Multirate Polyphase 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, without regard as to whether they are implemented using the 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 the FFT.

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 quite simple 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 passbands at higher frequencies.

Finally, the related subject of wavelet filter banks is briefly introduced in Chapter 12, and further reading is recommended at the end of this chapter.

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Subsections

• Upsampling and Downsampling
  • Upsampling (Stretch) Operator
  • Downsampling (Decimation) Operator
    • Example: Downsampling by 2
    • Example: Upsampling by 2
  • Filtering and Downsampling
  
  
• Polyphase Filtering
  • 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 Filterbanks (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
  • Paraconjugation
  • Lossless Filters
  • Lossless Filter Examples
  • Properties of Paraunitary Systems
  • Properties of Paraunitary Filter Banks
  • Paraunitary Examples
  
  
• Filter Banks Equivalent to STFTs
  • Polyphase Analysis of Portnoff STFT
  
  
• MPEG Filter Banks
  • Pseudo-QMF Cosine Modulation Filter Bank
  • Perfect Reconstruction 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, Any Window, Zero-Padded by 5
  
  
• Further Reading

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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.