelet Filter Banks

- Further Reading
- Conclusions

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