Critically Sampled Perfect Reconstruction Filter Banks

A *Perfect Reconstruction (PR) filter bank* is any filter bank
whose reconstruction is the original signal, possibly delayed, and
possibly scaled by a constant [287]. In this context,
*critical sampling* (also called ``maximal downsampling'') means
that the downsampling factor is the same as the number of filter
channels. For the STFT, this implies
(with
allowed for
Portnoff windows).

As derived in Chapter 8, the Short-Time Fourier
Transform (STFT) is a PR filter bank whenever the Constant-OverLap-Add
(COLA) condition is met by the analysis window
and the hop size
. However, *only the rectangular window case with no
zero-padding is critically sampled* (OLA hop size = FBS downsampling
factor =
). Perceptual audio compression algorithms such as MPEG
audio coding are based on critically sampled filter banks, for obvious
reasons. It is important to remember that we normally do not require
critical sampling for audio analysis, digital audio effects, and music
applications; instead, we normally need critical sampling only when
*compression* is a requirement. Thus, when compression is not a
requirement, we are normally interested in *oversampled filter
banks*. The polyphase representation is useful in that case as
well. In particular, we will obtain some excellent insights into the
*aliasing cancellation* that goes on in such downsampled filter
banks (including STFTs with hop sizes
), as the next section
makes clear.

- 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

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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University