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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. 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 $ R=M=N$ (with $ M>N$ for Portnoff windows).

As derived in Chapter 7, the Short-Time Fourier Transform (STFT) is a PR filter bank whenever the Constant-OverLap-Add (COLA) condition is met by the analysis window $ w$ and the hop size $ R$. However, only the rectangular window case with no zero-padding is critically sampled (OLA hop size = FBS downsampling factor = $ N$). Advanced audio compression algorithms (``perceptual audio coding'') are based on critically sampled filter banks, for obvious reasons.

Important Point: We normally do not require critical sampling for audio analysis, digital audio effects, and music applications. We normally only need it when compression is a requirement.



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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.


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