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 $R=M=N$ (with $M>N$ 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 $w$ and the hop size $R$ . However, only the rectangular window case with no zero-padding is critically sampled (OLA hop size = FBS downsampling factor = $N$ ). 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 $R>1$ ), as the next section makes clear.