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 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 = $N$). 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.