FBS and Perfect Reconstruction

An important property of the STFT established in Chapter 8 is that it is exactly invertible when the analysis window satisfies the constant-overlap-add constraint. That is, neglecting numerical round-off error, the inverse STFT reproduces the original input signal exactly. This is called the perfect reconstruction property of the STFT, and modern filter banks are usually designed with this property in mind [264].

In the OLA processors of Chapter 8, perfect reconstruction was assured by using FFT analysis windows having the Constant-Overlap-Add (COLA) property at the particular hop-size used (see §8.2.1).

In the Filter Bank Summation (FBS) interpretation of the STFT (Eq.(9.1)), it is the analysis filter-bank frequency responses that are constrained to be COLA. We will take a look at this more closely below.

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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.